OPTICAL ANALYSIS OF STRESS AND STRAIN IN PHOTOELASTIC ...

OPTICAL ANALYSIS OF STRESS AND STRAIN IN PHOTOELASTIC PARTICLE

ASSEMBLIES

H.G.B. ALLERSMA

TR diss) 1560


ASSEMBLIES


ASSEMBLIES

by

Henderikus G.B. Allersma

Delft University of Technology The Netherlands

Delft, 1987

TR dia 1560

to Janny Bart Allard

ACKNOWLEDGEMENTS

This research was carried out at the Geotechnical Laboratory of the Department of Civil Engineering of the Delft University of Technology, the Netherlands. I am grateful to the technical staff of the laboratory, A. Mensinga, J. van Leeuwen and J.J. de Visser, who modified ideas in real devices and took care for the preparation of several test models.

CONTENTS

1 INTRODUCTION 1 2 THEORY OF MEASUREMENT 7 2.1 STRESS TENSOR 7 2.1.1 Contact forces 7 2.1.2 Stress tensor in granular material 8 2.1.3 Stress distribution in a particle 10 2.1.4 Relation of interparticle stress and bulk stress 12 2.2 STRAIN TENSOR 15 2.2.1 Relative displacement of particles 15 2.2.2 Determination of the strain tensor 18 2.3 OPTICAL STRESS MEASUREMENT AT A MATERIAL POINT 19 2.3.1 Light 19 2.3.2 Polarisation 20 2.3.3 Double refraction 22 2.3.4 Optical filter system 23 2.3.5 Particle as sensors 31 2.3.6 Optical averaging inhomogeneous stress 33 2.4 DISPLACEMENT MEASUREMENT 36 2.4.1 Detection of marked particles 36 2.4.2 Determination of centre mark 38 2.4.3 Elimination of noise 40 2.5 DATA PROCESSING 41 2.5.1 Collected data 41 2.5.2 Principal stress trajectories 42 2.5.3 Stress distribution 45 2.5.4 Strain tensor 53 3 TEST SETUP 58 3.1 OPTICAL MEASURING DEVICE 58 3.1.1 Mechanical part 58 3.1.2 Optical system 60 3.1.3 Electronic circuit 63 3.1.4 Software and measuring procedure 66 3.2 MODELLING 73 3.2.1 Production of crushed glass 73

3.2.2 Mechanical properties of crushed glass 76 3.2.3 Optical properties of crushed glass 80 3.2.4 Preparation of a model 81 3.2.5 Loading systems and sensors 83 4 APPLICATIONS 85 4.1 SHEAR 85 4.1.1 Introduction 85 4.1.2 Theory of shear 87 4.1.3 Shear devices used 95 4.1.4 Measuring results 98 4.1.5 Discussion 115 4.2 EXPERIMENTS WITH LABORATORY-SCALE PENETROMETERS 121 4.2.1 Introduction 121 4.2.2 Models used 121 4.2.3 Measuring results 124 4.2.4 Discussion 141 4.3 EXPERIMENTS WITH A LABORATORY-SCALE HOPPER 148 4.3.1 Test setup 149 4.3.2 Measured results 150 4.3.3 Discussion 150 5 DISCUSSION 155 5.1 POSSIBILITIES AND LIMITATIONS 155 5.2 ACCURACY 157

NOTATION 160 REFERENCES 162 SAMENVATTING 168

1 INTRODUCTION

The development of material models and calculation procedures, to predict the mechanical behaviour of non-cohesive packed particle assemblies is still in full swing. In particular, numerical calculations based on the finite element method have proved very powerful. Contrary to the calculation techniques, however, little progress has been made in the development of new measuring principles. Sensoring and data collection have indeed been improved and more accurate devices are developed to investigate the behaviour of a sample of granular material. The measuring principle, however, is in general still based on the same concept as many years ago, namely: measurement of boundary loads or stress and displacement of boundary segments. The stresses and strains in the interior of a sample have to be estimated from the boundary conditions in this case, which is possible, only, if the stress distribution and deformation field are uniform. The information obtained by convential measuring methods is therefore very restricted if a granular material is subjected to complex boundary conditions. The boundary conditions in model tests relating to practical problems are in general too complex to yield useful information about the conditions in the interior of the granular material, so that it is not possible to analyse such problems in detail.

Because the actual mechanisms cannot be visualised in most tests, it is not properly possible to verify the results of calculations. Although the numerically calculation techniques that have been developed are highly advanced, the quality of the results greatly depends on the stress-strain relations used. The stress-strain relations are mainly based on the behaviour of samples of granular material which are subjected to a known uniform stress condition. A typical device for material testing is the triaxial apparatus, in which a sample can be subjected to very well defined boundary stresses. This good control over the boundary stresses, however, restricts the scope to simulate conditions which are more realistic in practice, such as stress

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rotation and continuous deformation. It is well known that the behaviour of a packed particle

assembly depends greatly on the stress history and stress path in the material. It is therefore not sufficient to investigate a sample under simple loading conditions only; but it is also necessary to know the mechanical behaviour of a sample when it is subjected to a stress path similar to the one occuring in reality. In the first instance, this requires knowledge about the stress path to which a granular material is subjected in a specific practical situation and, secondly, a device is required, which can simulate this stress condition. Several devices have been developed in which more realistic stress conditions can be simulated. Examples are the simple-shear apparatus (e.g. Roscoe, 1970) torsional-shear apparatus (e.g. Symes et al., 1982), directional-shear apparatus (Arthur et al. 1981) and multiaxial cubic test cell (Mould et al. 1982). For a particular problem a choice has to be made as to which material testing device gives the most useful results. Because clear information about the expected stress path is. often not available, many assumptions have to be made before the carefully measured material properties can be used in calculations.

To obtain more information about the stress-strain behaviour in practical problems and to verify the results of calculation procedures, a more advanced measuring method is required which would enable stress and strain components in the interior of the granular material to be visualised.

Up to now it was possible only to obtain detailed information about the strain in the interior of the granular material. Two methods are available for the strain measurements. The first method, developed by Roscoe et al.(1963), uses marked particles which are distributed in the granular material, such as lead shot. Because the density of lead is much more than, for example, that of sand the position of the lead shot in successive stages of a test can be located on a photograph by means of X-rays. This method has been used by e.g. Bransby et al. (1975) to investigate the flow of granular material in model tests of

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hoppers. The second method, which was developed by Butterfield et al., (1970), uses stereo-photogrammetry to determine the relative displacements in a granular material. In this method two successive photographs of the surface of a sand sample are used. The texture of the particle surface in combination with a non-uniform deformation show a mountainous area in a stereo viewer. The difference in altitude can be converted into displacements by means of a stereoscopic plotting machine. The strain tensor at a material point can be derived from the relative displacements of three points in a representative region.

The measurement of the stress distribution in the granular material, however, proved to be more complicated. As contrasted with elastic materials, there is no unique relation between stress and strain which can be used to determine the stress increment in a region with a known deformation. A more direct method is therefore required to obtain information about the stress distribution. It is not possible to realise this by means of electrical strain gauges because the large number of sensors, which would have to be distributed within the granular material to perform systematic measurements, would unacceptably disturb the behaviour.

At present only the optical method of stress analysis known as photoelasticity is available for investigating in detail the state of stress in a granular material. The fundamentals of this technique were established by Brewster, who in 1811 discovered the law which describes the phenomenon of polarisation by reflection. Another important step was achieved in 1852, when Herapath discovered that crystals of a complex salt containing quinine, hydriodic acid and sulphuric acid, possess the ability to absorb light which oscillates in a specific plane. The needle-shaped crystals, with a girth diameter which is considerably smaller than the wave-length of visible light, could be used to produce synthetic polarisation sheets. This technique was developed in the first decades of 1900. The production of large polarisation sheets has opened up the possibility to

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visualise double refraction in models of transparent materials. In 1930 Colter and Filon, both of the University of London, demonstrated that the doubly refractive property could be utilized to visualise stresses in elastic materials. Since then, photoelasticity has increasingly become a useful tool for solving problems in structural engineering practice. The theory of photoelasticity and many applications thereof are for example described in Frocht's book (1946). In the last ten years the use of the photoelastic method has declined dramatically because many complex boundary value problems in structural engineering can now be solved with powerful computers and numerical calculation procedures.

In 1957 it was demonstrated by Dantu as well as by Wakabayashi, that photoelasticity could be used to visualise the transmission of force in a packed particle assembly. The optical phenomenon in a granular material, however, had to be interpreted in a quite different way than was usual in elastic materials. It was not possible to create sharp isoclinics and isochromatics for analysing the stress distribution. Instead of isochromatics a pattern of clear stripes could be observed, which were assumed to represent major principal stress trajectories.

The transmission of force in a granular material was analysed in detail by De Josselin de Jong and Verruijt (1969). They performed tests with cylindrical disks of photoelastic material to simulate a two dimensional assembly of a granular material. The isochromatics were used to determine the magnitude and direction of the contact forces between the disks. This technique was later used by Drescher and De Josselin de Jong (1972) to investigate aspects of a mathematical model for the flow of granular material. They described a method for transforming the distribution of the discrete contact forces and displacements in a region into a second-rank tensor, so that the test results could be expressed in terms as usual be employed in soil mechanics. They found experimentally that a tensor describes the distribution of the contact forces in a region fairly satisfactorily. Much research on the distribution of the

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contact forces and the fabric structure in two-dimensional analogues of disks was performed by Konishi, Oda and Nemat-Nasser (1982). They used, for example, elliptical disks of photoelastic material to investigate the anisotropic behaviour of a granular material. The test technique with disks is not suitable for investigating the stress distribution in model tests of practical problems because an assembly of disks has a very discrete character and is restricted to two-dimensional assemblies only.

In 1976 , Drescher published an experimental study on flow rules for granular material in which he again used crushed glass as the optically sensitive material. Because attempts undertaken by Wakabayashi (1959) and De Josselin de Jong (1960) to determine the magnitude of stress in crushed glass, by means of compensators, did not yield satisfactory results only information about the stress direction could be obtained in samples of crushed glass. It was shown in a qualitative way by Wakabayashi (1957,1959), Drescher and De Josselin de Jong (1972) and Oda and Konishi (1974) that the average direction of the light stripes visible in circularly polarised light approximately coincide with the lines of action of the major principal stress. Manual measurement of the directions from photographs, however, was rather subjective and very time consuming, and the direction of the stripes established in photographs appeared to be dependent on deviations in the optical filters, on the wavelength distribution of the light source and on the sensitivity of the film to a particular wave-length. Because very realistic scale models can be prepared with crushed glass, it would be very 'useful if more information about stresses and strains could be obtained by using the optical properties of glass.

The main purpose of this study was to develop optical methods, for determining stresses and strains in assemblies of photoelastic granular material and to design a device for performing systematic measurements in scale models of practical problems and material testing devices. In the first instance a connection was established between the distribution of the

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contact forces at a material point and the optical phenomenon in the granular material. This resulted in D a method for determining two stress components in a three-dimensional plane strain sample, 2) a calculation procedure for determining absolute stresses, and 3) a prototype of a device for performing the measurements (Allersma, 1982a). At a later stage a computer-controlled optical device was developed which was equipped also with a digital camera. Stresses and displacements can now be measured simultaneously. The measuring method has been applied to analysing the mechanical behaviour of granular materials in several tests, which simulate practical problems and material testing devices.

The theoretical background of the photoelastic measuring method, the determination of the deformation and the data processing are described in Chapter 2. In Chapter 3 the layout, control and operation of the automated optical measuring device are described, and the production process of the crushed glass and the preparation, loading and sensoring of a model are explained. Some applications of the optical measuring method are presented in Chapter 4. The stress-strain behaviour under large shear deformations has been investigated, and several measurements of penetration tests are presented and discussed. Finally it is shown how the stress distribution changes in a hopper if some material flow has been taken place. In Chapter 5 the possibilities, accuracy and limitations of the measuring method are discussed, and suggestions for further development of the measuring method are proposed.

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2 THEORY OF MEASUREMENT

2.1 STRESS TENSOR

2.1.1 Contact forces

If a large number of particles are brought together there will be contact points between the particles. At each contact point there acts a force due to gravity and/or external loads. The direction of the forces and the friction and cohesion between the particles determine whether particles move with respect to each other or not. At micro scale the distribution of the contact forces is very inhomogeneous. Besides body forces and external load there are several other factors which influence the magnitude and distribution of the contact forces, such as;

- the density of the particle assembly, - the strength of the particles, - the shape of the particles, - the elasticity of the individual particles, - the particle size. The first four items influence the number of contact points

between the particles. If the assembly is packed more densely, more contact points between the particles are to be expected. If the averaged force transmission is the same, the magnitude of the individual contact forces will be smaller. The strength of the particles is responsible for the amount of crushing at the contact points, which results in a redistribution of the contact forces. The number of contact points between the particles is directly influenced by the shape and the elasticity of the particles. The magnitude of the contact forces is more or less inversely proportional to the particle size. If the particles are very small, the magnitude of the contact forces comes closer to that of the atomic forces. This explains, for example, why an initially non-cohesive granular material shows a cohesive behaviour when it is moulded to a fine powder.

It is usual in soil mechanics to describe the distribution of

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the contact forces with a second-rank tensor, because it is then possible to apply the fundamentals of continuum mechanics. It has to be realised, however, that a second-rank tensor gives a simplified image of reality. The discrete character of granular material and information on, for example, the number of contact forces and the magnitudes of the individual contact forces is not included in a tensor. At present, however there is no better way to symbolise the force distribution. Therefore the optical measuring method has been so designed, that parameters are obtained which can be used to derive the components of a tensor, so that the distribution of the contact forces is described in a similar way, as is usual in soil mechanics.

2.1.2 Stress tensor in granular material

In continuum mechanics, stress is defined as the resultant force acting on a unit area. This definition is based on homogeneous materials, which means that the elementary particles, i.e. atoms, are very small in comparison with the stress gradients. If continuum mechanics is applied to granular material, areas have to be considered which contain sufficient particles. On the other hand, it is not permitted to consider too large area units, because in that case information on the stress gradient is lost in samples with a non-uniform stress distribution. A unit area can be assumed to be representative if a small deviation in surface does not significantly change the average stress. However, it is not always suitably possible to define a representative area in a granular material, because large gradients in stress and strain may occur. The thickness of a shear band, for example, is about ten particle diameters, so that the contact forces of only a few particles are representative of the stress state. This is of course very little in comparison with homogeneous materials and it is not quite clear how such phenomena have to be interpreted.

If the macro stress in a region is not very inhomogeneous, the averaging procedure, which is described by Drescher and

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De Josselin de Jong (1972) can be used to transform discrete contact forces into stress compcAaents. If a region with volume V is considered with a non-uniform stress state o. ., which is in equilibrium, then the average stress o. . is defined by

«ij = è fv °ij d v ( 2 : l )

which, by using Gauss's divergence theorem, can be written as

*ij = hi x i ( m ) T j ( m ) ( 2 : 2 )

m=l in which

u = number of discrete forces intersected by the boundary of V

x. = i-co-ordinate of the intersection point of the m-th contact force (■ i=l,2,3)

T (m)= j_component of the m-th force ( j=l,2,3) Eq. 2:2 was used by Drescher and De Josselin de Jong to determine the stress tensor in a region of a two dimensional assembly of photoelastic disks. The contact forces were determined from the isoclinics in the disks, using a procedure developed by De Josselin de Jong and Verruijt (1969). Contrary to usual photoelastic measuring techniques, it was possible in this way to determine the complete stress tensor in a representative region. Since the distribution of the contact forces was known, it was possible to plot a Maxwell diagram of the forces which intersect a hypothetical circular boundary. An example of such a diagram is presented in Fig.2:1. The more closely the diagram approaches an ellipse, the better the force distribution can be described by a tensor. Taken into account that a relatively small number of particles were considered, fairly good agreement with an ellipse is observed.

The analysis of Drescher and De Josselin de Jong has demonstrated that the distribution of the contact forces can be described fairly well with a second-rank tensor, and agreement can be expected to be better if the region under consideration

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o 3 10 kg

Fig.2:1 Maxwell diagram of a circular region in a two-dimensional assembly of 150 disks (from Drescher and De Josselin de Jong, 1972).

contains more particles. Because the contact forces create stresses in the disks, the

stress tensor which describes the distribution of the contact forces in a region can also be derived by averaging all the internal stresses of all the disks in a region considered, as is shown by eq.2:l. It was found possible to determine two components of the average stress tensor in a body of photoelastic material by optical means. Because the whole volume of the body is considered in this method, the body may be of any shape. Since the optical averaging procedure is also valid over the thickness of a sample, the average stress components can also be derived in three-dimensional assemblies of arbitrarily shaped particles of photoelastic material. Because the optical averaging can be performed automatically, systematic measurements can be performed in scale models with more realistic particles.

2.1.3 Stress distribution in a particle

The particles play an important role in the optical measuring technique. In fact, they are used as gauges which translate the discrete contact forces into tensor components and they make these components measurable. In the optical measuring method

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employed the stress components are accumulated vectorially over the surface of a region. The stress distribution in the region considered therefore need not to be homogeneous. However, because the magnitude of the principal stress difference (a -a ), which can be measured is limited, it is necessary to know how the stresses vary at micro scale. Information on the stress distribution at micro scale can be obtained by calculating the stress distribution in a body which is loaded by discrete forces. If a particle is schematised as a circular disk the stress distribution can be calculated with an analytical solution based on elastic theory (Timoshenko and Goodier, 1951). In the case of a diametrical load the distribution of the principal stress difference and the principal stress direction, ty is given by

4P (a2-xz-y2) (a -a ) = (2:3) 1 2 ira Cx2+(a-y)z]Cx2+(a+y)23

2xy *-.-r-r-7 (2:4)

x -y +a

in which P = the applied diametrical load a = the radius of the disk

x,y = the co-ordinates of a point in the disk (-a<x<a and -a<y<a) The distribution of (a -a ) and ip in a diametrically loaded disk is represented in Fig. 2:2. As may be expected the distribution of (a -a ) and x|> is not uniform. It appears, however, that the gradients are smooth and that the variation of the principal stress difference is not very large in the most significant region. Only in a small region close to the contact points are too large principal stress differences likely to occur. However, this region is so small that no significant influence on the total result is be expected.

It is not necessary to consider disks which are subjected to more point loads, because the peak values of (a -a ) decrease in

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M

Fig.2:2 The distribution of the principal stress difference and the major principal stress direction in a diametrically loaded disk.

this case. Furthermore, it is not to be expected that the shape of the boundary has much influence on the variation in the principal stress difference, so that it is not necessary to investigate a great variety of particle shapes.

2.1.4 Relation of interparticle stress and bulk stress

A material point in a particle assembly is a volume which contains sufficient particles for obtaining representative averaged values of the stress components. In the case of a three-dimensional plane strain sample, of which the surface is large in comparison with the thickness, a material point is, for example, a cylinder with a length equal to the thickness of the

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Fig.2:3 Distribution of the contact forces at the boundary of a cylindrical material point.

sample. Such a cylindrical material point contains a number of particles over the thickness as well as over the surface. Each particle is loaded by a number of contact forces. In Fig.2:3 it is demonstrated schematically how the contact forces are distributed over the hypothetical cylindrical boundary. As will be the case in practice, the individual contact forces are not all plotted perpendicularly to the axis of the cylinder. Furthermore, a cohesionless granular sample has to be supported at the entire boundary, so that in reality the stress condition is three-dimensional. However, if the thickness of the sample is sufficiently large, the force components perpendicular to the plane of the sample are assumed to have no significant influence on the plane deformation. The averaged value of the components of the tensor, which describe the averaged stress in the cylindrical material point,

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Fig.2:4 Representation of the contribution of one contact force to the averaged stress in a cylindrical region.

can be calculated with eq.2:2. In Fig.2:4 it is shown, how a m-th contact force T has to be resolved into different components.

It is assumed that a sample can be loaded in such a way that one of the principal stresses, e.g. a , is perpendicular to the boundary of the sample and therefore a and a act in the plane of the sample. The state of stress at a point of a plane sample is represented in a Mohr diagram in Fig.2:5. It is to be noted that compressive stress has a negative sign. The averaging procedure is based on the whole volume of the material point. In reality, however, only a part of the volume is occupied by the granular material, while the pores are filled with air or a liquid. The average stress in the particle bodies is therefore larger than calculated with the averaging procedure. Although this point is debatable it is neglected, because a discussion leads to new details such as the phenomenon that some particles

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°"l

fr" \ ^

"Vx

"•«T

> \

"" / /

, '-"■ pole

°r*

°x*

°"YY

°»

Fig.2:5 agram.

Representation of the stress components in a Mohr di-

are not loaded at all or very little. An optical method can be applied to determine averaged stress

components which refer to a material point in a plane strain sample. It can be demonstrated with the theory, which describes two dimensional photoelasticity, that the optical averaging over the thickness and surface of a material point is almost the same as the averaging procedure with eq.2:l.

2.2 STRAIN TENSOR

2.2.1 Relative displacement of particles

If a granular material is loaded, the particles will move in relation to each other. The displacements of the individual particles cause some macro deformation of a sample. To investigate the strain behaviour of a sample in detail, it is necessary to know the displacements in a representative region. Because it is not convenient to use the displacements of all the individual particles in formulas, a method is required to characterise the displacements at a material point. In the same way as the stresses it is usual to describe the relative

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e2 Ux.x

Uy.y

Fig.2:6 Representation of the strain components in a Mohr diagram (relative displacement diagram).

displacements between the particles at a material point by means of a second-rank tensor. The components of the strain tensor at a material point are represented in a Mohr diagram in Fig.2:6. The strain tensor is in general not symmetrical, due to the rotation, u of the material.

In Fig.2:7 the distribution of the directions of the displacements of a tensorial deformation field is plotted. It is assumed that UJ=0 and that the volume remains constant: hence the principal strains E and E have the same absolute value. To demonstrate that the displacements in a granular material can be described reasonably well with a tensor, a similar pattern has been created by means of two negatives on which the particle structure of two successive stages of a plane sample are fixed (Fig.2:8). If the field is subjected to a homogeneous deformation the major directions of strain become visible. This technique was used by De Josselin de Jong, 1959, to analyse the deformation in a two dimensional assembly of rods. Although good averaging is obtained over a large amount of particles, this technique is not always suitable for obtaining detailed information on the deformation, because a large region with a homogeneous deformation is required.

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.• .' * l t t \ \ \ \

Fig.2:7 Calculated distribution of the directions of the displacements of a tensorial deformation field.

Fig.2:8 Distribution of the directions of displacements, visualised in a plane strain granular material.

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2.2.2 Determination of the strain tensor

Contrary to the optical stress measurement, it is not practicable to obtain strain components which are based on continuous averaging of the displacements over the volume of a material point. Therefore the determination of the strain tensor has to be based on the displacements of some discrete points. The displacements of only three points are required to calculate the strain components at a material point. At present there are two measuring principles available for determining the displacement of a discrete point. In the most advanced techniques two successive exposures of the particle structure at the surface of a sample are used to determine the displacement of a point by means of the stereo-photogrammetric method (Butterfield et al.,1970). The advantage of this method is that the displacement at the surface can be measured more or less continuously, so that large strain gradients can be observed. A disadvantage is, however, that the digitisation of the displacements requires expensive precision devices and is time consuming. Because the measuring procedure cannot be automated in a simple way, it is not practicable to apply this method to the interpretation of a large number of photographs. Further, it is not simply possible to follow the pattern of a single point, which is for example useful in tests with large deformations.

A simpler technique for determining the displacements of discrete points is to measure the co-ordinates of labelled particles in successive stages of a test. If a plane sample is not transparent, particles with a higher density e.g. lead shot, can be distributed in a plane in the sample and X-rays can be used to fix the actual position of lead balls on a photograph. This method was used, for example, by Bransby et al.,1973 to investigate the deformation in plane hoppers. This technique is very attractive in tests with crushed glass. Because the assembly of glass particles is made transparent by saturating the pores with a liquid which has the same refraction index as glass, black marks can be made visible simply by normal light.

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Furthermore particles, of the same density can be used such as black coloured glass. Because, due to small deviations of the refraction index of glass, the particle structure is visible in an assembly too, a stereo observation of successive photographs can be used to support the interpretation of the displacements of marked particles.

Several types of digitisers are available to determine the co-ordinates of the marks from photographs. In general, however, the digitisation of the co-ordinates is rather time-consuming, and if cross hairs are used it is not always easy to define a particular point on similar marks of successive tests. Furthermore, the photographic step is not convenient. To prevent errors due to strength of the film material, it is necessary to use glass plates with a light-sensitive layer. However, a disadvantage is that a camera which handles glass plates is not suitable for automation.

To eliminate the photographic step, a digital camera is applied in this research to determine the position of marks directly in a test model. The advantage of this method is that the procedure can be completely automated and the results are directly available, so that the progress of a test can be observed in real time.

2.3 OPTICAL STRESS MEASUREMENT AT A MATERIAL POINT

2.3.1 Light

Light is used as the medium for determining the stress distribution and deformation in a transparent assembly of optically sensitive glass particles. The major advantage of light is that it has no effect on the mechanical behaviour of the particles. Only secondary effects, such as heat production of a light source, can influence the measurement, because temperature gradients causes additional internal stresses in glass. For this reason a laser source appeared to be very suitable, because an intensive light beam is produced with minimum production of heat

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radiation. Furthermore, light with a very small range in wavelength is obtained, so that the mathematical description of photoelasticity is in very close agreement with the reality. For this purpose a He-Ne laser is used, which produces light with a wave-length of X=633nm. Light is a complex phenomenon and several theoretical models have been developed in order to describe this phenomenon completely. For this purpose, however, the classical ether-wave theory of Huygens is quite adequate. According to this theory, space is filled with a hypothetical perfect elastic medium, and light is a wave phenomenon caused by a disturbance in this medium. The disturbance consists of vibrations of the elementary particles. It is assumed that the vibrations take place in a direction perpendicular to the direction of propagation of the wave. In ordinary light the elementary particles are subjected to many harmonic waves, which vibrate in random directions and which differ in amplitude and phase. The motion of a particle is therefore entirely random as is schematised in Fig.2:9. It is not possible to describe ordinary light with simple mathematical formulas. To perform photoelastic measurements ordinary light has to be modified into a well-defined phenomenon which can be realised by means of a polarisation filter.

2.3.2 Polarisation

The random behaviour of ordinary light can be modified into a well-defined wave motion by means of polarisation. Polarisation can be achieved by dichroic crystals, such as herapatite (discovered by Herapath in 1852). The physical model assumes that ordinary light is converted into two mutually perpendicular plane-polarised beams when it enters the crystal. Because the absorptive capacity of dichroic crystals is much greater for one of these beams, only one plane-polarised beam is transmitted (Fig.2:10). In plane-polarised light the light vector vibrates harmoniously in parallel planes, as is shown schematically for

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Fig.2:9 Diagram of the motion of ordinary light, projected on a plane perpendicular to the light beam.

Unpolaiüad Inddint U(ht

CourUty M. Orabau and Polanid Corporation

Fig.2:10 Schematic representation of the production of polarised light by a dichroic crystal (from Frocht, 1946).

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Fig.2:11 Graphic representation of the wave motion of polarised light.

one plane in Fig.2:11. According to the wave theory polarised light'can be described by

Asin(ü)t) w = 2ir- rad/s (2:5)

were a = the actual amplitude at time t A = the maximum amplitude id = the angular velocity c = the propagation speed of light x = the wave-length of the light used

The intensity I of a plane-polarised light beam is defined by

I = 2A* (2:6)

2.3.3 Double refraction

Isotropic transparent materials, such as glass, become temporarily double-refractive when subjected to shear stresses. The physical background of this phenomenon is that the velocity of propagation of light which vibrates in the principal stress directions is different. The velocity of propagation decreases if the stress, acting on the plane of vibration, increases. To describe the double-refractive property the following physical model is assumed (Fig.2:12)

e*

i

c • r l

0> > ld 3

4 J Si fcn

•rH r H

<d ■4-1 0

i j

3 O

• H

> tri X! dl

Si

0) x! J J

&> c • H J J id I J

J J 10

3 i - i r - l • H

E td i j t * (Ö

■ H Q

es r H

• • CM

• tP • H [ i j

• r H

cd • H J j 0)

J J id B CU > • H

J j

u td i j

' M 0) i j l

01 r - i

X) 3 O x)

0) > ■ H

J j U id S J

' M 01 U 1

01 i-H

3 0 ■d

' M O

0) ■ P

ld r H CL

ld

B) I J

CU J J

c 0)

J j

Xl t f

• H ■ H

-d dl ( 0

• H I J

ttl i-H O a c a> si § i

10 01 > id 5 i j id

r H

3 u

■ H ■d c 01 CL i j dl CL

O 3

■ p

o J J

c • H

■d O) > i - i 0 ( 0 01 i j

U I ■ H

c 0

• H J J

o E O) > ld 3 0> XI J J

i H td

• H i j 01

J J

rt e

■ P

< ■

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- 24 -

POLARIZER 1/4- \ PLHTE

'. CIRCULRR "LIGHT

ELLIPTICAL LIGHT ROTATING FILTER

LIGHT SENSOR (2

Fig.2:13 Lay-out of the optical filter system.

In optics permanent double-refractive filters are used to produce light with particular properties. A typical example of such a filter is a quarter-wave plate. In this filter the retardation between the waves which vibrate in the direction of the principal axes is exactly a quarter of the wave-length of the light used.

2.3.4 Optical filter system

The application of stress optics to particle assemblies requires some properties of the optical filter system which are usually not of importance for stress analysis in bodies consisting of a homogeneous material.

Although the assembly of glass particles is made transparent with a liquid, the absorption of light due to refraction and reflection is not homogeneous over the surface of the plane sample and depends on the deformation. It is therefore necessary that the intensity of the incident light, including losses due to absorption by the optical filters, can be measured when the model is stressed.

The stress tensor at a material point has to be derived from

- 25 -

the micro stresses in the particles of a small region. The accumulation of the optical effects over the surface and thickness of the light beam should therefore be in agreement with the vectorial addition of the micro stresses.

The layout of the optical system which best meets the requirements is shown schematically in Fig.2:13. Monochromatic, linearly polarised light, produced by a He-Ne laser, is modified into circularly polarised light by means of a so called quarter-wave plate. If the circular-polarised light beam has been transmitted through a model of double-refractive material, an additional phase difference S causes an elliptical motion of the light vector. A rotating polarisation filter (analyser) and a light sensor are used to analyse the elliptically polarised light.

The initial polarised light beam, which can be described with eq.2:5, enters the quarter-wave plate in such a way that the plane of polarisation is incline at 45 with respect to the optical axes of the plate. The light wave is resolved in the directions of the optical axes q and q and can be described by

qx = iAJ~2cos(wt) (2:7a)

q2 = |Af2cos(ut) (2:7b)

When the light leaves the filter, the retardation has caused a phase difference of TT/2 between the perpendicular waves. The equations describing the vibration in the direction of the optical axes now become

q = |Af2cos(ujt) (2:8a)

q2 = |A4"2C os (ut-^r = |Af2sin(üJt) (2:8b)

Compounding these two perpendicular waves, a circular motion of

- 26 -

the light is obtained which can be written in two dimensions with

r = |AJ~2 (2:9a)

()> = ui t ■ (2:9b)

were r and $ are the length and the direction of the light vector at time t, respectively. The intensity of this circular polarised light, known as the initial light intensity, I , is

I = 2r2= A2 (2:10) o When the circular-polarised light enters the temporary double-

refractive material of the test model, the circular motion is resolved into two perpendicular waves again, which vibrate in the directions of the principal stresses, as represented by

a = iAJTsin(wt) (2:lla) 2 2

ai = |AJ~2cos(wt) (2:llb)

were a and a are the amplitudes of the light wave in the a and a direction, respectively. The propagation velocity of the two waves are not equal, and the velocity difference e, which is proportional to the difference of the principal stresses, is given by

E = K(a -a ) (2:12)

were K is an optical material constant. The velocity difference causes a phase retardation S between

the waves which is proportional to the thickness s of the model, the wave-lenght X of the light used and the average velocity c' of the light in the double-refractive material. The expression

- 27 -

for S becomes

S = ^r(a -a ) (2:13) XC 1 2

In in a test X, c', K and s are constants; this equation can therefore be written as

(a -a ) = MS (2:14)

were M is a factor taking account of all parameters influencing the retardation S.

The phase difference causes an elliptical motion of the light when it leaves the model, as is represented schematically in Fig.2:14. This motion can be described with

a2 = iAJ~2sin(0Jt+S) (2:15a)

8LI = AJTcosdjUt) (2:15b)

If the light enters the second polarisation filter (analyser) the light vector r (Fig.2:14) will be resolved into two perpendicular components, but only one component, which coincides with the polarisation plane will be transmitted. The amplitude of the transmitted wave depends on the angle y between the polarisation plane and the a -axis and can be described by

a = A.T2Csin(ut+S) sinY+cos(wt) cosyD (2:16)

The value of wt at which a is a maximum, A , can be found by y y

differentiating eq.2:16. After some rearrangement this gives

wt = arctgf—.co^S. A = $ (2:17) ^ (.cotgy+sinSj T

and thus

A = ?AJTCsin(0+6)sin>f+cos(t)cosY3 (2:18) y *

- 28

analyzer

Fig.2:14 Graphical representation of the light motion when it has been transmitted through the test model.

max

0 Pmax. Pmin. rotation analyzer

F i g . 2 : 1 5 L igh t i n t e n s i t y I as a

a n g l e p of t he a n a l y s e r .

180

function of the rotation

- 29 -

The light intensity as a function of y is now

I = 2A2 (2:19) Y Y

Because the maximum amplitudes in eq.2:15 are equal, it can be concluded that the position of the ellipse is always symmetrical with respect to the principal stresses; so in the case of Fig.2:14, I is a maximum if y= 45° and a minimum if -y= 135°. Substituting these extreme values into eq.2:17 gives respectively

<(,= 45°-| and <p = 135°-|

Using eq.2:18, the maximum and minimum amplitudes are found to be

A = iAf2(cos4+sin|) (2:20a) is 2 2 2

A _ = iAJ~2(cosf-sin4) (2:20b) 135 & A £»

Remembering that I =A , the maximum and the minimum light intensity can be written as

I = 2A2 = I (1+sinS) (2:21a) max is o I . = 2A2 = 1 (1-sinS) (2:21b) m m 135 o

If the directions of the principal stresses are not known, the maximum and the minimum light intensity can be found numerically by rotating the analyser through 180 and measuring, for example, at every degree the light intensity (Fig.2:15). The intensity of the incident light can be calculated from the average light intensity Ï as follows

I = Ï = (1 +1 . ) (2:22) o 2 max min

Sufficient data is now available to calculate the phase

- 30 -

Fig.2:16 Difference of the maximum and the minimum light intensity as a function of the phase retardation.

retardation, for example, with ,1 -I max r E +1 . , max mm' ,1 -l . .

o . _ f max min^ S = arcsinlj —= 1

(2:23)

If the initial position of the analyser is known, the direction of the principal stresses can be deduced from the rotation angle B of the analyser at maximum light intensity. In the case of Fig.2:15 the principal stress directions are

t|) = B -45 T l "max (2:24a)

\\> = B +45 2 max (2:24b)

A simplified equation can be derived to calculate I , because o I is apparently a harmonic function of y with a period of 180 .

An equivalent equation is then

(I -I)sin2Y+I max (2:25)

Substituting the expressions for the intensities gives

1 = 1 (sinSsin2Y+l) Y 0 ' (2:26)

- 31 -

were y may also be written as y = P-ip ■ The range of S is limited because I -I . decreases if S 3 max m m

becomes larger than —IT radians (Fig.2:16). However, a favourable circumstance is that the principal stress difference in particle assemblies is usually small in comparison with that found in tests on homogeneous cohesive materials. If a sample of glass particles with a thickness of 40 mm is used in a model test, a maximum principal stress difference of about 700 kN/m can be measured. In the tests described in this paper the maximum principal stress was in general not more than 250 kN/m (8<—ir).

2.3.5 Particles as sensors

The mathematical description of the optical determination of (a -a ) and the principal stress directions \|> and i|) is based on a plane stress model of photoelastic material, with a homogeneous stress distribution at the material point considered. In the case of a plane model of crushed glass the individual particles are used as an optical sensor which translate the distribution of the contact forces into optically measurable stress components. Apart from optical imperfections of the granular material, several other phenomena are not in agreement with the assumptions on which the mathematical description of the behaviour of light in double-refractive material is based. The following deviations can be noted; - the model is not homogeneous; - the stress distribution in the particles is not plane; - the stress distribution is not homogeneous over the surface and thickness of a material point. It is not to be expected that the inhomogeneous character of a

granular sample has a significant influence on the validity of the mathematical description of the optical measurement. Indeed, the stresses in the optically sensitive material of a particle assembly are larger than if the same space were filled with a homogeneous optically sensitive material. However, because the thickness of the model includes also the pore volume, the

- 32 -

Fig.2:17 Vectorial addition of two Mohr circles.

relation between the retardation S and (a -a ) in eq.2:13 is in 1 2 ^

agreement with the definition of stress in a granular material. Moreover, the linear relation between the parameters in eq.2:13 demonstrates that the optical constant for an assembly of crushed material is theoretically the same as for a homogeneous material.

In tests it was attempted to approach the plane stress condition as much as possible. However, the micro stresses in the particles cannot be prevented from deviating from the plane state of stress. This is not in agreement with the two-dimensional stress optics. Although there is no mathematical or experimental evidence, it is presumed that the optically determined stress components describe a two-dimensional stress state in the plane of the sample, which is in agreement with the averaged stress.

Contrary to the assumptions made for the theoretical analysis of the optical filter system, the stress distribution is not homogeneous over the surface and thickness of a material point considered. Since the volumetric averaged stress is obtained by vectorial addition of the components of the micro stresses, the optical measurement is required to produce a similar result. Due to this requirement several combinations of optical filters are not suitable for this application. A circular polariscope (Frocht, 1946), for example, adds the principal stress

- 33 -

differences over the surface as scalars, while in the case of polarisers on both sides of the model rotating synchronously (e.g. Redner, 1976),. equal stresses inclined at 45° are suppressed. It is demonstrated below that the filter system used in this research adds the stress components almost vectorially. An additional advantage of this filter system over other systems is that a separation is obtained between the directions of a and a . In particular, this is convenient in samples with a complex stress distribution.

2.3.6 Optical averaging inhomogeneous stress

The optically measured components of the stress tensor are the length S of a vector and its orientation 2tJ). In a Mohr diagram this vector is the line segment between the pole and the centre of the Mohr circle. The addition of stress tensors can be performed by vectorial addition of all the line segments of the Mohr diagrams. An example of a two-dimensional state of stress is shown in Fig. 2:17. The optically measurable parameters S, ijj and S', ij)' represent two regions with a different state of stress, and S", i))" is the result of the vectorial addition. The mathematical relation is

6" = JtS2+S'Z+2SS'cos2(ijj -U>' ) D (2:27a)

\|>" = i-arcos[ij-rrcos2ij) -»-|7rcos2tJ)'] (2:27b)

The optical addition of the tensor components can be separated into an addition over the thickness of a test model and an addition over the cross-section of the light beam.

The optical addition over the surface can be investigated with eq.2:26. If the surface of a material point is divided into two equal parts with different stresses eq.2:26 becomes

I_ = I cisinSsin2(p-i|) )+|-sinS ' sin2( p-\|>' )+13 (2:28)

- 34 -

The maximum and the minimum value of I and the values of 6 to match can be found numerically. The values for S" and t|j" can now be calculated from eq.2:23 and eq.2:24 and verified with the results of eq.2:27. Some numerical examples of the vectorial and optical addition are given in Table.2:1.

two regions with different stress

*i 0 0 0 0 0 0

s 10 20 2 5

20 20

6' 10 40 12 5 20 20

*; 0 10 45 45 45 30

optical addition surface S"

20 58.1 12.1 7.1

28.0 34.5

*; 0 6.7

40 23 23 15

thickness S"

20 59.1 12.3 7.1 28.0 34.5

<)>;• 0 6.0 40 23 21 14

vectorial addition S"

20 59.2 12.2 7.1 28.3 34.6

*; 0 6.7 40.3 22.5 22.5 15

Table 2:1. Comparison of optical and vectorial addition of optically measurable stress components.

It appears that the optical addition is in reasonable agreement with the vectorial addition. The error is a maximum if l -ijri = 45° and increases if S+S' or |S-S'| increases.

Also, it can be deduced from eq.2:28 that the average light intensity during the rotation of the analyser is independent of the stress distribution in the measuring region. The sum of the light intensity, over a rotation of 180° of the analyser, can be derived with

ïïr i ■> E I Q = <±I s i n S s i n 2 ( 8 - i | ) )+±I s i n S ' s i n 2 ( B - i | ) ' ) + I )dB ( 2 : 2 9 )

P 0 J 2 o r ^ l 2 o ^ T i o ^

Since 2P ranges from 0 to 2 IT, the first two terms do not contribute to the summation. The contribution of the last term corresponds with 1801 . The averaged light intensity during rotation is Ï = I , so that I can be derived from the averaged light intensity also if the stress distribution is inhomogeneous

- 35 -

2 \ d2 HNHLYSER

Fig.2:18 Behaviour of light transmitted through two layers of double-refractive material with different stresses.

over the cross section of the light beam. The optical averaging of the stress components over the

thickness is independent of the filter system used. To investigate the agreement between the vectorial and optical addition, it is assumed that the light described by eq.2:15 is transmitted through a second layer of double-refractive material with a different retardation and principal stress direction. The two perpendicular waves of eq.2:15 are transformed into

a' = a cosot-a sina 2 2 1

a.' = a cosa+a sina 1 1 2

were a' and a' are the vibrations in the a' and a' direction of 1 2 1 2

the second layer, respectively, and a. is the inclination between o and a' (Fig.2:18). When the light leaves the second layer an additional retardation S' has taken place. Substituting the functions of a and a , eq.2:30 becomes .

1 2 ^ a^ = 2A^2Ccosasin(wt+S+S')-sinacosCut+S')D (2:31a)

(2:30 )

(2:30b)

- 36 -

a' = ~A4~2Ccosotcos((jüt )+sin<xsin(wt+S) H

Assuming that a coincides with the x-axis the vibration in the direction of the polarisation plane can be described with

a = a'cos( v-oO+a' sin(Y-a) (2:32) y l ' z '

The intensity of the transmitted light is defined by

I = 2AZ (2:33) Y y

were A is the maximum amplitude of the light wave at the actual position of the polariser. I and the maximum and minimum light intensity during the rotation of the analyser are determined numerically and the values of 5' and i|)" are calculated with eq.2:23 and eq.2:24. If the results are compared with the vectorial addition (Table.2:1) it is shown that a similar deviation is found as with the addition over the surface. Also, it can be shown numerically that I = Ï if the stress

o distribution is not homogeneous over the thickness of the sample.

It is not necessary to investigate the addition over more than two regions or layers with different stresses, since the motion of the light will always be elliptical (harmonic waves with the same frequency are added). Because the distribution of the micro stresses is more or less random, positive and negative deviations can take place. It is therefore not to be expected that the optical summation will result in a large error. 2.4 DISPLACEMENT MEASUREMENT

2.4.1 Detection of marked particles

The deformation in a plane strain sample is determined from the co-ordinates of marked particles in successive stages of a test. Because the assembly of crushed glass is transparent, normal light can be used to observe the position of black

(2:31b)

- 37 -

LIGHT SOURCE LENS

IMRGE 11

>

MRRK

■■-¥■■

CRUSHED GLHSS

\

Fig.2:19 Diagram of the optical system to detect a black mark in an assembly of crushed glass.

particles which are situated in a plane halfway through the thickness of a sample. To eliminate the photographic step a digital camera is used to determine the co-ordinates of the marks directly in the plane sample. Since the area considered and the number of marks are rather large (e.g. 900 cm with 40 to 100 marks) it is not possible to analyse the total area accurately enough all at once. Therefore the position of the digital camera is controlled by an accurate x-y scanner, so that the area of the sample can be divided into small square elements which can be analysed consecutively.

The most important part of the camera is an integrated circuit with a window of about 6 mm square which contains a matrix of 128x128 light-sensitive pixels (diodes). Electronics in the camera converts the light intensity, received by each pixel, into a voltage which is connected to an output line for a short time. The voltage of the pixels can be read by a computer. A synchronisation signal, which is also produced by the camera is used to separate the signals of the pixels. If the image of a region with a black mark, which has a diameter of about 2 mm, is focused one-to-one on the window of the sensor (Fig.2:19), a number of pixels will receive a low light level. The output voltage of a pixel is converted into black (1) or white (0), depending on whether the voltage is respectively lower or higher than a reference value. The status of each pixel is read line by line and is stored sequentially in the computer memory. The

- 38 -

COLUMN

127

ROM

127

o 0

o 128

o 1

0 129

o 2

130

o 127

255

o 16383

Fig.2:20 Relation between the memory locations and the row and column number of a picture element.

relation between the row and the column number and the actual memory location is represented in Fig.2:20.

2.4.2 Determination of centre mark

The position of the centre of the image of a black particle, with respect to the upper left corner of the pixel matrix can be derived from the row and the column number of the dark pixels. If the number of the memory location of the first pixel is assumed to be zero, the co-ordinates of a pixel ( x corresponds with the memory location n can be calculated from

y ) which

xP = i n t S) s

y = n s - k y „ ' p J P

( 2 : 3 4 a )

( 2 : 3 4 b )

were k is the number of pixels in a row and s is the distance between the pixels. The centre of the image of a black mark ( x , y ) can be found by adding together the x and y co-

- 39 -

ordinates of all the black pixels and dividing the results by the total amount t as formulated by

x = ^Zx (2:35a) m t p y = ir£y (2:35b) -'m t Jp

The absolute position of a mark can be calculated from the known co-ordinates of the upper left-hand corner of the pixel matrix with respect to the origin of the scanner.

To prevent errors in the interpretation of the image of a mark several control parameters are used, such as - the number of dark pixels of the image of a mark; - the number of dark pixels at the boundary of the pixel matrix; - the measured displacement of a mark. The number of dark pixels which is covered by a mark can be

used to determine the surface of the black image, so that the computer can distinguish a mark from a small accidental dirty spot. The number of black pixels at the boundary of the pixel matrix indicates if a mark is close to a boundary segment of the model. If the measured displacement of a mark after a deformation step is unreasonably large, e.g. in comparison with measured displacements of boundary segments, it can be concluded that there is something wrong.

Because the whole image of a mark is used to determine the centre,, the accuracy is not significantly dependent on the shape of the mark. However, the assembly of crushed glass is not perfectly transparent due to small deviations in the refraction index and dirt, so that the image of a mark is often accompanied by more or less randomly distributed black spots (noise). To increase the accuracy a procedure has been developed to eliminate the noise before the co-ordinates of a mark are determined definitively.

- 40 -

b

O m~

O Fig.2:21 Example of an image of a mark and noise on the pixel

matrix of the image sensor.

2.4.3 Elimination of noise

A machine language routine has been developed to eliminate noise in an image of a black mark. An example of an image is represented in Fig.2:21. The mark is located by a circle, while some irregular indicated regions are supposed to be black fields due to dirt in the assembly. Because the surface of the mark dominates and the noise is distributed more or less at random, the main point of the black pixels is always situated within the boundary of the image of the mark. In Fig.2:21 the provisional centre of the mark is indicated by the intersection point of the lines aa' and bb'. To clean the image the program starts with a pixel in the first row which is located on line aa'. The pixel number is reduced until the left-hand part of the boundary of the matrix is reached. The contents of the corresponding memory locations are checked for black or white. If a white pixel has been observed once, all the following pixels in that segment of the row are also set to white. Next, the right-hand segment of the row is processed on the same way. This procedure is repeated

- 41 -

for all the rows, and the columns are similarly processed. The result is that all the black spots which are not connected with the black region of the mark are eliminated, so that the centre of the mark can now be determined definitively.

2.5 DATA PROCESSING

2.5.1 Collected data

During the measurement of the parameters which describe the condition in the granular material of a test model a considerable amount of data is produced. Since the digitised conditions of several tests are required to analyse the stress strain behaviour of the granular material, it is necessary that the data of each stage can be read easily by a computer. Therefore the measured data of the different stages is stored in files on a disk. The file name includes the name of the test and a number which represents the sequence. The following data are stored in a file - name, reference number of the measurement, date, time, number of strain gauges, thickness of the sample;

- co-ordinates of the angular points of the boundary, the location of strain gauges;.

- number of rows and columns of the nodal points at which the stress components are measured, the distance between the nodal points, and the co-ordinates of the first nodal point of the mesh;

- number of rows and columns of black marks in the field; - number of marks at the boundary; - a matrix containing the relative principal stress differences; - a matrix containing the principal stress directions; - a matrix containing the co-ordinates of the black marks;, - the output of the strain gauges. The data is collected by a single-board microcomputer which

controls the measuring device (Section 3.4.1). The digits are transmitted to a more powerful computer, which stores the data and graphically displays the elementary parameters of the

- 42 -

Fig.2:22 a) Graphical representation of the distribution of the principal stress directions, based on measurements at 100 discrete points, b) Visible pattern of light stripes, viewed with a circular polariscope.

stresses, displacements and strain gauges graphically. The course of a test can be conveniently monitored in this way. In a later stage the data of one or more tests are subjected to a more detailed analyses.

2.5.2 Principal stress trajectories

Since the digital,point information of the principal stress directions is not very convenient for visual inspection, a computer program has been developed which converts the discrete measurements into a continuous regular pattern. The pattern is formed by curved lines, principal stress trajectories, of which the tangent at any particular point represents the principal stress direction at that point of the sample. A computer plot of such a pattern is shown in Fig.2:22a. The plotting procedure is

- 43 -

started in points at' the boundary. The distance between the starting points determines the concentration of the trajectories. The points in the field of a trajectory are the end points of small line segments, which are the chords of circle segments describing the local curve. The direction of the chord is found by means of an iteration procedure. Due to this procedure the shape of a trajectory is not dependent of the plotting direction. A trajectory is assumed to have been completed when the boundary is reached again. To prevent a heavy local concentration of trajectories due to convergence, the co-ordinates of the starting point and end point of each trajectory are remembered. A new trajectory is plotted only if its end is not too close to one of the previous starting or end points. When a picture is almost complete, all the distances between the intersection points of the trajectories and boundary are checked. If a too large a distance is found, some concessions are made with regard to the minimum distance. The choice wether a concession is made or not is dependent on a few parameters. The value of the parameters, which depends on the shape of the model and the loading program, have to be changed manually. In the first instance the major principal stress trajectories are plotted. The starting point of the longest trajectory is remembered, because the points of this trajectory are used as starting points for the minor principal stress trajectories. The advantage of this procedure is that a very regular distribution of trajectories is obtained. Since not all boundary segments are reached in all cases in this way, a routine is used to find and fill large gaps. If a picture is not acceptable, an option can be activated to define start points of extra trajectories manually. Manual help, however, is in general not necessary.

Because the principal stress directions were measured at discrete points, an interpolation procedure had to be used to determine the direction at an arbitrary point. A simple linear procedure is used, which proceeds as follows (Fig.2:23)

- 44 -

xi-yi * " x2.y2 <

*,4

x3.y3 * r x ^ Fig.2:23 Graphical representation of the parameters used to

determine the averaged stress directions at an arbitrary material point.

^34= ^ » , V * ( ^ 3 ^ (2:36b) t 3

ï = tfi12-!—2- (ib3'*-tb12) (2:36 C) Ti vi y -y Ti Ti

Since it is not known in advance in which direction a trajectory is plotted, a procedure has to be used to adjust the given directions in the matrix. If a new starting point of a trajectory is determined, a reference direction, which is the angle of the perpendicular at the boundary in the direction of the field, is also calculated. This direction is compared with the directions tj) at the surrounding nodals, +180 and t|)-180 . The angle which is closest to the reference direction is used for further calculations. The last calculated direction of the line segment of a trajectory is used as a new reference value. The reference direction can also be used to detect isotropic points. An example of such a point is shown in Fig.2:22. A feature of a region close to an isotropic point is the large gradient in the principal stress direction. The interpolation procedure will fail in this region because it is not possible to adjust the directions at the four surrounding nodal points in good agreement with each other. This results in an unreasonable shape of the

- 45 -

trajectory, so that the plotting procedure has to be stopped if a trajectory is close to an isotropic point. In the program the detection of an isotropic point is based on the differences between the adjusted principal stress directions at the four surrounding nodal points. If one of the differences exceeds a given value the trajectory is assumed to be completed.

In some cases it is convenient to be able to label a region which cannot be traversed by trajectories. This option is realised by using real and integer values for the principal stress direction. If the principal stress direction has a real value at two of the surrounding nodals, the plotting procedure of the trajectory is stopped. Since a small increment is used to convert the integer value into a real one, no significant influence can be expected in regions where only one of the four surrounding nodals has an integer value.

The calculated trajectories, based on the optically measured data, were found to be in good agreement with the visible pattern in the sample. Fig.2:22b shows a photograph of the sample using a circular polariscope (Frocht, 1946). The visible pattern of light stripes are formed by those particles which are arranged in more or less straight chains in the direction of the local resultant force. Since the chains transmit the largest forces it can be reasonably supposed that the light stripes represent the principal stress trajectories (Wakabayashi, 1957, 1959, Drescher and De Josselin de Jong, 1972 and Oda and Konishi, 1974).

2.5.3 Stress distribution

Rectangular Cartesian co-ordinates x,y are used to describe the configuration of a plane strain body with an internal stress distribution a , a , a and a . Neglecting the body forces xx yy xy yx ' of the material, the equations of equilibrium are

_ x x + _ | i = o (2,37a,

3cr 3a — ^ H ^ = 0 (2*37 ) 3y 3x ^ . J / )

- 46 -

To process the optically measured data it is most convenient to express the stress components in terms of (a -a ) and i|). The stress components in terms of a , a and ty are

a = ^(a +cr -(a -a )cos2ü>) v (2:38a) xx 2 2 l z x T

a = 4(0 +o- +(o -o )cos2i|)) (2:38 ) yy 2 2 l 2 l ^

a = a =-^(or -o )sin2t|) (2:38c)) xy yx 2 2 l T

Substituting eq.2:38 into eq.2:37 gives

|~(^(a +a -(a -a )cos2i|))~f-(i(a -o )sin2ij)) = 0 (2:39a) oX Z 2 1 2 1 dy Z 2 1 |-(i(cr +a +(a -a )cos2x|)) -f-(^(a -a )sin2i|)) = 0 (2:39b) dy Z 2 1 2 1 dX Z 2 1

Differentiating eq.2:39 (all terms are functions of x and y) and rotating the x-axis in the direction of a ( i|)=0) the equilibrium equations reduce to

-Ji-(a -a )|f- = 0 (2:40a) dt 2 1 dt 1 2

2 1

were t and t are orthogonal curvilinear co-ordinates which coincide with the principal stress trajectories. The equations derived, which are known as the Lamé-Maxwell equations of equilibrium, can be used to calculate the increment of a or a along a principal stress trajectory, if (o'2"0r1) a n d ^ a r e known at each point of a test model. Since the principal stress difference and direction have to be approximated by interpolation (following the procedure of eq.2:36), the integration of eq.2:40 has to be performed numerically. An integration step along a ^-co-ordinate is shown graphically in Fig.2:24. The trapezoidal integration rule gives

47 -

c^-0",)

Fig.2:24 Integration step along a a -trajectory.

F i g . 2 :25 Sign conven t ion fo r dij>/dt ( = 1 / R ) .

Aa , ■ X{K-%>r)A*K-vU7)B} (2:41)

If At is always taken as positive, the sign of di|>/dt determines the sign of Aa ( a is negative for pressure). The sign of diji/dt depends on the curve of the trajectory, which is shown graphically in Fig.2:25. The increment of a along a t -coordinate can be similarly calculated.

For the calculation of the stress tensor at an arbitrary point of a model, the real value for a or a at some point and a

- 48 -

Fig.2:26 Mohr diagram of data measured at a material point close at the boundary.

multiplication factor M ( in eq.2:14) have to be known. These parameters can be derived if, besides the optically measured parameters 8 and i|), two normal stresses on a plane are also known. In Fig.2:26 a point on a boundary is shown at which the normal stress a , its direction p, the principal stress direction ij) and the relative principal stress difference S are known. It is shown graphically that the pole P and the origin of the Mohr diagram can be determined from these data. The principal stresses at the point considered can be expressed in M, 5, a ,4) and 0 with

o = a -±MS(cos2(B-i|) )+l) = or -l n 2 Ti n MT (2:42a)

a = or +±MS(cos2(B-il) )-l) = a +MT 2 n & l n 2 (2:42b)

To determine the multiplication factor M, the normal stress at a second point is used (Fig.2:27). The relative increment of a from A to B can be calculated using eq.2:41 and is assumed to be

Aa = MA ; thus (o )_= (a,).+MA l 1 XJ 1 A (2:43)

With eq.2:42 this gives

M = (VB-<VA

( T I ) B - ( T I ) A + A !2:44)

The absolute values of a in A and B are determined by substituting M into eq.2:42. In order to calculate the stress

- 49 -

Fig.2:27 Data used to calculate M, a and a .

tensor at an arbitrary material point of the model, either point A or point B can be used as starting point for the numerical integration with eq.2:41.

The calculation procedure is used in a computer program to derive stress maps from the measured data. Since the boundary of a model is in general not completely covered by the mesh of the measuring points, the first action in the program is to expand the matrix which contains the field information. The data in the ' new nodal points are set to the same value as the nearest points of the original matrix. Every time a pair of rows and columns is added, it is checked if the boundary of the model is located within the new boundary of the mesh. If the matrix is expanded sufficiently, the value of the multiplication factor M is determined, using all the possible combinations of points at the boundary where the normal stress is measured. If too large differences are found, e.g. due to a faulty load cell, the program asks for manual help. If the multiplication factor has been determined, the absolute values of the principal stress differences can be calculated. An example of the distribution

- 50 -

Fig.2:28 Distribution of the principal stress difference, based on 100 measuring points.

jM i im l i i f f l ^

, I , I

Fig.2:29 Calculated stress distribution in a sample, using optically measured data (1 scale = 100 kN/m or 1 cm).

- 51 -

of {a -a ) in a test is shown graphically in Fig.2:28. Next, the state of stress at the centre of the sample can be determined by calculating the increment of a or a from the points at the boundary, with known normal stress, to the centre. If the calculated stress components are in good agreement with each other, averaged values are used to set the parameters which describe the stress state at the centre. For further calculations the centre of the model is used as the starting point for the numerical integration procedure. In the first instance the distribution of the normal stresses at the boundary are calculated. Some of these stresses can be compared with the measured normal stresses to illustrate the accuracy of the stress map obtained. In Fig.2:29 the measured normal stresses are symbolised by arrows. Several other parameters can be calculated to visualise the stress condition of the model. In Fig.2:29 the two-dimensional state of stress at some material points is derived, and the distribution of the mobilised friction angle d>

m is shown in Fig.2:30.

Because two arbitrary points are in general not located on the same trajectory, a procedure has been developed to perform the numerical integration along straight lines. In Fig.2:31 point A is assumed to be a material point with a known state of stress and the unknown state of stress in point B has to be determined. The distance AB is divided into equal line segments with length As. At the start and the end point of each line segment a chord is drawn of the circle segment which represents the local curve of the major and minor principal stress trajectory, respectively. The intersection point C' of the line segments defines an integration step At and At along the t - and t -trajectories, respectively. First, the increment of a from A to C' is derived with eq.2:41. At point C' the local principal stress difference is used to calculate a . Next, the increment of cr is calculated from C' to A'. At point A' the value of a is derived again. This procedure is repeated until point B is reached.

Several routines which are used to plot the principal stress

- 52 -

x Fig.2:30 Calculated distribution of the mobilised friction an

ile (5°< <|> >43°).

Fig.2:31 Procedure for performing the integration along a straight line.

- 53 -

trajectories are applied in the calculation of the stress distribution. The principal stress difference at an arbitrary point is determined in a similar way as the principal stress direction, and the same procedure is used to adjust the directions at the four surrounding nodal points. The direction of the line AB is used as the reference direction in this case. Since the numerical integration will fail in a region close to an isotropic point, the procedure to detect such a region is used in each integration step. However, the integration procedure is not stopped. Instead, the results are labelled, so that if unreliable data is obtained this can be detected.

2.5.4 Strain tensor

The positions of marked particles in two successive tests are used in a computer program to derive the distribution of the strain tensor in a plane strain sample. If in a region with a homogeneous deformation the displacements of three marks are known, sufficient data is available to calculate the strain tensor in that region. To be able to measure the displacements between the marks accurately enough and to be sure that a region is considered which contains sufficient particles, a rather large distance between the marks is required. However, since the deformation of a granular material is often accompanied by local shear bands, there is no guarantee that the deformation is homogeneous in the considered region. For this reason the displacement of four marks is used to calculate the strain tensor at a material point, so that additional information is available to check if the strain can be approached reasonably well by a tensor.

Although the distances between the marks are not exact, the same the distribution is defined as a matrix by means of a number of rows and columns. The row and column numbers are used to define a particular mark and to divide the marks systematically into groups of four. The measured co-ordinates of four marks before and after a deformation step are shown in Fig.2:32. The

54

Fig.2:32 Example of the measured location of four marked particles, before and after a deformation step respectively.

components of the relative displacement diagram are determined four times, using all the possible combinations of groups of three marks, such as 1,2,3 ;1,2,4 and so on. If the derived tensor components are in good agreement with one another, the averaged values are used to describe the strain tensor at the centre of the four marks.

If the components of the relative displacement tensor are defined by e , e , e and e , the angular deformation y, the xx xy yy yx rotation w, the principal strains E , E and the direction of the ijor principal strain p are given by

Y = e +E (2:45) ' yx xy OJ = 0.5(Eyx-exy) (2:46)

e = 0.5(E +e )+0.5J(e -E ) Z+(E +E ) 2 (2:47) 12 xx yy xx yy yx xy

,E +E . P = 0.5arctg Y X x y (2:48)

1 Veirir vw/

In Fig.2:33 the displacement of three marks are considered. One of the marks is used as reference point, so that the displacement of two marks can be used to determine the unknown components of the relative displacement tensor. The displacement of the marks in the x- and the y-direction ( u and v respectively) can be

- 55 -

Fig.2.-33 Graphical representation of the parameters used to derive the components of the relative displacement tensor.

derived from the measured co-ordinates and can be expressed in terms of tensor components by

u = l(e cosot+e sina) xx xy

v = l(e COSOI+E sina) yx yy

were 1 and a are the length and direction, respectively, of the line segment between the mark considered and the reference point. The displacement of the two marks gives four equations, which is sufficient to determine the four tensor components.

A plot program has been developed to visualise the four relative displacement diagrams which are obtained if four marks are taken into account. As illustrated in Fig.2:34, the diagrams conveniently show whether the deformation can reasonably be described by a tensor or not. In the computer program the decision as to whether the deformation is acceptable is made dependent of the ratio of the radius of the smallest to the largest Mohr circle. If this ratio exceeds a given value, the principal strains are not plotted at that particular material point. If the deformation in a sample is very inhomogeneous, stereo observations of two photographs of successive deformation

(2:49a)

(2:49b)

- 56 -

Fig.2:34 Graphical presentation of Mohr diagrams at a material point with a non-uniform (a) and an uniform (b) deformation.

^ / - -T-

X X X A\ ,/G,

j i 1 1

Fig.2:35 Calculation example of the distribution of the strain tensor in a specimen.

- 57 -

steps in a simple viewer can be used to search for marks which are located in a part of the sample with homogeneous deformation.

In Fig.2:35 an example is shown of a specimen in which the strain is systematically determined. The direction of the principal strains are compared with the directions of the principal stress in this case. Since the data is available in digital form, many elaborations can be performed to analyse and visualise parameters which represent the mechanical behaviour of a granular material.

- 58 -

3 TEST SETUP

3.1 OPTICAL MEASURING DEVICE

3.1.1 Mechanical part

An important part of the measuring device is a computer controlled x-y scanner which moves the optical systems to any desired point of the surface of the plane model. The light source and the sensor are each fitted at the bracket of a fork (Fig.3:1), so that the model can be situated in the path of the light beam. The fork is connected to a beam by means of linear ball bearings and tempered steel shafts, to permit a translation (y-direction) of the fork along the beam (Fig.3:2). The centre of the beam is connected to a vertical frame by means of a shaft, so that the beam can be rotated in a vertical plane. Due to this construction, plane horizontal as well as plane vertical models can be investigated. The height of the beam can be adjusted, which is useful if plane horizontal models are used. A horizontal frame with axes and linear ball bearings make it possible to translate the optical systems in the x-direction. The horizontal frame is equipped with wheels, so that the whole device can be easily transported.

Adjusting screws can be used to fix the horizontal frame in a horizontal position to the floor.

The translation in the x- and y-directions is controlled by means of translation screws which are driven by step motors. The reduction ratio is such that one step is theoretically a displacement of 1/600 mm. In practice, however, a discrimination of about 1/200 mm could be measured. The bearings of the translation screws and the gear-wheels between stepper and translation screw cause some hysteresis if the translation direction is reversed. The effect of this phenomenon on the displacement measurements, however, can be eliminated by approaching each point from the same direction, which could easily be realised by means of software.

- 59 -

LIGHT SOURCE ROTRTING FILTER LIGHT" 'SENSOR

té CELL WITH CRUSHED GLRSS..-

FORK ] /

LRSJER .-••-'

.PIGiI6L..CRMERR/

F i g . 3 : l Diagram of the loca t ion of the o p t i c a l components on the fork of the scanner.

F ig .3 :2 Drawing of the x-y scanner.

- 60 -

The range of the scanner is about 1.5 metres in the x-direction and 1 metres in the y- direction. The particular reason for this large range is that a model can thus be made properly accessible to allow technical work to be carried out on it or a sample to be photographed.

3.1.2 Optical system

Two different optical systems are used to perform the stress and displacement measurements, respectively. The systems were developed for this particular purpose. Several physical phenomena and technical problems had to be analysed and solved to obtain a reliable measuring method. General problems in the past were deviations in the measuring results, caused by temperature gradients and fluctuations. Temperature gradients influence the properties of the optical filters and cause internal stresses in the glass wall of the test model. Although the experiments were performed in a temperature controlled room, temperature gradients and fluctuations could be caused by heat production of light sources and electronic equipment. This was one of the reasons why a laser was chosen as the light source for the stress measurements and light-emitting diodes as the light source for the digital camera. These sources produce very little heat radiation, so that a simple air cooling circuit could be applied to prevent temperature fluctuations in the optically sensitive materials by convection. The temperature of the stepper which rotates one of the optical filters is also controlled by this cooling circuit. Special attention has been paid to the construction of the rotation mechanism, to prevent heat transport to the optical filter by conduction.

The optical system used to detect the marked particles in the transparent assembly of crushed glass comprises a light source, a lens and an image sensor (type TN2200 of General Electric) with an active surface of 6mmx6mm containing a matrix of 128x128 light-sensitive pixels. The distance between the sensor and the

- 61 -

lens, and between the object and the lens, is twice the focal distance of the lens, so that a one to one image on the active area of the image sensor is obtained. Because the marks were not located exactly in the same plane in the sample of granular material, the distance between the marks and the lens was not precisely the same for all the marks. It appeared, however, that a lens with a focal distance of 40 mm still gave a sufficient sharp image of a mark, if the distance lens-object deviates 10 mm from the most ideal location. The marked particles can be positioned easily within this range in the sample. Of course some invisible variations in sharpness will take place. However, because the distance lens-object of a particular mark is always the same, this effect will not significantly affect the accuracy of the displacement measurement. An additional option of this optical system is that it can be used to detect the displacement of boundary segments. For this purpose the plane of the boundary, which is visible to the camera, was coated with a white layer, and black spots on the layer were used to schematise the shape of the boundary. Since the boundary was not transparent, a front light was used to make the black spots visible to the image sensor. The front light is controlled by the computer and is switched on only if the points at the boundary have to be detected.

In Fig.3:3 the lay-out of the optical system is shown, which was used to determine the averaged stress components in photoelastic granular material. A monochromatic polarised light beam of 3 mW, with a diameter of 1.26 mm and a wave-length of 632.8 nm, is produced by a He-Ne laser (model No. 3176H from Hughes). The diameter of the light beam is expanded 10 times by means of a collimator. The expanded beam of about 13 mm transmits through a dichroic polarisation filter (from Carl Zeiss). The light intensity of the laser can be controlled by rotating the polarisation filter with respect to the polarisation plane of the light. Next, the polarised light transmits through a mica retardation plate (type 02WRM013,

- 62 -

LIGHT SENSOR

ROTRTING POLRRISER GLRSS WRLL j J LENS

A A A CRUSHED GLRSS

¥ » y 1 f%«l DRI

-LIGHT VRLVE

-QUARTER WAVE PLRTE -POLRRISER -COLLIMRTOR -LRSER

Fig.3:3 Diagram of the optical system used for the stress measurement.

632.8 nm from Melles Griot) the optical axes of which are inclined precisely at 45 to the plane of polarisation. The retardation of the plate is exactly a quarter of the wave-length of the polarised laser light, so that circularly polarised light is obtained. Since the light intensity over the cross-section of the light beam of a laser source is not homogeneous, an aperture of 6 mm is used to transmit only the most homogeneous centre of the light beam. When the light has been transmitted through the model, a second aperture of 6 mm and two lenses, with a focal distance of 20 mm and 30 mm respectively, are used to reduce the transmission of dispersed light. Special attention has been paid to the mounting of the lenses to prevent internal stresses in the lens material. For this purpose a rubber compound was used to obtain smooth contact between the lens and the metal mount. Via a third aperture of 9 mm and a rotating polarisation filter the intensity of the light is measured by means of a photodiode ( type HAV-4000A from EG&G INC.). A photodiode with a large circular active area (11 mm diameter) is used because the position of the light

- 63 -

spot changes a little due to some refraction of the rotating polariser. The ratio of the focal distances of the lenses increases the diameter of the light beam up to 9 mm. This reduces the intensity of the transmitted light per unit area of the polarisation filter and so reduces the experimentally observed influence of the memory effect of the rotating polarisation filter on the measuring results. For the same reason a light valve has been used to prevent light transmission through the polarisation filter if the filter is not rotated. The position of the light valve is controlled by the computer. The rotating filter is mounted in the hollow axis of a gear-wheel. The gear-wheel is driven by a stepper controlled by the computer. Whenever a measurement was performed, the filter was rotated through 180 and the light intensity received by the photodiode was measured and digitised at every degree. When the first measurement of a series was started, the zero position of the rotating polarisation filter was found by the computer by detecting a hole in the gear-wheel by means of a light source and a photodiode.

3.1.3 Electronic circuit

Because a large number of data have to be collected and processed, the measuring procedure and data handling have been completely automated. An electronic circuit has been developed which permits programmable control of the various functions of the measuring device. This was very convenient during the development of the measuring method, since the measuring procedure could be easily changed by means of software. Moreover, because the electronic system has been developed in many details in the laboratory, it was often easy to expand the system with additional options. The brain of the system is an inexpensive single board micro computer (SYM 1 from Synertek Inc.) with a simple BASIC interpreter (computer I in Fig.3:4). The memory has been expanded up to 48 Kbyte. Since only 32 Kbyte of address space was free, two 16 Kbyte

- 64 -

REMOT CONTROL

I/O-PORTS

INTERFACE CRMERH

CRMERR

_^X-TRRNSLATION „^-TRANSLATION .^FILTER ROTATION

INTERFACE STEP MOTOR

COMPUTER

TELETYPE

COMPUTER

m DATA RQUISITION

COMPUTER

n T TV

.LIGHT SENSOR

.STRAIN GRUGES

.DISPLACEMENT TRANSDUCER

Fig.3:4 Diagram of the electronical system.

memory boards were page mapped. For communication with the measuring device the micro computer has been expanded with 48 input or output ports. The status of an input port can be read in a program and depends on the position of electronic and mechanical switches, the output of an AD-converter and the output of a digital camera. The status of the output ports can be set by the computer and are used to control a multiplexer, the AD-converter, step motors and, for example, a motor which displaces a boundary segment in a test.

The electronic and mechanical switches are used for several detection functions, such as the detection - of the zero point of the x- and the y-direction of the scanner

and the rotating filter; - whether the light sources and step motors are switched on; - of the switches of the remote control unit of the scanner. The switches are connected directly to the input ports of the computer, and the status of the switches can be read, for example, with a PEEK command in a BASIC program.

A 12 bit AD-converter (type ADC 80 of Micro Network

- 65 -

Corporation) was used to convert the analogue output signal of a sensor, which is chosen by the multiplexer, into a binary code. The data lines of the AD-converter are directly connected to 12 input ports, and one output port has been used to transmit a command which starts the conversion. The AD-converter is operated by a machine language routine.

The digital camera uses 4 input ports and has been connected via an interface to the computer. The most important components in the interface are a pulse generator, which controls the transfer speed of the pixel signals (12,000 pixels per second), and a comparator, which converts the voltage of the pixels into a 0 or 1, depending on the light intensity received by the actual pixel. Three lines have been used for the synchronisation and for the detection of the end of a line and the end of an image.

The multiplexer is a programmable switch which is used to connect the amplifiers of the different sensors to the AD-converter. Four output ports are used to select one of the 16 channels. Besides the photodiode of the optical system, several strain gauges and displacement transducers can be read out. The data were used to control the measuring device and were stored with the optically measured data to complete the digital reproduction of the stage of a test. The multiplexer is operated by a machine language routine.

The three step motors are controlled by six output ports. For each stepper one port has been used to define the direction of rotation and a second port transfers a pulse train which determines the number of rotations and the speed of rotation. An interface has been used to translate a pulse on the output port of the computer into one step of the stepper. The operation of the steppers is performed by means of a machine language routine.

Several outputs are used to control single functions via reed relays or solid-state relays, such as the front light of the digital camera the light valve in the optical system, switching of memory boards, an optional automatic film camera and motors to

- 66 -

operate a test. Since the control computer has no graphical options,

two additional computers have been used to produce graphical output. Computer II in Fig.3:4 (CBM64 from Commodore) visualises the actual image of the digital camera, which was convenient if a mark has to be searched for under remote control. A third computer (type HP9845 from Hewlett Packard) can be optionally connected to the control computer. This computer has advanced graphical options, so that the measuring results, such as the principal stress directions and the displacement of the marks, can be plotted on line. This appeared to be a very convenient option to monitor the course of a test.

3.1.4 Software and measuring procedure

A BASIC program and several machine language routines have been developed to control the measurement. The BASIC program organises the machine language routines into a logical sequence and elaborates and transfers the data of the routines and external devices. The machine language routines are used for fast operations, necessary to control the sensors and actuators. The following machine language routines are available. - Initialise This routine is called once when the program is

started. Several address locations are set to a default value and the ports of the computer are defined as an input or an output. - Digitise channel x This routine is entered with a parameter which is used to

define a particular channel by means of the multiplexer. A convert command is transmitted to the AD-converter, and the reflected 12-bit binary code, representing the output of the actual sensor, is stored in two 8-bit memory locations. The binary code is translated into a digit in the BASIC program. - Rotate filter

- 67 -

This routine performs the stress measurement at a material point of the sample. The amplifier of the light sensor is connected to the AD-converter by encoding the multiplexer, and the light valve is opened. The direction of rotation of the stepper, which rotates the polarisation filter, is defined and square waves are transmitted to the interface of the stepper. The output of the light sensor is converted every degree (=7 steps) into a binary code during the rotation of the optical filter through 180°. The binary code is stored sequentially in two 8-bit address locations. Before the measuring procedure is started, the initial position of the filter is checked by reading the output of an optical switch. If the position is not correct the filter is rotated until the optical switch detects a hole in a wheel. - Determine extreme intensities This routine determines the extreme and averaged values of the

previously measured light intensities. The corresponding rotation angle of the optical filter is deduced from the sequence of storage. If a peak; is defined by a number of measurements of the same value, the averaged value of the corresponding rotation angles is determined. The relevant data are stored and used in the BASIC program to calculate the stress components. - Control x-, y translation This routine performs the translation of the scanner in the x-

and the y-direction. Before this routine is called the number of steps in the x- and the y-direction and parameters to define the direction of the translation are calculated in the BASIC program and stored at specific addresses. 600 steps cause a translation of 1 mm. As far as possible, the steppers rotate simultaneously. One of the steppers is driven singly if the displacement in one of the directions has to be completed. Since it is not permissible to start or stop the steppers at the maximum speed, a procedure has been developed to increase and decrease the frequency of the pulse train linearly with time. The moment of delay is made dependent on the actual frequency and the remaining number of steps. The frequency ranges from

- 68 -

800 to 3600 Hz. - Find origin This routine initialises the position of the scanner and the

rotating polarisation filter. The routine is called once when the BASIC program is started. The steppers of the scanner and rotating filter are activated without a digital limit. The translation in the x- and y-direction of the scanner is towards the origin until an adjustable microswitch is touched, which is detected in the program. The rotation of the filter is stopped if the program detects a signal of a photo diode, which detects a hole in the wheel used to rotate the filter. - Remote control scanner This routine allows remote control of the position of the

scanner. The commands are received via an input port from the switches of a control unit. The program uses two three-byte counters to keep the actual co-ordinates. - Read and store image This routine handles the output of the digital camera.

First, two 16 Kbyte memory boards are exchanged under software control to be able to store the status of the pixels of the image without destroying the BASIC program. Since the camera operates continuously, the program waits in a loop until the end of frame signal is detected. From that point the status of the pixels are read via a comparator and are stored sequentially. A clock pulse is used to synchronise the program with the camera. The rate of transmission is 12,000 pixels per second. - Determine centre of image This routine determines the main point of the black image,

following the procedure described in Chapter 2.4.2. The row and column number of the main point are stored in two addresses. - Clear noise image This routine eliminates black regions which are not connected

to the boundary of the image of the mark, following the procedure described in Chapter 2.4.3. If the noise is eliminated, the main point is determined definitively with the previous routine and the number of black pixels representing

- 69 -

the mark were counted. - Check boundary frame This routines checks whether there are black pixels at the

boundary of the pixel matrix. The number of black pixels are stored in four addresses representing the four sides of the frame.

The measuring procedure described in the BASIC program can be separated into the following sections. - Start procedure - Change data - Stress measurement - Displacement measurement - Read external sensors - Compare sensors During the start procedure of a measurement several variables

can be set to define the operation of the scanner. In the first instance some practical data is entered, such as: name of test and a start number. A real-time clock takes care for the date and time of a measurement. The mesh of the stress measurement is defined by the number of rows and columns and the distance between the measuring points in the x- and y-directions. The location of the mesh in the plane of the model is defined by the co-ordinates of the first measuring point, which are determined by using the scanner as a remote-controlled co-ordinates digitiser. The program asks the operator to move the light spot of the laser to the desired position. If the operator is satisfied, a button is pressed to leave the remote control, routine after which the co-ordinates are stored in a variable. Similarly the co-ordinates of strain gauges and the angular points of the boundary can be digitised. The number of boundary points and strain gauges and the calibration factor are entered in advanced. The marked particles are also defined as a matrix. Since the distances between the marks are not exactly the same, an estimated average distance is entered. Next the program allows the user to move the camera under remot

- 70 -

control to the first mark of the mesh. If the mark is visible to the camera, a command is given by the operator to continue the program. The co-ordinates of the centre of the mark with respect to the origin of the pixel matrix of the image sensor are determined and the deviation from the physical centre of the pixel matrix is translated into steps which are used to displace the scanner, in order to centre the mark in the frame of the pixel matrix. Next, the program moves the scanner over the given average distance in the x- or the y-direction to search for the next mark. This procedure is repeated until the co-ordinates of all the marks are digitised. If a mark is not found, a searching procedure is used, which inspects an area of four frames (about 12mmxl2mm). If the mark is still not found, the program asks for manual help. The operator has to find the mark under remote control in a similar way as the first mark. Since a rather large variation in distance is allowed between the. particles and because the deviations do not accumulate due to the centre procedure, the co-ordinates off all the marks are found in general automatically. The initialisation of the co-ordinates of the black spots at the boundary has to be performed under remote control. Because the boundary marks are not numerous, this procedure does not take up much time. In this section of the program a front light is switched on to make the black spots at the boundary visible to the camera.

In the section "change data" the content of several variables can be changed. It is, for example, possible to displace the mesh of material points for the stress measurement and to adjust the co-ordinates of the strain gauges and angular points of the boundary. Furthermore, any particular mark which is lost for some reason can be searched for. This section of the program can be activated during the pauses between the measurements.

In the section "stress measurement" the measurement of the optical parameters at the material points of the mesh is organised. The measured data is used to calculate the stress components, which are transmitted to the external

- 71 -

recording devices. The actual material point is defined by the row and column number. The sequential rows are scanned in opposite directions. The new position of the scanner (x,y) is calculated with

x = x +(R-1)1 (3:la)) o x

y = yo+(K-l)ly (3:lb))

were x and y are the co-ordinates of the first material point, o -'o c

R and K are the actual row and column number, respectively, and 1 and 1 represent the distances between the material x y points. The required displacement of the scanner is translated into a number of steps, and the relevant machine language routine is called to displace the optical system. When the optical system is in position, the routines are called to rotate the optical filter, to measure the intensity of the transmitted light and to determine the extreme and averaged light intensities. To prevent errors due to accidental disturbances, the rotation angle of the filter between the extreme intensities is checked. If this angle deviates by more than 1° from the theoretical value of 90° the measurement is repeated! To prevent the possibility of an endless loop, the number of measurements at any one particular material point is limited to 10 times.

When the stress components are measured at all the points of the mesh, the program continues with the displacement measurement. The mesh of marks is defined as a matrix, so that a particular mark can be defined by a row and a column number. The successive rows are scanned in opposite directions. The serial number of a mark is used to find the last measured coordinates to match in an array in the memory. The number of steps necessary to reach that location is calculated, and the relevant routine is called to move the camera to that point. The new co-ordinates of the mark are determined and the mark is centred in the pixel matrix. The previous co-ordinates

- 72 -

are replaced by the most recent values, and the relevant data is transmitted to the external recording devices. If the displacement of a mark is so large that the entire image has goes beyond the boundary of the image sensor, the searching procedure is activated to inspect the surroundings. If the mark is still not found, the mark is presumed to be lost. To prevent errors in the interpretation of the image, the number of dark pixels of an image and the number of dark pixels at the boundary of the pixel matrix are checked. If the total number of dark pixels is smaller than 1000 it is presumed that the image does not represents a mark. If the number of pixels at the boundary of the pixel matrix is more than 10, it is presumed that the mark is too close to the boundary of the model and that the co-ordinates can no longer be determined. Because the mechanical part of the scanner contains gear-wheels and translation screws, some hysteresis could not be prevented. The error due to the hysteresis, however, can be eliminated by means of software. Before the co-ordinates of a mark are determined, the scanner is displaced 1 mm toward the origin and then back to the previous position. Thus a mark is always approached from the same direction, even when a row of marks is scanned in the opposite direction. When the co-ordinates of all the marks in the granular sample are digitised, the front light is switched on, so that the the co-ordinates of the black spots at the boundary can be digitised in a similar way.

To complete the measurement, the output of sensors which measure test conditions can be read. The sensors are read directly before and directly after the stress and deformation measurement. The data is transmitted to the external devices and all the data of a measurement are stored in a file.

The stress and displacement measurements can be operated in the normal and in the automatic mode. In the normal mode the program waits for manual intervention when the measurement is completed. In the automatic mode the decision that a new measurement has to be performed is made dependent on the output of strain gauges and displacement transducers.

- 73 -

When a measurement has been completed, the output of the sensors are stored. Next, the output of the sensors are read every 60 seconds and the differences are determined with the data in the memory. If the difference of one of the sensors exceeds a given value a new measurement is started. By using the signal of two sensors, e.g. a force and a displacement, it is possible to perform more measurements in the most interesting part of the test, e.g. when the force increases rapidly, while in a later stage when the force remains constant the measurement is dependent on the displacement. The dependence of the measuring frequency on the displacement of, for example, a boundary segment prevents too large displacements of the marked particles used for the displacement measurement.

3.2 MODELLING

3.2.1 Production of crushed glass

Since randomly shaped glass particles about 2 mm in size are not commercially available, a technique had to be developed to produce them. In:the first instance, sheets of glass were crushed with a hammer. The sheets were covered with a cloth to prevent flying splinters and the escape of dust. Because only small amounts of glass can be crushed on this way, another method had to be found. Crushing tests with mills failed, primary because it proved very difficult to prevent the escape of glass dust into the air. To eliminate the dust problem during the grinding process, a device has been designed which crushes the glass sheets under water. A diagram of the device is shown'' in Fig. 3:5. A perforated drum (with holes of 3 mm) with a diameter of 300 mm and a length of 600 mm rotates partly under water. The drum is filled with glass sheets and steel balls. The balls are lifted up by cogs, while the crushed glass sheets* remain at the bottom of the drum. When a steel ball is close to the maximum height the ball is dropped and crushes the glass at the bottom of the drum. When the

- 74 -

PERFORRTED DRUM

STEEL BRLL

Fig.3:5 Diagram of the glass crushing machine.

OVERFLOW

Fig.3:6 Device to separate dust from grains.

- 75 -

Fig.3:7 Diagram of the riddling machine.

effective diameter of the grains has been reduced to less than 3 mm, they leave the drum through the perforations and are caught by a curved metal sheet. Radial rubber slabs fixed to the outside of the drum transport the grains of crushed glass into a box with a perforated bottom, where they are washed with clean water. An overflow in the water container discharges surplus water with dust.

An extra treatment to remove dust can be performed with the device in Fig.3:6. A hopper is filled with crushed glass and water. A tube is placed in the centre of the hopper. The glass particles are transported upwards through the tube by means of a water jet. The large particles sink, while most of the dust is discharged through the overflow.

When the dust has been removed, the glass particles are dried in a centrifuge and sorted by means of a jolting screen (Fig.3:7). The sieve bottom of the riddling machine is fixed to a frame by parallel arms. Springs are used to hold the screen deck in the top position. The upward motion is limited by snaps. An eccentric disk moves the bottom smoothly downwards. A

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jump in the eccentric disk causes free upward motion of the screen deck, which is suddenly stopped when the snaps are touched. The particles on the screen, however, briefly continue their upward motion, which results in a small displacement of the particles. The advantage of this principle is that the holes of the screen do not become choked so soon, because the velocity between the particles and the sieve and so the forces are very small. The screen deck comprises three different screens. The first has slots 1 mm in width, to eliminate extremely flat and needle-shaped particles. Next, there are two screens with holes of 2 mm and 3 mm, respectively, which separate the particles into the fractions 1-2 mm, 2-3 mm and larger than 3 mm. The 2-3 mm fraction are most frequently used in the tests.

The useful fractions are treated with hydrochloric acid to remove organic substances and metal particles and to clean the particle surfaces. Finally, the glass particles are washed with water and dryed in a centrifuge so that they become suitable for tests.

3.2.2 Mechanical properties of crushed glass

Crushed glass is used in model tests to investigate the mechanical behaviour of a granular material under specific boundary conditions. Since many different granular materials with different mechanical properties exist in soil engineering and powder technology, it is impossible to replace all these materials by an optically sensitive substitute with the same mechanical properties. Moreover, glass is the only hitherto known material that performs suitable in the photoelastic experiments. Although the mechanical properties can be changed to some extend by polishing the particle surfaces or by using spherical particles, the material will never be precisely the same as a real soil or a particular powder. Therefore this testing technique can be used only to visualise general phenomena in non-cohesive particle assemblies.

- 77 -

Because glass is made from sand, the physical properties of glass and sand can be expected to be similar. The density of pure glass is 22.3 kN/m3. The manner of production of the grains, however, causes differences in the properties of the two materials. The glass particles are produced in a few minutes, whereas natural sand is shaped over periods of thousands of years. The difference in origin causes a difference in particle surfaces and particle shapes. The surface of the glass particles is rougher than that of sand and contains projections which crush easily. Furthermore, extremely flat particles and needle-shaped particles can be observed in crushed glass. These particles are removed by a screening machine. In sand the extremely-shaped particles are eliminated by erosion or separated in the rivers or by the wind.

In Fig.3:8 the response of a medium-dense sample of crushed glass of 2-3 mm (a) and 1-2 mm (b) particle size' , in a triaxial test is shown. The isotropic stress of the sample was about 81 kN/m . The calculated friction angle was $=40 and $=33 , respectively. In Fig.3:9 the response of a sample of fine gravel (a) and sand (b) with similar particle size distributions as the crushed glass is shown. It appeared that crushed glass was stronger than fine gravel ( $=33°) or sand ( <(> = 31 ). Contrary to the natural materials, the particle size of the crushed glass seems to have a considerable influence on the measured friction angle. This phenomenon can be used to investigate a problem involving materials having different friction angles.

A typical phenomenon in a triaxial test with crushed glass was the noise that occurred while the vertical load was being increased, sounding like crushing of the particles. It was found, however, that a similar noise, but less loud, occurred in triaxial tests on sand and gravel. Probably this noise was caused by local stick/slip effects, which sounds louder if the particles are larger. Although some crushing will take place during a test, no significant change in particle size distribution could be measured.

- 78 -

sample height * 137 mm sample diameter ~ 66 mm sample weight ■* 5.51 N density - 11.6 kN/m| Isotropic stress- 83.1 kN/m crushed glass 2-3 mm

j — ■* i t 1 1 i i >■■ * ■ - ■ ■ { -

10 — > s e t t l e m e n t Cmm3*0

n

E N 250 Z j * U l

■o

S 200" «— „ 0

Ü 150 ♦* C o >

1 0 0 -

l 50

/ samp ƒ samp / s amp

ƒ dens 7 I s o t

c r u s * ■■■■-« 1 - - - * — * --* 1—+..., . | —

l a l e l e I t y rop l e d

b h e i g h t - 140 mm d l uie

tc g

_*—

a m e t e r ■ 6 5 . 9 mm t g h t - 6 . 7 2 N

- 1 4 . 2 kN/n3 s t r e s s - 8 1 . ? kN/rn 2

l a s s l -2mm — i — i — t — t — i — t i — t — I 1

1 0 > s e t t l e m e n t [mnO 2 B

Fig.3:8 The response of crushed glass with particle size distribution of 2-3 mm (a) and 1-2 mm (b) in a triaxial cell.

- 79 -

300-n E i 250 z JC

■o g 200-

*id if 150 ♦> c. 0 >

100-A.

I 50

/ 3

/ sample height -/ sample diameter -ƒ sample weight — / density -1 Isotropic stress-

fine gravel 2-3mm —4 1 ( 1 ( ( 1 1 — 1 1 h — I 1 1

137 mm 87.5 mm 7.11 N 14.S kN/m| 81.7 kN/m2

- i — i — — i — t — i — | — i 10 s e t t l e m e n t Cmm3 30

300-r-i E \ 250 z JC u ■o S 200-0

° 150 *> c e >

100-

1 50

/ samp 1 samp / samp / dens f 1sot

sand — i 1 1 1 — — i 1 1 1 « —

Ie e e It ro 1

b height d1ameter weight

/

-= = -

Die stress— -2 mm — i — i i 1

145 mm 66 mm 7.79 N 15.8 kN/n| 80.4 kN/m i i i i 1 \ t

I0 > settlement Cmm]20

Fig.3:9 The response of fine gravel (a) and sand (b) in triaxial cell.

- 80 -

3.2.3 Optical properties of crushed glass

All transparent elastic materials are permanently or temporarily double-refractive. Various crystalline materials such as mica or calcite are permanently double-refractive. Isotropic materials such as glass or plastics become only double-refractive when subjected to shear stresses. This condition is temporary, since the double-refractive property disappears when when the shear stress is removed. The temporary optical sensitivity depends on the elasticity of the material. Elastic materials such as rubber or weak plastics are very sensitive, whereas rigid materials such as glass require a fairly large shear stress to mobilise the optical property. The sensitivity of a material is denoted by an optical constant K which relates the retardation S with the principal stress difference ia -a ) . The relation for K can be derived from eq 2:13, which gives

v - S Ac' pm3 radl ,,,,, K " s(ox-a2) li L W» s J l3-2)

Since the wave-length X and the isotropic propagation speed c' of light in a material are constant, the optical constant can be derived from the thickness s of a body and the actual retardation and principal stress differences.

The optical constant of the glass used was determined in a sheet of glass which was subjected to a known load. The phase retardation was determined at a point in the centre with the

— 9 9 optical measuring device. For x=633xl0 m and c'=0.204x10 m/s the optical constant was found to be K=0.81. The optical constant for homogeneous glass as reported in the literature (Frocht, 1946) ranges for all types from K=.62 to K=1.8.

Up to now no suitable method has been found for determining the optical constant of crushed glass in a direct way. The ratio S/(o -a ), symbolised by M, however, can be calculated from test results by using eq 2:44. The optical constant can be calculated by substituting M into eq 3:2. The averaged value of the optical constant of crushed glass, based on

- 81 -

several tests, was found to be K=l.l. It appears that an assembly of crushed glass is optically more sensitive than homogeneous glass. Physically there is no satisfactory explanation for this phenomenon. Since 6 is proportional to the thickness and the principal stress difference of the optically sensitive material in the cylindrical sample considered and (a -a ) is inversely proportional to the concentration of glass in the volume of the sample, an optical constant of the same value as for homogeneous glass was expected.

A second important optical property of the crushed material is refraction. To obtain a transparent particle assembly the pores have to be filled with a liquid which has the same refraction index as the crushed material. This technique is effective only if the refraction index of the crushed material is sufficient homogeneous. This was found not to be the case for several materials, such as plastics and ordinary glass. The best results have been obtained with heat resistant glasses, such as Pyrex. The refraction index of the glass used is 1.469. Several liquids with a similar refraction index are available, e.g. sunflower oil (1.468), peanut oil (1.469), rizella oil (1.475). Although the refraction index of the vegetable oils are closer to that of glass the synthetic rizella oil turned out to be most suitable from the practical viewpoint because the other oil types are chemically not stable in air.

So far, it has not been found possible to use particles of some other optically (more) sensitive material. The general problem was that the assembly could not be made transparent.

3.2.4 Preparation of a model

The assembly of crushed glass and liquid is loaded in a cell with parallel glass walls. To minimise stresses in the walls a smooth connection between the glass and the metal frame has to be achieved and glass with a low internal stress has to be used. In general, normal window-glass was found to be rather good for this purpose. A detail of a construction with 0-

- 82 -

GLASS WRLL

SILICON COMPOUND

/ WOOD

'.GAP BETWEEN GLASS AND METAL FRAME

a b

Fig.3:10 Detail of the construction of a cell with a metal frame (a) and a frame of wood (b).

rings is represented in Fig.3:10. Furthermore, a construction is shown were the glass walls, each 15 mm thick, are fixed in a wooden frame by means of silicon compound. The advantage of a wooden frame are its low cost the relatively low weight of the cell, and the speed into which it can be built. Cells of 60 cm square have been constructed in this way.

To obtain a transparent assembly an air-free mixture of liquid and glass particles has to be obtained. When a liquid rises in a dry particle assembly, air bubbles are trapped in the assembly due to differences in capillarity of the pores. This effect can be eliminated if a layer with a thickness of only a few particles is penetrated by a liquid. Fig.3:11 shows the arrangement used to obtain an air-free sample. A vibrating spout transports the particles in a thin layer. Since the spout ends under the liquid level, a continuous flow of particles is brought in contact with the liquid. Small air bubbles which sometimes adhere to the particle surfaces are merely separated if the particles sink in the liquid. A stream of liquid caused by the sinking particles transports the air

IVl GLASS WALL rJ

* 0-P.ING

- 83 -

TO VRCUUM PUMP

Fig.3:11 Schematical representation of the arrangement, used to fill a cell with crushed glass.

bubbles to the surface. The black markers used for the displacement measurement were

placed during the filling procedure. The cell was filled with equal layers of glass particles. When the thickness of a layer was sufficient the filling procedure was stopped and an array of marks was placed. The accuracy of the position of the marks could be improved by using a mould.

The filling procedure results in a loosely packed assembly. However, the density of the assembly can be changed to some extend by vibrating the cell during some time.

3.2.5 Loading systems and sensors

Loading systems and sensors were used to subject the granular material in a model test to specific boundary conditions. A tutorial arrangement of a test set-up is shown in Fig.3:12. In most tests the initial stress in the granular material was increased by means of platens. In several cases it was required that the initial load must remain constant during the test. If the expected displacements of the platens were small,

84 -JRCTURTOR WITH STEPPER

PNEUMRTIC CYLINDER

,•:: n STRRINGPUGED CANTILEVER

Fig.3:12 Diagram of a test set-up, to demonstrate the loading systems and sensors used.

a more or less constant force could be applied to a boundary segment by means of springs. If large displacements were expected, a pneumatic cylinder was used. The advantage of springs is that they are cheap and take up only little space. A pneumatic cylinder, however, is much more convenient for adjusting and controlling the force acting on a boundary segment. The displacement of boundary segments (e.g. a probe or a boundary in a shear cell) was performed by an electrically driven actuator. A continuous slow displacement could be achieved this way, which was necessary in several tests, to prevent the shear stress in the granular material from being reduced by creep.

The applied loads were measured by means of load cells, and displacement transducers were used to measure the displacements of particular boundary segments. Strain-gauged cantilever arms were used to obtain point information about the normal stress on a boundary segment caused by the particles. The output of the sensors was digitised and stored in a file on disk together with the optically measured data.

- 85 -

4 APPLICATIONS

4.1 SHEAR

4.1.1 Introduction

A typical feature of a granular material at failure is that the deformation is not distributed smoothly over the whole volume, but is concentrated in local shear bands. The shear bands divide a particle assembly into more or less rigid blocks, each of which contains a large number of particles. In general the network of shear bands is not stable but changes during the deformation and so changes the dimension and shape of the rigid blocks. The behaviour and distribution of the shear bands are strongly dependent on the boundary conditions of a granular material. If a boundary of a particle assembly is partially stress-controlled, very extreme shear bands can be observed, in which large blocks of granular material move with respect to each other. A typical example is the failure mechanism of a slope or in a biaxial compression test specimen. If the whole boundary is strain-controlled, extreme shear bands cannot be formed easily, since the motion of a particular rigid block is limited by the rigid boundary. In this case a shear band has a short life and new shear bands have to be formed to permit deformation of the particle assembly. An example of strain-controlled deformation is the flow of a granular material through the cone of a hopper. As demonstrated by Bransby and Blair-Fish (1975) by means of a radiographic technique, several shear zones can be observed which are not all active at the same moment.

As demonstrated by Lade (1982), for example, the failure type can be influenced in a triaxial compression test by changing the ratio between diameter and height of the sample. If the height of the sample is, for example, twice the diameter, the failure is concentrated in a shear band (called line failure by Lade) which separates two more or less rigid blocks. If the height

- 86 -

and diameter are the same, a homogeneous deformation can be observed, called zone failure by Lade. Lade observed that the stress-strain behaviour of dense sand depends strongly on the failure mechanism. He found that the measured peak friction angle in dense sand can deviate up to 10 degrees, depending on the failure mechanism. The consequence of this phenomenon is that the friction angle of a granular material must be determined with a failure mechanism similar to that which can be expected in the practical problem considered. The characterisation of failure in terms of zone failure and

line failure, however, appears to be rather idealized. In practice several gradations can be observed, so that it is not possible simply to subject a sample to a similar failure mechanism. Probably zone failure is also due to shear along discrete planes. The difference from line failure is that the sample is streaked with a mesh of shear bands. The area of the shear planes per unit volume is larger in this case, so that a larger volume is involved in the dilation process. Since dilation costs energy, larger forces are required per unit displacement to deform the sample than when the failure mechanism is based on only one shear plane. This difference is greater if the sample dilates more, which is generally the case with denser samples.

In some practical problems an unfavourable failure mechanism acts. In hoppers, for example, several shear planes are active when the material flows through the cone. The volume expansion due to dilation requires extra energy, which leads to higher boundary stresses and hampers the flow. In the case of an embankment a more uniform failure mechanism would improve the stability. Techniques to stimulate a more homogeneous deformation, are for example reinforcement of steep embankments by means of textile.

Since shear appears to be the elementary mechanism in deforming granular material, the optical measuring method was applied to investigate some aspects of this phenomenon in more detail. Because stress and strain could be determined

- 87 -

in the interior of the granular material it was possible to visualise some aspects of the double sliding free rotating model of De Josselin de Jong (1959). Furthermore, the effect of stress rotations and stress history on the mechanical behaviour of a sample could be visualised.

4.1.2 Theory of shear

Two extreme types of shear can be distinguished in granular material. In one type the shear is concentrated in a rupture layer with a thickness of about 10 to 20 particle diameters. In the other type a uniform shear deformation occurs in a layer of granular material of which the thickness is much more than 20 particle diameters.

Shear in a thin rupture layer is shown schematically in Fig.4:1a. A shear band is formed in a volume of non-cohesive granular material in which a homogeneous critical stress distribution is assumed. The shear band separates two more or less rigid blocks of particles. The strains in the blocks are considerably smaller than those in the shear band. Several theories have been developed to predict the angle between the shear band and the principal axes of stress. Assuming the relation between stress components and failure to be given by the Mohr-Coulomb criterion, the maximum ratio between shear stress T and normal stress o that can exist in a granular material is

jj = tan$ (4:1)

were $ is the internal friction. If it is assumed that a shear band is formed in the direction of a plane in which the ratio expressed by eq.4:l is reached, as symbolised by Si and S in the Mohrdiagram of planes in Fig.4:1b, the angle <x of the shear band with the major principal stress is given by

a = <45-|) (4:2)

- 88 -

°i PN

......:..

'A-"Vr

» ■ «

jT-""1

- 1 ' .

o;

T .

/S' 1

/ 1 *?\

'•"■ pole

« ■ «

«i,

""„

°"«

Fig.4:1 Shear in a rupture layer (a) and a Mohr diagram of planes (b) of the state of stress.

which is known as the Coulomb angle. If the deformation is completely concentrated in a shear band

compatibility requires the coincidence of the shear band with the zero extension lines. If an angle of dilation v, given by

sinu dv dY

(4:3)

(where dv and dy are increments of volumetric and shear strain respectively) is taken into account, then the angle n between the zero extension lines and the major principal strain is given by

T) = (45~) (4:4)

To determine the angle between the major principal stress and the shear band, an assumption has to be made with regard to the inclination between the principal axes of stress and strain. If coaxiality is assumed between the major directions of stress and strain, the angle a of the zero extension lines with the major

- 89 -

principal stress direction is equal to <x= n. This' angle is known as the Roscoe (1970) solution. In the case of non coaxiality, which possibility was demonstrated theoretically for a non-dilating material by De Josselin de Jong (1959), the angle i between the major directions of stress and strain is defined by the range

-|<j> < i < -tt» (4:5)

Combining this theory with the solution of Roscoe yields

4 5 - ^ < oc < 4 5 + ^ (4:6)

Several investigators have derived equations in which <x is dependent on both the friction angle and the angle of dilation. Arthur et al. (1977) derived.a relation, based on experimental observations, where a is described by

a = (45-^) (4:7)

They assume a shear band to consist of a number of local simple shear ruptures which are equally distributed along stress characteristics and zero extension directions. The observed shear band is in the direction of their bisector. Further, they suggest that an asymmetric distribution of the local simple shear elements is possible, e.g. as a result of constraints, which can lead to one of the. extreme solutions given by Coulomb and Roscoe.

Theoretical evidence for the solution of Arthur et al is produced by, for example, Vardoulakis (1980) and Vermeer (1982). In their analyses the shear band is considered as an instability, where the stress as well as the strain in the shear band differ from the homogeneous and constant stress and strain in the volume outside the shear band.

In Vardoulakis' analysis the sand is assumed to be a plastically hardening material obeying the Mohr-Coulomb failure criterion, were $ is independent of the stress level. In his

- 90 -

solution the shear band orientation is dependent on a hardening parameter h, defined by

d( sin<|> ) h = -,—— (4:8)

dy were 4> is the mobilised friction angle and y the shear strain. For h=0 the orientation of the shear band is that given by the Coulomb solution (eq.4:2). For small positive values of h the Roscoe solution (eq.4:4) is possible. For higher values of h a bifurcation solution exists, which results in an orientation which lies between the Coulomb and Roscoe solution. This solution is based on the assumption that the mobilised angles of friction and dilation are close to the peak values at the instant of bifurcation. The same solution (eq.4:7) of Arthur was found by assuming that the difference between the peak values of <t> and v are small. Vardoulakis and Graf (1985) found experimental evidence for this assumption.

A simpler analysis was produced by Vermeer (1982) and by Vermeer and de Borst (1984). Vermeer assumed that the shear band is formed in the zero extension direction. By taking the x-axis parallel to the shear band and by assuming the stress in the material outside the shear band to remain constant, the following stress and strain components are known

è = ê = ö = ö = 0 (4:9) xx zz xy yy

were the dots denote that the increments of the stress and strain components are used. Vermeer used a constitutive relation between stress and strain, referring to the hardening-softening model, to express the known strain components in terms of stress components. This finally resulted in the following expression for the angle between the shear band and the major principal stress

cos2<x = + (sin<|)+sinu)± J (sinifr-sinu)2-16(1~p } h (4:10)

- 91 -

were p is Poisson's ratio, E is Young's modulus and h the hardening modulus. In Vermeer 's study h was assumed to be dependent also on the stress level. Further, it must be noted that <)> and v refer to the peak values. It has been demonstrated by Vermeer that the inclination angle a depends strongly on the value of the hardening modulus. For large values of h, which is the case at the start of loading, the term under the root sign in eq.4:10 is negative, so that a. cannot be solved. Vermeer concluded that no shear bands can be developed in this case. If h causes the term under the root is zero and assuming <|>- \J<30 , a unique solution for a. exists which is the same as Arthur's equation 4:7. For smaller values of h the solution is not unique. Up to h=0, a has a value which lies between

45-|<)> < a < 4 5 - ^ (4:11)

and

4 5 - ^ < a < 45-^ (4:12)

Which solution occurs will, as Vermeer has suggested, depend on the boundary conditions and second-order effects. The orientation of the shear band in the post-peak stage can be deduced because the material has reached the state of constant volume and stress, hence u=h=0. With eq.(4:10) it was found that

a = 45-|<)> or a = 45° (4:13)

The first solution is a limit, which was also predicted by the double sliding free-rotating model of De Josselin de Jong (1959). Vermeer (1982) had doubts about the second result because he had not found experimental evidence for oc=45 for sand. In the following paragraphs it is visualised by a photoelastic experiment, however, that the second solution is also a realistic one.

- 92 -

In uniform shear it is assumed that the whole sample deforms uniformly, as is, for example, the case in a so-called simple-shear cell. It is not quite clear whether this type of deformation is really different from the deformation in a shear band. There is experimental evidence that a uniform simple shear deformation in a large sample is a result of several narrow local shear bands as, for example, observed by Fukushima and Tatsuoka (1982) in a torsional simple shear apparatus. However, this phenomenon cannot be observed in every test. Furthermore it was suggested, for example, by Arthur et al. (1977) that also in a narrow shear band the overall deformation is a result of small local shear planes. Simple shear deformation of a thick layer is not a very common phenomenon in practise. This mechanism probably acts in the bin of a hopper.

In the laboratory the simple-shear tests provides a better means of investigating a granular material in shear. Due to the thickness of the shear zone, better observation of the deformation is possible. Furthermore it proved possible to perform large shear deformations, so that the stress-strain behaviour could be observed in the steady-state phase.

A typical mathematical model for describing the steady-state behaviour of a granular material in simple shear was developed by De Josselin de Jong (1959, 1971, 1977 ). In this model it is supposed that the whole sample is in the limit state of stress and that the volume of the sample remains constant. The overall deformation is assumed to be due to slidings at micro scale, which can occur more or less simultaneously in the directions of planes of maximum shear stress ( S (a) and S (b) direction in Fig.4:2); and in addition, the rotation ft of the sliding elements is assumed to be entirely free. To obey the thermodynamic requirement that energy is dissipated during sliding, shear strain will be developed only in the direction of the shear stress in the plane of sliding. De Josselin de Jong formulated this as

a > 0 and b > 0 (4:14)

- 93 -

Fig.4:2 Failure mechanism during simple shear, as proposed by De Josselin de Jong.

The mathematical analysis of this mechanism has resulted in constitutive inequalities, from which it could be deduced that the angle between the principal direction of stress and the shear direction is in the range of

45—|(p < ex < 4 5 + i $ o r -|<}> ( 4 : 1 5 )

It was suggested by De Josselin de Jong that i depends on the stress history and that the extreme values are found if the simple-shear test was started with an initial major principal stress direction which is perpendicular or parallel to the overall shear direction. Several investigators have performed experiments to investigate the stress-strain behaviour in simple shear. However, since stresses and strains could not be measured systematically in the interior of a deforming particle assembly, no clear information could be obtained about the principal directions of stress and strain. It will be demonstrated in the next chapter that the

- 94 -

photoelastic measuring method can be applied to investigate the behaviour of granular material in shear.

Another important phenomenon in a granular material is the influence of stress rotation and stress history on the mechanical behaviour in the first stage of the shear deformation. It appears that a granular material behaves weaker in pre-failure when large stress rotations occur. Moreover this effect is influenced by the history of stress and strain. The microscopic deformation mechanism in simple shear was investigated by Oda and Konishi (1974ab) in an assembly of cylinders (made of photoelastic material) packed at random in a two-dimensional simple shear apparatus. They used the isochromatics to obtain information about the contact forces between the particles. The normals at the contact planes were used to analyse the modifications in the structur of the particles. It was observed that microscopic slidings occur in preferred contacts which results in the formation of a fibre structur. During shear the fibres prefer to orientate in the direction of the major principal stress (this mechanism is responsible for the light stripes if a loaded assembly of crushed glass is viewed in circularly polarised light). They suggested to separate the mobilised stress ratio -x/a in a simple shear test into two

n r

components, tgOO-iJj ) representing the granular fabric stucture and a constant factor K representing the interparticle frictional component. In simple shear with sand they found experimental evidence for the linear relation T/O = «tg(90-i|) ).

Several experiments have been carried out by Arthur et al (1980,1981) to investigate the behaviour of a granular material, subjected to a jump rotation of stress. The tests have been performed in a so called directional shear cell in which cubical samples could be subjected to a plane strain deformation. A complex system of pressure bags and shear sheets were used to control the boundary stresses. They found a drastical reduction in stiffness of the granular material when an initially loaded sample was loaded a second time on a similar way, but with a

- 95 -

large difference in the direction of the principal axes of stresses. Experiments in which the principal stress direction is the only variable in the loading program were performed by Ishihara and Towhata (1983) using a triaxial torsional shear test apparatus. They demonstrated that a plastic deformation could be observed if a sample was subjected to stress rotation only. Since stresses can be visualised in optically sensitive granular material, it was also possible to visualise some aspects concerning the stress rotation and stress history in shear tests.

4.1.3 Shear devices used

Several devices have been developed to investigate general phenomena in a photoelastic granular material in shear. Since stresses and.strains could be measured in the interior of the granular material no particular boundary constructions were required to enable the stress distribution to be estimated from known boundary conditions. It was therefore possible to subject the granular material to more extreme conditions than is usual with conventional testing methods. Diagrams of the most important devices used in this study are shown in Fig.4:3 The devices in Fig.4:3, a and b, are typical simple-shear apparatuses, in which the granular material is deformed by means of plane parallel boundary segments. The device in Fig.4:3a consists of two U-shaped segments which are connected by two hinged beams. The initial load P is applied by means of platens and springs, which produce an initial major principal stress direction perpendicular to the shear direction. The purpose of the U-shaped segments is, to minimise the influence of the rigid platens on the stress distribution in the shear zone. The normal stress is measured at four points on the boundary by means of strain gauged cantilevers. The device in Fig.4:3b is used to perform a simple shear test in which the initial major principal stress is parallel to the shear direction. Two U-shaped segments are so connected, that they can move only in the shear direction with respect to each other. Four rods at the corners and two parallel

- 96 -

r^

a

Pi p.

#

Fig.4:3 Schematic diagram of the devices used to investigate the behaviour of a granular material in shear.

- 97 -

platens make it possible to subject the sample to a simple shear deformation. The initial load is applied by compressing the sample in the horizontal direction, which is possible by displacing the legs of the U-segments in relation to each other by means of bolts. No springs are used in this case and no strain gauges installed to obtain point information about the normal stress at the boundary. The thickness of the plane sample in both devices is 50 mm and the particle size is about 2 mm. The shear deformation is performed by fixing one of the U-shaped segments and by displacing the opposite segment step by step. After each step an optical measurement is performed. Since the geometry of the sample changes during the deformation, the devices a and b are not suitable for investigating the material behaviour in large deformations. A shear cell which in principle allows■infinite deformations of the sample is shown in Fig.4:3c. The granular material is confined in a space enclosed by two parallel glass sheets 40 mm apart and a caterpillar. The initial load P is applied by a pneumatic cylinder and a platen. Parallel beams are used to permit a vertical displacement of the top platen only. A continuous shear deformation is obtained by displacing the bottom platen of the cell by means of a motor driven spindle with a speed of about 0.25 mm per hour. The top platen, bottom platen and caterpillar are provided with a profile to prevent slip between caterpillar, platens and particles. The device is equipped with two load cells to measure the total vertical load P and the total horizontal shear force F. A displacement transducer measures the displacement of the bottom platen. The data of the gauges is used in the program of the control computer to determine the pauses between the optical measurements. In this test a measurement is started if the increment of the displacement is more than 1 mm or if the total shear load increases by more than 100 N.

In Fig.4:3d the previously described test setup is used to investigate the formation of a shear band in a sample. In this test the sample is in direct contact with the platens; some horizontal support is provided by blocks of foam rubber.

- 98 -

4.1.4 Measuring results

A simple shear test with an initial major principal stress perpendicular to the shear direction was performed with the device of Fig.4:3a. The sample was compressed in the vertical direction until the strain gauges of the loading platens displayed a normal pressure of about 100 kN/mz. The initial stress distribution was measured optically and the position of the lead shot was fixed on a photograph (the digital camera was not yet available). Next, a boundary segment was distorted up to y=7 , divided over two steps of about dy=2 . After each step an optical measurement was performed and a photograph has been made of the plane sample.

A photograph of the initially loaded sample in circularly polarised light is presented in Fig.4:4a. The major principal stress direction is indicated by the light stripes. In Fig.4:4b the sample is presented after some deformation. It can be observed that a counter-clockwise rotation of the fibre structure has been taken place. The optically measured principal stress directions, based on 100 measuring points, are shown in Fig.4:5 a and b respectively . Fig.4:6 shows the calculated stress distributions of the initial loaded sample and after a distortion of 3°, 5 and 7 respectively. The averaged states of stress at the centre of the shear zone in the different stages of the boundary were found to be

Y=0° a -99 l a -41 r2

>m 24°

,0 Y=3 147 52 26° 29°

Y = 5 -152 - 53 33° 29°

-7° Y=7 -165 C kN/m2] -53 [ kN/mz3 36° 31°

The directions are determined with respect to the fixed bottom plate of the shear cell. The distribution of the mobilised friction angle in the final state is shown in Fig.4:7. Since the manual digitisation of the co-ordinates of the marked particles from photographs was not very accurate, a fairly large step was

- 99 -

Fig.4:4 Photograph in circularly polarised light, of the sample in the simple shear cell of Fig.4:3a.

- 100 -

\-f

rt

3

rp ( 1

1 I r r~ s

,

Fig.4:5 Computer plot of the principal stress trajectories, based on 100 measuring points (1 scale = 1 cm).

- 101 -

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Y=0"

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I I I I I I I I I l_J I I I I I L ' ' I ' ' I I I I I I I I I I I I I I I I I I 1

NJ l l l l l i n i , , " " " " " " " " " ï l l l l l l l l i n , ■

x X \ X

\ X X x

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Y=7" Fig.4:6 Stress distribution in a number of deformation steps (1 scale = 100 kN/mz or 1 cm).

- 102 -

Fig.4:7 Distribution of the mobilised angle of internal friction in the final stage.

required to obtain reasonable data for the determination of the strain rate tensor. In this test only the step from Y=3 to y=7 could be used. The measured displacements of the marked particles and the calculated distribution of the strain rate tensor are presented in Fig.4:8 a and b respectively. The averaged values in a region at the centre of the shear zone were

de =-0.030; de =+0.030; B =-49°; UJ=-0.027; dy= 3?4 1 2 1 T

In Fig.4:8b the major principal stress direction before and after the deformation step is also presented. A counter-clockwise stress rotation of about di|)=8° was observed in the shear zone. The average difference between the major direction of the principal stress and principal strain was found to be i=12 .

Fig.4:9 presents photographs of the photoelastic phenomenon in the particles and results of optical measurements as obtained with the shear device of Fig.4:3b, where the initial major principal stress coincides with the shear direction. In Fig.4:9a a photograph and a computer plot of the direction of the initial major principal stress are presented. The distribution of the

- 103 -

SS 56 57 SS 59 61 62

46 47 48 49 58 SI 52 53

37 38 33 48 41 «2 43 « "

?a J9 38 31 32 33 34 35 36

28 21 " " " " " "

II 12 13 14 " 16 17 | ' «

,C, de,

V y / y" d£' k y v \ - x Y x x y x \ x \ \ y x > < y \ y \ \ y X x x * \ \ \ X * X * \ \ * V X

\ V

LL Fig.4:8 Graphical representation of the measured displacements of marked particles (a) and the derived strain rate tensor (b) (1 scale= 1 cm).

- 104

SSPTO i—S*' -i A " * •»

* «• •- ■» »

'.«:#V- 2, «. * * *

** 4 -# «Jl

Fig.4:9 Simple shear test in which the initial major principal stress direction coincides with the shear direction.

- 105 -

óyj

\\xxx

Fig.4:10 Measured displacements during shear and the derived strain rate tensor (1 scale = 1 cm (boundary); 0.5 cm (field)).

principal stress direction after some shear is shown in Fig.4:9b. Fig.4:10 shows the measured displacement of marked particles (a) and the calculated strain rate tensors with the major principal stress direction to match (b). The data were measured over a distortion of the boundary of dy=4 , after an initial distortion of Y=2 . Averaged' data referring to a region at the centre of the sample are

de =-0.047 l de =+0;039 P =-46 2 *1

w=-0.033 V41 i = -5 l

In this case the device was not.equipped with strain gauges so that no clear information could be obtained about the magnitude of the principal stresses. However, assuming the maximum value of the mobilised (friction angle and the optical constant to be the same as in the previous test, the state of stress at a point

- 106 -

can be estimated with

1, .1+sind) a = -=-(a -or ) — : — - * 1 2 l 2 sin<)>

1. . l-sind> a = TT(Q -a ) :—r* 2 2 l 2 sinij)

The averaged stress at the centre of the sample before and after the deformation step under consideration was found to be, respectively

a =-302 a =-97 tji =-39° and a =-365 a =-117 <J) =-42° 1 2 Tl 1 2 1

were the stresses are expressed in kN/m • It appears that the stresses in this test are considerably greater than in the previous test, which can also be deduced from the clarity of the stripes in the photographs.

Large simple shear deformations are produced in the test setup of Fig.4:3c. A sample of crushed glass, with a height of 50 mm, a thickness of 40 mm and a length of about 110 mm was confined in a space formed by a caterpillar and glass walls. The sample was loaded in the vertical direction by a constant force P=700N. This load was measured by a load cell. The total load caused an averaged vertical pressure on the granular material

2 of cr =160 kN/m . The sample was subjected to shear by displacing the bottom platen in the horizontal direction at a constant speed of 0.25 mm/h. A second load cell was used to measure the total load needed to displace the bottom platen. Furthermore, a displacement transducer was installed to measure the displacement of the bottom platen. The load cells and displacement transducer were read by the control computer of the optical measuring device. The data of the gauges were used to determine the length of the pauses between the optical measurements and to produce graphical output to visualise the course of the test. A photograph in circularly polarised light of the sample, after some shear is presented in Fig.4:11. Knowing that the initial

(4:16a)

(4:16b)

- 107 -

S£

Fig.4:11 Photograph, in circularly polarised light, of the photoelastic phenomenon in crushed glass after a large shear deformation.

«""«»«

^ = 100 x Y l d e g G,-G2=1><Y Ikgf /cm' F / P z l x Y l - l

» 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 < < 1 1 1 1 1 1 1 i l l 1 1 1 1 1 1 > > 1 1 1 1 1 1 1 1 1

10 20 |f %

30 40 50 60

Fig.4:12 Graphical representation of data obtained from load cells and optical measurements.

- 108 -

major principal stress was vertical, it can be observed that some clockwise rotation has been taken place. In Fig.4:12 the data measured by the load cells and the optically measured data ( f0,-0",) a n d tlOr averaged over a region at the centre of the sample, are plotted as a function of the displacement of the bottom platen. The ratio F/P also includes effects which take place in the curved part of the caterpillar. This ratio can therefore be used only as an indication as to how the shear forces increases. Contrary to the previous tests, the sample could be subjected to such a large deformation that the stress became constant. Assuming the optical constant to be known, in order to calculate (a -a ) from the optically measured data, the stress distribution in the sample can be estimated from

a = a -t4(a,-a, )(cos2i|) +1) (4:17a) 1 n 2 l z l

or = a -ha -a )(cos2i|) -1) (4:17b) 2 n 2 l 2 l

where (a -or ) and t() are averaged values which refer to a region at the centre of the sample. In the steady-state phase of the test the state of stress was found to be <in kN/mz)

a =-190 a =-70 i|> =60° d>=28° 1 2 Tl T

In this test the position of 32 marked particles in several stages was determined automatically by means of the digital camera. The displacement pattern of the marked particles is plotted in Fig.4:13. Each dot represents the position of a mark after some deformation. In the upper region, which is close to the fixed top platen, the displacements in a deformation step are so small that the dots form a continuous line. The total distortion was about ^=40° (85%). The triangle in Fig.4:13 shows that the displacement is distributed with a linear gradient over the height of the sample. Furthermore, it can be shown, in particular if the upper row of marks is considered, that the thickness of the sample initially decreases. In a later stage, due to the dilation behaviour of the granular material, the thickness increases, which tends finally to a constant height of

- 109 -

Fig.4:13 Plot of the displacement pattern of 32 marked particles during shear.

*• X X X X X X\\ *« " = 18

dé, = -.022 dê,= .022 .* . X. X Xv yx . ix X

XX XX XX l |-.Ys '-S"

x A A X X X ,x-X XXXX

Fig.4:14 Calculated distribution of the strain rate tensor in two deformation steps (1 scale = 1 cm).

- 110 -

the sample. Fig.4:14 shows the distribution of the strain rate tensor of two deformation steps, as indicated by dy and dy in Fig.4:12. The direction of the major principal stress before and after a deformation step has also been plotted. In the first step a clockwise rotation of the principal stresses has been taken place. In the second step, however, the directions of the principal stresses have not changed, which demonstrates that the steady-state phase was reached. The calculation of the angle i between the major directions of stress and strain rate is based on data of 9 material points at the centre of the sample. The averaged direction of the principal stress before and after the deformation step was used. The angle of non-coaxiality in the first step was i=18 and stabilised at i=15° in the final state. This demonstrates that non-coaxiality is not a phenomenon which takes place only in the first stage of the shear deformation.

A reverse shear test was performed to demonstrate that the mechanical behaviour of a granular material depends not only on the stress path, but also on the stress history. As shown in Fig.4:15, a distortion of more than 40% was required in the reverse shear test to traverse the same range of stress rotation ( 90 to 55 ) as in the first test, in which a distortion of 20% was sufficient to reach the steady state phase. In both tests the final conditions are practically the same. In the first stage of the reversal shear test a very large deviation, i=30 was measured, which decreased to i=14 in the later stage.

Up to now, shear tests has been presented in which the shear was forced to occur uniformly in the sample. In the test setup of Fig.4:3c, however, the sample was confined in a box with soft boundary segments, so that the formation of an extreme shear band was possible. Contrary to simple shear-tests and plane strain biaxial tests (Vardoulakis et al,1985) the directions of both the shear band and of the principal stresses are not directly imposed by the boundary conditions during deformation. In this test a sample with a height of 30 mm, a length of 250 mm and a thickness of 42 mm was loaded in the vertical direction by means of two horizontal platens. In the horizontal direction some support was

- Ill -

> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 

Fig.4:15 Data of a reversed shear test, obtained from load cells and optical measurements.

provided by two foam rubber blocks. The total load in the vertical direction was P=2 kN, which caused an averaged vertical pressure of about a =190 lcN/mz. The sample was subjected to shear by displacing the bottom platen in the horizontal direction with a force F. The force transmission after some deformation, visualised in circularly polarised light, is shown in Fig.4:16. To visualise the course of the test, the optically measured data ( i() and (a -a )) and the ratio F/P have been plotted in Fig.4:17. The data was averaged over a region with a uniform state of stress, located at the centre of the sample. The initially loaded sample was distorted with a constant vertical load until the measured parameters remained constant. Next, the shear force was decreased to F=0 and then increased again until a steady state was reached. As can be observed, the changes in TJJ and (o -a ) are in agreement with the behaviour of the shear

1 2 force F. Due to friction in the device and in the granular' material the initial values were not reached when the shear force was reduced to F=0. The stiff behaviour when F was increased for

- 112 -

Hf.

4|>> i nfiïm«|^yi*i

^ ' * - - - - - - ü Fig.4:16 Force transmission a f t e r some shear deformation in the

device of F ig .4 :3d .

1.0 +

■5tf

^ = 100 K Y tdeg.] CT,-(J2 = 100*Y[kN/m2] F/P = 1 * Y H

H 1 1 1 1 h"

60 50 Y% >

Fig.4:17 Measured data during shear with the device of Fig.4:3d.

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3

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3 P r f ro ' S H -

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r t P ro

M l

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H -

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ï cn ro o M l

r f P" ro r f ro cn r t

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s p cn

r f P1

ro U I (B

n o 3 P r f H -

3 ro H -

cn r f

^ T3 H -

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p-H -U I

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l

- 114 -

\ _ _ ^ > « = r ~ - i — ■ — * _ — ^ «==: 1 X

Fig .4 :18 Displacement of marked p a r t i c l e s in a deformation step (1 sca le = 1 cm).

\ \ XAXX x X

Fig.4:19 Derived distribution of the strain rate tensor by combining several quartets of marked particles.

Assuming the shear band to be formed in the zero extension direction (which is in good agreement with stereo observations), the mobilised friction angle can be calculated from

sin<)> = sine tge = (4:18) sin(9+2<x) ii

where a. is the angle between the shear band and major principal stress (Fig.4:1). Although the absolute values for T and a are not known, an averaged value for the ratio can be derived with T la =F/P. The absolute values for the major principal n n stress can be obtained from eq.4:16, where § is replaced by $ . As can be observed, fairly large deformations can occur at low values of the mobilised friction angle. This is a typical phenomenon in tests in which large stress rotations take place. It appears that the major directions of stress and strain are in

- 115 -

close agreement in this test. The major directions of stress and strain are apparently coaxial if it is not attempted to fix the principal stresses or the shear band in an unnatural direction. This result agrees with Vermeer 's analyses (1982) of the possible orientations of a shear band in the steady-state phase of a shear deformation.

4.1.5 Discussion

The tests presented here visualise the behaviour of granular material in shear. In the devices a,b and c of Fig.4:3 it was attempted to create a uniform displacement field, so that measurements could be performed in a large shear zone. In device d a typical shear band was created. It appeared that an extreme shear band was formed if the boundary was partially stress controlled. A significant difference in friction angle in these two types of shear, as was found by Lade (1982), was not observed, because the samples were packed rather loosely. A friction angle of the order of <(>=30 seems to be rather low in comparison with the friction angle determined with the triaxial compression test, in which a friction angle of 0=40 was measured. However, the value of the friction angle was found to be strongly dependent on the device used. In a direct-shear cell, for example, a friction angle of $=30 was measured for the crushed glass also, so that the optically determined values appear realistic.

There are no clear indications that the failure mechanism of a granular material in a narrow shear band is different from the behaviour in a large shear zone. The displacement measurements in the caterpillar test (Fig.4:13) did not reveal the presence of local shear bands e.g. in the directions of the planes of maximum shear stress. The shear band mesh in this test probably has such a fine structure that observations at particle level are required to visualise this phenomenon. Such observations were performed by Drescher and De Josselin de Jong (1972) in an assembly of photoelastic disks. They found that clusters of about ten

- 116 -

particles form rigid bodies which slide with respect to each other. This concept is the basis of the double sliding free-rotating model of De Josselin de Jong. In this model it is assumed that sliding at micro-scale occurs in the planes of maximum shear stress. Which plane is predominant depends on the boundary conditions. In the extreme case, where sliding occurs in the direction of one of the planes of maximum shear stress, a deviation between the major directions of stress and strain of |i| = Y<1> was predicted. In the devices a and c it was attempted to subject the boundaries of the sample to such extreme conditions that sliding was expected to occur in the most nearly vertical plane of maximum shear stress (. (b) in Fig.4:2a). In the devices a and c the initial major principal stress was perpendicular to the shear plane. In this case a significant deviation of i=14 was observed between the major directions of stress and strain. With device c it was demonstrated that this deviation occurs not only in the first phase of the test, but also in the steady-state phase. Calculations of the shear rates a and b in the S and S directions (Fig.4:2) demonstrate that there was a significant difference between these shear rates. Averaged parameters referring to the device of Fig.4:3a are

a=0.011 b=0.052

More detailed test parameters were reported by Allersma (1982b). Apart from some non-significant exceptions a and b were found to be positive. If the sliding along one characteristic plane is dominant, a rotation & of the particle clusters is required to obtain a horizontal shear deformation at macro-scale (book row mechanism). The average value of the so called structural rotation was &=-2.8 . The value of the stress rotation di|>=80 in the deformation step under consideration demonstrates that there is not always a simple one-to-one relation between these parameters, as was assumed by Spencer (1964), for example. In the device of Fig.4:3b the initial direction of the major principal stress was horizontal. It was expected in this case

- 117 -

that sliding occurs predominantly in the most nearly horizontal plane of maximum shear stress, which is indicated by a negative angle i. In this test a small negative deviation of i=-5° was measured, where the average values for the sliding in the S and S directions and the structural rotation are respectively

a=0.066 b=0.037 il=-0.9

The deviations are not so extreme as in the first test, probably because the horizontal load was not kept constant by means of springs or a dead weight. Furthermore, it could not be demonstrated in this test whether the major directions of stress and strain are coaxial or non-coaxial in large deformations.

Up to now it has not been found possible to design a device for crushed glass which permits large shear deformations with a horizontal initial major principal stress direction. That a large negative angle i between the major directions of stress and strain can exist was, however, demonstrated by experimental and theoretical analyses, for example, of Arthur et al. (1977), Vermeer (1982) and Vardoulakis and Graf (1982), as discussed in Chapter 4.1.2. In the plane strain biaxial tests of Vardoulakis and Graf a tabular-shaped soil sample is surrounded by a thin rubber membrane and is placed in a biaxial apparatus (Fig.4:20). The bottom part of the apparatus is mounted on roller bearings and can move horizontally. After a confining pressure had been applied, the sample was subjected to a vertical load. Due to a soft inclusion a sharp shear band was developed after some displacement of the top platen. Vardoulakis and Graf found directions of the shear band of the order of 63 with respect to the horizontal. Assuming that the major principal stress is in the direction of the applied vertical load and that the principal strain axis forms an angle of 45° with the direction of the shear band, a non-coaxiality angle of more than i=-15 can be deduced. This test can probably be regarded as a shear test in which the initial direction of the major principal stress is more or less in the shear direction. The extreme value of i=-.5 <t> has not

- 118 -

a-er, = p

Fig.4:20 Plane strain biaxial apparatus (from Vardoulakis and Graf, 1982).

been reached in this test ( a peak mobilised friction angle of 45 was measured). According to Vermeer's theoretical analyses it may be expected, however, that the extreme angle can be reached if the correct boundary values are chosen. It would seem that non-coaxiality occurs if one or both of the major principal stress or the shear is compelled to occur in a direction which is not preferred by the material. The results of measurements with the test device of Fig.4:3d (Table 4:1) demonstrate that stresses and strains are apparently coaxial if both the rotation of the principal stress and the formation of the shear band are free to occur. The angle between the shear band and the principal stress is 45 in this case. This angle was also predicted as a possible solution in Vermeer's analysis. The range of eq.4:6, where the dilatation and non-coaxiality are combined, appeared to be realistic also, because large positive angles i were observed in the first phase of a shear test.

To decide which solution is predominant in a particular practical problem is by no means simple. The stress distribution and boundary conditions are in general so complex that different mechanisms are active in a considered volume, so that it is not simply possible to subject a sample to a failure mechanism

- 119 -

similar to that which occurs in the practical problem.

The angle between the principal stress axis and shear direction is important in the interpretation of the results of shear tests. The relation between the ratio T/O^, ty and $, which can be derived from the Mohr diagram, is

T _ sin2i|)sin0 cr l-cos2\l>sin<t>

where a horizontal shear direction is assumed, Only if <x=45 is the simple relation

- = sin$ (4:20) a n valid. When the non-coaxiality angle i is positive in a sample, eq.4:20 yields too low a friction angle. In Fig.4:21 it is shown that the determination of the friction angle is not possible at all in a shear test, if the non-coaxiality angle is close to the positive extreme value in the steady-state phase of shear. In this diagram the ratio T/cr is plotted as a function of i|> for different friction angles. Three lines indicated by a, b and c show the relation between T/O and $ in the case of i = - — <t>, i = 0

n 2 and i= —d>, respectively. The curve b shows that T/Q is more or less independent of the friction angle if Q> exceeds 30°. For this reason the device of Fig.4:3c is, for example not suitable for determining friction angles. Also, devices such as direct shear cell and a hollow cylinder apparatus have to be considered critically, since it is not quite clear if the axes of stress and strain are coaxial during shear. The friction angle can be determined most significantly in a device in which i=- —$, as is shown by the line a in Fig.4:21. However, it proved to be by no means simple to realise this extreme condition in a sample. The angle between the principal axes of stress and strain was best defined in the device of Fig.4:3d. Furthermore, the ratio T/O

n is known, since the ratio is independent of the direction of the shear band. In the case of coaxiality the theoretical relation

(4:19)

so that <x= i|j.

- 120 -

tf > Fig.4:21 Graphical representation of the relation

between T/O , <t> , i|) and i.

between x/a and <|> is given by line c in Fig.4:21 or by eq.4:20. The boundary conditions have been shown to have a considerable

influence on the steady-state shear behaviour of a granular material. However, considering the pre-steady-state phase, the behaviour was found to be dependent on more variables, such as the stress rotation and the stress history. In the reverse shear test the sample behaves differently, while similar boundary conditions are applied. The much,weaker response of the sample in the pre-steady-state phase is notable. In this phase large stress rotations occur. Up to now it appeared not to be possible to describe this phenomenon in a fundamental way. The linear relation x/a = Ktg(90-ib ), as proposed by Oda and Konishi

n i (1974ab), did not yield a constant factor K for tests with crushed glass. The photoelastic tests demonstrate that the relation between x/o and y in reverse shear was dependent not

n only on the actual principal stress direction, but also on the

- 121 -

maximum rotation angle of the principal stress direction with respect to the direction in the steady-state phase. More research has to be performed to analyse the effect of stress rotation in greater detail. Here the photoelastic measuring technique can can be expected to make a valuable contribution to a better understanding of the pre-steady-state phase.

4.2 EXPERIMENTS WITH LABORATORY-SCALE PENETROMETERS

4.2.1 Introduction

The cone penetration test is widely used to investigate the mechanical properties of soil layers in the field. In this test a probe with a wedge shaped tip is driven vertically into the soil, while the cone resistance and the friction of a shaft segment are measured continuously. The cone resistance increases with depth, and the resistance in sand is much higher than, for example, in clay. The measured cone resistance and shaft friction are used to determine soil types and soil parameters of deep layers, which information is used in engineering practice, e.g. to design the foundations of buildings. Much research has been performed on the behaviour of a soil during penetration of a probe in more detail. The information obtained from laboratory tests, however, is restricted because very little is known about the stress and strain in the soil during penetration. Since the stresses and strains are not '^uniformly distributed in this problem, no techniques were available to estimate the stress distribution in the interior of the sample from the boundary conditions. In this section it will be shown that the optical measuring method can be applied to visualise stresses and displacements simultaneously in 'scale models of penetration tests.

4.2.2 Models used

Several scale models were used to perform laboratory tests with crushed glass in order to visualise stresses and

- 122 -

displacements during penetration. The plane samples in a test were confined in cells with plane-parallel glass walls at a distance varying from 50 to 70 mm apart. Crushed glass with a particle size of 2 to 3 mm was used. Photographs of two test setups are shown in Fig.4:22. The dimensions of the sample in Fig.4:22a were 50mmx280mmx280mm. To simulate a greater depth, an initial load was applied by means of platens and springs. The models were placed in a frame, so that a load could be applied to the probe. A probe with dimensions of 50mmxl5mm was made to penetrate into the sample by means of an air cylinder. The total load on the probe was deduced from the air pressure. Strain-gauged cantilever arms were used to obtain point information about the normal stress at some boundary segments. To approximate to a two dimensional stress state as closely as possible, the thickness of the probe was made equal to the thickness of the plane sample. This test setup was used for exploratory experiments, and a series of tests were performed to investigate the influence of the shape of the tip of the probe on the stress distribution in the granular material. Furthermore, some tests were carried out to show the influence of the initial stress on the stress distribution during penetration, and special probes were designed to visualise the influence of the tip and shaft separately. The probes in these tests were too small to visualise the stress and strain close to the tip. In Fig.4:22b the setup for a large scale test is shown. The dimensions of the sample are 70mmx600mmx600mm and a wedge-shaped probe of 70mmx50mm was made to penetrate into the granular material. The initial stress in the sample was increased by means of dead weights (iron blocks) placed on the free surface of the sample. The probe was displaced by a motor-driven jactuator at a speed of 1 mm per day. The total force on the probe and the displacement were measured by means of electrical gauges.

- 123 -

Fig.4:22 Photograph of two test setups for investigating the behaviour of granular material during penetration.

- 124 -

4.2.3 Measured results

Shape of tip Probes with three types of tip were driven into the granular material to investigate whether the shape of the tip has a significant influence on the stress distribution. In the first instance the cell was filled with crushed glass and liquid. The preparation technique resulted in a rather loose packing. The probe was placed in the centre of the free surface in such a way that the distance between cell bottom and tip was about 0.75 times the height of the sample. Next, the platens were placed on the free surface and loaded by means of springs. The actual force was estimated from the compression of the springs and from two strain-gauged cantilever arms mounted on the platens, which measured the normal pressure at the particle boundary. It was attempted to perform each test under similar conditions. A total load of about 1 kN was applied to the platens, which theoretically produce an averaged stress of 71 kN/m at the particle boundary. After adjusting the initial load, an optical measurement was performed in which the distance between the measuring points in x- and y-direction was 20 mm. The stress distribution calculated from the optically measured data of an initially loaded sample is shown in Fig.4:23. Due to boundary friction the stress distribution was not perfectly uniform. Finally the probe was displaced over a distance of about 5 mm and the stress distribution was determined again. This test procedure was performed with a probe with a wedge-shaped, a rounded and a flat tip. The visible phenomenon of the different tests, viewed through a circular polariscope, are presented in Fig.4:24. In Fig.4:25 plots are shown of the principal stress trajectories, which are based on the optically measured directions. It appears that large stress rotations take place in the sample during penetration of the probe. There is found to be good agreement with the pattern of light stripes visible in circular polarised light. It seems, however, that the optical measurement shows more detail in regions with a low stress level.

- 125 -

i i i i int i i i i i +

-wN , niflimiiiii i i i i i i i

- ^ + + -v

f t t ♦ V .

* ♦ + \ *

+ + + + + + + • . • ' i l i i i i l i i i i i i i i i i i i i i i i i i i i i i i i l l l lU l l l l l l l l l i n im

JL ■ ' ' ' ' ' ' _i_

Fig.4:23 Calculated stress distribution in an initially loaded sample, based on 182 measuring points. (1 scale = 100 kN/m2 or 1 cm)

In Fig.4s26 the calculated distribution of the major principal stresses are presented. To indicate the test conditions, some test parameters are given in Table 4:2.

shape of probe Fig. No. load on probe ( kN )

2 gauge 1 ( kN/m ) gauge 2 ( kN/m2 ) e at centre ( kN/m ) <t> (max. ) < deg. >

wedge -0

-120 -100 -110 24

4:26a 1.8 -110 - 90 -180 30

rounded 4:23

0 - 85 -100 - 80 24

4:26b 1.6 -100 -125 -172 34

flat -0

-100 -100 - 80 20

4:26c 1.9 -100 -120 -222 35

Table 4:2 Test parameters during penetration of probes with different shapes of the tip.

The position of the gauges 1 and 2 for measuring normal stresses are indicated by arrows in Fig.4:26. It seems that the shape of the tip has no major influence on the stress distribution in the granular material. The visible differences in the patterns of the principal stress trajectories and stress level may be a result of small differences in the initial conditions.

- 126 -

Fig.4:24 Photographs of penetration tests, viewed in circularly polarised light; a) wedge-shaped tip, b) rounded tip, c) flat tip.

127

; ^ g" ■ ^Q'TW—f—4—H ^ ^^^^T' 1 1 1 "tr é v '^^f^J~^/^^l -Jr*\,S\ /

— i — i — i — i _ i — i — i _ i > i i i . . i . . . . i . . , . i .

Fig.4:25 Computer plot of the principal stress trajectories, based on 182 measuring points of the test in Fig.4:24 (1 scale = 1 cm) .

- 128 -

" I I I I H A I I I l l l 1

u t

- X

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x / -fl \ x

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+ + + + \ "ni l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l lM

• M i l l l l l l l l l l l l l l l + / X

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H f

i

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JT7 * i ■ i * * i * * ■ ■ i ■ ' ■ i ! ' ' ■ ' i ■ ■ ■ ■

Fig.4:26 Calculated stress distribution, based on 182 measuring points of the tests in Fig.4:24. (1 scale = 100 kN/mz or 1 cm).

- 129 -

Fig.4:27 Principal stress trajectories in a sample during penetration, with a large initial stress (left-hand side) and a low initial stress (right-hand side) (1 scale = 1 cm).

Influence of stress level The influence of the initial stress level on the stress distribution is demonstrated by the test results in Fig.4:27. The initial state of this test was similar to the test of Fig.4:25b. However, the load on the platen on the right-hand side was decreased to a very low value. As a result, the principal stress trajectories at a large and at a low initial stress level are visible in the same sample. In the case of a lower level of the initial stress, the stress field caused by the probe is noticeable in a larger region. The stress field obtained is comparable with the stresses in the problem solved by

- 130 -

- x x / -f. \ x x z

1111111lAl 11111 l l l l l , '

- / /

■Al 11111111111111

+ X

it \ H x z

t t \

I i + +

1 I I l l l l l l l l III II II Uil l l l l l l l II I I NU >n i I i i i i I i i i i I i i i t I i i 1 ' ' * ' '—I—I—I—L

Fig .4 :28 Penetra t ion of a probe in to a granular m a t e r i a l in which f r i c t i o n between probe and p a r t i c l e s i s e l imina ted ; a) p r i n c i p a l s t r e s s t r a j e c t o r i e s , b) s t r e s s d i s t r i b u t i o n (1 sca le = 1 cm).

- 131 -

Prandtl (Sokolovski, 1960). Assuming that the, initial major principal stress was vertical, it can be observed that stress rotations of 90° take place in a large region.

Elimination of friction Penetration of a probe into a granular material is a complex phenomenon. Friction and point resistance causes some change of the stress distribution in the granular material. Due to stress increments some elastic, or elastic as well as plastic, deformation takes place. The photoelastic measuring technique makes it possible to visualise the influence of a particular aspect. Fig.4:28 demonstrates the penetration of a probe into a granular material in which the friction between probe and particles was eliminated. The elimination of friction was achieved by inserting the probe between thin metal sheets, which were fixed in the vertical direction. The sheets were introduced into the sample during its preparation. The reduction of the friction can be deduced from the fact that the major principal stress is perpendicular to the boundary of.the wedge-shaped tip, as visible in the photograph of Fig.4:28a. The difference in the stress field is particularly manifest when this photograph is compared with the photograph in Fig.4:24a. The principal stress trajectories and the stress distribution are shown in Fig.4:28b and c, respectively. It seems that the additional stress caused by the probe is transmitted more in the horizontal direction. This causes large stress rotations, so that a softer response of the granular material can be expected.

Separation of tip and shaft Two types of probe were designed to investigate separately the influence of the tip resistance and shaft friction, on the stress distribution in the granular material. The shaft of the probe in Fig.4:29a was masked by a tube which was fixed in the vertical direction. Only the tip was in contact with the granular material during penetration. In Fig.4:29b a probe is shown where the tip is masked, so that the particles are in interaction with

- 132 -

i-,

a b

Fig.4:29 Two types of probes to investigate the tip resistance and shaft friction separately.

the shaft only. The probes where introduced into the sample during preparation. After an initial load was applied, the probes penetrated some distance into the granular material. In Fig.4:30 a series of test results are shown representing penetration of the tip only. In Fig.4:30a the principal stress trajectories of the initially loaded sample are plotted. In Fig.4:30b the probe has been displaced 3 mm; in this case the total force on the probe was F=450N. In Fig.4:30c the principal stress directions are shown; here the tube which protects the probe has accidentally been displaced a little. Contrary to the previous stage, the principal stresses are not parallel to the boundary of the tube. This phenomenon demonstrates that the displacement of the tube has caused a shear force between tube and particles. The stress field caused by the shear force, however, has no significant influence on the stress field caused by the tip of the probe. In Fig.4:30d the force on the probe has a similar value (F=1650N) as in the tests shown in Fig.4:25. The friction between shaft and particles was not completely

- 133 -

- J — I — I — I I I ■ ■ ■ ■ I i I ■ I I I l _ J — I — L - l — I — I — 1 - J — I — ■ — ■ — I — I — I — I — I I ■ ■ ■ ■ ' ■ ■ ■ ■ I

Fig.4:30 Principal stress trajectories during penetration of a tip only; a) initial loaded sample, b) F = 450N, c) F = 600N, d) F = 1650N (1 scale = 1 cm).

- 134 -

CRUSHED GLHSS V.

CONSIDERED 14 C H ! REGION

ZDD

Fig.4:31 Test setup of the large-scale penetration test.

eliminated in this test. However, the section of the shaft close to the tip shows that the friction was not mobilised so strongly as in the tests of Fig.4:25. -On comparing the pattern of the trajectories in Fig.4:30d with the patterns of the tests in Fig.4:25, it can be concluded that the stress field due to shaft friction has no major influence on the stress field due to tip resistance. This observation has been confirmed in tests with the probe of Fig.4:29b. The shaft friction appeared to cause very little change in the initial stress. Only close to the shaft could some stress rotations be observed, and the measured displacements in the granular material were very small.

Observations close to the tip In this test a region close to the tip of the probe (see Fig.4:31) is considered. Lead markers (black pixels in Fig.4:22b) were distributed in a plane at mid-thickness of the sample to monitor the deformation. The front face of the probe was covered with a white layer on which the shape of the tip was symbolised by means of three black dots, so that the displacement

- 135 -

# * 2

r •*

*-4| Fig.4:32 Photograph in circularly polarised light, showing the photoelastic phenomenon during penetration.

' " i ' ' " i i l n m i i n | n i n i i i i | i i i t i n i i | I I I H I I M | I m i n n [ i n H I M I [ I I H i n n | i i n 10 20 30 40 50 60 70 80

S [ mm] ^, Fig.4:33 Graphical presentation of the total load on the probe during penetration.

p. H-in T3 M P n ro 3 (T) 3 c t

H-3 r t 3" ro T P ' 4 3' c t l

3" P 3 p.

^ ro i f l H-

o 3

H-3

P

Ul 3 P M M

P. H-U) 13 t~> 9> n (V 3 ID 3 r t

P. H-01 73 M P n fl> 3 ro 3 c t

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- 137 -

Fig.4:34 Presentation of the optically measured displacement of the tip of the probe and the marked particles. The displacement of the tip was about 100 mm.

Fig.4:35 Graphical presentation of a displacement step; the probe has been displaced 9 mm. The displacements of the marks are enlarged (1 scale is a displacement of 2 mm).

- 138 -

step. However, if a large displacement interval is considered, there is no significant difference. In Fig.4:36 the displacements of the marked particles are plotted with respect to the tip of the probe. This diagram shows how the granular material flows around the probe. In Fig.4:37 only the top row and bottom row of the marks are considered. Since the bottom row is more or less undisturbed a straight pattern of these marks can be observed. On comparing the vertical displacement of the marks of the top row with that of the bottom row, it is seen that marks located at a distance of more than half the width of the probe from the shaft are additionally lifted up. The resultant displacement of the material close to the probe, however, is in the downward direction.

The stress distribution in the granular material derived from the optical measurements is shown in Fig.4:38. The plot refers to test No. 73, which is indicated in Fig.4:33. Since no point information was available about the stresses at a boundary segment, the friction angle was used to derive the absolute stress level at the centre of the sample. A friction angle of 35 was assumed. A typical phenomenon is that the mobilised friction angle is largest in a region at some distance below the tip of the probe. The observed shear zone in Fig.4:35, however, does not pass through this region. This behaviour can be explained by the fact that a granular material behaves more weakly if stress rotation takes place, so that the shear band prefers to go through a region with large stress rotations and a relatively low value of the mobilised friction angle. The tip of the probe was assumed to be flat in this calculation example, so that it was easy to determine the average vertical stress on the tip of the probe. In this stage a vertical stress of 340 kN/m was found. The vertical stress derived from the applied load on the probe was found to be 390 kN/m2. If it is taken into account that the total load includes some shaft resistance, a good agreement between the mechanically measured load and the load derived from the optically measured data is found to exist.

- 139 -

Fig.4:36 the probe.

Displacement of the marks with respect to the tip of

i

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Fig.4:37 Observation of the displacement of two rows of marks with respect to the tip. The bottom row is more or less undisturbed.

- 140 -

X

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Fig.4:38 Stress distribution in the granular material. (1 scale = 100 kN/m2 or 1 cm)

* a

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x x % *

Fig.4:39 Distribution of the strain rate tensor. The principal direction of strain is compared with the major principal stress direction.

- 141 -

The Fig.4:39 the distribution of the strain rate tensor and the major principal stress direction are plotted, based on a displacement step of 10 mm of the probe (No.60 to No.70 in Fig.4:33). The principal stress directions of both stages were plotted. It seems, however, that there was very little rotation in the displacement step considered. It is shown that the major directions of stress and strain do not coincide in general, which appeared to be a normal phenomenon in a granular material. This plot also demonstrates that the region below the tip behaves very rigidly: large strains occur in regions with a relative low value of the mobilised friction angle. The volume was found to increase at several material points. However, on considering a large region during the passage of the tip of the probe, a small decrease of the volume was observed.

4.2.4 Discussion

In the literature very little material is available which can be used to verify the optically measured stress and strain fields in penetration tests. Small scale penetration tests showing the photoelastic phenomenon in an optically sensitive granular material during penetration of a probe were reported by Dantu (1957) and Wakabayashi (1959). These tests showed qualitatively similar results, but dit not yield details about stresses and strains. Several experiments have been reported in which the deformation of the soil during penetration is made visible quantitatively by means of coloured layers or by means of radiographs. The slip lines under footings, which have been made visible by Kimura et al. (1985) by means of X-rays radiographs, show a curve similar to the slip lines which can be observed in the large-scale penetration test in Fig.4:35. Many calculation procedures have been developed to predict the ultimate bearing capacity of soil masses. A finite element analysis of static penetration tests in clay, presented by de Borst and Vermeer (1982) and de Borst (1982), shows the

- 142 -

Fig.4:40 Schematisation of the interaction between probe and granular material by means of a point load.

displacement field around the tip of the probe. These results, however, are not properly comparable with the optically measured displacements because they refer to cohesive materials with a low friction angle. In general it seems that most analyses refer to cohesive materials. Several analyses of a pile foundation are based on the assumption that the granular material behaves like an elastic medium, as for example the method described by Poulos and Mattes (1969). However, no analyses were found which yield a detailed map of the stress distribution and strain in the granular material. To obtain some evidence of the validity of the optically measured stress fields, an elastic calculation method has been developed which can produce a detailed map of the stress distribution. Since plane strain samples were used, the system can be considered as a two dimensional problem. If it is furthermore assumed that the interaction between probe and granular material can be schematised by a series of point loads, each of which causes a radial stress distribution; a simple elastic solution is

- 143 -

available for calculating stresses at an arbitrary point. In Fig.4:40 the interaction between a probe and the granular material is schematised by a point load P acting at a point x , y on the face of the wedge-shaped tip with inclination a. The angle between the forces and the normal to the plane is y. The initial stress at a material point of the granular material is given by a , a , a and a .

xx yy xy yx The stress components at point x,y, caused by the load P at point x , y can be calculated with o o

°xx = crrcosZ<e+a> (4:21a)

a = or rsinZ(6+oc) ( 4 : 2 1 b )

a =-or = a sin(9+<x)cos(e+<x) ( 4 : 2 1 c ) xy yx r

where a at point x,y is (Timoshenko, 1951)

2P ar =~cos(Y+6) (4:22)

The state of stress, at a material point is found by summing similar stress components of the initial stress and the stress components caused by the point loads. This calculation method was used in a computer program, in which a series of point loads could be defined by the parameters x , y , P, y and oc. In Fig.4:41 four numerical examples are shown in which the conditions are more or less similar to those of Fig.4:30,a, b, c and d. A homogeneous initial stress was assumed (Fig.4:41a) with a =-160 kN/m2, a =-70 kN/m2 and t|> =90°. The interaction between probe and particles was schematised by two point loads, perpendicular to the flat tip of the probe (the locations are indicated by X in Fig.4:41). The value of the point loads represent the measured total load on the probe in the test of Fig.4:30. In the diagrams b and c of Fig.4:41 a force on the probe of 450N and 600N respectively has been simulated. Just as in the the actual test represented in Fig.4:30b and c, the

- 144 -

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Fig.4:41 Stress distribution in an elastic medium, caused by two point loads. The initial stress in the elastic medium was o =-160 kN/mz, a =-70 kN/mz.

1 ' 2

- 145 -

principal stress direction is influenced in a small region. in the case of Fig.4:30d and Fig.4:41d the load on the probe was increased to 1650N. It seems that in both samples the stress in a large region is influenced by the tip. However, quantitatively there is little agreement with regard to the distribution of the principal stress direction. The reason for the deviation is that in the elastic calculation no deformation is taken into account during penetration. In an actual test the granular material is stretched in the horizontal direction in a region beneath the tip. This phenomenon is simulated in Fig.4:42 by directing the point loads in a more horizontal direction. It appears that in this case curves of the major principal stress trajectories similar to those in the actual test were obtained.

A simple elastic approach did not yield a general solution for this problem. The elastic calculations, however, show the same tendency at low stress increments as in the actual test, which gives some alternative theoretical support for the reliability of the optical measuring method.

Since the deformation in the large-scale penetration test (Fig.4:35) shows agreement with Prandtl's failure mechanism (Sokolovski, 1960), it was investigated whether the solution given by Prandtl could be used to estimate the total load on the tip of the probe in this test. The principle of Prandtl's problem is shown in Fig.4:43. Two different line loads P and P r ^ c v act on the surface of a semi infinite granular medium. In the case of failure it is assumed that slices move along the slip lines, as indicated in the diagram. Prandtl's equation, describing the ratio between P and P as a function of the friction angle in a non-cohesive material is p = p I+sinj irtgq» c v l-sin<|> v«*.4z>

When a friction angle of <t> = 35° is assumed, the ratio between the line loads is found to be P / P =33. At a depth of 30 cm the measured maximum load at the probe was 1 kN. This results in an average stress at the tip, with a dimension of 5cmx5.7cm,

- 146 -

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t +

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Fig.4:42 Simulation of plastic deformation by directing the point loads in a more horizontal direction.

Fig.4:43 Failure mechanism in Prandtl's solution of a footing problem.

- 147 -

of P = 350 kN/m2. Hence P = 11 kN/m2. The vertical stress in the c v field calculated from the specific weight of the granular material and the weight of the iron blocks was P = 5.1 kN/m2, which is lower by a factor of 2 approximately than the field stress calculated with Prandtl's equation. In spite of the difference, the result seems reasonable because the granular material at the considered depth is lifted up (see Fig.4:37) so that a passive pressure has to be taken into account. Axially symmetric tests with anchors in sand which are loaded in the vertical direction (reported by Kiewiet, 1982) also show a larger effective stress. It appeared that the ratio between the passive and active stress was proportional to the area of the

2 anchor. An anchor of 0.02 m , for example, gave a ratio of 3 if the peak loads were considered. In the penetration test the uplift of the sample was small, so that the peak stress was probably not reached. A smaller ratio in this test is therefore reasonable. If eq.4:23 is used to calculate the bearing capacity of a pile with a cross-section of 0.25mx0.25m which is installed in a sand layer at a depth of 14 metres (for which <J> = 35°, 7=12 kN/m3 and the ratio between the passive and active pressure is 2), a total tip load of 693 kN is found. This appears a reasonable result when compared with a case study described by Heijnen and Janse (1985), in which a total tip load of 700 kN was found. A cone penetration test performed at the same location measured a cone resistance of 16300 kN/mz at a depth of 14 metres. With eq.4:23 the ratio between the passive and active stress in the field was found to be equal to 3. It must be mentioned that the result of eq.4:23 is strongly dependent on the friction angle used and that the maximum tip load of the actual pile was deduced from the total load on the pile. Furthermore, the cone resistance in the cone penetration test varies strongly over a short distance. In order to obtain more general results, field tests in a well defined soil layer have to be performed and the point resistance of a pile has to be measured by a direct method. Comparison of model tests with

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Fig.4:44 Photograph of a scale model of a hopper.

stress- strain behaviour of a granular material in a hopper. In this chapter some results of those experiments are presented.

4.3.1 Test setup

A two-dimensional scale model of a hopper was used to visualise the stress distribution in the interior of a granular material. A photograph of the test setup is shown in Fig.4:44. A model of a hopper was placed in a cell of 300mmx300mm, with plane-parallel glass plates at a distance of 50 mm apart. The boundary of the hopper was equipped with four strain gauged cantilever arms to obtain point information about the normal stress at the boundary. The outlet of the hopper was closed by a hinged plate which could be removed very simply by pulling out a small supporting beam. Since the self-weight of the granular material was not sufficient to visualise the stress by optical means, an additional load was applied on the free surface of the material to simulate a greater height. The load was applied by means of a rigid metal plate and

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a pneumatic cylinder. Crushed glass with 2-3 mm particle size was used as the granular material. Black markers were distributed in a plane at mid-thickness the sample to monitor the deformation.

4.3.2 Measuring results

The purpose of this test was to visualise the stress distribution after filling and after some of the material had flowed out.

Photographs of the optical phenomenon in the two test stages, viewed in circularly polarised light, are shown in Fig.4:45. In Fig.4:46 two computer plots are presented showing the principal stress trajectories, based on data of the optical measurements. As can be observed, the principal stress directions change significantly when some flow has been taken place. In Fig.4:47 the absolute values of the major principal stresses at some points in the field and the normal stresses at the boundary have been plotted. The calculation is based on the optically measured data, using the mechanically measured normal stresses at the boundary (the labelled arrows) as boundary values. It is seen that the normal stress at the upper part of the boundary of the cone has increased significantly when some flow has been taken place. The displacement of the marked particles during the flow step are represented in Fig.4:48. The co-ordinates of the marked particles are derived from photographs by means of a remote controlled digitiser.

4.3.3 Discussion

In Fig.4:47 it is demonstrated how the stress distribution in the interior of the granular material and the boundary stresses change when the outlet is opened. It appears that the boundary stress has significantly increased in the upper part of the cone section after some material has been flowed out. The local

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Fig.4:45 Photograph, in circularly polarised light, of the optical phenomenon in a hopper; a) after filling, b) after some flow.

J_i_ _i_L Fig.4:46 Computer plot of the optically determined principal stress trajectories (1 scale = 1 cm).

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i<

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t=3

i I _L

1 = > / + + *xx\ / X x X \ \

J_i_

Fig.4:47 Presentation of the stress distribution in a hopper after filling (a) and after some flow (b) (1 scale = 100 kN/m2 or 1 cm).

I I

Fig.4:48 Displacement of marked particles after some flow.

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boundary stresses in this test were not of such extreme magnitude as was found in several theoretical models. It must be noted, however, that only one particular case has been considered. As observed by Bransby and Blair-Fish (1974), the boundary stresses change with time. To investigate this phenomenon in more detail by optical means, the stress distribution in more stages has to be considered.

The isotropic points, marked by A and B in Fig.4:46b are a typical phenomenon. At an isotropic point the principal stress difference is {a -a )=0. If the stress distribution in the cone x 2 section is in the active state, the two isotropic points neutralise each other and disappear, so that a stress distribution as shown in Fig.4:46a is obtained. When the hopper starts to flow A and B are separated, and B moves to the outlet. Point A indicates the transition between the active stress in the bin and the passive stress in the cone section. Only at this point does a jump rotation of the principal stresses of 90° take place. In other regions the transition appeared to be smooth. The shear stress at the isotropic point is zero, and the value of the mobilised friction angle in the region close to the isotropic point also has a low value. It would seem that the isotropic points are general phenomena in a flowing two-dimensional hopper, so that there is always a region which is not in the limit state of stress. Since only one deformation step has been considered, there is no experimental evidence to indicate that the position of the isotropic point is fixed at one particular place during flow.

The results of the displacement measurements are presented in order to give some idea of the flow in the hopper. A more or less radial flow can be observed in the cone section, which was also found in the experiments of Bransby et al. (1973). The stress distribution in the deformation step changes too much to allow aspects of the stress-strain behaviour to be investigated in greater detail. It can be observed in Fig.4:47b that the rigid plate is not very suitable to apply the additional load. The plate did not follow

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the differences in settlement of the particle surface, so that the stress decreased at the centre of the plate. This phenomenon influences the stress distribution in the whole hopper. In a future device a more flexible loading plate will therefore have to be used.

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5 DISCUSSION

5.1 POSSIBILITIES AND LIMITATIONS

The optical measuring method can be applied succesfully to examine the behaviour of several types of system the mechanical properties of which are strongly dependent on the stress-strain behaviour of a granular material. Since the measured material state is expressed in terms of stress and strain rate tensors, a good comparison between experiments and theoretical solutions is possible in principle . The optical method is particularly suitable for investigating problems in which the granular material is subject to a reasonably large shear stress, as is, for example, the case in penetration tests and shear devices. There are several other problems which can be investigated by this measuring method, e.g. : - the behaviour of ground anchors - friction between a granular material and a rigid boundary - interaction between granular material and pressure transducers - horizontally loaded piles

At present it is not possible to visualise stresses in systems in which the stress originates from the body force only. One way to solve this problem consists in using a more sensitive material, such as particles of plastic. Although several kinds of transparent plastic particles are available, it has hitherto not proved possible to find a material with a sufficient homogeneous refraction index necessary for making transparent samples. A more complicate method is to increase the body force of the granular material, so that a reasonable stress level will be reached. Exploratory experiments have shown that the body force can be sufficiently increased by means of a liquid flow. Since the application area of the optical measuring method would be dramatically increased if self-weight systems could be investigated, a device based on this principle may be a worthwile development.

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Up to now crushed glass has proved to be the most suitable material in the photoelastic test technique. Since glass is not a material commonly employed, in engineering practice, the optical measuring technique has to be used mainly to investigate general mechanisms in samples. However, the mechanical properties of glass are close to those of sand, so that this material can be simulated in a realistic manner. In the present study particles of 2-3 mm size have been used. It is possible, however, to use particles of different size, and the shape of the particles can be changed. The minimum grain size which can be used depends on the homogeneity of the refraction index of glass. If glass of a good quality is used, the particles can perhaps be made so small that the material possesses some cohesive behaviour. If the particles are small, the mechanical behaviour of the assembly will be influenced by gradients in the liquid pressure during deformation, so that this phenomenon can be investigated also by optical means.

The optical measuring technique is restricted to two-dimensional problems in which a principal stress difference can be measured in the range of approximately 20 < (or2~°f1) < 7 0 0 kN/m2. The minimum value is restricted by the optical sensitivity of the glass, while the maximum value is limited by the wave-length of the light used in the optical system for stress measurement (see Chapter 2.3.4). At the present stage there seem to be no possibilities for the generalisation of the method to test models with a three dimensional stress distribution. Furthermore it is not possible significantly to increase the measuring range of the principal stress difference. However, since a large number of different problems can now be examined, the technique appears to be a valuable measurement technique.

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5. 2 ACCURACY

The accuracy of the measuring method is influenced by a large number of factors. If the properties of the optical measuring device are considered, it can be concluded that the reproducibility of the optical measurement is ample. If the light beam has not been transmitted through a model, a constant light intensity is measured during the rotation of the analyser. Deviations of about 0.5% can be detected. The reproducibility error in the determination of the principal stress direction is about 1°, and the deviations of the retardation measurements are less than 1%. Measurements in homogeneous elastic material show a linear relation between (a -a ) and the retardation 6. The

2 1 error in the reproducibility of the determination of the coordinates of the marked particles (Allersma, 1984) is about 0.01 mm, which includes the linearity error of the translation screw.

Several errors may be introduced by the test model. It appeared for example that the glass particles are not quite free of stress in the unloaded condition, and internal stresses in the glass wall may also affect the measurement. Theoretically the internal stress in the particles has no influence on the measurement because the average stress is zero if no external load is applied. However, in practice some initial stress is measured. The internal stress in the glass wall is strongly dependent of the quality of the glass. Normal window-glass with a thickness of 14 mm appeared to be most suitable. It is of course possible to use special stress-free glass, as is for example used in the optical industry. However, this would increase drastically the cost of the tests. Since the glass wall confines the particle assembly, it cannot be prevented that the particles cause some stress in the glass sheets by lateral confinement pressures. Because the friction angle between glass particles and a glass sheet is very small (about 10 ) no large shear stresses can be expected. It is therefore assumed that the glass sheets are mainly subjected to pure bending. In this case

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the tensile and the compressive stress is symmetrically distributed over the thickness, so that the sheet behaves in an optically neutral manner. Since the glass particles and the glass walls are always loaded at the same time, it is not simply possible • to verify if the glass wall was neutral in an actual test. A test with a single glass sheet, however, demonstrated that little change in the optical signal could be observed if the sheet was subjected to pure bending only.

There are some other factors which can cause errors in the measurement, such as - the assumption that the state of stress is two-dimensional; - gradients in the light intensity over the surface; - reflections at the particle surfaces; - measurement of normal stress at the boundary. It has not been found possible to analyse the influence of all these factors separately. Furthermore it has not been possible to determine the accuracy in a direct way, because no method is available to calculate the true stress distribution in a granular material. However, several aspects can be checked in a test to decide whether or not the results are reasonable. One of them is the comparison between the calculated values of the major principal stress at some point in a sample, by using different measuring points at the boundary. If the normal stress is measured at two or more points at the boundary of a test model, a surplus of data is available, so that good agreement is obtained only if the measured data is reasonable. As an example, the major principal stress was derived four times at the centre of the model of the test series of Fig.4:6, in which a value of K=l.l was used for the optical constant. The values of o derived from each of the four measuring points at the boundary are presented in Table 5:1. It appears that the deviation is larger if the stress has a lower level. In the test the deviation between the most extreme value and the averaged value is 26%. The deviation decreases if the stress increases. In tests b, c and d the deviation is 11, 9 and 4%, respectively. The deviations may be caused by, the mechanical strain gauges as well as by the optical

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s t r a i n gauge No.

F i g . 4 : 6 a b c d

1 a

l - 89 -144 -152 -168

2 a

l -116 -159 -172 -172

3

1 - 72 -134 -157 -176

4 a

l - 92 -137 -152 -166

Table 5:1 Major principal stress ( kN/m ) at the centre of a model, derived from four different points at the boundary.

measurement. A deviation of less than 10% was considered a sufficiently good result, which was attainable in most tests. Other check parameters, of a more qualitative kind, are the maximum value of the mobilised friction angle and the value of the minor principal stress. In the test of Fig.4:2 the maximum value of the mobilised friction angle was 0 =36°, which is a reasonable value for crushed glass. Since crushed glass is not cohesive, the minor principal stress must not be positive anywhere in the field. A positive value of a was very exceptional in the tests hitherto, performed. The accuracy can be increased by using a better quality of glass and more advanced strain gauges to determine boundary stresses. However, it is expected that the accuracy of the measuring method is finally limited by the discrete behaviour of the granular material, so that the accuracy cannot be improved unrestrictedly by improving the sensoring.

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NOTATION

The principal symbols are presented in the list below. Since the meaning of each symbol is not unique a separation has been made between the optical section and sections where stresses and strains are discussed. Symbols not listed below are clearly defined in the text.

Optical section A maximum amplitude a actual amplitude c, c' propagation speed of light <jj angular velocity A wave-length of light S retardation e difference in propagation speed q , q optical axes in double-refractive material r length light vector t time I light intensity P rotation angle analyser K optical material constant s thickness of sample

Stress and strain de strain rate uj assymmetric part of the deformation tensor 0 direction of the principal compression strain Y shear distortion a stress tj) major principal stress direction ot angle between major principal stress and shear band T) angle between principal compression strain and shear band <b mobilised friction angle ^m ^ v angle of dilation p ,. Poisson's ratio

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F total shear force P total load S , S planes of maximum shear stress a, b shear rates in planes of maximum shear stress i non-coaxiality angle D. structural rotation

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SAMENVATTING

OPTISCHE BEPALING VAN SPANNINGEN EN VERVORMINGEN IN FOTO-ELASTISCHE KORRELSTAPELINGEN

In conventionele beproevingsapparatuur voor onderzoek naar het gedrag van korrelig materiaal, moeten de spanningen en meestal ook de vervormingen in het inwendige van een korrelstapeling worden afgeleid van de condities op de wand. Daar dit alleen mogelijk is bij zeer specifieke randvoorwaarden, is in de meeste experimenten weinig bekend over de spanningstoestand en de vervorming in de korrelstapeling zelf. Hierdoor wordt beperkte informatie verkregen over het korrelgedrag in modelproeven en is het niet goed mogelijk om de details van rekenmodellen te vergelijken met de proefresultaten. In de jaren vijftig werd aangetoond dat met behulp van korrels van foto-elastisch materiaal het transport van krachten in een korrelstapeling zichtbaar gemaakt kon worden. In dit proefschrift is beschreven hoe met behulp van deze proeftechniek een optische meetmethode kan worden ontwikkeld om spanningen te kwantificeren in het inwendige van een drie-dimensionale foto-elastische korrelstapeling, die in een vlakke vervormingstoestand verkeert. Door middel van een mathematische analyse is aangetoond dat met behulp van optische filters de gemiddelde waarden van twee componenten van de spanningstensor kunnen worden gemeten in een gebiedje met een inhomogene spanningsverdeling. Door metingen te verrichten in een groot aantal punten van een model is de verdeling van de spanningscomponenten min of meer continu bekend. Een rekenprocedure is beschreven waarmee met behulp van de optisch gemeten data en enkele randvoorwaarden de volledige spanningsverdeling in een belaste korrelstapeling kan worden bepaald. Omdat de gebruikte korrelstapelingen transparant waren kon een optische techniek toegepast worden om de verplaatsingen van gemerkte deeltjes rechtstreeks in het proefmodel te bepalen.

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Beschreven is hoe dit nauwkeurig en automatisch kan geschieden met een digitale camera. Een rekenmethode is beschreven waarmee uit de gemeten verplaatsingen van een aantal deeltjes de vervormings tensor in een stoffelijk punt kan worden bepaald. Naast het meetprincipe is het ontwerp van een computer gestuurd meetapparaat beschreven, waarmee de spanningscomponenten en verplaatsingen in korrelstapelingen volledig automatisch kunnen worden bepaald. Ook wordt aandacht besteed aan de optische en mechanische eigenschappen van het gebruikte korrelig materiaal en aan de proeftechnieken voor het produceren van korrels en het belasten van korrelstapelingen. De optische meetmethode is gebruikt voor onderzoek naar het gedrag van korrelig materiaal onder diverse omstandigheden. Een aantal schuifproeven zijn beschreven, die laten zien dat het materiaalgedrag zeer afhankelijk is van de randvoorwaarden en dat de hoofdrichtingen van spanningen en vervormingen meestal niet samenvallen. Omdat de bepaling van de spanningscomponenten niet afhankelijk is van de randvoorwaarden kon het materiaalgedrag bij extreem grote vervormingen worden bestudeerd. Verder is de meetmethode gebruikt om spanningen en verplaatsingen zichtbaar te maken in schaalmodellen van funderingspalen en silo's. Deze proeven laten zien dat grote spanningsrotaties kunnen optreden, waardoor grote deformaties kunnen plaatsvinden bij een relatief lage schuifspanning. Ook in deze proeven kon worden zichtbaar gemaakt dat de hoofdrichtingen van spanningen en vervormingen niet altijd samen vallen. Tenslotte is uiteengezet wat de mogelijkheden en beperkingen van de meetmethode zijn en een aantal aspecten met betrekking tot de nauwkeurigheid zijn behandeld.

OPTICAL ANALYSIS OF STRESS AND STRAIN IN PHOTOELASTIC ...

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