Chapter 6 INVESTIGATION OF BULK FOAM BEHAVIOUR FOR MODELLING · 77 Chapter 6 INVESTIGATION OF BULK...

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Chapter 6 INVESTIGATION OF BULK FOAM BEHAVIOUR FOR MODELLING

This chapter describes a brief investigation in to the uniaxial compressive and tensile

properties of Alporas foam with the aim of retrieving material characteristics for use in a

finite element material model. The optical full-field strain measurement system was also

used to provide further analysis of the aluminium foam behaviour. The compressive and

tensile stress-strain responses were used to generate FE material model input parameters.

The behaviour of an FE model of the compressive loading state was then compared with

experimental observations.

6.1 COMPRESSION TESTING The behaviour of aluminium foam under uniaxial compression loading is often used as

an initial characterisation method. For many constitutive models, the compressive response

is important in describing the failure surface. In particular, the Deshpande-Fleck model uses

the uniaxial compressive stress strain curve. The Reyes [103, 104] implementation of the

Deshpande-Fleck model was used for the finite element modelling in this project, requiring a

study of the compressive behaviour for the material model input parameters. The optical

full-field strain measurement system was used to investigate the compressive response of the

Alporas foam, in particular to explore the variation of the response with respect to the spatial

orientation. For foam structures, the manufacturing process can result in some variation in

cell structure according to the direction of production. This can result in anisotropic

mechanical behaviour [105]. The Alporas manufacturing method uses a batch process, and

has been found to have significantly less directionality than other aluminium foams which

are manufactured using continuous production methods [35].

Chapter 6 Investigation of bulk foam

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6.1.1 SAMPLE MANUFACTURE Cubes of 30 × 30 × 30 mm dimensions were cut from a 30mm thick sheet of 0.23

g/cm3 Alporas foam, as supplied by the manufacturer. Samples were cut using a diamond

tipped saw and visual inspections found no damage to the foam structure from the cutting

process. Figure 6.1 shows a typical cubic sample. Each sample was marked to record the

orientation of the cube with respect to the large sheet – showing through thickness, length,

and width of the panel. Figure 6.2 shows the three orientations with respect to the purchased

panel. The orientation of this panel with respect to the manufacturing process is not known.

The z-direction is most relevant in this study as it is the orientation that was used in the

sandwich beam production. A paint speckle pattern was applied to each face, as described in

Chapter 3.

Figure 6.1: Cube compression sample

Figure 6.2: Diagram showing orientation labels for cubes cut from bulk foam panel

6.1.2 TEST CONDITIONS Samples were tested using a universal testing machine (Instron 4505), at a crosshead

velocity of 2mm/min. A vacuum bag teflon sheeting was placed at the interface between the

sample and the loading platen to reduce friction. The tests were continued to a crosshead


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displacement of about 18mm (approximately 60% strain). Tests were repeated for each

direction for 4 samples. Load displacement curves were recorded for each sample. Figure

6.3 shows a test in progress.

Figure 6.3: Cube compression test in progress

The Aramis system was used to calculate full-field strain contours for each sample

orientation. The system set up was as described in Chapter 3, with a measuring volume of

40 × 50 × 50 mm and a sampling time of 2s. Movie files showing images from one of the

Aramis cameras and the calculated von Mises strain distributions are provided in the

appendix.

6.2 COMPRESSION TESTING RESULTS Figure 6.4 shows the typical compressed shape of the aluminium foam cube after a

crosshead displacement of 6mm. Generally the foam crushes progressively, with very little

expansion. The individual cells crush and collapse into each other. The position of the

deformation initiation varies irregularly between the samples. The collapse appears to start

where ever the weakest cell exists, which may be in the form of a large cell, or a defect in the

cell wall shape, thickness or microstructure. The failure then spreads through the sample in a

deformation band. The deformation band typically remains within the general horizontal

plane of the initial collapse, although there may be some deviations out of this plane as the

fracture follows any weaker cell regions. These deformation bands have been observed in

various aluminium foams [90, 91, 106, 107] and are typical in the Alporas compression

failure. A movie showing the typical compression progression from images taken with the

optical strain measuring system is provided in the appendix.


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Figure 6.4: Typical deformed sample under compression loading, at 6mm crosshead

displacement

6.2.1 LOAD-DISPLACEMENT CURVES A typical load-displacement curve for each orientation is shown in Figure 6.5. The

general curve shape follows that observed in many cellular structures, and as discussed in the

literature review chapter. There is initial elastic behaviour up to a first peak load. This is

followed by a plateau region before densification begins as the foam completely crushes and

the load increases. There is not a very large variation in magnitude between the orientations,

with the first peak and plateau loads around 1-1.3kN. The average measured first peak load

and stress values are given in Table 6.1. These values are slightly below the values quoted

by the manufacturer who reports a compressive strength of 1.9 ± 0.3 MPa [33]. Generally,

the literature also reports higher values. Ashby et al. [12] lists a first peak compressive

strength of 1.3-1.7MPa for Alporas foam . Ramamurty and Paul [35] found an average first

peak stress of 1.93MPa for 25mm cubes of density 0.226-0.284g/cm3. Andrews et al. [39]

reported a first peak compressive strength for two directions of 0.25g/cm3 Alporas foam to

be 1.84 and 1.46 MPa. A first plastic collapse stress of 1.615MPa was recorded by Jeon and

Asahina [50]. The variations in the measured values may be related to the reported size

effect where the ratio of sample size to cell size can become influential on various properties.

The general inhomogeneous nature of these materials as well as the presence of voids and

defects can also influence the measured properties. The stress-strain curve suggests that the

densification strain would occur at around 0.65-0.8. This agrees with Ashby et al. [12].

Whilst this study is limited by a small sample size, the results from the study are

deemed sufficient for developing input parameters for the foam material constitutive model.

The use of the full-field strain analysis will also be advantageous in evaluating the

performance of the finite element model.


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Figure 6.5: Typical load-displacement plot for each sample orientation

Table 6.1: Compression curve characteristics; average properties with standard deviation in parentheses

Orientation Peak load

Stress at peak load

Strain at peak load

(kN) (MPa) (mm/mm) X 1.068 1.206 0.045 (0.168) (0.169) (0.007)

Y 1.006 1.132 0.068 (0.084) (0.089) (0.026)

Z 0.980 1.121 0.048 (0.339) (0.377) (0.021)

6.2.2 ENERGY PLOTS A significant property of aluminium foam is its ability to absorb energy. Figure 6.6

shows the typical energy absorbed for each orientation of the cube. The energy absorbed

was calculated by the area under the force displacement curve. There is not a significant

difference between the three directions; however the z orientation samples did display

slightly greater energy absorption. Cell shape isotropy within the foam structure has been

found to be highly influential on mechanical performance with greater strength found in

structures with minimal ellipticity and with cells oriented parallel to the loading direction

[105]. Alporas has been described as significantly isotropic with minimal observed

ellipticity [35]. The small variations in mechanical properties measured in this project are


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supported by these reported observations which suggest that perhaps the z-direction may

have slight increased ellipticity compared to the other directions.

Figure 6.6: Typical energy absorbed during compression for each sample orientation

6.2.3 STRAIN DISTRIBUTION The optical full-field strain measurement system was used to record strain

distributions throughout the compression testing for a sample of each orientation. Figure 6.7

shows the strain distributions recorded for each orientation at a crosshead displacement of

1.3 and 2.7mm. The first displacement corresponds to the first peak load in the load-

displacement curve while the second displacement corresponds to the initial part of the first

plateau region. There are some noticeable differences between the differently oriented

samples. In the early displacement contours, the strain distribution is relatively uniform,

with some dispersed regions of higher strain. There are only a few isolated regions of strain

concentration in the y-direction contour (see Figure 6.7(c)) visible as light blue and red

areas. The number of these regions is increased in the x-direction contour, in Figure 6.7(a).

The z-direction contour (see Figure 6.7(e)) shows even more regions of strain concentration.

These differences correspond to the small variation in the displacement at which peak load

was reached for each orientation. The z-direction sample reached the first peak load at an

earlier displacement than the other orientations, and the failure had progressed further at the

comparison displacement of 1.3mm.

The strain distributions for the second displacement investigated, corresponding to the

beginning of the plateau region of the load-displacement curve, show more variation


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between the sample orientations. The y-direction contour in Figure 6.7(d) shows a few more

regions of strain concentrations than at the earlier displacement; however the magnitude has

not increased significantly. The main strain concentration region from the earlier

displacement has actually reduced in magnitude. In contrast, the x-direction contour (see

Figure 6.7(b)) has significantly more regions of high strain concentrations. These regions

have developed in the area of the strain concentrations that were visible at the earlier

displacement. Similarly in Figure 6.7(f), the z-direction contour shows several regions of

high strain values which extend across the image. These regions probably correspond to the

deformation bands widely reported in the literature [90, 91, 106, 107].

Figure 6.7: Typical compression strain distributions for each sample orientation at two

crosshead displacements; x-orientation at a) 1.3mm and b) 2.7mm, y-orientation at c) 1.3mm and d) 2.7mm, z-orientation at e) 1.3mm and f) 2.7mm

6.2.4 SECTION LINE PLOTS Section planes were taken through the measured strain contours and used to produce

line plots of the strain magnitude along the sections. Five sections were taken across the


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strain distribution, symmetrically dispersed throughout the image with a spacing of 5mm, as

shown in Figure 6.8. Figure 6.9 shows the section line plots for the z-direction sample at

crosshead displacements of 1.3 and 2.7mm. The development of the increased strain

magnitude in the region of section 3 can be seen. As the high strain region expands

diagonally upwards this is reflected in the increase in the strain magnitude of section 2.

These plots further demonstrate the bands of high strain values and hence bands of

deformation previously discussed.

Figure 6.8: Cube strain contour showing position of 5 section planes

Figure 6.9: Section strain line plots of Z-direction sample at a) 1.3mm (stage 20) and b) 2.7mm

(stage 40)

The development of a deformation band and associated strain contour is shown in

Figure 6.10. Here, the strain line plot of section 3 in the z-direction sample is displayed at

increasing crosshead displacements, ranging from 0.7mm to 4mm. The increasing strain is

clearly visible, particularly around the 10mm x position. At the maximum x position there is

very high strain magnitude in the curve for the greatest crosshead displacement (stage 60).

This may correspond to a region in the contour where there was high distortion in the photo

image. This may have caused miscalculations in the image analysis.


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Figure 6.10: Section strain line plot of the Z-direction sample showing the typical progressive

strain distribution of Section 3 across a displacement rang of 0.7-4m

6.3 TENSILE TESTING Uniaxial tensile testing was conducted to estimate the critical value of strain to

determine element failure in the finite element model [108]. The optical full-field strain

measurement system was used to record the tensile tests. Movie files showing images from

the tensile testing and the calculated von Mises strain distributions are provided in the

appendix.

Cubes of 30 × 30 × 30 mm dimensions were cut from a 30mm thick sheet of 0.23

g/cm3 Alporas foam, as supplied by the manufacturer. Steel sample mounts were adhered to

the cube using a two part epoxy adhesive (Araldite® Kit K138, Vantico Pty Ltd), cured at

room temperature for 24 hours. Samples were painted with a speckle pattern as described in

Chapter 3. Tensile tests were conducted using a universal testing machine (Instron 4505), at

a crosshead velocity of 1mm/min. Four samples were tested for each cube orientation. Load

displacement plots were recorded for each sample. Figure 6.11 shows a typical tensile test in

progress with a crack propagating through the foam sample. A failed sample is shown in

Figure 6.12.


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Figure 6.11: Typical tensile test illustrating a propagating crack

Figure 6.12: Typical failed tensile sample

All samples failed in similar manner, with a crack initiating in a weak cell and steadily

propagating completely across the sample. This is reflected in the typical stress-strain curve,

as illustrated in Figure 6.13. The peak load averaged 0.89kN (standard deviation 0.41)

across the samples, though there was significant variation with a few samples failing at much

lower loads. This is expected due to the inhomogeneous nature of the material. The average

tensile peak stress was measured as 1.013MPa (standard deviation 0.39). The strain at peak

stress averaged 0.033 (standard deviation 0.012), with some samples displaying values of

less than 0.02.


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Figure 6.13: Typical tensile stress strain curve

Strain distributions were recorded during the tensile tests. Figure 6.14 shows a typical

strain distribution development as the crosshead displacement increased. The regions of

high strain visible in the middle right area of Figure 6.14(a) correspond to a crack initiating

near the peak load. This is followed by the growth of the crack within the sample, as

illustrated in Figure 6.11.

Figure 6.14: Typical tensile strain distribution for progressive crosshead displacements at a)

0.7mm (stage 20), b) 1mm (stage 40) and c) 1.3mm (stage 60)

6.4 MODELLING An aim of this project was to investigate the ability of an existing finite element

material model to represent the behaviour of aluminium foam within a sandwich structure.

The model used was an implementation of the Deshpande-Fleck constitutive model in the

finite element code LS-DYNA. Although this study is limited to quasi-static behaviour, the

possible applications of this material system are likely to involve dynamic loading, making

the use of the explicit FE code LS-DYNA appropriate. The data recorded from the

compression testing was used to calculate input parameters for the material model. This


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material model was then used to represent the uniaxial compressive test and the response

was compared with the experimentally observed behaviour. In particular, the experimentally

measured full-field strain contours were compared with the model predictions. This

compression model was investigated before the material model was used in the sandwich

structure under flexural loading.

6.4.1 NUMERICAL IMPLEMENTATION OF DESHPANDE-FLECK MODEL Deshpande and Fleck [81] proposed two models for the plastic behaviour of metal

foams: the first model was based on a self-similar yield surface while the second model is

more complex due to the inclusion of differential hardening effects from hydrostatic stress

on the shape of the yield surface. The hydrostatic stress term is important in a porous

material model as the volume can change when cells of the foam collapse under compression

[104]. The models are an extension of the von Mises yield criterion with the hydrostatic

stresses included as part of the equivalent stress term [104]. This model was implemented

by Reyes et al. [103, 104] as a material model in LS-DYNA [109]. LS-DYNA was chosen

for its capabilities in highly non-linear dynamic loading conditions which may be

appropriate for potential applications of these sandwich structures.

The model is described by Reyes et al. [104] as follows. The yield function (Φ) is

defined by:

0ˆ ≤−=Φ Yσ (Eq 6.1)

where σ̂ is the equivalent stress and the yield stress Y can be expressed as:

)ˆ(εσ RY p += (Eq 6.2)

Here, pσ is the plateau stress, ( )ε̂R is the strain hardening term and ε̂ is the

equivalent strain [104]. Deshpande and Fleck [81] define the equivalent stress, σ̂ , as

( )[ ][ ]2222

2

311ˆ mvm σασα

σ ++

≡ (Eq 6.3)

Here, the von Mises effective stress is vmσ , mσ is the mean stress and the shape of

the yield surface is defined by the parameter α . This parameter α is defined as:

( )( )p

p

υυα

+−

=1

21292

(Eq 6.4)

where pυ is the plastic coefficient of contraction.


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Details of the method used to implement this model as an integration algorithm within

LS-DYNA are provided in Reyes [110]. The material model requires some general material

parameters input such as the density and Young’s modulus of the foam as well as some

parameters obtained from compression testing. The yield stress is expressed as:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

++= β

εε

αεεγσ

D

DpY

ˆ1

1ˆ2

(Eq 6.5)

where pσ , 2α , γ and β are material parameters from a curve fit of the stress-strain

data from the uniaxial compression. The densification strain Dε is determined from the

density of the foam ( fρ ) and virgin metal material ( 0fρ ).

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

lnf

fD ρ

ρε

(Eq 6.6)

The model has been implemented as Mat_Deshpande_Fleck_Foam in LS-DYNA

[109].

6.4.2 CURVE FITTING A typical stress strain curve from the uniaxial compression test was analysed to

produce the material model input parameters. The z-direction orientation was used as this

orientation corresponds to the typical orientation of the foam for flexural loading

applications. The data was fitted to a curve described by Equation 6.5, to find the parameters

of plateau stress (σp), α2, γ, and β. The value of εD was calculated using Equation 6.6.

Figure 6.15 illustrates the experimental stress strain curve and the curve fit used to obtain the

model parameters. The values found for each parameter are given in Table 6.2. The

parameter α was found using Equation 6.4 with an assumption of a plastic coefficient of

contraction of zero as used by Hanssen et al. [71]. The parameter Cfail is the value of the

failure strain of foam and is used to remove failed elements during simulation. This was

determined by the tensile failure strain measured in the uniaxial tensile tests discussed

earlier. A value of 0.02 was chosen as this was the lower limit in the range of observed

failures.


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Figure 6.15: Deshpande Fleck yield surface curve fit of experimental compression data

Table 6.2: Material input parameters for foam model Mat154

ρf E υp α γ εD α2 β σp CFail

[g/cm3] [GPa] [MPa] [GPa] [MPa]

0.23 1.1 0.0 2.12 1.47 2.4629 0.245 4.00 0.777 0.02

6.4.3 FINITE ELEMENT MODEL A 30mm edged cube of 1000 brick elements was generated using the HyperMesh pre-

processor. Figure 6.16 illustrates the cube mesh geometry for the compression loading. The

default eight-node brick element was used with a one point reduced integration scheme and

the LS-DYNA stiffness based hourglass control. A rigid body was generated with 400 shell

elements to simulate the top plate of the compression rig. The shell elements were given the

material properties of tool steel using the LS-DYNA *MAT_RIGID material model. Each

element was assigned a thickness of 10mm. The shell elements were slightly offset from the

top surface of the cube to prevent initial penetration issues. An automatic surface to surface

contact definition was used between the foam cube brick elements and the rigid shell

elements, utilising a soft constraint-based formulation. This contact formulation is

recommended when the material constants of the surfaces in contact have large differences

in elastic bulk moduli values. The bottom surface nodes of the cube were fully constrained

and a stationary geometric rigid wall was used as a bottom plate.


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Figure 6.16: Typical cube compression mesh geometry

The load was applied through motion of the top plate shell elements. These elements

were constrained to move only in the z-direction and a prescribed velocity was applied using

the *BOUNDARY_PRESCRIBED_RIGID_MOTION keyword. To replicate the quasi-

static loading experienced during testing, the following velocity field was applied [104]:

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−

−= t

TTd

tv2

cos12

max πππ

(Eq 6.7)

Here, T is the total time of the loading and dmax is the maximum displacement of the

top plate. This velocity field produces an initial acceleration of zero, ensuring that the

loading takes place gradually. For this study, the total loading time T was 100ms and the

maximum displacement was 24mm. This velocity field ensured that quasistatic conditions

were simulated in an explicit finite element formulation.

6.5 COMPRESSION MODELLING RESULTS The behaviour of the finite element model was compared with observations from the

physical uniaxial compression tests. Several aspects were explored such as the general

deformation response, stress-strain curves, and the full-field strain contours. This study was

used to confirm the suitability of the material model input data as retrieved from the

experimental results.

6.5.1 DEFORMATION BEHAVIOUR The deformation of the foam cube model was consistent with the displacement of the

top plate. Figure 6.17 shows the general deformed shape of the cube after the top plate has


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moved 13mm in the negative z direction. There is very little expansion in the x-y plane as

the cube deforms. This follows the behaviour observed in the experimental testing, and is

associated with the zero plastic coefficient of compression used to generate the α model

parameter. Brick elements in the top two thirds of the cube appear to progressively crush

and reduce in height as the top plate displaces. In contrast elements in the lower third show

very little compression or change in volume.

Figure 6.17: Deformation of foam cube model after 13mm of top plate displacement

The stress-strain response of the model was investigated by looking at the von Mises

stress and strain of individual elements within the cube. Figure 6.18 shows the stress strain

curve from a single element #2567 which was located near the centre of the cube. The

model curve closely follows the experimental data, and continues smoothly past the

maximum strain of the physical testing.

Figure 6.18: von Mises Stress-strain response of single element compared with experimental

data

6.5.2 STRAIN DISTRIBUTION The strain contours produced by the FE model were compared with the strain

distributions measured by the optical full-field strain system during the compression test.


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Figure 6.19 shows the strain contours for a typical z-orientation sample for progressive

crosshead displacements. The coloured legend shows strains from 0 to 50%. The general

magnitude of the strain in the model corresponds well with the measured experimental strain.

The maximum strain in the model is less than in the experiment as the homogeneous nature

of the model does not generate any high strain concentrations such as those caused by cell

variation in the experiment. Similarly, the distribution of strain is uniform in the model

contours. The region in which the model strain starts to vary is approximately one third from

the bottom of the simulated cube. One of the main regions of high strain magnitude

(deformation bands) in the experimental distributions also occurs at this approximate

position in the cube.

Figure 6.19: Typical z-orientation strain distributions from the experiment and model data

respectively at crosshead displacements of a) and b) 0.76mm, c) and d) 1.76mm, e) and f) 3.4mm and g) and h) 5.75mm


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6.6 SUMMARY The uniaxial compression and tensile behaviour of Alporas aluminium foam has been

investigated using the optical full-field strain measuring system. The variation in properties

according to orientation relative to the manufacturing process was observed. Deformation

bands were identified as part of the compressive failure mechanism. The compressive stress-

strain response was analysed to produce parameters to fit the curve to the Deshpande-Fleck

constitutive model. These parameters were used as input to a FE material model in LS-

Dyna. The behaviour of a uniaxial compression FE model using this material model was

considered to verify the material input data. This model generally reflected the experimental

behaviour, with the full-field strain contours corresponding well. The correlation between

the experimental and model behaviour supports the validity of the retrieved material data and

suggests it is suitable to use in further modelling studies. The performance of the foam

material model within a sandwich structure under flexural loading is discussed in the next

chapter.

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Chapter 7 FE MODELLING OF FLEXURAL BEHAVIOUR OF ALUMINIUM FOAM SANDWICH STRUCTURE

This chapter describes a study of a finite element model of the flexural behaviour of

the composite sandwich structure using the finite element analysis code LS-DYNA. Two

core thicknesses, 5 and 20 mm, were investigated. The effect of increasing the skin

thickness was also studied. The aluminium core material model input parameters were

retrieved from the uniaxial compression and tensile testing described in the previous chapter.

The FE results were compared with full field strain results from the previous experimental

work.

7.1 INTRODUCTION Currently, most research activity on aluminium foam sandwich structures has focussed

on the potential for significant impact energy absorption and damage tolerance in structural

applications. However, the main in-service attribute of a sandwich structure is high bending

stiffness with minimal increase in weight due to the low density core. Therefore, the quasi-

static flexural behaviour of a sandwich structure that includes an aluminium foam core must

be fully understood when designing for future applications. In particular, it is important for

FE models to capture this complex flexural behaviour.

The cellular nature of commercially available aluminium foam presents a significant

problem when attempting to predict mechanical behaviour [111]. The properties can depend

on the relative magnitude of the average cell size and the geometry of the specimen. The

bulk material behaviour of aluminium foam has exhibited a distinct size effect [46, 48],

where the compressive and shear strength properties were found to reach a plateau level as

the ratio of specimen size to cell size increased [46]. This is of particular relevance to the

Chapter 7 FE Modelling of Flexural Behaviour

96

use of the foam as thin cores within sandwich structures. Chen and Fleck [94] also found

constraints on the foam core from skin sheets also resulted in a size effect.

One widely used constitutive model for metal foams, the Deshpande-Fleck model, has

been implemented as a material model within a number of FE packages and has been used in

a range of modelling studies [58, 86, 112]. While various studies using this material model

have investigated impact energy absorption behaviour in blast loading or compressive

crushing in foam filled energy absorbers, there has been a small amount of work done on

flexural behaviour. These studies have focussed on metal skinned sandwich structures [47,

85]. One such study reported the flexural load response was underestimated and was

dependent on the size of the testing geometry used to measure the input parameters [54]. A

study by McKown and Mines [113] used an alternative foam material model to investigate

the simulation of a sandwich structure with Alporas foam and composite skins; however the

study required the development of a specific Arcan test fixture for retrieving the foam

material model input data. Further investigation is required into the applicability of the

implementation of the Deshpande-Fleck model within a FE package to predict the flexural

behaviour of aluminium foam sandwich structures with composite skins. In particular, it is

important to understand the presence and consequence of a size effect in effectively

modelling the behaviour of the sandwich structure under in-service conditions.

The FE model of the current study utilises an existing LS-DYNA material model

[104] developed for aluminium foam energy absorbers and based on the Deshpande-Fleck

constitutive model [81]. Although this study is limited to quasi-static behaviour, the possible

applications of this material system are likely to involve dynamic loading, making the use of

the explicit FE code LS-DYNA appropriate. The results are compared with load-

displacement behaviour, failure modes and full-field strain contours obtained from the

previous experimental study of flexural behaviour of the same sandwich structures discussed

in Chapter 5 and published [114]. This strain data is particularly useful for FE model

validation.

7.2 FINITE ELEMENT MODEL The results from the finite element model were obtained using an SGI Altix UNIX

platform at the Australian Partnership for Advanced Computing, National Facility. The

sandwich beam was modelled using brick and shell elements for the core and skins

respectively. Figure 7.1 shows a typical mesh for the 20mm core structure. The 4-point


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bending loading conditions were replicated using four rollers of rigid shell elements with the

material properties of tool steel. The sandwich beam was modelled with symmetry

conditions along the width to reduce the computational time. The FE pre-processing package

Hyperworks, from Altair, was used to develop the mesh and input deck for LS-DYNA.

Figure 7.1: Typical mesh geometry for sandwich structure FE model

7.2.1 MATERIAL MODELS The material parameters for the aluminium foam were obtained by performing

compressive tests on cubes of Alporas foam as discussed in Chapter 6. The values used in

this study were taken from the z-orientation samples. A range of sample responses were

observed during the compression testing due to small inconsistencies in the foam cellular

structure and variations in the samples. The results presented here use values from the upper

range of observed sample properties as these provided the best match for the finite element

results. Use of the upper range also minimised the influence of any weakening of the cube

samples due to cutting damage during preparation. Figure 7.2 illustrates the experimental

stress strain curve and the curve fit used to obtain the material model parameters of plateau

stress (σp), α2, γ, and β. The values used for the different parameters are given in Table 7.1.

The damage progression in the composite skin was modelled using the composite

material model Mat22 provided by LS-DYNA [109]. This is a model for orthotropic

composites and can model matrix cracking, compressive failure and final failure due to fibre

breakage. The mechanical properties for the composite skin were obtained from

manufacturer’s data sheets for Twintex®. The material model parameters are shown in

Table 7.2. Here, S1, S2 and S12 are the longitudinal tensile strength, transverse tensile


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strength and shear strength respectively, while C2 is the transverse compressive strength of

the composite material.

Figure 7.2: Deshpande Fleck yield surface curve fit of experimental compression data used for

flexural modelling

Table 7.1: Aluminium foam material model input parameters using sample Z1

ρf E υp α γ εD α2 β σp CFail [g/cm3] [GPa] [MPa] [GPa] [MPa]

0.23 1.1 0.0 2.12 3.12 2.4629 0.368 4.47 1.35 0.02

Table 7.2: Composite material model input parameters

ρ Ea, Eb Ec υba υca υcb Gab Gbc, KFail S12 S1, S2 C2

[g/cm3] [GPa] [GPa] [GPa] [GPa] [GPa] [GPa] [GPa] [GPa] 1.5 15.0 8.0 0.0 0.12 0.1 0.0 1.6 40.0 0.018 0.27 0.15

7.2.2 ELEMENT AND CONTACT DEFINITIONS The default eight-node brick element was used for the core with a one point reduced

integration scheme and the LS-DYNA stiffness based hourglass control. Skin layers were

modelled with shell elements using the Belytschko-Tsay formulation. The interface between

the core and skin materials was replicated using a tied contact type with an offset. This

contact definition used a soft constraint-based formulation. This contact formulation is

recommended when the material constants of the surfaces in contact have large differences


99

in elastic bulk moduli values. No failure criterion for the interface was implemented, to

replicate the lack of any delamination observed experimentally.

7.2.3 LOAD APPLICATION The load was applied through constrained motion of the top load rollers. The bottom

rollers were constrained in all degrees of freedom to make them stationary. A prescribed

velocity in the z direction was applied to the load rollers to simulate the experimental

conditions. To replicate the quasi-static loading experienced during testing, a velocity field

similar to that described in Chapter 6 was applied (see Equation 6.7)[104]. For this study,

the total time of loading was 750ms and the maximum displacement of the load rollers was

30mm. This velocity field produces an initial acceleration of zero, ensuring that the loading

takes place gradually. This replicates quasistatic conditions while using an explicit finite

element formulation.

7.3 RESULTS AND DISCUSSION The deformation and failure of sandwich structure models having 20 mm and 5 mm

aluminium foam cores was compared with the observed experimental behaviour [114], using

both load-displacement curves and full-field strain distributions. Load-displacement curves

were generated using the measured load roller displacement and the contact definition

reaction force between the load rollers and the top skin elements. The load-displacement

response of the models is useful in examining the flexural and energy absorbing behaviour,

while investigations of the full-field strain distributions may help to develop failure theory

useful in designing for future applications.

7.3.1 DEFORMATION BEHAVIOUR OF 20MM CORE STRUCTURE The general deformation shape and failure mechanisms of the 20mm core model

compared closely with that observed in the physical testing. Figure 7.3 shows a comparison

of the final deformed shapes after approximately 25mm of crosshead displacement. The pre-

dominant deformation mechanisms observed in the tested sample were core crushing and

indentation damage under the loading rollers. There was some minor skin failure in the form

of slight fracture and wrinkling. A similar deformation shape was produced by the FE

model. Core crushing was observed under the top rollers with little distortion of core

elements elsewhere in the beam. There was some minor deformation of the skin elements

following the core indentation but no significant skin wrinkling was observed.


100

Figure 7.3: Typical deformation in the 20mm core structure; a) FE model and b) observation

from experimental work

Figure 7.4 depicts the load displacement curves recorded during the physical testing

and as produced by the numerical model. The general shape of the model curve matches the

experiment with an initial linear elastic region followed by a decrease in slope up to a first

peak load point. In the curve from the physical testing, this peak load point is followed by a

plateau region. In this region the load level is reasonably constant with some small variation

towards the end of the test. This curve agrees with the deformation mechanisms observed;

the initial peak corresponds to the first significant failure of foam cells followed by the

progressive crushing and densification of the core. The fluctuation in the load magnitude

may relate to the inconsistency in the cells; for example, as larger or weaker cells fail the

load will drop considerably. The second part of the model curve also shows some small

fluctuations throughout a semi-plateau region. The complete profile of the model curve

matches well with the experimental curve.


101

Figure 7.4: Comparison of the load-displacement curve from FE model with the curve from

experimental work for the 20mm core structure

While the general shape of the curve produced by the model is in agreement with the

experiment, there is a difference in the magnitude of the load. The peak load predicted by

the model is 0.8kN compared to the experimental value of 1.2kN. This deformation

behaviour has not been reported in other studies where the major emphasis of the work is on

bulk compressive behaviour [58, 115].

There are two likely reasons for the underestimation of the load by the model. The

first is related to the size effect in the core material. Previous studies of metal foams have

found a number of potentially significant size effects on material properties, with respect to

the ratio of cell size to specimen size. In particular, Chen et al. [47] reported that shear

response is sensitive to the thickness of the specimen, with a stronger response displayed by

specimens of smaller thickness. Similar results have been discussed by Kesler et al. [111] as

very important in considering sandwich panel design. The core thickness used in this

investigation is less than the sample size used to generate the input parameters for the

material model, and as such, a size effect may be involved. As discussed by Chen et al. [94]

in an investigation of constrained deformation, the material model appears to be unable to

predict the sample size effect on the strength. The inclusion of this effect is essential in

developing an accurate model for sandwich structure applications. A second factor that may

be contributing to the lower load prediction by the model is related to the Saint-Venant’s

principle. The experimental flexural testing can be influenced by the concentrated loads of

the rollers on the sample. These point loading conditions can lead to elevated stress values

in the region around the points of loading or support, and thus can result in an amplified

recorded load magnitude. In contrast, the compression testing for deriving the model input


102

parameters, involves a relatively uniform stress distribution. It is suggested the combination

of these issues of size effect and stress concentrations from point loads may have caused the

difference in load magnitudes between the simulation and experimental results.

The effect of some of the material model parameters was investigated in an attempt to

match the experimental results more closely. The magnitude of the parameters used to

describe the Deshpande-Fleck yield surface was increased and the resulting load

displacement curves are illustrated in Figure 7.5. The parameters of plateau stress (σp), γ and

α2 were increased by factors of 1.5 and 2 for two repeat tests. The magnitude of the load

displacement curve increases accordingly, with the experimental curve most closely matched

by the model with parameters increased by a factor of 1.5. The effectiveness of simply

increasing these parameters supports the suggestion that the initial model underestimation of

peak loads is, at least partially, related to the size effect.

Figure 7.5: Comparison of the load-displacement curves after modifying material parameters (plateau stress (σp), γ and α2 magnitude) for the 20 mm thick aluminium foam core in the FE

model

The effect of parameters in the skin material model was also investigated. The

compressive, tensile and shear strength parameters were varied around the initial value

without any significant effect on the behaviour of the model. Similarly the shear modulus

parameter was found to have minimal effect on the model. The parameters relating to the

longitudinal modulus were found to have the most effect on the magnitude of the curve.

Figure 7.6 illustrates load displacement curves from models with Young’s modulus values

having a very low value (5 GPa) or higher values (20 GPa) compared to manufacturer’s

reported values of 15 GPa. As expected, these changes affect the initial slope of the curve


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and have only minimal effect on the overall magnitude of the curve. The minimal effect of

varying the skin model properties suggests that the foam core material model dominates the

overall behaviour of the sandwich structure for this particular geometry.

Figure 7.6: Load-displacement curves for the 20mm core structure after modifying Young’s

modulus for the composite skin in the FE model

7.3.2 STRAIN DISTRIBUTION OF 20MM CORE STRUCTURE A full-field strain distribution of the region of the sandwich structure between the load

rollers was recording throughout the flexural testing. Figure 7.7 provides a comparison of

von Mises strain contours between simulation and experiment at a crosshead displacement of

2.7mm. This value of crosshead displacement corresponds to the initial peak load. The

experimental results exhibit isolated regions of slightly higher strain dispersed throughout

the sample, which can be associated with the cellular structure of the core. In the regions

beneath the load rollers, small, more concentrated regions of high strain have appeared. On

the right side there is a significant region of high strain in the centre of the thickness beneath

the load roller. This is likely to be the site of a weak cell where initial crushing is beginning.

The simulation results illustrate regions of increased strain directly under the load rollers.

This region is also where the first cell failure and crushing was observed in the experiment.

The remainder of the beam displays uniform regions of strain level unlike the dispersed

higher strain regions seen in the experiment. This is a result of the use of the continuum

material modelling method which does not include any variation in properties between

elements. More importantly, the magnitude of the strain levels agrees well with the

experimental strain values for most parts of the structure.


104

Figure 7.8 provides a comparison of von Mises strain contours between simulation

and experiment at a crosshead displacement of 10mm. The simulation and experimental

results indicate that the regions under the load rollers have concentrated high strain values.

These regions correspond to the observed regions of core crushing. The maximum strain

value in the model at this crosshead displacement was 0.436 which compares well with the

maximum strain value of 0.47 observed in the experiment. The correlation of overall strain

distribution between experimental and simulation results is very good. This study is the first

of its kind to validate the constitutive model for composite sandwich foam structures through

experimental observation of the structures experiencing non uniform strain fields.

Figure 7.7: Typical strain distribution at peak load (~2.7mm displacement) for the 20mm core

structure; a) FE model and b) real-time experimental measurement

Figure 7.8: Typical strain distribution at 10mm displacement for the 20mm core structure ; a)

FE model and b) real-time experimental measurement

7.3.2.1 STRAIN SECTION LINE PLOTS FOR 20MM CORE STRUCTURE The distribution of strain in the FE model was further examined by producing section

strain line plots similar to those generated from the experimental data in Chapter 5. Three

sections were taken through the model full-field strain image, showing the strain magnitude

along the length of the model. For comparison with the experimental data, the line plots are

restricted to the region between the load rollers as was captured by the optical strain


105

measurement system. The three sections were taken along near the top, centre and bottom

regions of the strain distribution as shown in Figure 7.9. Figure 7.10 shows the section strain

line plots from the 20mm core model at load roller displacements of 2.7 and 10mm. To

enable direct comparison with the experiment results, the colour of each curve matches the

colour of the corresponding section curve in the line plots presented in Section 5.2.4.2 of

Chapter 5. For the model line plots, the strain magnitude of the top section is dominant in

both curves, corresponding to the core compression failure mode. This compression

deformation has extended further into the core at the later displacement with the centre and

bottom section curves also showing increased magnitude at each end of the plot. There is

very little strain in the central region of the plots. The general shape of the curves agrees

well with the experimental section strain plots; however the magnitude of the model curves

is lower. The maximum strain at the peak load is 60% (top section) for the model while the

experiment data shows maximum strains of 80-120% (sections 1 and 2) as illustrated

previously in Figure 5.9(b). The localised large deformation and inhomogeneous aspects of

the experimental measurement is likely to have caused the variation.

Figure 7.9: Schematic showing the position of sections taken through the model strain contour

for strain line plots

Figure 7.10: Section von Mises strain line plot for 20mm 1ply model at crosshead displacements

of a) peak load 2.7mm and b) 10mm


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7.3.3 DEFORMATION BEHAVIOUR OF 5MM CORE STRUCTURE The effect of reducing the core thickness on the model performance was investigated

using a 5mm core sample. Figure 7.11 provides a comparison between simulation and

experiment on the failure behaviour of this reduced core thickness. The main failure

observed for this structure was skin wrinkling and fibre fracture with minor core cracking.

There was no apparent crushing within the core structure. Instead, the structure exhibits

plastic hinge type deformation behaviour beneath each load roller. The simulation results for

the overall deformation behaviour matches experimental results. The simulation results also

exhibit some minor compression of the core directly beneath the load rollers, and some

element rotation at each beam hinge.

Figure 7.11: Deformation shape for the 5mm core structure; a) FE model and b) observation

from experimental work

Figure 7.12 illustrates a typical load-displacement curve for this structure. The curve

recorded from the experiment shows initial linear elastic behaviour followed by a decrease in

slope up to a maximum load magnitude. This is followed by a sharp drop in load before

reaching a plateau. This progression agrees with the deformation mechanisms observed of

skin wrinkling and fibre fracture. The curve produced by the simulation follows the general

shape of the experimental curve closely though there are some differences in the load

magnitude. The initial stiffness response of the structure has been overestimated by the

model. In contrast, the peak load produced by the model is significantly lower than

experimentally measured value. In this thinner core structure where the deformation appears

to be dominated by skin failure mechanisms, the initial slope of the curve is highly

dependent on the skin properties. An overestimated skin thickness is a possible cause for the


107

model’s high initial stiffness. This model used the manufacturer’s nominal thickness for a

single ply of consolidated Twintex. This nominal thickness may be greater than the effective

thickness achieved by the metal foam sandwich manufacturing process used in this study,

causing the overestimated stiffness. The underestimated load magnitude was also observed

in the sandwich structure with 20mm core and can be attributed to a combination of a core

size effect and a Saint-Venant’s principle effect. The core size effect in constrained

deformation as observed by Chen et al. [94] is likely to be highly significant to this thin core

geometry. The load-displacement response of the model can be increased to match the

experiment data by increasing the core input parameters by a factor of 2. This is higher than

the factor of 1.5 used with the 20mm core, indicating the importance of size effect in

developing constitutive models for the foam material.

Figure 7.12: Load-displacement curves for the 5mm core structure; from the experimental

work, the initial FE model, and after modifying material parameters (plateau stress (σp), γ and α2 magnitude) for the aluminium foam core in the FE model

7.3.4 STRAIN DISTRIBUTION OF 5MM CORE STRUCTURE Figure 7.13 provides a comparison of von Mises strain contours between simulation

and experiment at a crosshead displacement of 10mm. There are regions of higher strain

beneath the load rollers corresponding to the regions where skin failure, core cracking and

plastic hinging were visually observed. The model contour displays a region of increased

strain in the central area at the bottom of the beam. This region of the beam is under tension.

The experimental measurement did not capture a similar region of high strain in this area.

The model interface definition between the skin and core may have transferred a greater


108

proportion of this tensile loading to the model core than was transferred to the core in the

experiment. While generally there is good agreement between simulation and experiment,

the model was not as successful at predicting the strain distribution for this reduced core

thickness.

Figure 7.13: Typical strain distribution at 10mm displacement for the 5mm core structure; a)

FE model and b) real-time experimental measurement

7.3.4.1 STRAIN SECTION LINE PLOTS FOR 5MM CORE STRUCTURE The presence of tensile strain in the model strain contour is clearly displayed in the

strain section line plots. An early strain section line plot is shown in Figure 7.14 at a

displacement of 1.5mm. This plot shows peaks at each end corresponding to the region

below the load rollers. The central region of the plot shows that at this displacement the

centre section has the lowest strain magnitude corresponding to the neutral axis of the beam.

The top and bottom sections have greater strain of similar magnitudes in this central region

which matches the compression and tension strain distribution expected in the case of pure

bending. This behaviour was not observed at the same 1.5mm displacement in the 20mm

core model where the localised loading from the rollers was more dominant in the strain

distribution. Figure 7.15 shows the section line plots for load roller displacements of 8mm

(at peak load) and at 10mm. The greatest strain magnitude is in the top section in the regions

under the load rollers while the other sections also have strain peaks in these regions. The

magnitude of these regions does not change significantly as the displacement increases. The

behaviour in the central region of the plots is quite different. The region in the centre of the

x-position shows greatest strain in the bottom section. This area is under tensile load during

the flexural test. As the displacement increases, the strain magnitude increases and extends

further into the beam with both the centre and top sections displaying increased strain. The

section line plots from the experimental data do not show this central increased strain (see

Figure 5.7), instead being dominated by localised deformation peaks and compression under

the load rollers.


109


of 1.5mm


of a) peak load 8mm and b) 10mm

Overall, this study has illustrated that an existing constitutive material model for

aluminium foam can be effectively utilised to model the behaviour of a complex sandwich

structure with two different core thicknesses under flexural loading. The model

underestimated the peak load magnitude for both thicknesses. However, the general

deformation behaviour and load-displacement curve shapes were well matched. The

discrepancy between the load magnitudes and its possible relationship to a core size effect

needs to be further investigated. The von Mises strain distributions produced by the model

were generally in good agreement with those recorded from the experiments, though the

experimental measurements included strain concentrations and localised deformation from

the inhomogeneous structure which was not included in the model. Future investigations are

needed to further examine the behaviour of the model across a greater range of core

thicknesses to further explore the variation in behaviour. The effect of the skin thickness is


110

also of interest, and the ability of the model to represent an increased skin thickness is

considered in the next section.

7.3.5 DEFORMATION BEHAVIOUR OF 20MM CORE STRUCTURE WITH 4PLY SKIN

The model was varied to investigate the effect of an increased skin thickness. In the

experimental observations, the increased ply samples displayed significantly different

deformation compared to the single ply samples. In particular considerable core cracking

occurred. The FE model was modified to increase the thickness of the shell elements to

represent the 4 ply skin. The initial model used the same material input parameters as listed

in Table 7.1 however there were instabilities in the FE computation. Error terminations

occurred as a result of out of range velocities. These instabilities occurred following the

erosion of elements as part of the CFail foam material model parameter. The value of CFail

was varied and found to have little effect on the model behaviour, apart from the

displacement at which elements were first eroded. For this section, a large value (0.2) was

used for CFail to minimise the related instabilities. This was necessary as investigations into

a more detailed solution were outside the scope of this study.

The general deformation shape of the 20mm core 4ply structure as predicted by the

model is compared with the experiment behaviour in Figure 7.16. The model shows a

similar degree of core indentation below the load rollers to that displayed by the experiment.

There is some bending at the support rollers although not as much the plastic hinges formed

in the experiment. This may be in part due to the lack of element failure present in the

model. The experiment showed significant core shear cracking which was not represented in

the model. The large element failure parameter may have increased the rigidity of the beam

compared to the experiment in the later stages of the flexural test.


111

Figure 7.16: Typical deformation in the 20mm 4ply core structure; a) FE model and b)

observation from experimental work

The load-displacement response of the 4ply model compared with the experimental

curve is shown in Figure 7.17. Both curves show initial elastic behaviour however the model

significantly underestimates the peak load. This is similar to the magnitude differences

observed in the 20mm core single ply model and experiment; however the 4ply model

displays a greater discrepancy. This may be due to an underestimation of skin thickness in

addition to the previously discussed thickness effect. Following the peak load, the

experiment curve shows a gradual drop in load to a plateau corresponding to the initiation,

growth and arrest of the significant shear core cracks. The model does not predict this shape,

probably as a result of the effective absence of solid element failure from the large CFail

value. Instead the model curve displays a gradual increase in load, perhaps corresponding to

core crushing and densification.

Figure 7.17: Comparison of the load-displacement curve from FE model with the curve from

experimental work for the 20mm core 4ply structure


112

7.3.6 STRAIN DISTRIBUTION OF 20MM CORE STRUCTURE WITH 4PLY SKIN

A comparison of the strain distributions for the 20mm core 4ply structure is shown in

Figure 7.18 at a load roller displacement of 10mm. The model strain contour shows regions

of high strain below the load rollers which extend through the thickness of the beam with

decreasing magnitude. The remainder of the distribution shows uniform low strain. This

corresponds well with the experimental strain distribution which also shows regions of high

strain beneath the load rollers. There is less uniformity in the experimental contour due to

localised deformations and inhomogeneity which is not captured by the model. The

experiment contour does not show the gradual decrease in strain magnitude toward the

bottom of the beam. This variation in the model may be caused by the model displaying

more indentation from the load rollers than occurs in the experiment measurement. This

might be caused by an underestimation of the skin thickness, as well as the lack of element

failure. Without element failure, there is no weakening from shear failure in the region

between the support and load rollers which was observed experimentally, perhaps

concentrating more strain at the point of load application.

Figure 7.18: Typical strain distribution at 10mm displacement for the 20mm core 4ply structure

centred on the load rollers; a) FE model and b) real-time experimental measurement

This region between the support and load rollers was captured by the optical strain

measurement system on a second sample. Figure 7.19 shows a comparison between the

model and experiment strain distribution for the region at a displacement of 10mm. The

experimental distribution shows concentrated high strain across the length of the image,

corresponding to the visually observed cracking of the core. The model distribution also

shows a region of higher strain in a similar area; however it is not as concentrated as in the

experimental distribution. The model also shows maximum strain directly below the load

roller where there was indentation, which is not apparent in the experimental results. Again

these differences are likely due to the lack of element erosion and hence lack of “cracking”

in the model failure behaviour.


113

Figure 7.19: Typical strain distribution at 10mm displacement for the 20mm core 4ply structure

centred on the shear crack; a) FE model and b) real-time experimental measurement

7.3.6.1 STRAIN SECTION LINE PLOTS FOR 20MM CORE STRUCTURE WITH 4PLY SKIN The presence of concentrated strain under the load rollers is further demonstrated in

strain section line plots. Figure 7.20 shows the strain section line plots generated for the

region between the load rollers at the peak load (4.5mm displacement) and at a displacement

of 10mm. The top section shows the greatest magnitude with peaks around 6 and 60mm of

x-position. The centre and bottom sections have far smaller peaks in these regions compared

to the top section, with similar magnitudes. The top section peaks are at least four times the

magnitude of the other sections at 10mm displacement, showing the domination of

indentation and crushing in this region of the model strain distribution. The top two sections

of the experiment strain section line plots (Sections 0 and 1) show similar dominant peaks at

10mm displacement (see Figure 5.17b). The strain magnitude is greater in the experimental

curves and there is greater variation due to localised deformation.

Figure 7.20: Section plot for 20mm 4ply model central region at crosshead displacements of a)

around peak load 4.5mm and b) 10mm


114

Strain section line plots were also taken for the region outside the load roller as shown

in Figure 7.21. Figure 7.22 shows the strain section line plots for the area between the left

support and load roller for two displacements. At the peak load displacement the three

sections have similar strain magnitudes through the central x-position region. The top

section shows a small peak at an x-position of 56mm which appears to correspond to the

strain concentration from indentation under the load roller. The bottom section also shows a

small peak at the left end which corresponds to the region above the support roller. These

general trends continue in the later displacement plot; however the top section peak has

significantly increased. This contrasts with the curves displayed in the experimental strain

section line plots where the centre sections show the greatest strain magnitude and the top

section is considerably lower in magnitude (see Figure 5.19). There is also no obvious peak

corresponding to the region directly beneath the load roller.

Figure 7.21: Schematic showing the position of sections taken through the model strain contour

for strain line plots centred on the shear crack

Figure 7.22: Section plot for 20mm 4ply model crack region at crosshead displacements of a) around peak load 4.5mm and b) 10mm


115

The performance of the FE model in predicting the change in behaviour associated

with an increase in skin thickness has been considered using full-field strain measurements

from physical testing. The model captured the overall deformation behaviour although the

element failure criterion (CFail) used was not accurate. On closer inspection the model

predicted more indentation deformation than was observed in the experiment, and did not

show the significant strain concentrations associated with the shear crack failure. It is likely

the performance would be improved with a more physically realistic element erosion

algorithm; however this is outside the scope of this project. This study highlights the

importance of this criterion in capturing the wide range of deformation behaviours observed

across a range of sandwich structure geometries.

7.4 SUMMARY A finite element model of the aluminium foam composite sandwich structure

undergoing 4-point flexural testing was produced using an existing foam material model

based on the Deshpande-Fleck yield surface. The material model parameters were

determined using simple uniaxial compression and tensile testing. The ability of the model

to replicate the behaviour of structures for two different core thicknesses was investigated.

The performance of the model was evaluated using full-field strain data from experimental

observations. Several core and skin thicknesses were considered.

The damage progression and deformation of each of the single ply skin models

generally reflected the physical testing results although the load-displacement response was

underestimated. This underestimation can be attributed to the non-inclusion of the size

effect in the constitutive material model. The strain predicted by the FE model also agreed

reasonably well with the distribution and magnitude of strain obtained experimentally. A

simple modification of the FE model input parameters for the foam core subsequently

produced good agreement between the model and experimental load response. The model

with the increased skin thickness captured the overall deformation shape reasonably well.

However the shear crack deformation was not depicted as a result of difficulties with the

element failure criterion. This was further elucidated in the strain distributions when

compared with the experimental data.

The use of experimental full-field strain distributions was valuable in providing an

alternative method of validating the deformation performance of the models. Comparisons

of the strain contours in combination with the load-displacement response demonstrated the


116

FE model was able to accurately capture some aspects of deformation, while also elucidating

some deficiencies. The complex composite structure displayed a wide range of failure

mechanisms across the investigated geometries. Development of the model is necessary to

more fully encapsulate this variation in behaviour, particularly an inclusion of the size effect.

Further investigations and improvements are required to advance the model to produce a

more useful FE design tool.

117

Chapter 8 PARAMETRIC STUDY OF ALUMINIUM FOAM MATERIAL MODEL

The implementation of the Deshpande-Fleck constitutive model as a material model in

LS-Dyna requires the definition of several input parameters, as discussed in Chapter 6. The

study in Chapter 7 found a simple increase in the magnitude of the foam material parameters

increased the load-displacement response of the flexural model to more closely match the

experimentally observed behaviour. This method used an across-the-board increase in

magnitude for all of the material curve parameters (including σp, γ, and α2,) and has initially

appeared effective as a possible approach to take account of the size effect. To further

develop this approach, it would be useful to understand in more detail the effects of each of

the individual material parameters on the flexural behaviour of the composite sandwich

structure. This, in combination with a better understanding of the magnitude of the size

effect can enable the development of a simple magnification factor procedure to allow more

accurate FE flexural simulations to be performed with the Deshpande-Fleck model. This

chapter describes a study of individual foam material model parameters and their effect on

the flexural behaviour of the sandwich structure.

8.1 DESIGN OF EXPERIMENT STUDY The influence of the individual material input parameters for the LS-Dyna Deshpande-

Fleck foam model was investigated using a fractional factorial experiment. This approach is

used to reduce the number of experiments conducted, instead of a full factorial experiment

which involves investigating all the combinations of all the factor levels. The treatment

combinations used in the fractional factorial method are chosen to provide sufficient

information to determine the factor effects using the analysis of means (ANOM). The

experimental set of treatments was defined using a modified Taguchi L16-5 array [116].

Chapter 8 Parametric Study

118

This array uses five factors over four levels for a total of 16 runs. Table 8.1 shows the

orthogonal array used. This factorial analysis enables the use of defined metrics to study the

main effects of each of the experimental factors. A more detailed understanding of the

effects of each of the material input parameters on the behaviour of the FE model under

flexural loading conditions may help elucidate approaches to improve the modelling

predictions. In particular, it may help to devise simple modifications that would improve

predictions in various geometries and minimise any errors due to unmodelled size effect

consequences.

Table 8.1: L16 Array

Run Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 1 1 1 1 1 1 2 1 2 2 2 2 3 1 3 3 3 3 4 1 4 4 4 4 5 2 1 2 3 4 6 2 2 1 4 3 7 2 3 4 1 2 8 2 4 3 2 1 9 3 1 3 4 2

10 3 2 4 3 1 11 3 3 1 2 4 12 3 4 2 1 3 13 4 1 4 2 3 14 4 2 3 1 4 15 4 3 2 4 1 16 4 4 1 3 2

8.1.1 FACTORS The material input parameters determined from curve fitting the uniaxial compressive

stress strain data to the Deshpande-Fleck constitutive model were used as the factors in this

study. The five factors chosen were σP ,εD, α2, γ and β which are variables in the yield stress

model (see Equation 6.5, repeated here for clarity).


119

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

++= β

εε

αεεγσ

D

DpY

ˆ1

1ˆ2 (Eq 6.5)

Each factor affects the strain-hardening yield function with the plateau stress (σP)

primarily determining the y-intercept. The densification strain (εD) sets the strain at which

the curve has rapidly increased and has reached a maximum slope. The parameters of α2, γ

and β are curve shape variables that determine the curvature of the yield surface between the

plateau stress at zero strain and the densification strain at maximum stress. Figure 8.1 shows

the effects of α2 and β on the curve shape as described by Reyes et al. [104]. The purpose of

this study was to investigate the effects of these parameters on the way the model behaves

under the complex flexural loading situation.

Figure 8.1: Effect of parameters on strain-hardening curve [104]

All other factors were kept constant to analyse the effects of the five factors

considered here. The study used the 20mm core single ply skin model, as described in the

previous chapter. In particular, foam material parameters of ρf, E, υp, and α remained

constant for all of the 16 runs. A value of 0.2 was used for CFail to minimise the occurrence

of any instabilities from element erosion. Constraints and loading conditions were as used

previously.


120

8.1.2 LEVELS The factorial experiment used 4 different levels for each factor. These were chosen

based on the values determined from the initial curve fitting analysis. The spread of levels

was chosen to represent values below and above the initial values, with two levels in the

upper range. This reflected the previous brief investigations in Chapter 7 where it was found

that increasing the magnitude of the material parameters increased the force-displacement

response to more closely match the experimental results. Table 8.2 shows the four levels of

the five factors used in the parametric study. Level 2 values were those previously

determined from the compressive testing curve fit.

Table 8.2: Factor levels

Factor Level 1 Level 2 Level 3 Level 4 1 σP (GPa) 0.001 0.00135 0.0015 0.002 2 α2 (GPa) 0.3 0.368 0.43 0.5 3 γ (GPa) 0.002 0.00312 0.004 0.005 4 β - 4 4.47 4.75 5 5 εD - 2.2 2.4629 2.75 3.0

8.1.3 QUALITY MEASURES To investigate the effect of the study factors, a number of quality measures were

chosen. The load-displacement response for each run was examined and an analysis of

means (ANOM) was conducted on two metrics. This allowed the generation of factor level

effect plots for the peak load and the displacement at the peak load. The effective plastic

strain contours were also investigated. Further detail was obtained from analysis of means

studies on data from strain section line plots. In particular, the strain behaviour of the centre

section was studied, focussing on the region below the load roller. Factor level effect plots

were produced for the strain at the peak load, and at a displacement of 10mm.

The load-displacement response was chosen as a quality measure as it provides a good

overview of the progressive flexural behaviour and energy absorption characteristics which

are important in designing for in-service applications. The effective plastic strain was also

investigated as the strain behaviour is important in understanding the different deformation

mechanisms and in defining failure criterion for the model.


121

8.1.3.1 ANALYSIS OF MEANS Analysis of Means (ANOM) is a statistical method used to compare the effects of the

factors on the chosen quality measures. The method uses group means for each level of each

factor to elucidate the relationship between the factor and the quality measure. The group

means are calculated as follows.

For each level of each factor, a group mean is calculated using the quality measure

values for each run that used the factor at that level (see Table 8.1). For example, for the

plateau stress factor, the group mean for level 1 is the average of the response from runs 1, 2,

3, and 4 as shown in Equation 8.2. Similarly, the group mean for the εD factor at level 4 uses

the response from runs 4, 5, 11, and 14 (see Equation 8.3). These group means are presented

as factor level effect graphs plotting the group means against the factor levels. This method

was repeated for each of the quality measures investigated.

( )4

4321)1(1

yyyyy levelFactor

+++= (Eq 8.2)

( )4

141154)4(4

yyyyy levelFactor

+++= (Eq 8.3)

where y is the group mean and yi is the quality measure response from Run i.

8.2 FACTOR EFFECTS The influence of each of the material model input parameters was investigated using

both qualitative and quantitative methods including an analysis of means on several quality

measures. These are presented in the following sections.

8.2.1 LOAD DISPLACEMENT BEHAVIOUR The load-displacement response for each of the 16 runs was recorded. Figure 8.2

shows each curve as well as a typical curve recorded during the experimental testing.

General observations of these load curves can reveal some of the effects of the material

model factors, particularly the influence of the plateau stress factor as this factor increases

consistently with every 4 runs. All curves display a similar shape with an initial elastic

region before slowing to a peak load. The behaviour following the peak load is more varied

across the 16 runs with earlier runs appearing to have a large second peak. The later runs

have an increased peak load magnitude which corresponds to the increase in plateau stress.


122

The displacement at which peak load occurs also appears to increase with increasing plateau

stress. The slope of all of the curves reduces after the initial elastic region before reaching

the first peak load. This differs from the experimentally observed behaviour. Further factor

effects cannot be easily determined from this figure and additional analysis was conducted.

Figure 8.2: Load-displacement curves from each run of the L16 array

The peak load magnitude for each of the 16 runs was used to perform an analysis of

means to produce factor effects. This method calculates an average of the peak load from

each run that uses the factor at a particular level. These averages are used to plot the trends

in the peak load as the individual factor levels increase. Figure 8.3 shows the factor level

effects on the peak load for each of the five factors. As observed from the load-displacement

curves in Figure 8.2, there is a clear trend as the plateau stress levels increase. This appears

to be a linear relationship between the peak load and the plateau stress. The plot also

displays the total average peak load of the 16 runs as well as the average peak load recorded

from the experimental testing. It is likely that the plateau stress is the dominant parameter


123

that caused the increased response discussed in Chapter 7 where after the magnitude of each

factor was doubled the match with the experiment data was greatly improved.

The effect of the plateau stress is much greater than the effect of the other four factors.

Figure 8.3(b) shows the level effect plot for the α2 parameter. This factor appears to have

very little effect on the peak load remaining relatively constant across the four levels. There

is slightly more variation in the peak load when the γ factor averages are considered, as

shown in Figure 8.3(c). The peak load average increases as γ increases, however the

dependence is not as strong as that observed with the plateau stress. Figure 8.3(d) shows

very little variation in the peak load as the β level increases. There is more variation in the

peak load as the εD level varies (see Figure 8.3(e)) with the greatest average peak load

occurring when the εD level is at 2.4629, which is the value initially calculated from the

relative densities. These factor level effects suggest the only parameter that will have a

significant impact on the peak load of the flexural test is the plateau stress.

A similar analysis was completed for the average displacement at peak load for the 16

runs. The displacement at peak load relates to the flexural rigidity of the structure and the

degree of deformation the beam can withstand before yielding. The factor level effects are

shown in Figure 8.4. There is very little effect of the plateau stress factor on the

displacement at peak load. Figure 8.4(a) shows this displacement remains close to 4.25mm

as the plateau stress increases. The average displacement at peak load measured during the

physical testing is well below the model value at 2.7mm. Figure 8.4(b) shows there is some

minor variation as the level of α2 increases. The displacement is smallest in the middle

region of the α2 levels. The average displacement has a more linear relationship with the γ

factor, with an increase in displacement as the γ levels increase. Figure 8.4(c) shows the

displacement varies by approximately 0.3mm across the range of investigated γ levels.

There is slightly less variation in the displacement as the β level varies (see Figure 8.4(d)).

Again, the displacement is smallest in the middle region of the β levels. Figure 8.4(e) shows

the effect of the εD factor. This factor had the greatest effect of all the factors on the

displacement at peak load. The average displacement was at a maximum at the second level,

again corresponding to the εD level calculated from the relative densities.


124

Figure 8.3: Factor effect level plots for the peak load for a) plateau stress, b) α2, c) γ, d) β, and e)

εD


125

Figure 8.4: Factor effect level plots for the displacement at peak load for a) plateau stress, b) α2,

c) γ, d) β, and e) εD

None of the factor levels investigated show a great improvement towards matching the

displacement at peak load that was recorded experimentally. This suggests there are other

influences that may have a greater affect on this quality measure that were not included in

this investigation. The displacement at peak load is dependent on the overall effective

stiffness of the structure. The Young’s Modulus (E) of the core material will certainly be


126

influential on this quality measure, and may be the most dominant factor. The properties of

the skin and therefore the composite material model will also be significant.

8.2.2 STRAIN CONTOURS The strain behaviour of the model for each of the 16 runs was investigated. Figure 8.5

shows a von Mises strain contour at 10mm of crosshead displacement for Run01. This strain

distribution was typical of all of the runs, with only very slight variation in the shape or

magnitude of the higher strain regions. Section strain line plots were utilised to study any

variation in strain in more detail.

Figure 8.5: Typical von Mises strain contour for each of the 16 runs

8.2.3 STRAIN SECTION LINE PLOTS Strain section line plots were taken from each of the 16 runs for comparison. The

sections were taken in the same central region of the beam as described in Chapter 7 (see

Figure 7.9) with the results presented here showing the strain behaviour of the centre section.

The strain section plots were taken at displacements of 2.7mm and at 10mm. Figure 8.6

shows the section strain line plot for the 16 runs at 2.7mm which corresponds to a point in

the load-displacement curve (see Figure 8.2) before the peak load. All of the runs have

similar strain curve shapes as previously observed in the full-field strain contours. The

magnitude of the curves is generally similar to that observed in the experimental full-field


127

measurements. For example, the experimentally measured centre strain section at this

displacement displayed strain below the load roller in the order of 2%. The range of

magnitude across the 16 runs for the same region is approximately 1.1 to 2.5%. There are

some obvious trends in the effect of the plateau stress factor. As the plateau stress increases

(from Run 1 to Run 16), the strain decreases.

Figure 8.6: Strain section line plots for each run at 2.7mm displacement before peak load,

showing the centre section for the region bordered by the load rollers

Similar trends were observed in the strain section plots for the displacement of 10mm,

shown in Figure 8.7. Again the general shape of all of the strain curves agrees well with the

experimental observations. The magnitude corresponds reasonably well, though is generally

lower for the simulations. The strain range for the 16 runs for the region below the load

roller was approximately 10-15% whereas the experimentally measured value was in the

order of 15-20%. Similarly to the 2.7mm section plots, the strain magnitude decreases with

increasing plateau stress. Further examination of factor effects was undertaken using the

analysis of means method previously described.


128

Figure 8.7: Strain section line plots for each run at a displacement of 10mm, showing the centre

section for the region bordered by the load rollers

The strain values at a single x-position were averaged for each factor level to produce

factor level effect plots. The x-position used was 7.5mm which corresponds to directly

below the left load roller. The factor effect plots were generated for both displacements of

2.7mm and 10mm. Figure 8.8(a) shows a clear linear effect of the plateau stress on the

strain. Increasing the plateau stress from 1MPa to 2MPa resulted in a large monotonic

decrease in strain magnitude. Compared to the other four factors, this factor had the greatest

influence. Figure 8.8(b) shows the level effects of the α2 factor. There is less variation than

seen for the plateau stress, with the lower α2 levels resulting in greater strain averages. A

similar range of strain is seen in the γ factor level effect plot (see Figure 8.8(c)). This factor

also has a minimum strain in the centre level region with maximum strain averages at the

lowest γ level. Figure 8.8(d) shows the factor level effect plot for the parameter β. As the β

level increases, the strain magnitude decreases. This is a trend similar to that seen with the

plateau stress, however the effect is not as strong and the change is not as monotonic. The

fifth factor, εD, has the second greatest effect on the strain magnitude. Shown in Figure


129

8.8(e), the strain is lowest at the εD second level. In summary, the plateau stress has a strong

linear effect, while the other factors have weaker or non linear effects on the strain

magnitude for the range of levels examined. As seen in the factor level effect plots for the

peak load and the peak load displacement, it appears the plateau stress is the factor with the

most influence on the early displacement strain magnitude.

Figure 8.8: Factor effect level plots for the strain in the centre section at x-position 7.5mm at a

displacement of 2.7mm for a) plateau stress, b) α2, c) γ, d) β, and e) εD


130

The factor level effect plots for the strain at the displacement of 10mm are shown in

Figure 8.9. There is generally a greater range of average strain values across the levels for

this later displacement than for the displacement of 2.7mm. Again, the largest variation is

seen by varying the plateau stress (see Figure 8.9(a)). As the plateau stress increases, the

strain reduces by approximately 13% of the DOE average strain value.

Figure 8.9: Factor effect level plots for the strain in the centre section at x-position 7.5mm at a

displacement of 10mm for a) plateau stress, b) α2, c) γ, d) β, and e) εD


131

The effect of the α2 factor is shown in Figure 8.9(b). There is no clear trend for this

factor, with the highest average strain calculated for the third level while the other 3 levels

display reasonably consistent magnitudes. Figure 8.9(c) shows the strain decreases with

increasing γ factor levels by approximately 7% of the average DOE strain value. The β

factor has less effect on the strain varying by 3.5% of the average DOE strain value across

the 4 levels, as seen in Figure 8.9(d). This factor also has the opposite effect with strain

increasing for increasing β levels. Figure 8.9(e) shows the average strain for the εD levels.

The effect of εD is slightly greater with the strain varying by 5% of the average DOE strain

value. The trend is nonlinear with the values around the third level having the greatest

magnitude.

The only factor that shows a consistent affect on the average strain for both the

displacements analysed is the plateau stress. Increasing the plateau stress decreases the

magnitude of the strain. The γ factor displays generally similar trends with the minimum

average strain occurring towards the middle of the range of levels investigated for both

displacements. The other three factors of α2, β and εD were found to have opposite trends for

the two displacements. The plateau stress was the main influential factor having a consistent

and most significant effect on the strain in the centre section under the load roller. It appears

this factor is the most appropriate to facilitate any manipulation of the strain magnitude

behaviour.

8.3 SUMMARY The influence of five foam material model parameters on the flexural behaviour of the

composite sandwich structure was investigated using a parametric study. A fractional

factorial experiment was conducted using the factors σP, α2, γ, β and εD at four levels. The

load-displacement response and strain behaviour were used as quality measures to study the

effects of each of the factors.

Factor level effect plots were generated for the peak load, and for the displacement at

which peak load occurred. The plateau stress was the only factor to have a significant effect

on the peak load, with an increase in plateau stress resulting in an increase in peak load

magnitude. There were no clear or considerably dominant effects of the five factors on the

displacement at which peak load occurred. This behaviour is perhaps more dependent on

other parameters in the model such as core material E and the composite skin model which

will determine the effective stiffness of the structure.


132

The strain behaviour was investigated using strain section line plots. The sections

were taken through the centre of the beam for displacements before and after peak load. The

shape of the strain curves remained relatively consistent as the factor levels varied, with the

main variation being the magnitude. Factor level effect plots were generated using single

data points below the load roller for each displacement. Once again, the plateau stress factor

had the greatest effect on the strain magnitude. The other factors had less influence and the

factor trends varied between the two displacements before and after peak load.

This thorough study on the effect of varying five foam material model parameters has

shown the load-displacement response of the model structure can be adjusted to converge on

the experimental observations. In particular, the peak load magnitude is adjustable using the

plateau stress. The displacement at which peak load occurs was found to be mainly

unaffected by the five factors studied, and could not be modified to more closely match

experimental results. It is noted that one of the main attractions of using these aluminium

foam sandwich structures in structural applications is their energy absorption properties. The

area under the load-displacement curve is an effective indicator of this energy absorption

capacity. The displacement at which peak load occurs represents a small percentage of the

overall energy absorption of the structure, and hence over prediction of this displacement by

the model is going to have minimal effect on the energy absorption response. Instead,

matching the peak load magnitude is most important in predicting the energy absorption

behaviour of the structure. The failure of structures made of these foam materials is likely to

be predicted by a strain based failure criterion. The relative insensitivity of the effective

plastic strain on four of the five factors studied implies that the constitutive model can be

quite robust in predicting the failure of these structures. It was found that the plateau stress

is the most influential factor on the strain response of these structures and by using suitable

values for this stress it is possible to use this constitutive model as an effective design tool

for both energy absorption and failure investigations.

This study has shown the plateau stress is the most important foam material parameter

in determining the general flexural behaviour of the composite sandwich structure. While

further studies involving other quality measures could perhaps elucidate the role of the other

factors in more detail, it is suggested the plateau stress is the most influential factor. This is

of particular interest in developing a method to manage the variation in behaviour due to the

size effect, and in particular, the use of a single sample size in determining the model input


133

parameters for a range of model geometries. Future investigations may develop a process

that could involve simple magnification of the single plateau stress factor. This work has

provided initial observations on the influence of material parameters on the behaviour of the

LS-Dyna implemented Deshpande-Fleck foam model under flexural loading conditions

which could be valuable for future modifications or improvements in the utilisation of this

material model for in-service applications.

134

Chapter 9 CONCLUSIONS AND FUTURE WORK

9.1 THESIS CONCLUSIONS This study has investigated the flexural behaviour of aluminium foam composite

sandwich structures using a combination of experimental and modelling techniques. The

project was motivated by a need to develop further understanding of the mechanical

behaviour of these structures, as a step in improving the predictability of their response under

flexural loading situations. This is necessary to advance their implementation in a range of

structural applications where the multifunctional characteristics of these materials could be

highly advantageous. This work has revealed aspects of the various deformation

mechanisms of the sandwich structures using full-field strain measurements. These were

used for comparison in investigating the performance of FE simulations in modelling

flexural loading. This research has considered aspects of an existing foam material model

when used within a sandwich structure model in bending. It is suggested that with careful

consideration of input parameters, particularly in regard to the observed “size effect”, the

material model could have potential for use as a design tool for applications that involve

bending loads.

9.1.1 EXPERIMENTAL WORK The experimental component of this project featured the use of a 3D optical measuring

technique to generate detailed full-field strain distributions of the physical testing. This is of

particular interest for the aluminium foam sandwich core, which can display a range of

different failure behaviour characteristics. This study has provided a valuable observation of

the progressive strain behaviour of this material which will assist in developing accurate

material models for typical service applications.

Chapter 9 Conclusions and Future Work

135

The flexural behaviour of the aluminium foam composite sandwich structure was

compared with the more traditional polymer foam structure. The two material systems

displayed significantly different deformation mechanisms with the aluminium foam showing

a steady progressive core deformation. The polymer structure had a more rapid brittle failure

and decrease in load after the initial core failure. The differences in load-displacement

curves resulted in significantly greater energy absorption in the aluminium foam structure.

The aluminium structure was also found to have equivalent or improved flexural properties.

The full-field strain measurement system showed an initially irregular distribution consistent

with the inhomogeneous cellular structure of the aluminium foam. A region of strain

concentration developed in the central area where final core fracture occurred. In contrast,

the initial strain distributions from the polymer foam structure were more even; then high

strain concentrations developed beneath the load rollers. These observations suggest the

aluminium foam structure has comparable or improved characteristics compared to the

polymer foam structure in a flexural loading situation. In particular, the progressive steady

deformation and higher energy absorption properties suggest the system could be a valuable

material choice in a variety of applications compared to polymer foams.

The existence of a size effect related to the cellular structure of the aluminium foam

was observed by examining the effect of the core thickness on the flexural behaviour. The

strain distributions illustrated several different deformation mechanisms. Skin failure was

significant in the thinner samples, while increasing the core thickness instigated greater core

deformation with the thickest sample displaying substantial core indentation. Section strain

line plots revealed complex strain behaviour which was influenced by localised

concentrations and did not display pure plastic bending characteristics. Increasing the skin

thickness of the thickest core sample reduced the incidence of core indentation and produced

core shear cracking. This was reflected in the full-field strain contours which provided a

unique insight in to the development of the complex strain characteristics. It is important

that finite element modelling is able to capture the observed variation in flexural behaviour.

9.1.2 MODELLING WORK The performance of an existing foam material model was investigated to determine its

suitability for use as a core within a sandwich structure under flexural loading. The

Deshpande-Fleck constitutive model has previously been implemented in LS-Dyna and has

primarily been used within energy absorbing and compression dominated loading situations.

Analysis of experimental uniaxial compressive stress-strain curves was conducted to deduce


136

the material input parameters. The load-displacement response and strain distribution of the

models were compared with the physical four-point bend testing.

The finite element model was able to correctly predict the general deformation shape

and progression of the single ply skin structures. The model underestimated the magnitude

of the load-displacement response. It is believed that the underestimation may be related to

the non-inclusion of the size effect in the constitutive material model. A simple

magnification of the input parameters subsequently produced an improved match between

the model and experimental load response. The model strain distribution was in good

agreement with the experimental measurements. The model with an increased skin thickness

was able to capture the general deformation shape reasonably well, however difficulties with

the element failure criterion did not allow the model to predict the shear crack that was

observed experimentally. This was supported by the corresponding high strain regions in the

model which matched the experimental strain contours. Section strain line plots showed a

complex distribution which did not closely follow theoretical pure plastic bending. The

section line plots from the model matched those from the experimental results reasonably

well in general shape and magnitude however they did not capture the localised cellular

effects.

A fractional factorial study of five of the material model input parameters was

completed, using factors of the plateau stress, the densification strain and three stress strain

curve shape parameters. The plateau stress was found to be the most influential factor,

having a direct linear effect on the peak load. There was no clear effect from any of the

factors on the displacement at which peak load occurred. The peak load was able to be

increased to match the experimental results, while the displacement at peak load was

overestimated by the model at all factor levels. The strain behaviour was also dependent on

the plateau stress, and relatively independent of the other factors. This study has shown that

it is possible to make simple modifications to the material input parameters to improve the

load-displacement response and hence energy absorption, to more closely match

experimental observations. The strain behaviour of the model remains relatively consistent

with the experimental full-field strain contours. These findings suggest the Deshpande-Fleck

constitutive model, implemented as Mat154 in LS-Dyna, has potential as a suitable foam

model for flexural loading situations. In particular, the ability to predict the load-

displacement response is necessary to predict energy absorption characteristics. Similarly,

the general correspondence of the strain distributions with the experimental contours


137

suggests the model is appropriate for strain based failure criterion, and with some

development, could be effective in predicting failure. This would be extremely valuable in

designing aluminium foam sandwich structures for a range of potential applications.

9.2 FUTURE WORK This thesis has shown the value of using an optical full-field strain measurement

system in elucidating the complex flexural behaviour of the aluminium foam composite

sandwich structure and allowing direct comparison with FE simulations. This study used

strain information calculated for a single face of the sandwich beams in flexure. Further

work could expand the use of this analytical tool to investigate the strain behaviour of the

composite skin. Additional physical testing could be conducted such as residual strength

testing following flexural loading. The effect of varying the core thickness should be

examined in more detail, using a greater range of dimensions and skin thicknesses. This

would be useful in further understanding the importance of the size effect on the behaviour

of the sandwich structure in flexure, particularly with the use of the full-field strain

measurement analysis.

Future work needs to continue to develop the finite element model for use as an

accurate design tool. Further work is needed to more fully investigate the potential of a

simple parameter magnification factor method as a technique to overcome the size effect in

generating appropriate material input parameters. This could allow material characteristics

from a single experimental geometry to be utilised for a large range of model geometries.

More detailed work on the effect of all of the material input parameters is also needed. This

method could be an alternative to generating a more complex constitutive model

incorporating the size effect. Further developments of the FE model should include a

statistical variation of material properties throughout the elements as described by Reyes et

al [103]. Including the inhomogeneous character of the cellular structure provides a more

realistic replication of the foam material.

138

APPENDIX

The enclosed cd-rom contains the following movie files produced using the Aramis

optical strain measurement system during this project:

4-POINT FLEXURAL TESTING OF SANDWICH STRUCTURES: Movie Core material Core thickness Skin thickness Image region

1A, 1B, 1C Divinycell H100 10mm 1 ply Between top load rollers 2A, 2B, 2C Alporas 5mm 1ply Between top load rollers 3A, 3B, 3C Alporas 10mm 1ply Between top load rollers 4A, 4B, 4C Alporas 20mm 1ply Between top load rollers 5A, 5B, 5C Alporas 20mm 4ply Between top load rollers

6A, 6B, 6C Alporas 20mm 4ply Between left support and load roller

*A = movie shows left camera image, B= movie shows calculated von Mises strain, C= movie shows calculated von Mises strain overlayed on camera image

BULK MATERIAL TESTING: Movie Material Test type Orientation

7A, 7B, 7C Alporas Compression Z-direction 8A, 8B, 8C Alporas Tensile Z-direction

*A = movie shows left camera image, B= movie shows calculated von Mises strain, C= movie shows calculated von Mises strain overlayed on camera image

139

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