3D Crack p Ropagation in Unreinforced c Oncrete

8/3/2019 3D Crack p Ropagation in Unreinforced c Oncrete

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3D crack propagation in unreinforced concrete

Physical modeling and numerical analyses

Christian T. Gasser and Gerhard A. Holzapfel

Institute for Structural Analysis Computational Biomechanics, Graz University of

Technology, Schiesstattgasse 14/B 8010 Graz, Austria, e-mail: [email protected] e-mail: [email protected]

Published in

C.T. Gasser and G.A. Holzapfel: 3D crack propagation in unreinforced concrete.Physical modeling and numerical analyses. In: G.A. Holzapfel, W. Moser and G.Reichard (eds.), Advanced Numerical Analyses of Solids and Structures, and Be-

yond, Verlag der Technischen Universitt Graz, 2004, ISBN 3-902465-01-8, 61-79.


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3D crack propagation in unreinforced concrete

Physical modeling and numerical analyses

Christian T. Gasser and Gerhard A. Holzapfel

Institute for Structural Analysis Computational Biomechanics, Graz University ofTechnology, Schiesstattgasse 14/B 8010 Graz, Austria, e-mail: [email protected] e-mail: [email protected]

Abstract. Concrete is a quasi-brittle material, where tensile failure involves progressivemicro-cracking, debounding and other complex irreversible processes of internal damage.Strain-softening is a dominate feature and advanced numerical schemes have to be appliedin order to circumvent the ill-posdness of the boundary-value problem to deal with. Wepursue the cohesive zone approach, where initialization and coalescence of micro-cracksis lumped into the cohesive fracture process zone in terms of accumulation of damage.We employ a (discrete) constitutive description of the cohesive zone, which is based on atransversely isotropic traction separation law. The model reflects an exponential decreasingtraction with respect to evolving opening displacement and is based on the theory ofinvariants. Non-negativeness of the damage dissipation is proven. Based on the concepttwo numerical examples are studied in detail, i.e. a double-notched specimen under tensileloading and a pull-out test of unreinforced concrete. The computational results show mesh-independency and good correlation with experimental results.

Keywords. Unreinforced concrete, strain-softening, cohesive zone model, PU finite ele-ment method, double-notched specimen, pull-out test.

1 Introduction

Prevention of failure of structures is a major concern in engineering. To designmeaningful constitutive models for capturing failure of structures and to implementthem in an efficient, fast and numerically accurate way is a major task in science,industrial engineering practice and in biomechanics. Reliable constitutive and com-putational modeling of cracks in, for example, rocks and rock faults are of pressingneed in tunneling and geotechnical enginnering (for an overview of numerical sim-ulation in tunneling see, for example, Beer and Watson (1992), Beer (2001), Beer(2003)). In soft and hard tissue biomechanics efficient finite element formulations,suitable for large scale computation with clinical relevance, are required. Examplesare the numerical analyses of bone fracture and fracture of plaques, which occur,for example, during therapeutical treatments of arteries such as balloon angioplasty(see, for example, Holzapfel et al. (2000), Holzapfel and Ogden (2003), Holzapfel(2004)).

The numerical modeling of damage and failure in concrete, which is the aim ofthe present work, has been under extensive research interest in the past two decades.


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2 Christian T. Gasser and Gerhard A. Holzapfel

Different approaches have been adopted, i.e. different constitutive models and nu-merical approaches. Many constitutive models consider concrete as a quasi-brittle

material, where strain-softening is a dominate feature. Based on this typical char-acteristic of concrete, the standard description within polar continuum mechanicsfails; energy at failure is incorrectly predicted to zero (Bazant and Pijaudier-Cabot(1988)). In order to avoid this physically meaningless solution, more advanced the-ories have to be adopted, for example, application of non-local damage models,Cosserat models, rate-dependent models, gradient-enhanced models among others.From a numerical point of view these developments circumvent the ill-posdness ofthe boundary-value problem (BVP) to deal with.

For a first approximation to mixed mode situations, fracture parameters for theopening mode (mode I) can be used if the amount of shear stress compared withthe tensile stress is moderate. If shear stresses become dominant, shear friction andaggregate interlocking can no longer be neglected. Tensile failure of unreinforced

concrete involves progressive micro-cracking, debounding and other complex irre-versible processes of internal damage. The associated softening can coalesce into ageometrical discontinuity, which separates the material. The discrete crack-conceptis the approach that reflects this phenomena most closely.

In order to model concrete failure in terms of strong discontinuities, we followthe pioneering works by Dugdale (1960) and Barenblatt (1962) for elasto-plasticfracture in metals, and Hillerborg et al. (1976) for quasi-brittle failure of concretematerials and introduce a cohesive fracture process zone. Initialization and coales-cence of micro-cracks is lumped into the cohesive fracture process zone in terms ofaccumulation of damage and other inelastic effects and, accordingly, in which highstrain gradients prevail. Cohesive theories regard fracture as a gradual process inwhich separation between incipient material surfaces is resisted by cohesive trac-

tion. Cohesive traction is governed by a number of material-dependent mechanismssuch as cohesion at the atomistic scale, bridging ligaments, interlocking of grainsand others. The cohesive theory is well-known in engineering science and based ondifferent (phenomenological) particularizations of the traction separation laws, forexample, cubic, exponential and trilinear traction separation laws are discussed inNeedleman (1987), Needleman (1990) and Tvergaard and Hutchinson (1992). Anextension of cohesive models is given in Nguyen et al. (2001), where unloading-reloading hysteresis for fatigue crack growth is taken into account. Recently, itis shown that every continuum constitutive model includes a discrete model in aconsistent manner (Oliver (2000), Oliver et al. (2002)). For the description of com-plex materials, the more classical approach, i.e. the ad hoc definition of cohesiveconstitutive models, is sometimes advantageous (one gets more freedom in fittingexperimental data), which we will follow throughout this work.

Alternative to the description with a strong discontinuity, the works Ortiz et al.(1987), Belytschko et al. (1988), de Borst et al. (1993), Sluys and Berends (1998),among others, dealt with a weak discontinuity, where a jump is added to the strainfield directly and a width of the localization band meets the formulation. The strongdiscontinuity problem can be regarded as a limit case of the weak discontinuity


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3D crack propagation in unreinforced concrete 3

Figure1. Body at the reference configuration separated by a strong discontinuity 0 d.

problem as the width of the localization band tends to zero (see, for example,Oliver et al. (1999)).

In this work we employ a (discrete) constitutive description of the cohesive zone(see Gasser and Holzapfel (2004)), which is based on a transversely isotropic tractionseparation law. The model is associated with isotropic damage and reflects an ex-ponential decreasing traction with respect to evolving damage. The model enforcescrack-closure as a contact constraint for which the penalty method is employed.

Based on this concept two numerical examples are studied in detail, i.e. a double-notched specimen under tensile loading and a pull-out test of unreinforced concrete.The computational results show mesh-independency and good correlation with ex-

perimental results.

2 Continuum mechanical framework

In this section we briefly derive and introduce the basic continuum mechanical rela-tions. Particular emphasize is given on the kinematics of strong discontinuities. Aninvariant based and thermodynamical consistent formulation of the discrete consti-tutive law is introduced. Mainly, it is based on a transversely isotropic formulationof the traction separation law (TSL) with isotropic damage in the cohesive zone.

2.1 Kinematics

We provide the fundamental kinematical quantities for the (finite) deformation of asingle discontinuity embedded in a continuum body (see Fig. 1). The discontinuity0d separates the body 0 into two sub-bodies occupying the sub-domains 0+and 0, so that 0d 0+ = , 0 d 0 = and 0+ 0d 0 = 0.The orientation at an arbitrary point Xd of the discontinuity is defined by the unit


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normal vector N(Xd). We choose N such that it points into the sub-domains 0+.Dirichlet and von Neumann boundary conditions define the BVP in the classical

sense.In order to describe the discontinuous displacement field u(X) at a referential

position X, we use an additive decomposition into compatible uc and enhanced Hueparts. Thus,

u(X) = uc(X) + H(X)ue(X), (1)

where

H(X) =

0, X 0,

1, X 0+,(2)

denotes the Heaviside function. Note that both displacement fields uc and ue are

continuous.Partial derivative of the displacement field u with respect to a referential position

X and use of the property GradH(X) = dN(Xd) the material gradient of thedisplacement field is given by

Grad u(X) = Grad uc(X) + HGrad ue(X) + d(X)ue(X) N(Xd). (3)

Herein d denotes the Dirac-deltafunctional with the properties

d(X) =

0, X 0d,

, X 0dand

0

d(X)(X)dV =

0d

(Xd)dS, (4)

where denotes any scalar-valued function, dV is the infinitesimal volume elementdefined in the reference configuration 0 and dS is the infinitesimal surface elementdefined on the discontinuity 0d. The material gradient operator is subsequentlydenoted by Grad () = ()/X. Hence, by means of relation (3) the deformationgradient is defined to be F(X) = I + Grad u(X), with det F = J > 0. Throughoutthis work, we use Cartesian coordinates, therefore, I denotes the identity tensorwith the property (I)ij = ij , where ij is the Kronecker delta.

Kinematics at the discontinuity

In order to study the (finite) deformation of the discontinuity 0 d we introduce thenecessary kinematical quantities. We introduce a fictitious discontinuity d, whichis the map of 0d to the current configuration placed between the two surfaces

created by the crack formation (see Fig. 2). We follow Wells et al. (2002), Gasserand Holzapfel (2003b), Gasser and Holzapfel (2003a), and introduce an averagedeformation gradient Fd at Xd according to

Fd(Xd) = I + Grad uc +1

2ue N. (5)


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Figure2. Discontinuous kinematics: the three deformations Fe = I + Grad uc + Grad ue,Fd = I + Grad uc + ue N/2 and Fc = I + Grad uc represent the currentconfiguration of the body separated by a strong discontinuity.

Based on (5) the unit normal vector n onto the fictitious discontinuity is defined by

n =NFd

1

|NFd1|, (6)

which can be interpreted as a weighted push-forward operation of the covariantvector N. The knowledge of the spatial orientation n at any point xd of the discon-tinuity will be required for the cohesive material model (traction separation law) inorder to capture rigid rotations.

Hence, the introduced kinematical description of a strong discontinuity requiresthree deformations, as illustrated in Fig. 2. The compatible deformation gradientFc = I + Grad uc (with det Fc = Jc > 0), as known from standard continuummechanics, which maps 0 into , and the enhanced deformation gradient Fe =I+Grad uc+Grad ue (with det Fe = Je > 0), which maps 0+ into +. In addition,the average deformation gradient Fd = I + Grad uc + ue N/2 (with det Fd =Jd > 0), which maps the referential discontinuity 0 d into the (fictitious) spatialdiscontinuity d.

3 Constitutive models

In this section we particularize the employed constitutive formulation and intro-duce specific continuous and cohesive (discontinuous) material models. Although


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where t0, a, b and are (non-negative) material parameters to be determined fromexperimental data. In particular denotes the ratio between the transverse and

normal stiffness of the cohesive zone. For the special case = 1, the formulation (15)describes an isotropic interface, which is independent of the invariant i4. Basedon definitions (11)1 and (11)4 the introduced potential (15) is quadratic in theenhanced displacements. Hence, keeping the state of damage fixed ( = const),(15) is an anisotropic linear elastic relationship between the traction t and the gapdisplacement ue.

Remark 1. The present discrete constitutive model distinguishes from that previ-ously introduced by the authors (see Gasser and Holzapfel (2003b) and Gasser andHolzapfel (2003a)). The proposed model (15) has the advantage over the previ-ous model that the initial elastic stiffness is infinite and falls within the regimeof initially rigid damage models in which an interface is inactive until the trac-tion across it reaches a critical level. Problems regarding time continuity of initially

rigid damage models may lead to oscillatory behavior, non-convergence in time anddependence on nonphysical regularization parameters (see Papoulia et al. (2003)).In order to circumvent this problem, we modify the proposed model such that theinitial opening displacement is perturbed by the positive value , which is a smallmachine dependent number depending on a characteristic geometric length of theproblem. The initial opening displacement is then = + . For the computednumerical examples we have used = 1.0 1008. This type of model we call quasi-initially rigid damage model.

4 Numerical examples

The concept of embedded discontinuities is now applied to two representative prob-lems extensively discussed in the literature. The considered examples embrace thesimulation of a double-notched specimen under tensile loading and an anchoragestructure. The numerical examples have in common that in the localization zonethe cohesive properties of the material dominate over the frictional properties, andthe corresponding failure mode exhibits fracture in the form of separation bandsrather than friction along shear bands (Rots (1988)). The loading process was suchthat crack closing never occurred.

A crack-propagation model and the used assumptions for the constitutive mod-els are briefly introduced. All examples model the brittle failure of unreinforcedconcrete. The presented computations are performed on a PC with a PENTIUM 4processor and 1.0 Gb RAM.

4.1 Capture of brittle failure of unreinforced concrete

Concrete is a type of material in which softening localizes in a crack (or crackband). Hence, softening leads to a typical discrete phenomenon, see, for example,the experimental results that were obtained from deformation-controlled uniaxial


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tests under static and cyclic loading of leightweight and normalweight concrete,as documented in Reinhardt et al. (1986). Servo-controlled experiments indicate

that concrete is not a perfectly brittle material in the Griffith sense. Typically, inthe uncracked material, the stress increases linearly with deformation up to about60% of the maximum attainable stress, then the stress decreases slightly more thanproportionally with respect to the deformation up to the tensile strength. Afterreaching the tensile strength concrete shows some load carrying capacity (stresstransfer is still possible) the stress drops down rapidly with increasing (large)deformation until the specimen is separated into two parts. All nonlinearities arecomprised at the crack band. This typical behavior is well-suited to be modeled bythe introduced discrete cohesive law, either in an isotropic or transversely isotropicfashion.

The following crack-propagation model and assumptions for the constitutivemodels serve as a basis for the numerical examples accomplished subsequently.

Crack-propagation model. We use a crack-initialization (crack-activation) crite-rion of Rankine type, in which it is assumed that crack growth takes place if themaximum tensile strength is reached. Crack-initialization is then perpendicular tothe axis of the maximum principal tensile stress. For concrete this is accepted if nosignificant lateral compression is present, see Rots (1988) and references therein.Hence, we start the computation with standard displacement finite elements andperform an eigenvalue decomposition of the stress tensor. If the crack-initializationcriterion is met in a particular element a discontinuity is introduced in the next loadstep. Thereby, the orientation N of the discontinuity is associated with the orien-tation of the maximal principal stress in a non-local sense, i.e. including directionalinformation of the surrounding elements. We follow Wells (2001) and use

N =

nelemi=1 Niwinelemi=1

Niwi

, wi = exp

r2

2l2

, (16)

where wi denotes a Gaussian-like weight function of the ith-element, r is the lengthbetween the center of the actual element and the center of the ith-element and l isa parameter determining the shape of a Gaussian-like bell-curve. For the discussedexamples we have used four times the characteristic length of the actual element,

i.e l = 4V1/3e , where Ve denotes the referential element volume.

Assumptions for the constitutive models. The constitutive response of the con-tinuous (bulk) material, i.e. the uncracked concrete, is modeled by the uncoupled

neo-Hookean model. The elastic material parameters are chosen to be = 16.67 103 (MPa) and = 12.5 103 (MPa), which correspond to normalweight concretewith maximum aggregate size of 8 (mm) (compare with con8 in Schlangen (1993)).A (cohesive) tensile strength is assumed to be t0 = 3.0 (MPa), and the evolution ofthe damage of the cohesive zone is captured by the parameters a = 11.323(mm1)and b = 0.674. These values were obtained by a least square fit of the shape of


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0 0.02 0.04 0.06 0.08 0.10 0.120

0.2

0.4

0.6

0.8

1.0

1.2

Normalizedstress/t0

(-)

Crack opening displacement (mm)

Cohesive model (15)

Experiments: Reinhardt et al. (1986)

Figure3. Fitting of the softening response of normalweight concrete with the introducedcohesive model (15). Experimental data are based on tensile tests by Reinhardtet al. (1986) and indicated by the scatter band (hatched area).

the experimental traction separation data documented in Reinhardt et al. (1986)for normalweight concrete on the basis of the introduced cohesive model (15). The

normalized stress /t0 with respect to the crack opening displacement for a ten-sile specimen, which is based on these material parameters, is plotted in Fig. 3,and compared with the experimental data (see Reinhardt et al. (1986)) indicatedby a scatter band (hatched area). The model results for the stress-displacementresponse nicely falls within the experimental scatter. The fitted response leads toa mode-I fracture energy ofGIf = 0.106 (Nmm

1), which renders the data used byother authors, see, for example, Rots (1988), Schlangen (1993) and Wells (2001).Furthermore, we have assumed an isotropic cohesive law, i.e. we have chosen = 1such that the mode-II fracture energy is GII

f= GI

f= 0.106 (Nmm1). Since > 0,

the typical feature of concrete, i.e. its capability to transfer shear across rough cracksurfaces due to interlocking of the aggregate particles, is considered. Note that Rotsand de Borst (1987) achieved realistic numerical results considering shear softeningby taking GIIf = 0.075 (Nmm

1).

4.2 Double-notched specimen under tensile loading

This example investigates a direct tension problem, where a rectangular block ofconcrete with dimensions 50.0 60.0 250.0 (mm) is considered under tensile load,


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see Fig 4. In order to initialize failure of the specimen, the bar is weakened by twonotches with cross-sections 5.05.0 (mm). Reinhardt et al. (1986), for example, ob-

tained experimental data from deformation-controlled tests under static and cyclicloading. The use of a closed-loop electro-hydraulic testing machine allowed to recordthe complete softening response.

From the numerical point of view this example leads to several solutions, whichstrongly dependent on the applied boundary conditions, see, for example, Chaves(2003). For the sake of simplicity, we restrict our consideration to the symmetricsolution, where two planar cracks propagate from the notches and join each otherwithin the specimen. In order to obtain this type of solution we constrain the unitnormal vector of the crack to coincide with the axial direction of the bar. Hence,the crack path is pre-defined in this case.

The computation is based on two different unstructured finite element meshes,refined in the region where the crack is expected. The meshes are generated with

the software package NETGEN vers. 4.2, see Schoberl (2002), and shown in Fig. 5.The two discretizations required 2951 and 12878 tetrahedral elements. In orderto capture the post-peak regime in a stable way, we have employed the indirectdisplacement control scheme, which involves a constraint equation based on a fewdominant nodal displacements (see de Borst (1986)). We have used a nodal displace-ment that represents the crack-mouth-opening leading to a well-suited numericalproblem.

Figure 6(a) illustrates the propagating cracks based on the coarse mesh. Thedifferent stages are associated with different levels of loads. The first four levels arepre-peak loading and the last levels belongs to post-peak loading. Note the com-plex crack pattern, resulting from the non-smooth stress field. As pointed out inSection 4.1, the computation starts with standard displacement elements and thepartition of unity finite elements are dynamically generated if required. Hence, dur-

ing the computation the number of PU finite elements dynamically increases. Forthe coarse mesh, this is shown in Figure 6(b), where only the PU finite elementsare plotted. Figure 7(a) illustrates the propagating cracks based on the fine meshconsisting of 12878 elements. Again, the different stages are associated with differ-ent load levels. Note that the crack pattern are now more smooth because of themesh refinement. The dynamically increase of PU finite elements for the fine meshis shown in Figure 7(b). Figure 8 depicts the maximum principal Cauchy stressdistribution in a representation where the displacements are scaled by a factor 100.

Figure 9 finally shows the load displacement curve of the problem, where thedisplacement is the total enlargement of the bar. Note that the peak load occurssudden, whereafter a rapid drop of the load is observed in terms of a snap-backeffect. In addition, Fig. 9 indicates a good agreement of the computations based

on the coarse and the fine mesh. The computed results are in qualitative agree-ment with those shown in Chaves (2003). The quantitative deviation from Chaves(2003), however, is obvious since different material parameters have been used forthe present computation.


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Figure4. Geometry (dimension in millimeter) and boundary conditions of a double-notched specimen under tensile loading.

Figure5. Unstructured finite element meshes for the uniaxial tensile test. Mesh (a) con-sists of 2951, while mesh (b) consists of 12878 tetrahedral elements.


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(a)

(b)

2981.0 (N) 5590.7 (N) 6472.8 (N) 6876.5 (N) 6832.1(N)

Figure6. (a) Crack-growth and (b) evolution of the partition of unity finite elements

based on the coarse mesh shown for different levels of loads.

(a)

(b)

3430.7 (N) 5434.6 (N) 6893.5 (N) 7055.5 (N) 6917.4 (N)

Figure7. (a) Crack-growth and (b) evolution of the partition of unity finite elementsbased on the fine mesh shown for different levels of loads.

4.3 Anchorage structure

This example models the pull-out of a steel anchor embedded in concrete. Steelanchors are often used as connection in concrete structures and roof bolts in rocks.In particular, we simulate a pull-out test of a steel disc with 400 (mm) in diameterand 40 (mm) thickness out of a massive cylindrical concrete block with 1400 (mm) indiameter and 600 (mm) thickness. Detailed geometrical data and applied boundary

and loading conditions are shown in Fig.10, where just a quarter of the problemis illustrated. In the center of the steel disc a vertical load F is applied, and theembedded disc is pulled against a counterpressure placed concentric with the discon the surface (shaded area on the top of the block, fixed in the x3-direction),until failure of the concrete occurs. The pull-out test is used for reliable (in situ)estimation of the compressive strength of the concrete in question. It relates the


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5434.6 (N) 7055.5 (N) 6259.4 (N) 5425.9 ( N) 2432.4 (N) 52.7(N)

-1

-0.5

0

0.5

1

Figure8. Evolution of the maximum principal Cauchy stress during failure of the double-notched specimen under tensile loading. Stress distributions at six different loadlevels; before the peak (5434.6 (N), 7055.5 (N), 6259.4 (N)) and three in the post-peak regime (5425.9 (N), 2432.4 (N), 52.7 (N)). The displacements are scaled bya factor 100.

0 0.05 0.10 0.15 0.20 0.250

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Load(kN)

Displacement (mm)

2951 Elements12878 Elements

Figure9. Load-displacement curves showing mesh-independent results. The displacementrepresent the total enlargment of the bar.


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1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

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1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

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1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

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1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

1 2 3 4 5 6 7 8 9 0 1 2 3 4

Figure10. Geometry, applied boundary and loading conditions of the anchorage structure(dimension in millimeter).

Figure11. Unstructured mesh for the pull-out test consisting of 18173 tetrahedral elements.


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measured pull-out force to the strength by means of a calibration curve. The problemis similar to the so-called Lok-Test, the Danish name of punch-out test. For

example, the anchorage structure problem has been investigated by de Borst (1986)and Rots (1988) using axi-symmetric finite elements. The study by Rots (1988) isbased on a smeared fracture model for concrete.

As shown by Rots (1988) failure of the anchorage structure can happen either ina conical pull-out mode or a splitting mode. In the present study we are interestedin the conical pull-out failure, where the discontinuity propagates from the upperedge of the steel disc into the concrete, and the failure surface forms a cone. Fromexperimental studies of the Lok-Test it is known that failure evolves in three differentstages (see Krenchel and Bickley (1985)): at a level of about 3040% of the ultimateload, tensile cracking starts at the upper edge of the disc forming a cone withan angle of about 100 135 degrees. In the second stage, at higher loadings, theformation of stable microcracks running from the top of the disc to the bottom of

the counterpressure. Finally, in a third stage, tensile/shear cracks occur (sometimesafter reaching the ultimate load) running from the outside edge of the disc to theinside edge of the counterpressure ring. The present computation is based on ageometry of the anchorage structure, which differs from that of the Lok-Test. Inaddition, it is assumed that only tensile cracking is present.

For computational purposes, we discretize a quarter of the problem and applysymmetric boundary conditions. The investigated unstructured mesh consists of18173 tetrahedral elements, as shown in Fig. 11. For simplicity we assume that thedisc is infinitely stiff so that displacements can simply be prescribed on the disc-concrete interface. For a more realistic modeling of the interaction problem contactmechanics has to be applied. Although the results showed that high compressivestresses appear (failure pattern of experimental tests showed typically compressionfailure), for simplicity we used the neo-Hookean material to describe the continuousmaterial and do not consider the nonlinearity in compression of concrete.

In Fig. 12(a) the propagating cone-shaped crack is shown. Again, the stages arelabeled by the associated loading, where the first stage refers to a pre-peak load,the second is at the peak-load and the third is past the peak load. The dynamicallyincrease of PU finite elements for the mesh is shown in Figure 12(b). Figure 13 illus-trates the evolution of the pull-out load F versus the displacement , defined to be inthe x3-direction of the steel disc. The load-displacement response is (almost) linearuntil the ultimate load of about 520.0 (kN) is reached, which is at a displacementof about 0.23 (mm). By going beyond the displacement of 0.23 (mm) we observedsuccessive crack growth. Figure 14 shows the maximum axial Cauchy stress distribu-tion at four different load levels (179.10 (kN), 468.39 (kN), 515.45 (kN), 640.91 (kN))during failure of the anchorage structure in a setting where the displacements are

scaled by a factor 500.

The applied tensile failure criterion causes the discontinuity running below thecounterpressure ring, which leads to a re-increase of the load at a displacement ofabout 0.26 (mm). In more refined modeling shear and/or compression failure areconsidered, and full separation of the cone-shaped part consequently occurs. This


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(a)

(b)

297.99 (kN) 515.45 (kN) 640.91 (kN)

Figure12. (a) Crack-growth and (b) evolution of the partition of unity finite elements forthe anchorage structure.

0 0.05 0.10 0.15 0.20 0.25 0.30 0.350

100.0

200.0

300.0

400.0

500.0

600.0

700.0

LoadF

(kN)

Displacement (mm)

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

Figure13. Load F versus vertical displacement of the steel disc.


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179.10 (kN) 468.39 (kN)

515.45 (kN) 640.91 (kN)

-1

-0.5

0

0.5

1

Figure14. Maximum axial Cauchy stress at four different load levels during failure of theanchorage structure. The displacements are scaled by a factor 500.

leads to the experimentally observed drop of the load. Our computation did notshow a snap-back of the load-displacement response, as obtained in, for example,Rots (1988). The qualitative response, however, is comparable with the results in,for example, de Borst (1986) and Etse (1998), which indicate a more plastic-likeresponse.

5 Conclusion

The typical discontinuous fashion of unreinforced concrete failure is modeled withinthe framework of strong discontinuities. A cohesive fracture process zone is intro-duced, where the initialization and coalescence of micro-cracks is modeled in termsof a phenomenological traction separation law. We have employed a transversely


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isotropic traction separation law within the framework of finite elasticity and thetheory of invariants. The model is associated with isotropic damage and allows to

capture an exponential decreasing traction with respect to evolving damage. Theinternal dissipation of the model was proven to be non-negative.

The constitutive framework is numerically represented by the partition of unityfinite element method, which allows to include a priori knowledge of the local be-havior of the solution in the finite element space. This technique leads to an embed-ded representation of non-interacting failure zones having several advantages overtraditional smeared and discrete approaches. The finite element model has been im-plemented for tetrahedral elements using the multi-purpose finite element analysisprogram FEAP; an external software package, from the authors developed, handlesthe representation and progress of the crack in 3D.

The introduced traction separation law is fitted to experimental data of tensiletests of normalweight concrete. Two representative numerical examples of unrein-forced concrete have been investigated, i.e. a double-notched specimen under ten-sile loading and an anchorage structure. The computations are based on irregularmeshes and have shown mesh-independent results and a good correlation betweencomputational and experimental results.

The presented approach is well capable for modeling tensile failure of concrete,where no intersection of the discontinuities appears. An extension to intersectingdiscontinuities is desirable but in 3D accompanied with complexities.

6 Acknowledgement

Financial support for this research was provided by the Austrian Science Foundationunder START-Award Y74-TEC. This support is gratefully acknowledged.

References

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3D Crack p Ropagation in Unreinforced c Oncrete

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