Master Thesis

Abstract

In this thesis, an embedded crack model based on the finite element method for thesimulation of fracture processes in quasi-brittle materials and structures is presented.The model is formulated within the framework of the strong discontinuity approachunder the assumption of small strains.

The nonlinear (softening) process is modeled with the statically and kinematically op-timal nonsymmetric formulation. The continuum part of the material is considered tobe linear elastic, while a specific damage-based traction-separation law is included inthe model which links the traction transmitted by the discontinuity to the displacementjump.

The detailed numerical implementation strategy of finite elements with embedded cracksis described in this thesis and a brief overview of implementing the user subroutine inAbaqus is presented. In the present model, a constant crack opening is introduced intwo dimensional finite elements, including constant strain triangle element and quadri-lateral element. To cover the physical phenomenon as much as possible, numericaltreatment of closed crack is introduced in the model; an implicit/explicit integrationscheme is used to improve the robustness of the computation of softening materials andstructures; in order to deal with the crack locking problem in the standard algorithm,crack adaptation is adopted.

The proposed model is first validated with different test cases and then applied to realstructures; the results are analyzed and compared with the experimental data. The nu-merical work is mainly done in Matlab and the finite element program Abaqus makinguse of user subroutines.

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Acknowledgement

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Contents

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . 1

1.2 Goal of the work . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . 5

2 Basic formulation 7

2.1 Standard finite element method . . . . . . . . . . . . . . . 7

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Principle of virtual work . . . . . . . . . . . . . . . 12

2.2 Strong discontinuity approach . . . . . . . . . . . . . . . . 16

2.2.1 Three-field variational formulation . . . . . . . . . 16

2.2.2 Finite element discretization . . . . . . . . . . . . . 18

2.2.3 Basic types of enhancement . . . . . . . . . . . . . 21

2.2.4 Governing equations . . . . . . . . . . . . . . . . . 24

3 Elements with embedded displacement discontinuity 27

3.1 Localization band two dimensions . . . . . . . . . . . . . . 27

3.2 Statically optimal symmetric formulation . . . . . . . . . . 29

3.2.1 Element with a localization band . . . . . . . . . . 29

3.2.2 Element with a discontinuity line . . . . . . . . . . 30

3.2.3 Constant strain triangle element . . . . . . . . . . 31

3.3 Kinematically optimal symmetric formulation . . . . . . . 32

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ii CONTENTS

3.3.1 Element with a localization band . . . . . . . . . . 32

3.3.2 Element with a discontinuity line . . . . . . . . . . 35

3.3.3 Constant strain triangle element . . . . . . . . . . 36

3.4 Statically and kinematically optimal nonsymmetric formu-lation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Derivation from the principle of virtual work . . . . . . . . 38

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Numerical Procedure 43

4.1 Damage-based traction-separation law . . . . . . . . . . . 43

4.2 Evaluation of internal forces and element stiffness matrix . 48

4.3 Numerical treatment of closed crack . . . . . . . . . . . . 53

4.4 An implicit/explicit integration scheme for non-linear model 54

5 Computational implementation 59

5.1 Algorithm for local iterative solution procedure . . . . . . 59

5.1.1 1D iterative solution procedure . . . . . . . . . . . 60

5.1.2 2D iterative solution procedure . . . . . . . . . . . 63

5.2 Implementation of the user element in Abaqus . . . . . . . 68

5.2.1 Pre-processing and Solver . . . . . . . . . . . . . . 68

5.2.2 Post processing . . . . . . . . . . . . . . . . . . . . 73

5.3 Crack adaptation . . . . . . . . . . . . . . . . . . . . . . . 75

6 Validation and Application 79

6.1 Single element test . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Three-point bending test of a notched beam . . . . . . . . 89

6.3 L-shape panel test . . . . . . . . . . . . . . . . . . . . . . . 94

7 Conclusion and outlook 99

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Chapter 1

Introduction

1.1 Background and Motivation

Fracture mechanics is the field of mechanics concerned with the studyof the propagation of cracks in materials. It is a method for predictingfailure of a structure containing cracks. It is an important tool in im-proving the mechanical performance of materials and components. It isfact that most failure in structures began with cracks. These cracks maybe caused by material defects (dislocation, impurities), discontinuities inassembly and/or design (sharp corners, grooves, nicks, voids), harsh en-vironments (thermal stress, corrosion) and damages in service (impact,fatigue, unexpected loads). Most microscopic cracks are arrested insidethe material but it takes one run-away crack to destroy the whole struc-ture. The straightforward design consideration to structural failure isthat maximum stress local to the crack may exceed the strength of thematerial and thus fracture can occur.

Figure 1.1: Overview of fracture mechanics [10].

Basically fracture mechan-ics was introduced to ana-lyze the relationship amongstresses, cracks and frac-ture toughness(ability ofmaterial containing a crackto resist fracture). Fracturemechanics is primarily usedto prevent fracture but themethodology is of great use

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2 CHAPTER 1. INTRODUCTION

to understand the forma-tion and nature of natural fractures in the structures. In recent years,many numerical models have become very famous to calculate the driv-ing force on a crack and to characterize the material’s resistance to frac-ture.

Particularly in case of brittle fracture which involves little or no plas-tic deformation, usually associated with flaws or defects in the materialwhere bulk stresses concentrate. Crack propagation in brittle materialsis very fast and it is nearly perpendicular to the direction of the appliedstress. Crack is unstable which propagates rapidly without increase inapplied stress. With increasing tensile stress, the brittle material showsfirst a linearly ascending stress-strain curve representing the elasticity ofthe material, followed by a progressively descending branch indicatingthe reduction of the stiffness and strength. In the case of compression,the material shows a more or less ductile material behavior, which israther different from that under tension. Brittle materials such as con-crete, rock, glasses, mortars, polymers and necking in metals are justfew examples which show nonlinear elastic and inelastic behavior underoperating conditions that involve large loads.

Traditionally smeared crack approach or the discrete crack approach arethe classical material models aiming at a macroscopic description of frac-ture processes characterized by strain localization in brittle materials. Inthe approaches of the smeared crack models, the crack is smeared out ina continuum fashion, it is not discrete. The presence of a crack affectsthe stress and the material stiffness in each material point. The soft-ening behavior described by an appropriate stress-strain relation withinthe framework of continuum mechanics is characterized by reducing thematerial stiffness. According to the [24] and other researchers, smeared-crack models for brittle fracture suffer by stress locking, i.e., by spuriousstress transfer across a widely open crack. Stress locking is mainly dueto shear stresses generated by a rotation of the principal strain axes afterthe crack initiation for fixed crack models with a nonzero retention fac-tor. In [22] the source of this phenomenon was analyzed. It was shownthat the poor kinematic representation of the discontinuous displacementfield around a macroscopic crack is the cause of spurious stress transfer.To avoid the stress locking by improving the kinematic representation of

1.1. BACKGROUND AND MOTIVATION 3

highly localized strains, discrete approach has been developed.

The discrete models aim at the incorporation of discontinuous displace-ment fields to properly capture the strong discontinuity kinematics of adiscrete crack [5, 2]. The softening behavior as a consequence of thefracture process in a material is not described using a stress-strain rela-tion but rather with tractions, depending on displacement jumps alongthe discontinuity. In these models a discontinuity interface is introducedwithin the solid and its behavior is governed by a discrete traction–separationlaw. Hence, these models are also known as cohesive crack models [25].Discrete models most often make use of adaptive procedures and exploitthe concept of interface elements that allow the numerical representationof the displacement discontinuity [3].

During the last few years the development of elements with embeddeddiscontinuities increased the popularity of cohesive crack models leadingto the strong discontinuity approach (SDA). The concept of elements withembedded discontinuities is based on the idea of enriching elements withelement level additional degrees of freedom representing the displacementjump across the discontinuity or crack. The displacement discontinuity isintroduced within the domain of an element but is rather not constrainedto the element boundaries anymore. The enriched elements have beendesigned to model the displacement field inside solid elements as thesum of two contributions: a regular (continuous) part, and a possiblydiscontinuous one which is introduced after a suitable activation criterionhas been satisfied.

Hence in SDA approach, without the need for adaptive procedures themacroscopic crack path may cross a given spatial discretization in a moreor less arbitrary way. For the efficient modeling of regions with highly lo-calized strains, this idea of incorporating strain or displacement disconti-nuities into standard finite element interpolations has become a powerfultechniques and showed to be a valuable approach for numerical predic-tion in failure mechanics.

Beside the concept of elements with displacement discontinuities themore recently emerged extended finite element method (X-FEM) alsoaims at the computational resolution of the strong discontinuity kine-matics. The X-FEM is based on the partition of unity concept with the


C0-continuous standard basis of interpolation enriched by a discontinu-ous part. Since the nodes serve as the support for this enhanced finiteelement interpolation the X-FEM is based on a nodal enrichment. In con-trast to the elements with embedded discontinuities the discontinuouspart of the displacement field is represented by additional global degreesof freedom. It is shown in [21] that the X-FEM offers a better and moreversatile kinematic description of solids exhibiting displacement discon-tinuities. However, since within the X-FEM displacement discontinuitiesare interpolated by means of additional nodal degrees of freedom withthe increasing number of equations in the course of an analysis appears asits major drawback for the implementation in a commercial FE-program[3].

In this thesis we focus on the computational modelling of failure of brit-tle materials by SDA approach which is based on element enrichment(in contrast to an nodal enrichment). According to the work by Jirasekand Zimmermann [19], basically there are three fundemental class ofelements with embedded displacement discontinuity. They reffered thisfinite element formulations as statically optimal symmetric (SOS), kine-matically optimal symmetric (KOS) and statically and kinematically opti-mal non-symmetric (SKON) formulations. This classification is based onkinematic enhancement and of the stress contunity condition.

The drawback of the first two approaches are, SOS formulation cannotproperly reflect the kinematics of a completely open crack but it givesa natural traction continuity condition, while the KOS formulation de-scribes the kinematic aspects satisfactorily but it leads to an awkwardrelationship between the stress in the bulk of the element and the trac-tions across the discontinuity line. The development of the nonsymmet-ric SKON formulation, which combines the optimal static and kinematicequations and leads to an improved numerical performance [19].

The present work is based on the implementation of SDA approach inparticularly with SKON statically and kinematically optimal nonsymmetricformulation according to the work by [19]. Implementation is done inthe Abaqus user subroutine and Matlab softwares. Abaqus is one of theleading commercial programs for finite element analysis(FEA). Besidesa large number of built-in nonlinear material models, it also providesthe possibility to program (with FORTRAN) one’s own model through

1.2. GOAL OF THE WORK 5

user-subroutines in this case user element subroutine and Matlab is a pro-gramming environment for numerical computation, algorithm develop-ment, data analysis and visualization. Using MATLAB, technical com-puting problems can be solved faster than with traditional programminglanguages such as C, C++ and FORTRAN.

1.2 Goal of the work

The main purpose of this thesis is to implement and improve the em-bedded crack model for constant strain triangle element and quadrilat-eral element based on statically and kinematically optimal nonsymmet-ric (SKON) formulation with reference to [18] with the constant crack-ing opening, including crack closure. Considering robustness issue im-plcit/explicit integeration scheme [11] is used. In order to avoid thecrack locking, the standard algorithm is modified and crack adaptationis adopted with reference to [15]. Finally compare and validate the re-sults with expermental results in Matlab and Abaqus user subroutine soft-wares.

1.3 Outline of the thesis

This thesis is divided into seven chapters, the remainder are organized asthe following:

• Chapter 2 concerns on the variational formulation. Initially basicmathematical expressions of standard finite element method is pre-sented including principle of virtual work and then the techniqueof enriching standard finite element interpolations by strain or dis-placement discontinuities is proposed and derived from simple phys-ical considerations, i.e., from the extended principle of virtual work,from the Hu Washizu variational principle (Three field variationalprinciple), using an EAS format and a B-bar format with reference[19].

• According to the work by [19], this chapter 3 concerns on the for-mulation of three fundemental class of elements with embedded dis-


placement discontinuity. The classification is based on kinematic en-hancement and of the stress contunity condition. Strong and weekpoints of individual formulations are critically evaluated, with thespecial attention on brittle fracture.

• Chapter 4 focuses on the numerical aspects of implementation ofdamage based tracion seperation law, numerical treatment of closedcrack, evaluation of element internal force/stiffness matrix and ro-bustness issue(computability) are presented.

• In Chapter 5, the detailed implementation strategy of the standardmodel is presented. The basic idea of implementing the user sub-routine UEL in Abaqus is explained. The problem of crack lockingwith the standard algorithm is summerised and to resolve the prob-lem, the concept of crack adaptation is exlained and adopted to themodel.

• In Chapter 6, the model is first investigated with single element ex-amples, i.e., for CST element and quadrilateral element. Then themodel is applied into structures and response of the structure withdifferent meshes are simulated. The results are analysed and com-pared with the experimental data in Abaqus and Matlab softwares.

• Chapter 7 summarizes the work and proposes some future researchdirections.

Chapter 2

Basic formulation

To introduce a displacement discontinuity within the domain of an finiteelement, it is necessary to introduce a discontinuity into the kinematicfields of an element. In this chapter, the effect of a displacement jump isadded to finite elements as an incompatible strain mode. The displace-ment jump is embedded in the element formulation, with the orientationof the discontinuity determined by the local stress and the constitutivemodel. By including the effect of a displacement jump only in the strainfield of an element as an incompatible mode, the procedure can be easilyimplemented in existing finite element codes.

Significant work has been carried out into so-called embedded disconti-nuity elements where the effect of a displacement jump is included in thestrain field of an element. The approach followed here is based on thework of [18, 19]. According to the [19] the exact form of the incompat-ible strain modes is determined by equilibrium and kinematic considera-tions. Through careful consideration of the three field variational state-ments with assumed strain method, a finite element with incompatiblemodes was developed based on SKON formulation.

2.1 Standard finite element method

2.1.1 Introduction

The finite element method (FEM) has been developed into a key indis-pensable technology in the modelling and simulation of various engineer-ing systems. The FEM was first used to solve problems of stress analysis

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8 CHAPTER 2. BASIC FORMULATION

after then it has been applied to many other problems like thermal analy-sis, fluid flow analysis, piezoelectric analysis and many others. Basically,the analyst seeks to determine the distribution of some field variable likethe displacement in stress analysis, the temperature or heat flux in ther-mal analysis, the electrical charge in electrical analysis and so on. TheFEM is a numerical method seeking an approximated solution of the dis-tribution of field variables in the problem domain that is difficult to obtainanalytically.

Figure 2.1: Illustration of basic concept of FEM[27]

Basic idea of finite element methodis that, dividing the problem do-main into several elements, (asshown in Figures 2.1,2.2). Knownphysical laws are then applied toeach small element, each of whichusually has a very simple geome-try. A continuous function of anunknown field variable is approx-imated using for instance piece-wise linear functions in each sub-domain, called an element formedby nodes. The unknowns are then the discrete values of the field variableat the nodes. Next, proper principles are followed to establish equationsfor the elements, after which the elements are ‘tied’ to one another. Thisprocess leads to a set of linear algebraic simultaneous equations for theentire system that can be solved easily to yield the required field variable.

The behaviour of a phenomenon in a system depends upon the geometryor domain of the system, the property of the material or medium, andthe boundary, initial and loading conditions. For an engineering system,the geometry or domain can be very complex. Further, the boundary andinitial conditions can also be complicated and important consideration inmechanics. There are displacement and force boundary conditions forsolids and structures. It is therefore, in general, very difficult to solve thegoverning differential equation via analytical means. In practice, mostof the problems are solved using numerical methods. Among these, themethods of domain discretization championed by the FEM are the mostpopular, due to its practicality and versatility.

2.1. STANDARD FINITE ELEMENT METHOD 9

Depending on the property of the material, solids can be elastic, meaningthat the deformation in the solids disappears fully if it is unloaded. Thereare also solids that are considered plastic, meaning that the deformationin the solids cannot be fully recovered when it is unloaded. Elasticitydeals with solids and structures of elastic materials, and plasticity dealswith those of plastic materials.

Basic mathematical expressions of FEM:

The motion of continuum in three-dimensional space is completely de-fined by the position vector of a material point X = [X1, X2, X3]

T andits change of position at deformation under arbitrary internal or exter-nal influence. This motion of the material point from the undeformedto the deformed state is described by means of the displacement vectoru = [u1, u2, u3]

T as a function of the position of the material point [27].

A basis for this description is the consideration of a body as an ensem-ble of material points as well as the characterization of their initial andcurrent position by means of the position and displacement vectors. Byconsidering the immediate vicinity of material points one finally gets tothe concept of strains which describe the deformation of a material body.The equation represents the strain state of continuum which is calledGreen Lagrange Strain Tensor E (for complete description see; [27]) as

E =1

2[∇u+∇Tu+∇Tu · ∇u] (2.1)

The displacement gradient ∇u is decomposed into a symmetric and askew-symmetric part, i.e., ∇u = ∇symu +∇skwu. Based on this decom-position, the Green Lagrange strain tensor can be written in the followingcompact form

E = ∇symu+1

2[∇Tu · ∇u] (2.2)

where ∇symu = 12 [∇u + ∇Tu] which is the linear function of displace-

ment gradient and the second term is the non linear term of ∇u whichis 1

2 [∇Tu · ∇u]. The strain measure of the geometrically linear theory isdefined by the symmetric part of the displacement gradient ∇u whichis also described as the infinitesimal strain tensor, is denoted with ε torepresent the theory of small strains.

ε = ∇symu (2.3)


The components of the symmetric strain tensor ε can be described bythe definitions of the symmetric part of a second order tensor and thegradient where εij = εji. In the context of the finite element method, thestrain state is characterized by means of the strain vector.

ε =1

2[ui,j + uj,i]ei ⊗ ej (2.4)

The differential operator has to be developed as a basis for the direct cal-culation of the strain vector from the displacement vector. The desiredkinematic relation of the strain and the displacement vectors is derivedfrom the definition of the strain components in equation (2.4), wherebythe components of the differential operatorDε represent rules for deriva-tives.

ε11

ε22

ε33

2ε12

2ε23

2ε13

=

∂∂X1

0 0

0 ∂∂X2

0

0 0 ∂∂X3

∂∂X2

∂∂X1

0

0 ∂∂X3

∂∂X2

∂∂X3

0 ∂∂X1

u1

u2

u3

; ε = Dεu (2.5)

Kinetics describes the relation between external and internal forces act-ing on a material body. According to the stress principle of Cauchy, atensor field of stresses σ exists in a material body as a consequence ofthe external forces. Together with the static and dynamic loads actingthroughout the volume, these stresses form the local balance of momen-tum or the equilibrium of forces. The balance of momentum must be sat-isfied throughout the deformed configuration. According to the CauchyLemma, the stress vector in the interior of the body as a function of theoutward directed normal is balanced with the stress vector of the inwarddirected normal. The orientation of the surface is characterized by meansof its normal vector n. The theorem of Cauchy demands that a tensor fieldσ related to the vector t exists, which satisfies a linear mapping as follows

t(X,n) = σ(X) · n. (2.6)

The so-postulated symmetric stress tensor σ is known as Cauchy’s stresstensor where σ = σi,jei ⊗ ej and σ = σT .

The balance equation of the linear momentum describes the equilibriumof the internal forces and the stresses. The local balance of momentum


can be defined in accordance with continuum mechanics, based on theintegral balance of momentum and under consideration of Cauchy’s the-orem and some mathematical simplifications [27] as

ρu = divσ + ρb (2.7)

where divσ symbolizes the divergence of the Cauchy stress tensor σ.

Constitutive equations in the classical sense presume the existence of arelation between forces and deformation, respectively between stressesand strains, which is exclusively local, i.e., at the considered materialpoint. A material law sets the relation between stresses σ, strains ε andstrain rates ε, which describe the velocity dependence of the stress tensorand internal variables α, which represent the dependence of the stresseson the history (plastification or damage).

σ = σ(ε, ε,α) (2.8)

This generalized material law contains a number of material models forthe description of nonlinear material behaviour taking into account mi-crostructural damage, residual plastic strains and time-dependent effects.For the modelling of reversible, time-independent, elastic processes, thestress state can be defined only based on the strain state with the stresstensor turning into a null tensor in the undeformed configuration.

σ = σ(ε) (2.9)

Furthermore, it is to be assumed that the material is homogeneous andthat the material properties are not dependent on the direction. Elastic-ity means that the stress state only depends on the instantaneous strainstate and not on the stress path. The desired path-independence is onlyguaranteed, if the stress tensor can be derived by differentiation of anelastic potential function W (ε) with respect to the strain tensor.

σ(ε) =∂W (ε)

∂ε(2.10)

If the deformation is independent of the path, the corresponding materiallaws are hyperelastic. Derivation of the stress tensor with respect to thestrain tensor yields the tangential modulus of elasticity, constitutive tensor


or material tensor D. The material tensor represents the linear mappingof the strain tensor onto the stress tensor.

D =∂σ

∂ε; σ = D : ε (2.11)

As a consequence of the symmetry of the stress and strain tensors, theconstitutive tensor satisfies the symmetry properties. The constitutivematrix can be described by means of the modulus of elasticity E andthe Poisson ratio ν. For the deformation analysis of two-dimensionalcontinua, the plane stress and the plane strain states are of interest. Planestress states are structural members of small depth. In the case of aplane stress state it is assumed that the stress components σ33, σ13 andσ23 vanish,

σ33 = σ13 = σ23 = 0. (2.12)

where the corresponding ε33 is different from zero but ε13 and ε23 be-comes zero. By summarizing linearly dependent terms one obtains thelinear elastic material law of the plane stress state in the formσ11

σ22

σ12

=E

1− ν2

1 ν 01 0

sym 1−ν2

ε11

ε22

2ε12

. (2.13)

The plane strain state is mostly used in cases where the dimension inone direction is very big with the loading in this direction remainingunchanged. For the generation of the plane strain state it is assumed thatthe strain components ε33, ε13 and ε23 vanish

ε33 = ε13 = ε23 = 0. (2.14)

Here stress components σ13 and σ23 become zero, the stress σ33 on thecontrary, is different from zero. The constitutive equation of plane strainturns intoσ11

σ22

σ12

=E

(1− ν)(1− 2ν)

1− ν ν 01− ν 0

sym 1−2ν2

ε11

ε22

2ε12

. (2.15)

2.1.2 Principle of virtual work

The principle of virtual work is the fundamental for the formulation ofthe finite element method. The basis of the so-called weak formulation


of the initial boundary value problem of elastodynamics is characterizedby the description of the deformation of a material body by means of thedisplacement field and the corresponding strains (Kinematics), the forceequilibrium of stresses on a differential volume element (Kinetics), geo-metric and static boundary conditions and the constitutive relationshipbetween stresses and strains (Material Law) [27].

The change from the strong form of the partial differential equation andits boundary conditions to the weak form gives in the end the principleof virtual work. In the weak form the geometric boundary conditionsare strongly satisfied, whereas the balance of momentum and the staticboundary conditions must only be satisfied in an integral form. This inte-gral formulation hence allows the exact solution of the initial boundaryvalue problem to be replaced by an approximated solution, which satis-fies the integral but not the local form of the corresponding differentialequation.

All three components (in tensor notation equations (2.7,2.11,2.3)) to-gether form the second order partial differential equation of linear elas-todynamics with the displacement field as the solution variable. In gen-eral, the solution of this differential equation is not possible analyti-cally. Therefore approximation methods, in particular the finite elementmethod, is used in order to find an approximate solution. For the gen-eration of the principle of virtual work, this differential equations arescalarly multiplied by a vector-valued test function and integrated overthe volume, respectively over the neumann boundary condition. As testfunction the virtual displacements δu are chosen. This test function hasthe special properties see [27].

The weak formulation of the balance of momentum (2.7) and of the staticboundary condition (2.6) results from the reformulation of fundamentalequations and multiplication by the test function δu, integration overthe volume, respectively over the neumann boundary and addition of theintegral terms.ˆ

Ω

δu(ρu− ρb)dV +

ˆΩ

δu · divσdV −ˆ

Γσ

δu · (σ · n− t∗)dA = 0 (2.16)

For simplification of above equation, product rule for divergence is ap-plied. Additionally, the interchangeability of the order of application of


variation with the symbol δ and differentiation with the symbol ∇ is uti-lized for further simplification. Furthermore, the Gauss theorem for thedivergence of a first order tensor is applied see [27] for detail formula-tion. Finally, the principle of virtual work is derived in its usual form,with the scalar product of the variation of the strain tensor and the stresstensor.ˆ

Ω

δu · uρdV +

ˆΩ

δε : σdV =

ˆΩ

δu · bρdV +

ˆΓσ

δu · t∗dA = 0 (2.17)

The first term in above equation is described as virtual work of the inertialforces δWdyn, second term as internal virtual work δWint and right sideterms as virtual work of the external forces or external virtual work δWext.

δWdyn + δWint = δWext (2.18)

The definition of stresses and strains as vectors is used for the generationof finite elements and additionally the kinematic equation (2.5) and theconstitutive law (2.11) are taken into account. At the end one obtainsthe internal virtual work as function of the displacement vector u, theconstitutive matrix D and the differential operator Dε.

δWint =

ˆΩ

δε : σdV =

ˆΩ

δu ·DTε D DεudV (2.19)

Let us consider plane finite elements, the finite element discretizationand analysis of plane continua consists of the partitioning of the struc-ture or the domain into finite elements and the approximation of con-tinuously distributed physical quantities (e.g. displacements) by discretenodal degrees of freedom and the assumption of their distribution overthe element area. This assumption is associated with the choice of shapefunctions, which depend on the variables ξ1 and ξ2 (called natural coor-dinates in isoprametric mapping) for the case of plane elements.

The topological element structure, formed by the subdivision into subdo-mains Ωe or finite elements e is called finite element mesh (Figure 2.2)and the process of its generation is meshing or mesh generation.

Within the scope of isoparametric approximation of geometry and ele-ment variables, the continuous position vector X can be approximated bymeans of shape functions N i(ξ) in natural coordinates and with the help


Figure 2.2: Finite elemenet mesh (a) discretization with four-noded elements, (b) discretization withthree-noded elements [27].

of discrete positions of element node Xei

X(ξ1, ξ2) = X(ξ) ≈ X(ξ) =NN∑i=i

XeiN i(ξ) (2.20)

where NN generally stands for the number of element nodes. Equation(2.20) describes the approximation of physical coordinates as function ofnatural coordinates. Within the scope of the isoparametric element con-cept we can approximate the continuous displacements, variation andsecond time derivative of displacements, analogous with the approxima-tion of the position vector in above equation.

u(ξ) ≈ u(ξ) = N (ξ) ue (2.21)

δu(ξ) ≈ δu(ξ) = N (ξ) δue (2.22)

u(ξ) ≈ ˜u(ξ) = N (ξ) ue (2.23)

With the approximation of continuous displacements according to equa-tion (2.21), we get the approximation of the strain vector from equation(2.5). Here strain vector components ought to be described in naturalcoordinates. This is accomplished by applying the derivation rule to thisdisplacement strain relation to get approximation as

ε(ξ) ≈ ε(ξ) = Dεξ u(ξ) = Dεξ N (ξ)ue = B(ξ) ue. (2.24)

We get the approximation of the strain vector from equation by means oflinear mapping of the element displacement vector and the differentialoperator definition called B-opertor B(ξ).


For plane elements, the volume element is replaced by the area elementof the middle surface dA and found to be as such with the kinematicand kinetic assumptions and the pre-integration over the thickness co-ordinate X3 is depicted, see [27] for detailed description. The internalvirtual work is developed with equation (2.19) is substituted by the ap-proximation of strain and by modifying the surface element dA and byadjusting the integration boundaries to natural coordinates ξα ∈ [−1, 1].Approximation of internal virtual work comes from substitution of theexact strain vector by the approximated strain vector in equation (2.24).

δW eint = δue ·

1ˆ

−1

1ˆ

−1

BT (ξ) D B(ξ) |J(ξ)|h dξ1 dξ2 ue = δue ·Keu

e (2.25)

Where J is the Jacobi matrix used for transformation of a physical co-ordinates to natural coordinates and |J(ξ)| is Jacobi determinant. Here,we have defined the element stiffness matrix of a plane element of anarbitrary shape.

Ke =

1ˆ

−1

1ˆ

−1

BT (ξ) D B(ξ) |J(ξ)|h dξ1 dξ2 (2.26)

The above expression ended with integral expressions defining the stiff-ness matrix of an element. In order to conduct the finite element method,it is necessary to solve the respective integral expressions. It is recom-mended to apply numerical integration because the analytical integrationis very demanding or not at all possible for the entire shape diversity ofan element type. Here Gauss-Legendre integration is used to solve theintegral in equation (2.26), see [27] for detail description.

2.2 Strong discontinuity approach

2.2.1 Three-field variational formulation

Embedded discontinuities with the finite element method is representa-tion by powerful type of a Hu–Washizu variational principle. The Hu-Washizu principle is used to develop mixed finite element methods. The

2.2. STRONG DISCONTINUITY APPROACH 17

Hu-Washizu variational principle [8, 28], which deals with three inde-pendent fields yields the so-called three-field variational statements. Thetheorem may be written as

I(u, ε,σ) =

ˆΩ

1

2εTD εdΩ +

ˆΩ

1

2εTD ε0dΩ +

ˆΩ

σT (∇su− ε)dΩ−ˆ

Ω

uT b dΓ−ˆSt

uT t dΓ−ˆSu

tT (u− u)dΓ = stationary

(2.27)where ε0 is initial strain (constant), b are the prescribed body forces andt are the prescribed tractions (surface forces). The variational is appliedto the displacement field u, the strain field ε, and the stress field σ.Except for the kinematic boundary conditions, these fields are definedin a domain V whose boundary consists of two parts, Su and St, withprescribed displacements and tractions, respectively,

u = u on Su (2.28)

A variational theorem is stationary when the arguments (e.g. u, ε, σ)satisfy the condition where the first variation vanishes and for certainrestrictions on regularity, all other governing equations can be replacedby the variational equality shown in (2.29) where the assumed fields arecompletely arbitrary and mutually independent.ˆ

V

δεT σ (ε) dV + δ

ˆV

σT (∂u− ε) dV =

ˆV

δuT b dV +

ˆst

δuT t dS

(2.29)

This above equality must hold for any admissible variations δu, δε , δσ.In (2.29) the symbol δ denotes variation, σ (ε) is the stress computedfrom the assumed strain ε using the constitutive equations, ∂ is the kine-matic operator transforming displacements into strains (the engineering-notation counterpart of the symmetric gradient operator).

Variation of the second integral is applied, by applying the Green theoremto the term containing ∂δu, and grouping the terms δu, δε, δσ together,we can derive from (2.29) the strain displacement equations

∂u = ε (2.30)

constitutive equationsσ(ε) = σ (2.31)


equilibrium equations∂∗σ = b (2.32)

and static boundary conditions

nσ = t (2.33)

in which ∂∗ is the adjoint operator of σ and n is the outward unit normalto the boundary [19].

2.2.2 Finite element discretization

Kinematic description: Based on the regularity of the displacement fieldu(x), generally we can distinguish three types of kinematic descriptions.The present work is the first one incorporates strong discontinuities, i.e.,jumps in displacements across a discontinuity curve (in two dimensions)or discontinuity surface (in three dimensions). The strain field ε(x), thenconsists of a regular part, obtained by standard differentiation of the dis-placement field and a discontinuous part having the character of a multi-ple of the Dirac delta distribution. The basic idea is that the displacementfield is decomposed into a continuous part and a discontinuous part dueto the opening and sliding of a crack. This is schematically shown for theone-dimensional case in (2.3 a).

Figure 2.3: Kinematic description with (a) strong discontinuity, (b) weak discontinuities, (c) nodiscontinuities [20]

Another possible kinematic description represents the region of localizeddeformation by a band of a small but finite thickness, separated from the


remaining part of the body by two weak discontinuities, i.e., curves orsurfaces across which certain strain components have a jump but the dis-placement field remains continuous. This is illustrated in (2.3 b). Sincethe displacement is continuous, the strain components in the plane tan-gential to the discontinuity surface must remain continuous as well, andonly the out-of-plane components can have a jump. In physical terms, theband between the weak discontinuities corresponds to a damage processzone with an almost constant density of microdefects [20].

The most regular description uses a continuously differentiable displace-ment field, and the strain field remains continuous. Strain localizationis manifested by high strains in a narrow band, with a continuous tran-sition to much lower strains in the surrounding parts of the body. Atypical strain profile of this type is shown in (2.3 c). In physical terms,this corresponds to a damage process zone with a continuously varyingconcentration of defects [20].

Finite element implementation: The weak form of equations (2.30-2.33) are solved by finite element method. The simulation of fractureprocesses within the finite element (FE) context requires special provi-sions for dealing with displacement discontinuities. The displacementdiscontinuity is incorporated into standard finite element interpolationsby an additional degree of freedom. Discretised forms of the initialvalue problem are expressed in matrix-vector form where the stress andstrain tensors are represented as column vectors σ = (σx, σy, τxy)

T andε = (εx, εy, γxy)

T in 2D case. When discretizing the problem variationalstatement (2.29), representing the weak form of Eqs. (2.30-2.33) can beutilized . We interpolate the unknown fields displacement, strains andstresses as

u ≈Nd+Ncdc (2.34)

ε ≈ Bd+Ge (2.35)

σ ≈ Ss (2.36)

where vectors d, dc, e, and s are the degrees of freedom correspond-ing to nodal displacements, enhanced displacement modes, enhancedstrain modes and stress parameters, respectively. Nc and G are ma-trices containing some enrichment terms for displacements and strains,


respectively, S is a stress interpolation matrix. N is the standard dis-placement interpolation matrix (containing the usual shape functions),B is the standard strain interpolation matrix (containing the derivativesof the shape functions).

Techniques such as the EAS method [13] or the B-bar approach [9] areused for the present interpolation which is fairly general . Substitutingapproximations equations (2.34-2.36) into the variational identity (2.29)and taking into account that ∂(Nd) = Bd we obtain

δdTˆV

BT σ(Bd+Ge)dV+δeTˆV

GT [σ(Bd+Ge) − Ss]dV

+ δsTˆV

ST(Bcdc −Ge)dV + δdcT

ˆV

BcTS s dV = δdTf ext + δdc

Tf c

(2.37)where Bc is a strain interpolation matrix that would correspond to thedisplacement interpolation matrix Nc (i.e., Bc is defined by the identity∂(Ncd) = Bcd),

f ext =

ˆV

NT b dV+

ˆSt

NT t dS (2.38)

is the vector of (standard) external forces and

f c =

ˆV

NcT b dV +

ˆSt

NcT t dS (2.39)

is the vector of nonstandard external forces. For simplicity we will as-sume f c=0, that the loads are applied outside the region with enhancedinterpolation.

By taking into account the independence of variations, we obtain thediscretized equations asˆ

V

BT σ(Bd+Ge)dV = f ext (2.40)

ˆV

GT σ(Bd+Ge)−ˆV

GTS dV s = 0 (2.41)ˆV

STBc dV dc −ˆV

STG dV e = 0 (2.42)


ˆV

BcTS dV s = 0. (2.43)

The rate (incremental) formulation and differentiate (2.40-2.43) withrespect to time is applied in order to linearize the dependence of σ onparameters d and e. Any incrementally linear stress-strain equations canformally be written in the rate form for a given state as

˙σ = Dε ≈D(Bd+Ge) (2.44)

where D = ∂σ∂ε is the tangential stiffness matrix of the material. Substi-

tuting (2.44) into (2.40-2.43) we obtain a set of linear equations

ˆV

BTDB BTDG 0 0

GTDB GTDG −GTS 0

0 −STB 0 STBc

0 0 BcTS 0

dVde

s

dc

=

f ext

000

(2.45)

The interpolations of stress and strain can be discontinuous, so we canselect the interpolation functions such that each stress or strain param-eter is associated with only one finite element. The same holds for theenhanced displacement parameters. Parameters e, s and dc can thereforebe eliminated on the local (element) level, so that the global equationscontain only the standard displacement degrees of freedom d. From nowon, we consider Eqs. (2.40-2.43) and (2.45) written for one finite ele-ment occupying a certain volume Ve. Of course, the external force vectorf ext is then replaced by the contribution of the current element to theinternal forces f eint [19].

2.2.3 Basic types of enhancement

The basic three types of finite element models with embedded discon-tinuties known from the literature are present them as particular casesor modifications of the general formulation equations (2.40-2.43). Thedetailed formulation and behavior of this three models are analysed inchapter (3) with the finite element.

1. In the total formulation, equations (2.42) and (2.43) are linear al-ready, and so it is better to inspect them first. Let us assume that we


do not introduce any displacement enhancement terms, i.e., we en-rich only the strain interpolation. Then all terms containing dc, Nc,or Bc can be deleted from the formulation. Equation (2.43) thendoes not exist (it has been derived by taking the variation of dc) andequation (2.42) becomesˆ

Ve

STG dV e = 0 (2.46)

The element must be able to reproduce a constant stress field ex-actly in order to pass the generalized patch test and so the minimumchoice for the stress interpolation matrix is S=I=unit matrix. Thecompatibility conditions (2.46) now reads asˆ

Ve

G dV e = 0. (2.47)

The matrix G can always be constructed such thatˆVe

G dV = 0. (2.48)

We could modify G by subtracting from each entry its mean valueover the element, if this condition of zero mean was not satisfied.As the standard strain approximation ε = Bd always contains allconstant-strain modes, this modification does not affect the space offunctions described by the enhanced approximation ε = Bd+Ge.We can therefore assume that the enhanced strain interpolation ma-trix G satisfies condition (2.48). Then, equation (2.42) is satisfiedfor any e. As the matrix multiplying s in (2.41) is now a zero matrix,the stress parameters s completely disappear from the formulationcite9. We finally end up with a set of nonlinear equationsˆ

Ve

BT σ(Bd+Ge)dV = f eint (2.49)

ˆVe

GT σ(Bd+Ge)dV = 0 (2.50)

which, at a given state can be linearized intoˆVe

[BTDB BTDG

GTDB GTDG

]dV

de

=

f int

0

(2.51)


Remark if D is symmetric then the linearized system of equations(2.51) is also symmetric.

2. The other way is that, we can construct some suitable enhanced dis-placement interpolation matrix Nc and evaluate the correspondingmatrix Bc for which ∂(Ncd) = Bcd and set G = Bc. Equations(2.42) can be satisfied (independently of the choice of S) by set-ting dc = e. Equation (2.43) now imposes some restrictions on thestress parameters s but the important point is that by adding (2.43)to (2.41), we eliminate the stress parameters and construct exactlythe same equations as before, i.e., (2.49) and (2.50). Thus both ap-proaches lead to the same form of governing equations, with the onlydifference that the matrixG either has to satisfy condition (2.48), orhas to be constructed by applying the strain-displacement operatorto a suitable displacement enhancement cite9.

3. Modification of the governing equations was proposed by Simo andOliver [12] that does not follow from a variational principle, con-ceptually it corresponds to a weak form with test functions differentfrom trial functions, similar to the Petrov-Galerkin method. This ap-proach represents a compromise between the preceding two casesbecause they use G = Bc in the strain interpolation (2.35) whilereplacing GT in (2.50) by a matrix G∗ that is not the transpose ofBc but satisfies the condition of zero mean (2.48). Later it will beexplained why this choice leads to an improvement of the elementperformance. The resulting linearized equations

ˆVe

[BTDB BTDG

G∗DB G∗DG

]dV

de

=

f int

0

(2.52)

are in general nonsymmetric even if the material stiffness matrix Dis symmetric.

In chapter 3 it will be shown that elements derived from the firstabove mentioned formulation nicely satisfy the traction continuitycondition but they cannot properly represent the kinematics of adisplacement or strain discontinuity. On the other hand, elementsderived from the second formulation nicely reflect the discontinuitybut they lead to an awkward approximation of the traction continuity


condition. The above cases 1 and 2 will be referred to as the stati-cally optimal symmetric (SOS) formulation and the kinematically op-timal symmetric (KOS) formulation, respectively. The third approachhas the potential of representing both the kinematic and the staticaspects properly, so it will be called the statically and kinematicallyoptimal nonsymmetric (SKON) formulation [19].

2.2.4 Governing equations

Assumed-strain format: The basic idea is that, this is the mixed assumedstrain method within the context of physically non-linear elasticity. Thisclass of mixed methods includes, as a particular case, the classical methodof incompatible modes. For any of the three formulations introduced inthe preceding subsection, the governing equations can be written as[

Kbb Kbg

Kgb Kgg

]d

e

=

f int

0

(2.53)

whereKbb =

ˆVe

BTDBdV, (2.54)

Kbg =

ˆVe

BTDGdV, (2.55)

Kgb =

ˆVe

G∗DBdV, (2.56)

Kgg =

ˆVe

G∗DGdV. (2.57)

For the SOS formulation, G is a matrix with zero mean over the elementandG∗ = GT . For the KOS formulation,G = Bc andG∗ = BT

c . Finally,for the SKON formulation, G = Bc and G∗ is a matrix with zero mean.

e = −K−1ggKgbd (2.58)

f int = Kbbd−KbgK−1ggKgbd = Kd (2.59)

whereK = Kbb −KbgK

−1ggKgb (2.60)


is the condensed element stiffness matrix to be used in the assembly pro-cess. This is the usual procedure exploited by EAS methods [14].

B-bar format: The symmetric formulations (SOS and KOS) can easily becast into the B-bar format. Substituting (2.58) into the rate form of thestrain approximation (2.35) we obtain

ε = Bd+Ge = Bd−GK−1ggKgbd = (B −GK−1

ggKgb)d = Bd (2.61)

whereB = B −GK−1

ggKgb (2.62)

is the so called B-bar matrix. A straightforward calculation shows thatthe element stiffness matrix (2.60) can be alternatively be defined as

K =

ˆVe

BTDBdA (2.63)

and that

ˆVe

BT ˙σdA =

ˆVe

BT ˙σdA (2.64)

so that the equivalent internal forces

f eint =

ˆVe

BT σdA (2.65)

that (after assembly) have to be in equilibrium with the external forcescan alternatively be computed as

f eint =

ˆ t

0

ˆVe

BT ˙σdAdτ. (2.66)

Note that the B-bar matrix defined by (2.62) changes throughout thecalculation (because it depends on the current tangential stiffness of thematerial) while the standard B-matrix reamins constant.

Chapter 3

Elements with embedded displacementdiscontinuity

According to the kinematic relationships presented by Jirasek and Zim-mermann in [19], the Dirac-delta distribution is included in the incom-patible part of the strain field. Through the use of the Dirac delta dis-tribution, the amplitude of the incompatible mode can be interpretedas a displacement jump. This chapter concerns on the formulation ofthree fundemental class of elements with embedded displacement dis-continuity. According to the work from [19], the three finite elementformulations are referred as statically optimal symmetric (SOS), kinemat-ically optimal symmetric (KOS) and statically and kinematically optimalnon-symmetric (SKON) formulations. The effect of the discontinuitiesare characterized by additional degrees of freedom on the element level.This classification is based on kinematic enhancement and of the stresscontinuity condition. Strong and week points of individual formulationsare critically evaluated and presented.

3.1 Localization band two dimensions

We specially focus on a plane problem, discretized by triangular or quadri-lateral elements. Suppose that a certain element, occupying an area Ae,is crossed by a localization band of a constant width k. The element canbe divided into a region of localized strain L and its complement N, whichusually consists of two disjoint regions, N− and N+ ; see Figure (3.1 a) .

When dealing with a single element, we can work in a local coordinate

27

28 CHAPTER 3. ELEMENTS WITH EMBEDDED DISPLACEMENT DISCONTINUITY

Figure 3.1: Element with a localization band [19]

system (x,y) aligned with the localization band to be captured. Axis x isassumed to be normal to the band and axis y parallel to it. If AL is thearea of the localization band, we can define an equivalent length of theband l = AL/k ; see Figure (3.1 b).

It is also useful to define an equivalent ’element width’ (characteristicsize) h = Ae/l see Figure (3.1 c), so that AL/Ae = kl/hl = /k/h. Thearea of the nonlocalized region is AN = Ae − AL = (h− k)l.

As it is already explained in chapter 2.2.2, the interpolations of stress andstrain can be discontinuous so we can select the interpolation functionssuch that each stress or strain parameter is associated with only one finiteelement. The same holds for the enhanced displacement parameters.Parameters e, s and dc can therefore be eliminated on the local (element)level, so that the global equations contain only the standard displacementdegrees of freedom d. From now on, we consider equations (2.40-2.43)written for one finite element. Of course, the external force vector f ext isthen replaced by the contribution of the current element to the internalforces f eint.

Now we will introduce three basic techniques as particular cases or mod-ifications of the general formulation equations (2.40-2.43).

3.2. STATICALLY OPTIMAL SYMMETRIC FORMULATION 29

3.2 Statically optimal symmetric formulation

3.2.1 Element with a localization band

As presented in section (2.2.3):(1), we enrich only the strain interpo-lation, i.e., we do not introduce any displacement enhancement terms,then all terms containing dc, Nc, or Bc can be deleted from the formu-lation. Generally standard displacement interpolation would lead to astrain ε = Bd. Due to the presence of a discontinuity, a part of the strainis relaxed by crack opening and sliding. The SOS formulation starts fromthe standard strain and partially relaxes the strain component normal tothe discontinuity line and the shear component.

From linearized system of equations (2.51), D is symmetric then the lin-earized system of equations (2.51) is also symmetric, hence the namestatitally optimal symmetric formulation. The approach of Belytschko etal. [26] is recovered from the SOS formulation if we use a strain enrich-ment that allows a constant jump in strains εx and γ on the boundariesof the localization band. This can be achieved by setting

G =1

kAe(ANχL − ALχN)P , (3.1)

where

P =

1 00 00 1

(3.2)

and χL and χN are the characteristic functions of the localization bandand of its complement, respectively. Matrix G is thus piecewise constantand can be presented as

G = P LχL + PNχN , (3.3)

hereP L =

AN

kAeP , PN = − AL

kAeP (3.4)

The factors at χL and χN in (3.1) are chosen such that G satisfies thecondition of zero mean (2.48). Indeed,ˆ

Ae

(ANχL − ALχN)dA = ANAL − ALAN = 0 (3.5)


and soÁeGdA = 0. The common scaling factor 1/kAe is chosen such

that the components of e have the meaning of normal and shear straindifference between the localized and nonlocalized region multiplied bythe width of the band. In the limit for a localization band width ap-proaching zero, they represent the opening and sliding component of astrong (displacement) discontinuity.

According to (2.49) , the internal forces are evaluated as

f eint =

Âe

BT σdA =

ÂL

BTσLdA+

ÂN

BTσNdA (3.6)

where σL is the stress in the localization band, computed from the strain

εL = Bd+ P Le = Bd+AN

kAePe (3.7)

and σN is the stress outside the localization band, computed from thestrain

εN = Bd+ PNe = Bd− AL

kAePe (3.8)

The stresses must satisfy the internal equilibrium condition (2.50), whichfor the matrix G defined by (3.1) leads to

AN

kAe

ÂL

P TσLdA−AL

kAe

ÂN

P TσNdA = 0 (3.9)

after simple arrengenet we obtain

1

AL

ÂL

P TσLdA =1

AN

ÂN

P TσNdA. (3.10)

Note that the multiplication by P T selects the first and third componentfrom σL or σN , i.e., the normal stress σx and the shear stress τxy. Equa-tion (3.10) can therefore be interpreted as a weak stress continuity con-dition, stating that σx and τxy averaged over the localization band mustbe the same as σx and τxy averaged over the nonlocalized region.

3.2.2 Element with a discontinuity line

The band collapses into a curve SL, if we consider the limit case whenthe thickness of the localization band tends to zero. From the section

3.2. STATICALLY OPTIMAL SYMMETRIC FORMULATION 31

(3.1) we know the definiton of AN and AL and from the definition ac-cording to equation (3.1) of the enhanced strain interpolation matrix canbe rewritten as

G =1

khl[(h− k)lχL − klχN ]P =

(h− khk

χL −1

hχN

)P (3.11)

As k → 0, the term χL/k tends to Dirac distribution δL and the G tendsto

G =

(δL −

1

h

)P . (3.12)

The contribution of the localization band to the internal forces vanishesand (3.6) reads

f eint =

ˆAe

BTσdA, (3.13)

where we write σ instead of σN because the nonlocalized region N nowextends over the entire element with the exclusion of the discontinuitycurve SL. For the same reason, we will drop the subscript at εN .

In equation (3.10), the product P TσL has to be replaced has to be re-placed by the cohesive tractions t, and the resulting internal equilibriumcondition

1

l

ˆSL

t dS =1

Ae

ˆAe

P TσdA (3.14)

means that the tractions averaged over the discontinuity line SL must beequal to the first and third component (σx and τ) of the stress averagedover the bulk of the element. This is the weak form of the condition oftraction continuity across the localization line.

3.2.3 Constant strain triangle element

To understand further insight into the structure of the discretized equa-tions, consider the simplest finite element (CST) with a finite localiza-tion band [19]. Strains εL inside the localization band are constant, andstrains εN outside the band are also constant (but in general differentfrom εL). Consequently, the stresses are also piecewise constant and theinternal forces can be evaluated as

f eint = BT (ALσL + ANσN). (3.15)


The internal equilibrium condition (3.10) for the CST element reads

P TσL = P TσN . (3.16)

Due to the fact that the stresses are piecewise constant, the traction con-tinuity condition on the boundary between L and N is enforced in thestrong sense.

The structure of the governing equations is shown in the diagram in Fig-ure (3.2 a). The top part describes to the equilibrium equations, themiddle part to the constitutive relations and the bottom part to the kine-matic equations. Dashed arrows indicate that the source is added to thetarget. For example, the strain εN is given by the sum of of Bd andPNe. Symbol σ denotes the constitutive operator, i.e., σN = σ(εN) andσL = σ(εL).

In the limit case for k → 0 we obtain a CST element with an embeddeddisplacement discontinuity line, described by the equations schematicallyshown in Figure (3.2 a). Symbol s denotes the constitutive operator forthe interface. Solid arrows mean that the source must be equal to thetarget. For example, the tractions t are not the sum of P Tσ and s(e) butthey are equal to either of these expressions, i.e., t = P Tσ = s(e).

3.3 Kinematically optimal symmetric formulation

3.3.1 Element with a localization band

In this technique as described in chapter (2.2.3):(2), KOS formulation isclearly superior from the kinematic point of view. As stated earlier, dueto the presence of a discontinuity, a part of the strain is relaxed by crackopening and sliding. The KOS formulation first subtracts the contributionof the displacement jump from the nodal displacements and only then ap-plies the kinematic operator.

The difference between the SOS and KOS technique is that, the matrixG either has to satisfy condition (2.48) of zero mean (SOS formulation),or has to be constructed by applying the strain-displacement operator toa suitable displacement enhancement; i.e., Kinematically optimal sym-metric formulation. The approach of Lotfi and Shing [17, 7] is recovered

3.3. KINEMATICALLY OPTIMAL SYMMETRIC FORMULATION 33

Figure 3.2: Structure of the equations describing the SOS formulation (a) with a localization bandand (b) with a localization line [19].

from the KOS formulation if we use an enhanced displacement interpo-lation matrix

N c =

[Nc 00 Nc

](3.17)

with the special shape function

Nc(x) = HL(x)−N∑i=1

HL(xi)Ni(x), (3.18)

where N is the number of nodes per element, Ni,i = 1, 2, ....., N , are thestandard shape functions; xi, i = 1, 2, ..., N are the local coordinates vec-tors; and HL is a ramp function assuming a constant value HL = 0 inN− and a constant value HL = 1 in N+ with a linear transition from0 to 1 in L. The corrective term

∑HL(xi)Ni(x) is a linear combina-

tion of the standard shape functions while the function HL supplies theactual enrichment of the displacement interpolation. Inorder to makethat the standard degrees of freedom d keep the meaning of nodal dis-placements , this correction makes sure that the value of the nonstan-dard shape function Nc at every node is zero. Taking into account that


Ni(xj) = δij = Kronecker delta, we get

Nc(xj) = HL(xj)−N∑i=1

HL(xi)Ni(xj) = HL(xj)−N∑i=1

HL(xi)δij = 0.

(3.19)We obtain the enhanced strain interpolation matrix by applying the strain-displacement operator to the enhanced displacement interpolation

Bc =

HL,x 00 HL,y

HL,y HL,x

− N∑i=1

HL(xi)

Ni,x 00 Ni,y

Ni,y Ni,x

. (3.20)

Here subscript after a comma indicates partial differentiation, e.g.,HL,y =∂HL/∂y. If the coordinate system is aligned with the localization bandfigure (3.1a) , we have HL,x = χL/k and HL,y = 0.

For isoparametric finite elements, each of the matrices

Bi =

Ni,x 00 Ni,y

Ni,y Ni,x

(3.21)

is a certain submatrix of the standard strain interpolation matrix

B =[B1 B2 ... BN .

](3.22)

Therefore, the enhanced strain interpolation matrix from (3.20) can bepresented in a compact form

Bc =1

kχLP −BHL (3.23)

where P is the Boolean matrix defined by (3.2) and

HL =

HL(x1) 00 HL(x1). .. .

HL(xN)0

0HL(xN)

. (3.24)

3.3. KINEMATICALLY OPTIMAL SYMMETRIC FORMULATION 35

Formula (3.6) for the internal forces remains valid but this time the stressσL is computed from the strain

εL = Bd+

(1

kP −BHL

)e = B(d−HLe) +

1

kPe = BdN +

1

kPe

(3.25)and the stress σN is computed from the strain

εN = Bd−BHLe = B(d−HLe) = BdN , (3.26)

wheredN = d−HLe. (3.27)

For the present choiceG = Bc, the internal equilibrium condition (2.50)readsˆ

AL

(1

kP −BHL

)TσLdA−

ÂN

(BHL)TσNdA = 0 = 0, (3.28)

which can be transformed into

HTL

(ÂL

BTσLdA+

ÂN

BTσNdA

)=

1

k

ÂL

P TσLdA. (3.29)

According to (3.6) the expression in parentheses on the left-hand sideis equal to the vector of internal forces. Summing the internal forcescorresponding to the nodes that are located in N+ and adding a weightedcontribution of the nodes that happen to be in the localization band L isequivalent to multiplication by HT

L. Thus, rewriting (2.56) as

HTLf

eint =

1

AL

ÂL

P TσLdA. (3.30)

We obtain a condition with the following interpretation, sum of inter-nal forces acting on the ‘positive’ side of the band must be equal to theaverage values of stress components σx and τxy in the localization bandmultiplied by the length of the band.

3.3.2 Element with a discontinuity line

The internal forces are evaluated according to (3.13), in the limit fork → 0, and the weak stress continuity condition (3.30) reads

HTLf

eint =

ˆSL

t dS (3.31)


This means that the cohesive tractions integrated along the discontinuityline must be equal to the sum of internal forces acting on the ‘positive’side of the band. In other words, if we cut the element along the disconti-nuity line and replace the interaction between the separated parts by thecohesive tractions, either of the separated parts of the element must sat-isfy the conditions of force equilibrium (but not necessarily the conditionof moment equilibrium); see Figure (3.3).

Figure 3.3: Equilibrium of an element split by a discontinuity line [19].

3.3.3 Constant strain triangle element

The structure of the governing equations for a CST element based on theKOS formulation with a finite localization band is shown in Figure (3.4 a)and a discontinuity line is shown in the Figure (3.4 b). Symbol I denotesthe identity operator, i.e., dN is equal to the sum of Id ≡ d and −HLe.

3.4 Statically and kinematically optimal nonsymmetric formulation

As it is already described in chapter (2.2.3):(3), this approach is a com-promise between SOS and KOS formulation. A general version of theSKON formulation for any type of parent element was proposed by Simoand Oliver [12]. They useG = Bc in the strain interpolation (2.35) whilereplacing GT in (2.50) by a matrix G∗ that is not the transpose of Bc butsatisfies the condition of zero mean (2.48). The linearized equations

´Ve

[BTDB BTDG

G∗DB G∗DG

]dV

de

=

f int

0

are in general nonsymmetric. In SKON formulation we use the enhancedstrain interpolation matrixBc given by (3.23) but in the internal equilib-rium condition (2.50) we use the transpose of the matrix G from (3.1).

3.4. STATICALLY AND KINEMATICALLY OPTIMAL NONSYMMETRIC FORMULATION 37

Figure 3.4: Structure of the equations describing the KOS formulation (a) with a localization bandand (b) with a localization line [19].

Consequently, the stresses σL and σN are calculated from the strains εLand εN given by (3.25) and (3.26), same as for the elements due to Lotfiand Shing [17, 7], but the stress continuity condition (3.10) or (3.14) istaken from the approach due to Belytschko et al [26].

The SKON formulation of a CST with a discontinuity line leads to theelement constructed based on simple physical considerations. The nodaldisplacements are decomposed into a part due to uniform strain in thebulk of the element and a part due to crack opening. In our notation,these parts correspond to dN andHLe, respectively. The strains ε = BdNare related to stresses σ using a linear elastic law, and the crack openinge to the cohesive tractions t using a traction seperation law.

Finally, a natural traction continuity requirement t = P Tσ was imposed,and the internal forces were calculated as f eint = AeB

Tσ. The structureof the governing equations for this element is shown in Figure (3.5 b)and (3.5 a) presents a element with a finite localization band.


Figure 3.5: Structure of the equations describing the SKON formulation (a) with a localization bandand (b) with a localization line [19].

3.5 Derivation from the principle of virtual work

With reference [19]; The derivation of the symmetric formulations fromthe PVW is as follows. When looking at a single element, we may considerthe forces f eint as external ones and express the external virtual work as

δWext = f eintT δd. (3.32)

For a model with a strong discontinuity, the internal virtual work is doneby the stresses in the continuous part on the corresponding virtual strainsand by the tractions transmitted by the discontinuity on the virtual open-ing and sliding components of the displacement jump. Hence we canwrite

δWint =

ˆAe

σT δε dA+

ˆSL

tT δe dS. (3.33)

According to the PVW, the internal virtual work must be equal to the ex-ternal one for any virtual change of the kinematic state. By definition, thevirtual change must be kinematically admissible. If we postulate a certainkinematic assumption with reference to kinematic equation (3.26), e.g.,

ε = B(d−HLe) (3.34)

3.5. DERIVATION FROM THE PRINCIPLE OF VIRTUAL WORK 39

then the virtual change is uniquely described by δd and δe, which maybe regarded as independent parameters that determine the virtual strain

δε = B(δd−HLδe) (3.35)

Substituting (3.35) into (3.33) and setting the resulting expression equalto (3.32) we obtainˆ

Ae

σTB(δd−HLδe)dA+

ˆSL

tT δe dS = f eintT δd. (3.36)

This equality holds for arbitrary δd and δe if and only ifˆAe

BTσdA = feint (3.37)

and−ˆAe

HTLB

TσdA+

ˆSL

t dS = 0. (3.38)

The first condition is the standard formula for the evaluation of equiva-lent nodal forces while the second one can be rewritten asˆ

SL

t dS = HTLf

eint. (3.39)

This is condition (3.31) which enforces (in the weak sense) equilibriumbetween the tractions and the nodal forces acting on the solitary node.It was derived from the requirement that, if the strains in the continuouspart of the element are kept constant and the displacement discontinu-ity is subjected to a virtual change, the virtual work done by the nodalforces must be equal to the virtual work done by the tractions across thediscontinuity. So a derivation based on the PVW that starts from a ‘nat-ural’ kinematic assumption (3.34) automatically leads to static condition(3.39) that is work-conjugate with the kinematic one but differs from the‘natural’ condition of traction continuity.

In an similar manner, it is possible to proceed from a static assumption tothe work-conjugate kinematic condition derived from the complementaryPVW, which states that the work done by virtual nodal forces on theactual displacements,

δW ∗ext = δf eint

Td (3.40)


must be equal to the work done by virtual stresses on the actual strainsand by virtual tractions on the actual displacement discontinuity,

δW ∗int =

Âe

δσTε dA+

ˆSL

δtTe dS (3.41)

for an arbitrary virtual state that is statically admissible. The postulatedstatic assumptions consist of the standard relation between stresses andnodal forces, ˆ

Ae

BTσ dA = f eint (3.42)

and a condition that links the stresses and tractions across the disconti-nuity. It is natural to require that the tractions be equal to the projectedstress components,

t = P Tσ. (3.43)

Relations similar to (3.42) and (3.43) must also hold for any virtualchange of the static state. We might therefore regard the virtual stressδσ as independent and express the virtual nodal forces and virtual trac-tions as δf eint =

ÁeBT δσdA and δt = P T δσ, respectively. Substituting

this into (3.40) and (3.41) and applying the complementary PVW weobtain ˆ

Ae

δσTε dA+

ˆSL

δσTP e dS =

Âe

δσTB dA d. (3.44)

The virtual stress field δσ is not completely arbitrary but has to be self-equilibrated inside Ae. This condition holds in particular for constantstress fields. In this case, we might take δσ out of the integrals, and bystandard arguments we arrive atˆ

Ae

ε dA+

ˆSL

P e dS =

Âe

B d dA. (3.45)

For the special case of a CST, this may be further simplified to

ε = Bd− l

AePe (3.46)

which is the kinematic condition work-conjugate with the ‘natural’ con-dition of traction continuity equation (3.10); with εN → ε and AL/k → l.

3.6. DISCUSSION 41

3.6 Discussion

A number of techniques enriching the standard finite element interpola-tion by additional terms corresponding to a displacement or strain dis-continuity have been presented within a unified framework and criticallyevaluated. It has been shown that there exist three major classes of thesemodels, called here SOS, KOS and SKON. The SOS formulation cannotproperly reflect the kinematics of a completely open crack but it givesa natural traction continuity condition, while the KOS formulation de-scribes the kinematic aspects satisfactorily but it leads to an awkwardrelationship between the stress in the bulk of the element and the trac-tions across the discontinuity line [19].

These properties of the symmetric formulations were the driving forcebehind the development of the nonsymmetric SKON formulation, whichcombines the optimal static and kinematic equations and leads to an im-proved numerical performance. This formulation deals with a very natu-ral traction continuity condition and is capable of properly representingcomplete separation at late stages of the fracturing process, without anylocking effects (spurious stress transfer). The price to pay is the loss ofsymmetry of the tangential stiffness matrix. Results reported in the lit-erature show that the nonsymmetric model can be used with success innumerical simulations of localized cracking. It is also worth noting thatthe SKON formulation does not require any specification of the ‘length’of the localization band. This is an important advantage because suchlength is in general not an objective quantity and its value depends onthe (partially ambiguous) rule for the positioning of the discontinuityinside the element [19].

Chapter 4

Numerical Procedure

In this chapter, numerical aspects of the implementation are presented.This chapter complements the practical implementation of a specific modelwith a strong (displacement) discontinuity embedded in a finite elementbased on the statically and kinematically optimal non-symmetric formu-lation. The constitutive description of a damaging interface is presented.Algorithms for the evaluation of internal forces and stiffness matrix ofthe element have been presented and also extended to special cases suchas a closed crack.

4.1 Damage-based traction-separation law

The type of the discontinuity affects the choice of an appropriate constitu-tive model suitable for implementation in an element with an embeddeddiscontinuity. It is sufficient to postulate a continuum stress-strain lawfor models incorporating weak (strain) discontinuities , but the modelswith strong discontinuities require, a traction-separation law governingthe behavior of the discontinuity (crack or plastic slip surface) in additionto a stress-strain law for the bulk material [19]. In the present study wewill focus on the strong discontinuity model. It is, therefore, necessaryto present a law that links the traction transmitted by the discontinuityto the displacement jump. Based on the thermodynamic approach thismodels can be conveniently derived , starting from an expression for thedensity of Helmholtz free energy. For example, the free energy densityper unit volume for the isotropic continuum damage model with a single

43

44 CHAPTER 4. NUMERICAL PROCEDURE

scalar damage parameter is given by

ψ(ε, ω) =1

2(1− ω)ε : D : ε (4.1)

where ε is the strain tensor, D is the elastic stiffness tensor, and ω is thedamage parameter growing from zero (virgin material) to one (completeloss of integrity). For simplicity, we neglect the effect of temperature,assuming that the process is isothermal. The potential (4.1) describesonly the energy stored in the elastic deformation of the bulk materialbetween microdefects such as cracks or voids.

We have to start from an expression for free energy per unit area, whenconstructing a traction-separation law. The strain tensor ε is replaced bya vector e characterizing the displacement jump(separation). This vec-tor describes only the inelastic part of deformation (it is identically zerobefore the cracking). The strain tensor corresponds to the sum of theelastic and inelastic deformation. Before a crack is initiated, it does notcontribute to the deformation, and its ‘initial’ stiffness has to be consid-ered as infinite. The surface density of free energy can be expressed as

ψ(e, γ) =1

2γe · D · e (4.2)

as a function of the separation vector, e, and a new internal variable γwhich is called the compliance parameter and varies from zero to infin-ity. In the continuum model defined by equation (4.1), the complianceparameter would correspond to ω/(1 − ω). Symbol D in equation (4.2)denotes a second-order tensor describing the stiffness of the discontinu-ity (crack) at an intermediate reference state when γ = 1. This state setsthe scale for γ and its choice does not affect the response of the model(in the sense that after proper rescaling of the compliance parameter thesame response is obtained with any choice of the reference state).

For any possible process, the model must satisfy the dissipation inequality

D ≡ t · e− ψ ≥ 0 (4.3)

where D is the dissipation rate (per unit area), and t is the tractiontransmitted by the crack. In the absence of dissipative (viscous) stresses,standard thermodynamic arguments [16, 6] lead to the state equations

t =∂ψ

∂e=

1

γD · e (4.4)

4.1. DAMAGE-BASED TRACTION-SEPARATION LAW 45

Γ = −∂ψ∂γ

=1

2γ2e · D · e (4.5)

where Γ is the dissipative thermodynamic force associated with γ. Thedissipation inequality (4.3) now reads

D ≡ Γγ ≥ 0 (4.6)

The thermodynamic force Γ defined by Equation (4.5) is always non-negative if the reference stiffness D is assumed to be positive definite.The rate of γ must not be negative according to condition (4.6), i.e., thatcan only increase or remain constant but can never decrease. Symmetryarguments lead to the condition that, in local co-ordinates for which thefirst axis is aligned with the crack normal n, the reference stiffness mustbe represented by a diagonl matrix

D =

Dnn 0 00 Dss 00 0 Dss

(4.7)

because, e.g., normal opening of the crack should not generate shear trac-tions on its faces, and sliding in a given tangential direction should notproduce shear tractions in the perpendicular direction (unless the mate-rial is anisotropic). In view of Equation (4.7), the traction-separation law(4.4) can be written as

tn =Dnn

γen (4.8)

ts =Dss

γes (4.9)

where tn = t · n is the normal traction transmitted by the crack, ts = t−tnis the tangential traction, en = e · n is the normal component of theseperation vector (crack opening) and es = e − enn is the tangentialcomponent of the seperation vector (crack sliding).

It is necessary to postulate an evolution law for the compliance param-eter γ. Using the formalism of generalized standard materials [16] , weassume the existence of a (dual) dissipation potential φ∗(Γ, γ) such that

γ =∂φ∗(Γ, γ)

∂Γ(4.10)


If the potential is nonnegative, equal to zero for Γ = 0, and convex withrespect to the thermodynamic force Γ (for any admissible value of theinternal variable ), the dissipation is guaranteed to be non-negative

D ≡ Γγ = Γ∂φ∗

∂Γ(4.11)

Similar to continuum damage models, let us define a loading functionf ∗(Γ, γ) such that tha inequality f ∗ < 0 characterizes the elastic domain.If f ∗ < 0 the deformation process is reversible (elastic), i.e., the dissipa-tion rate must be zero. If f ∗ = 0 damage grows which is accompanied byenergy dissipation.

For rate-independent models the damage increment is assumed to be in-stantaneous and states for which f ∗ > 0 can never be reached. All this isreflected by a dissipation potential defined as the indicator function [6]of the elastic domain F = (Γ, γ) |f ∗(Γ, γ) ≤ 0. The indicator functionis equal to zero in F and equal to infinity outside F . Its gradient withrespect to Γ that appears in equation (3.10) must be interpreted as asubdifferential [6].

The resulting evolution law is described by

γ = λ∂f ∗(Γ, γ)

∂Γ(4.12)

along with the Kuhn-Tucker conditions,

λ ≥ 0, f ∗ ≤ 0, λf ∗ = 0 (4.13)

from which we can derive the consistency condition,

λf ∗ = 0 (4.14)

Equations (4.13) and (4.14) are formally the same as for standard plas-ticity, where f ∗ would be the yield function and λ would be the plasticmultiplier. The response of the model is fully determined by specifyingthe loading function, f ∗. Inspecting the state law (3.5) we observe thatthe force Γ is related to e · D · e , which can be interpreted as a scalarmeasure of the displacement jump across the discontinuity. The state lawcan be rewritten as

Γ =Dnn

2γ2

(e2n +

Dnn

Dsse2s

)=Dnne

2

2γ2(4.15)

4.1. DAMAGE-BASED TRACTION-SEPARATION LAW 47

where es = ‖es‖ is the Euclidean norm of the tangential component ofthe separation vector (crack sliding), and

e =

√e2n +

Dnn

Dsse2s =

√e · D · eDnn

(4.16)

can be called the equivalent separation (in analogy to the equivalentstrain in continuum damage mechanics).

Let us assume that the traction-separation curve for fracture under pureMode I has been identified from experiments and described by an explicitrelation tn = g(en). This relation should be reproduced by the constitu-tive law (4.8), which means that the compliance parameter under mono-tonically increasing crack opening must satisfy the relation γ = F (en)where

F (en) = Dnnen/g(en) (4.17)

For decreasing functions g(en) that characterize softening, F (en) is alwaysan increasing function. This is closely related to the fact that the compli-ance parameter (inversely proportional to the slope of the line connect-ing the current point on the traction-separation curve with the origin) ismonotonically increasing as long as the crack opening en keeps growing.If en drops below its maximum previously reached value, unloading takesplace and the compliance parameter is temporarily frozen.

As an example, consider the exponential traction-separation law, (pureMode I)

tn = g(en) ≡ ftexp

(−eneu

)(4.18)

where ft is the tensile strength and eu = Gf/ft , Gf being the Mode-Ifracture energy.

In 2D case, consider the exponential traction-separation law with isotropicdamage

te = g(e) ≡[tnts

]=fte

exp

(− e

eu

)[enes

](4.19)

Denoting the equivalent traction by t = ftexp(− eeu

)and where e equiv-

alent seperation, tn and ts are the normal and tangential components oftraction vector, en and es are the normal and tangential components ofseperation vector.


4.2 Evaluation of internal forces and element stiffness matrix

In an incremental-iterative analysis of a structure discretized by finite ele-ments with embedded discontinuities, the nodal displacements are com-puted iteratively from the global equilibrium equations, and the maintasks on the level of one finite element are to evaluate the internal forcesand the tangent stiffness matrix for a given increment of nodal displace-ments.

As already presented in the introduction, incorporating displacement dis-continuities into the finite element interpolation can be substantially im-proved by the kinematic representation of highly localized fracture . Itis shown there that the optimal performance is achieved if the kinematicand static equations are constructed independently, based on their phys-ical meaning. This so-called statically and kinematically optimal nonsym-metric formulation (SKON) shall be used here.

For simplicity we assume that the bulk material surrounding the discon-tinuity remains linear elastic and that the crack initiation is controled bythe Rankine criterion of maximum principal stress and for easy under-standing we describe for CST element. For constant opening, extensionto quadrilateral element is in the similar way but we need to takecare ofgauss points . The basic idea is that the displacement field is decomposedinto a continuous part and a discontinuous part due to the opening andsliding of a crack; see Figure (4.1(b)). The same decomposition appliesto the nodal displacements of a finite element. we represent the discon-tinuity by additional degrees of freedom collected in a column matrix e.The effect of crack opening and sliding is then subtracted from the nodaldisplacement vector d = (u1, v1, u2, v2, u3, v3)T , and only the nodal dis-placements due to the continuous deformation serve as the input for theevaluation of strains in the bulk material, ε; see Figure (4.1(c)).

This leads to the kinematic equations from (3.26) in the form (strongpoint of KOS formulation)

ε = B(d−He) (4.20)

where B is the standard strain-displacement matrix, and H is a matrixreflecting the effect of crack opening on the nodal displacements. Inthe context of finite elements we make use of the engineering notation,

4.2. EVALUATION OF INTERNAL FORCES AND ELEMENT STIFFNESS MATRIX 49

Figure 4.1: CST element with an embedded displacement discontinuity: (a) global co-ordinate sys-tem x,y and local co-ordinate system n,s aligned with the crack; (b) normal and tangen-tial component of the displacement jump; (c) element deformation due to strain in thebulk material, with the contribution of crack opening and sliding to nodal displacementssubtracted; (d) equilibrium between tractions across the crack and stresses in the bulkmaterial [18].

i.e. in 2D case we denote column matrices σ = (σx, σy, τxy)T and ε =

(εx, εy, γxy)T instead of second-order tensors.

Normally, the displacement jump is approximated by a suitable function,for example a polynomial one. It is easy to show that the approximationneed not be continuous. For triangular elements with linear displacementinterpolation, the strains and stresses in the bulk are constant in each el-ement, and so it is natural to approximate the displacement jump alsoby a piecewise constant function. In each element, the jump is describedby its normal (opening) component, en, and tangential (sliding) compo-nent, es figure . These additional degrees of freedom have an internalcharacter and can be eliminated on the element level, which means thatthe global equilibrium equations are written exclusively in terms of the


standard unknowns nodal displacements. From Figure (4.1(b)) it is clearthat the crack-effect matrix is given by

H =

0 00 00 00 0c −ss c

(4.21)

means that the discontinuity line separates node 3 from nodes 1 and 2(in local numbering). In Equation (4.21), c = cosα and s = sinα, whereα is the angle between the normal to the crack and the global x-axis; seeFigure (4.1(a)).

Strains in the bulk material generate certain stresses, σ, which are herecomputed from the equations of linear elasticity,

σ = Dε (4.22)

but the constitutive law for the bulk material could be non-linear in gen-eral. The tractions transmitted by the crack (damaging surface), t, can becomputed from the displacement jump using the traction-separation lawfrom the preceding section from equation (4.4). The constitutive relationfor an opening or partially closing crack,

t =1

γD · e (4.23)

The stresses in the bulk and the tractions across the crack must satisfycertain conditions that express internal equilibrium and serve as staticequations associated with the internal degrees of freedom, e. The mostnatural requirement is that the traction vector be equal to the stress ten-sor contracted with the crack normal, similar to static boundary condi-tions. This internal equilibrium (traction continuity) condition can bederived from equilibrium of an elementary triangle with one side on thediscontinuity line; see Figure (4.1(d)). In the engineering notation thetraction continuity condition reads from equation (3.14) (strong point ofSOS formulation)

P Tσ = t (4.24)

4.2. EVALUATION OF INTERNAL FORCES AND ELEMENT STIFFNESS MATRIX 51

where

P =

c2 −css2 cs2cs c2 − s2

(4.25)

is a stress rotation matrix. For linear triangles, both t and σ are constantin each element, and so condition (4.24) can be satisfied exactly.

Finally, the nodal forces are evaluated from the standard relation

f int =

ˆAe

BTσdA = AeBTσ (4.26)

where Ae is the area of the element. Substituting into the traction con-tinuity condition (4.24) Equations (4.22) and (4.20), we obtain a usefulexpression for the traction vector in terms of the kinematic variables,

t = P TDB(d−He) = A(d−He) (4.27)

where we have denotedA = P TDB (4.28)

Comparing Equation (4.27) with Equation (4.23),

A(d−He)− 1

γD · e = 0 (4.29)

we obtain after simple manipulation

(D + γAH)e− γAd = 0 (4.30)

Combined with the evolution equation for the compliance parameter γ,Equation (4.30) makes it possible to express the separation vector interms of the nodal displacements and eliminate it from the formulation.

The tangent stiffness of an element with embedded discontinuity can beconstructed by expressing the separation rate in terms of the displace-ment rate and substituting into the rate form of the basic equations in-troduced above. Differentiation of equilibrium equation (4.30) leads tothe rate equation

Ad+

(−AH − D

γ

)e+

(De

γ2

)γ = 0 (4.31)


we know that γ = ∂γ∂e e and with simple modification, we have

Ad =

[(AH +

D

γ

)+

(De

γ2⊗ ∂γ

∂e

)]e (4.32)

From above equation , we express M as

M =

(AH +

D

γ

)+

(De

γ2⊗ ∂γ

∂e

)(4.33)

Here Dnn = Dss = D0 and with details

M = AH +

[D0

γ 0

0 D0

γ

]−

[D0enγ2D0enγ2

]⊗

[D0ente + D0en

teuD0este + D0es

teu

](4.34)

where t is equivalent traction, and the realtion between d abd e is ex-pessed as

Ad = Me (4.35)

For growing damage, it is possible to express the above equation as theseperation rate as

e = M−1Ad (4.36)

The rate of nodal forces is now obtained from the rate form of equations(4.20) , (4.22) and (4.26) as

f int = AeBTDB(d−He) (4.37)

where Ke = AeBTDB is the elastic element stiffness matrix and from

the rate form of e (4.36) , we have

f int = [Ke −KeHM−1A]d (4.38)

The present tangent element stiffness matrix is in the form

Ktan = Ke −KeHM−1A (4.39)

Remark: If damage does not grow (the crack is unloading), there is noincrease of damage parameter , i.e in (4.31) γ = 0, consecquently

M = AH +

[D0

γ 0

0 D0

γ

](4.40)

The tangent stifness matrix becomes actually secant stiffness.

4.3. NUMERICAL TREATMENT OF CLOSED CRACK 53

4.3 Numerical treatment of closed crack

Unto now we have considered only cracks that are opening or partiallyclosing. The model can be improved by taking into account the unilateralcharacter of damage manifested as a stiffness recovery after a completecrack closure. It is clear that the crack faces cannot overlap, and so thenormal component of the separation vector should never become neg-ative. At the moment when the normal separation vanishes, the crackfaces establish contact and become capable of transmitting compressivetractions without a further change of the normal displacement jump, en.

The normal part of the traction-separation law (4.8) describes only thecase when en > 0. Upon crack closure, it has to be replaced by conditionsen = 0 and tn ≤ 0. Both cases are simultaneously covered by conditions

tn −Dnn

γen ≤ 0, en ≥ 0,

(tn −

Dnn

γen

)en = 0 (4.41)

having again the KuhnTucker form. For a closed crack (en = 0), it isnecessary to modify equation (4.9) governing the evolution of the slidingcomponents of the displacement jump vector. Sliding can take place evenif the crack is closed, provided that the shear traction is suffciently largeto overcome the residual cohesion of the crack by dry friction. Considera two-dimensional model with a line crack, for which the traction andseparation vectors have only one shear component, respectively denotedas ts and es. The residual cohesive resistance in shear is the shear tractioncomputed from the damage model (4.9).

To recovery stiffness after a complete crack closure, it is necessary tocheck whether the converged separation vector has a positive normalcomponent. When a negative value of en is detected, the crack is closed,and a different algorithm must be applied. Most equations remain validbut the constitutive relation (4.23) must be replaced by en = 0.

Numerical treatment of a closed crack always starts from the assumptionthat the crack separation vector e remains constant. This corresponds tothe ‘sticking’ mode, in which the crack surfaces are in contact and do notexperience any relative motion. Setting e = (0, (es)

n)T and γ = (γ)n, wecan evaluate the trail tractions according to equation(4.27)

t = A(dn+1 −He) (4.42)


Here (n) and (n+1) are the step numbers and check the conditions fromreference [18]

tn ≤ 0 (4.43)∣∣∣∣ts − Dss

γes

∣∣∣∣− µ 〈−tn〉 ≤ 0 (4.44)

where 〈.〉 are the McAuley brackets(’positive part’) and µ is the coefficientof friction between the cracked surfaces, which is neglected in this case.

If both conditions are satisfied the solution is admissible, and we canaccept e as the value en+1 at the end of the step. If the first condition isviolated, the crack starts re-opening and the standard algorithm shouldbe applied. If the second condition is violated, the crack starts sliding andthe trial values should be corrected. Formula (4.27) makes it possible toexpress the tractions in terms of the sliding relative displacement, es. Thenormal component of e is in the sliding mode equal to zero, and so wehave

tn+1 = Adn+1 −Ah2en+1s (4.45)

where h2 is the second column of H matrix.

4.4 An implicit/explicit integration scheme for non-linear model

Let us consider a typical, displacement driven,non-linear solid mechanicsproblem. The computability to those solid mechanics problems were ro-bustness is an important issue. The properties of material failure modelsequipped with strain softening, soft materials, etc., in terms of computa-tional cost, robustness and accuracy are the important issues to be con-sidered. Due to the softening behavior of the material, the most of theintegration schemes suffers from loss of computability when the globaltangent stiffness approaches negative-definite. To improve the robust-ness (computability) of the computation of strain-softening structures,an implicit/explict integration scheme for isotropic continuum damagemodel and elasto-plastic model is introduced in [11].

Generally explicit integration schemes yield robust but expensive (in termsof the computational cost) solving algorithms, whereas implicit integra-tion schemes lead to accurate results, even for large time steps, but at the

4.4. AN IMPLICIT/EXPLICIT INTEGRATION SCHEME FOR NON-LINEAR MODEL 55

cost of a loss of robustness of the resulting numerical algorithm which, forcases of practical interest, can also dramatically affect the correspondingcomputational cost. Implicit integration schemes are generally uncondi-tionally stable. Therefore, there is no intrinsic limitation on the lengthof the time step, other than the control of the integration error, whichuses to be small, and the number of required time steps, is small whencompared with explicit algorithms [11].

In this context, non-linear solid mechanics problems involving crack for-mation and propagation particularly discrete SDA approach, when mate-rial failure is modeled via traction–separation laws, which must be alsoequipped with strength softening which involves computability difficul-ties. To examine the reasons for this, let us consider the problem, dis-cretized in time and in space in a finite element mesh and from the sec-tion (4.2) the tangent stiffness matrix reads as shown in equation (4.39),Negative values of the softening parameters, (for example: H and M inthis equation (4.39)), translate into loss of the robustness of the algorith-mic problem through the following process

1. At initial stages of the analysis, the tangent stiffness matrix Ktan iselastic, is Ktan = Ke and, therefore, positive definite. So is theglobal tangent stiffness matrix.

2. In subsequent stages, the tangent stiffness matrix for loading cases,in equation (4.39) loose positive definiteness at those points wherematerial failure occurs and, therefore, exhibit negative eigenvalues.

3. Consequently, the general integration scheme in equation (2.52) lossespositive definiteness, and exhibits negative eigenvalues as well.

4. As material failure propagates through the solid, those local negativeeigenvalues deteriorate, via the assembling process, the conditionnumber of the global tangent stiffness matrix Ktan, whose smallesteigenvalues become progressively closer to zero.

5. Eventually, Ktan, becomes singular and the convergence fails. Ingeneral, there are no simple remedies for this, and the simulationprocess cannot be continued beyond that point.

Remark: Through the preceding reasoning it appears that the lack of pos-itive definiteness of the algorithmic tangent stiffness, as a consequence


of including strain softening in the model, is responsible for the observedloss of robustness.

This work proposes a combination of implicit and explicit integrationschemes that exploits the advantages of both, while overcoming some oftheir drawbacks. In essence, it is a combination of a standard implicit in-tegration scheme of the stresses, σ, in the constitutive model in equation(4.22) with an explicit extrapolation of the involved internal variables inthis case γ which depends on equivalent seperation e. This method im-plicit/explicit algorithm, renders relevant benefits when it is convenientlyexploited in computational mechanics. we now on shortened the nameas as IMPL-EX method. They can be summarized as follows:

• The algorithmic tangent stiffness matrix becomes symmetric and semi-positive definite even in those cases as the analytical one is not. Thisleads to dramatic improvements of the robustness in problems whereimplicit integrations result in singularity or the negative character ofthe algorithmic tangent operators.

• In many cases, the algorithmic tangent stiffness matrix becomes con-stant. Therefore, in absence of sources of non-linearity other thanthe constitutive model, the complete non-linear problem reduces to asequence of linear (at every time step) problems. The classical New-ton–Raphson procedure takes a unique iteration to converge and theproblem becomes step-linear. The effects on the computational costsare also dramatic.

• The good stability properties of the implicit integration algorithm areinherited by the proposed IMPL-EX integration algorithm.

• The order of accuracy of the IMPL-EX integration algorithm, withrespect to the size of the time step, is, at least, linear; the same asmany classical backward-Euler implicit algorithms. Nevertheless, theabsolute error is larger for the same time step length.

• The method can be exploited to render robust, steplinear and com-plex non-linear problems.

Usually, the numerical simulation of a structural made of strain-softeningmaterial, under quasi-static and small deformation conditions, can becarried out with displacement-controlled loading steps.

4.4. AN IMPLICIT/EXPLICIT INTEGRATION SCHEME FOR NON-LINEAR MODEL 57

At the beginning of the time interval [tn, tn+1], all the necessary data isgiven: the internal force vector f intn , the tangent stiffness matrix Ktan

n

and the nodal displacement vector Un. A stepwise increment of the load-ing condition is applied: Un+1 = Un + ∆U . Applying Newton-Raphsoniterative solution procedure, one can calculate the unknown state valuesof the structure at tn+1.

Applying the essence of [11] to the present Embedded Crack Model, theImplicit/explicit integration scheme is summarized in the following.

In a certain element (at the constitutive or local level), given loadingdn+1, considering the results of the last step en and γn, do the implicitcomputation and obtain the variables en+1 and γn+1.

Instead of returning implicit internal nodal force f int,n+1 and element tan-gent stiffness matrixKtan

n+1 as in Section 4.2, they are calculated explicitly.Based on the already known implicit values of internal parameter, whichis in the present model the maximum equivalent opening (α ≡ emax), ofthe previous two steps: αn−1 and αn, the unknown internal parameter ofthe current step is calculated directly by means of extrapolation [11] andshown in Figure (4.2):

Figure 4.2: Internal variable extrapolation [11].

αn+1 = αn −tn+1 − tntn − tn−1

(αn − αn−1) (4.46)

Then, the explicit damage parameter is calculated by comparing the ex-


pressions of te from the exponetial law and the damage law, i.e., fromequations (4.4) and (4.19) one obtains

γ =D0emax

ftexp(−emax/eu)(4.47)

γn+1 =D0αn+1

ftexp(−αn+1/eu)(4.48)

Unlike the tangent stiffness matrix for a loading step in (4.39), the ex-plicit algorithmic stiffness is given in the form of secant stiffness, as if itwere an unloading step:

Kalg

n+1 = Ke −KeHM−1

n+1A (4.49)

with

Mn+1 = AH +

[D0

γn+10

0 D0

γn+1

](4.50)

Finally, the explicit internal nodal force is obtained

f int,n+1 = Kalg

n+1dn+1 (4.51)

The explicit nodal force and stiffness are returned for the assembly ofglobal internal force vector and algorithmic stiffness matrix. Since thestiffness is only dependent on γn+1, which is constant during the step, theglobal solution should converge within one single iteration.

Chapter 5

Computational implementation

In this chapter, the detailed implementation strategy of the standardmodel is presented. The local Newton-Raphson solution tecnnique isexplained in detail, the problem of crack locking with the algorithm issummerised and to resolve the problem, the concept of crack adaptationis exlained and adopted to the model.

5.1 Algorithm for local iterative solution procedure

Suppose that the values of all variables at the end of a certain compu-tational step number n are prescribed, and our task is to calculate theirvalues at the end of the subsequent step number n+1. The approxima-tions of nodal displacements dn+1 are supplied by the iterative solutionof the global equilibrium equations, and during each evaluation of theelement nodal forces they can be considered as given. Before crack ini-tiation, it suffices to compute the stresses from the equations valid forlinear elasticity,

σn+1 = DBdn+1 (5.1)

and then check the crack initiation condition, formulated here as the sim-ple Rankine criterion, σ1 = ft , where σ1 is the maximum principal stressand ft is the tensile strength. The direction of maximum principal stressat crack initiation provides the normal to the discontinuity, n, and alsodetermines the matrices P and H that depend on the crack orientation.

After crack initiation, relations (4.30) written at the end of step n+1contain only the crack opening parameters en+1 and the compliance pa-

59

60 CHAPTER 5. COMPUTATIONAL IMPLEMENTATION

rameter γn+1 as basic unknowns. Every finite element which violates thecrack initiation condition has to satisfy the internal equilibrium (tractioncontinuity) condition with one side on the discontinuity line (4.24). Thedetailed iterative solution procedure using Newton-Raphson algorithm,in order to achieve the local equilibrium condition is described in both1D and 2D cases.

5.1.1 1D iterative solution procedure

The 1D case (pure Mode-I fracture) can be investigated as a basic exam-ple.

Assuming a 1D element with length 1.0 and the elastic modulus E. Adisplacement condition d is applied. An embedded “crack” with openingwidth e will appear, if d exceeds the elastic limit.

With a “crack”, having an unknown width e, the stress calculated fromthe continuum is from equation (4.27)

tσ = E(d− e) (5.2)

and that from the assumed exponential law within the crack from equa-tion (4.18).

te = ftexp

(−eeu

)(5.3)

The inequality between them, which is called the “residual”

R(e) = tσ − te = E(d− e)− ftexp

(−eeu

)(5.4)

which is a nonlinear relation with respect to e, should be eliminated at theend of the time step tn+1, using the Newton-Raphson iterative solutionmethod:

R + ∆R = 0 (5.5)

∆R =

[−E +

fteu

exp

(−eeu

)]∆e (5.6)

With the expressions above, the iterative increment of the unknown, ∆e,can be calculated from the following formula[

−E +fteu

exp

(−e(i)

n+1

eu

)]∆e = −R(i)

n+1 (5.7)

5.1. ALGORITHM FOR LOCAL ITERATIVE SOLUTION PROCEDURE 61

and used for the updating

e(i+1)n+1 = e

(i)n+1 + ∆e (5.8)

If the residual is found to be zero, i.e. |R| 6 TOL, the solution procedureis considered as converged. Then EXIT, and set en+1 ← e

(i+1)n+1 .

The algorithm above does not contain damage. To take also the damageof the material within the crack into account, one can assume an internalparameter γ ∈ [0,∞]. Then the stress in the crack can be expressed fromthe equation (4.23)

te =D0e

γ(5.9)

with D0 a chosen constant, whose value is not important, indicating areference state of the stiffness. The residual takes this form:

R(γ, e) = E(d− e)− D0e

γ(5.10)

the differentiation becomes

∆R =

(−E − D0

γ

)∆e+

(D0e

γ2

)∆γ (5.11)

Comparing the expressions of te from the exponetial law (4.18) and thedamage law (4.23), one obtains

γ =D0e

ftexp(−e/eu)(5.12)

∆γ =

[D0

ftexp(−e/eu)+

D0e

ftexp(−e/eu)eu

]∆e (5.13)

The equation (5.11) now has the detailed form:

∆R =

[(−E − D0

γ

)+D0e

γ2

[D0

ftexp(−e/eu)+

D0e

ftexp(−e/eu)eu

]]∆e

(5.14)After some simple manipulation, the same expression as (5.6) is ob-tained. Therefore, the identical formula for the iterative solution (5.7)can be used.

Therefore, with the already known state values en and γn, given a dis-placement condition dn+1, the algorithm for the iterative solution proce-dure of the unknown en+1 and γn+1 (considering damage), can be sum-marized as the following:


Without crack ,the element has elastic behaviour, the stress in the el-ement is calculated and checked with the Rankine criterion,i.e a crackis detected, if the Rankine criterion is violated, in the present work weassume the crack is fixed once it is detected.

1. If the element is already cracked in the previous step:

Crack closed check:

(a) If the element is cracked but open (not closed) in the previousstep, solve the linear equations ts = te, with γ(i)

n+1, as if unloading,to find etrial.

tσ = A(dn+1 −Hetr) =Detrial

γn= te (5.15)

If normal component of etrial is negative, then the open crackcloses again, make the normal component of etrial as zero andupdate the calculated values of e(i+1)

n+1 with out local newton iter-ations,ElseGo to next step to find the e(i+1)

n+1 based on iterative technique.(b) If the cracked is already closed in the previous step, find the trial

traction from the continuum i.e, equation (4.27), to find ttrial.If the normal component of ttrial is positive, then the closed crackopens again. Go to next step to find e(i+1)

n+1 based on local newtoniterations,ElseUpdate the previous values with out the iterative technique, i.e.,the crack is still closed.

Remark: Crack closed condition is checked only during the first globaliteration, if the closed crack is detected during this iteration, then itis closed for the entire step.

2. Loading/unloading check: assuming unloading (of the existing crack),compare etrial =‖ etrial ‖ with emax; if etrial 6 emax, it is an unloadingstep, therefore en+1 ← etrial, γn+1 ← γn, EXIT; else, it is a loadingstep, the nonlinear governing equation system (tσ,n+1 − te,n+1 = 0)have to be solved iteratively as the following.


3. Initialization: i = 0, e(i)n+1 = en and γ(i)

n+1 = γn.

(a) Residual:

t(i)σ,n+1 = A(dn+1 −He(i)

n+1), t(i)e,n+1 =

[De

γ

](i)

n+1

=

[D0enγ

D0esγ

](i)

n+1(5.16)

R(i)n+1(γ, e) = t

(i)σ,n+1 − t

(i)e,n+1 (5.17)

Remark: for the singular case of the initiation of the crack asγ = 0, use t(i)e,n+1 = ft simply.

(b) Check convergence: if |R(i)n+1| 6 TOL, set en+1 ← e

(i)n+1 and

γn+1 ← γ(i)n+1; EXIT.

(c) Solution: if not converged, solve for ∆e using (5.7).

(d) Update: e(i+1)n+1 = e

(i)n+1 + ∆e and calculate directly:

γ(i+1)n+1 =

D0e(i+1)n+1

ftexp(−e(i+1)n+1 /eu)

(5.18)

for the next iteration.Remark: the internal parameter γ can also be calculated incre-mentally, using (5.13), then update γ(i+1)

n+1 = γ(i)n+1 + ∆γ. The test

program shows the same final results for the problem, althoughwith more iterations and worse intermediate results.

4. Based on the updated values, the element stiffness matrix by usingimplicit-explicit integation scheme is calculated from the equations(4.49) and the internal force vector is calculated from the equation(4.51) as already described in previous chapter (4.4).

5.1.2 2D iterative solution procedure

In the present model, the “maximum equivalent opening in the loadinghistory” emax, plays the role of an internal parameter α:

α ≡ emax =

eold if enew 6 eoldenew if enew > eold

(5.19)


The damage parameter is determined by the internal parameter, see also(5.12):

γ =D0emax

ftexp(−emax/eu)=

D0α

ftexp(−α/eu)(5.20)

Based on the known state en and γn at time tn, given loading condition inthe form of nodal total displacement dn+1, we are to calculate the statevalues for tn+1.

Assume without crack, the stress in the element can be simply calculatedas

σn+1 = DBdn+1 (5.21)

which is checked with the Rankine criterion, σ1 6 ft. The direction ofthe maximum principal stress provides the normal to the crack, and thesubsequent matrices P and H, if the Rankine criterion is violated, i.e. acrack is detected. In the present work, the crack is assumed to be fixed.

For an inelastic load step, the conditions that must be satisfied at the end(tn+1) are:

Rn+1 = tσ,n+1 − te,n+1

= A(dn+1 −Hen+1)−

ftexp(− eeu

) (ene

)ftexp

(− eeu

) (ese

)n+1

= 0 (5.22)

And the internal parameter

D0en+1

tn+1= γn+1 (5.23)

γn+1 − γn > 0 (5.24)

The nonlinear equation system (5.22) is solved iteratively. The lineariza-tion of the iterative residual reads simply:

R(i)n+1 + ∆R = 0 (5.25)

in detail:

∆R =

−AH − [ ∂tn∂en

∂tn∂es

∂ts∂en

∂ts∂es

](i)

n+1

∆e = −R(i)n+1 (5.26)


is used for the iterative increment ∆e, where

∂tn∂en

= t

[− e2

n

eue2+e2s

e3

]∂tn∂es

= t

[−eseneue2

− enese3

]∂ts∂en

= t

[−eneseue2

− esene3

]∂ts∂es

= t

[− e2

s

eue2+e2n

e3

](5.27)

Implementation Procedure:

Similar to the 1D case, given the nodal displacement condition dn+1, thesolution procedure for the unknown separation vector en+1 and the dam-age parameter γn+1, can be summarized as the following:

1. If the element is already cracked in the previous step:

Crack closed check:

(a) If the element is cracked but open (not closed) in the previousstep, solve the linear equations ts = te, with γ(i)

n+1, as if unloading,to find etrial.

tσ = A(dn+1 −Hetrial) =Detrial

γn= te (5.28)

If normal component of etrial is negative, then the open crackcloses again, make the normal component of etrial as zero andfind the shear component corresponding to the intermediate stateand update the calculated values of e(i+1)

n+1 with out local newtoniterations,ElseGo to next step to find the e(i+1)

n+1 based on iterative technique.(b) If the element is already closed in the previous step, find the trial

traction from the continuum i.e, equation (4.27), to find ttrial.If the normal component of ttrial is positive, then the closed crackopens again. Go to next step to find e(i+1)

n+1 based on local newtoniterations,


ElseUpdate the previous values with out the iterative technique, i.e.,the crack is still closed.

Remark: Crack closed condition is checked only during the first globaliteration, if the closed crack is detected during this iteration, then itis closed for the entire step.

2. Loading/unloading check: assuming unloading (of the existing crack),compare etrial =‖ etrial ‖ with emax; if etrial 6 emax, it is an unloadingstep, therefore en+1 ← etrial, γn+1 ← γn, EXIT; else, it is a loadingstep, the nonlinear governing equation system (tσ,n+1 − te,n+1 = 0)have to be solved iteratively as the following.

3. Initialization: i = 0, e(i)n+1 = en and γ(i)

n+1 = γn.

(a) Residual:

t(i)σ,n+1 = A(dn+1 −He(i)

n+1), t(i)e,n+1 =

[De

γ

](i)

n+1

=

[D0enγ

D0esγ

](i)

n+1(5.29)

R(i)n+1(γ, e) = t

(i)σ,n+1 − t

(i)e,n+1 (5.30)

Remark: for the singular case of the initiation of the crack (mustbe Mode-I opening), as γ = 0, use t(i)e,n+1 = [ft, 0]T simply.

(b) Test: if ‖ R(i)n+1 ‖6 TOL, set en+1 ← e

(i)n+1 and γn+1 ← γ

(i)n+1; EXIT.

(c) Solution: solve for ∆e using (5.26).Remark 1: for the singular case, considering e → 0, en → e,es → 0, t→ ft, etc. the terms in (5.27) have the initial values:

∂tn∂en

= t/eu,∂tn∂es

= 0,∂ts∂en

= 0,∂ts∂es

=∞ (5.31)

For the programming, one can set an arbitrary value for ∂ts∂es

, sincein the special situation that in the following linear equation sys-tem for ∆e [

a bc d

] [∆en

0

]=

[−Rn

0

](5.32)

only the component ∂tn∂en

, related to a, plays a role.


Remark 2: although the residual above is expressed also with γ(damage), but the detailed components of the linearized terms,considering

∆γ =∂γ

∂e·∆e =

[D0ente + D0en

teuD0este + D0es

teu

]·∆e (5.33)

and after some transformation, are the same as those in (5.27).See also the comparison between (5.6) and (5.14) in the 1D sam-ple case in Section (5.1.1).

(d) Update: e(i+1)n+1 = e

(i)n+1 + ∆e and calculate directly:

γ(i+1)n+1 =

D0e(i+1)n+1

ftexp(−e(i+1)n+1 /eu)

(5.34)

for the next iteration.Remark 1: similar to the note an the end of Section (5.1.1), theinternal parameter γ could also be calculated incrementally, us-ing (5.33), then updated as γ(i+1)

n+1 = γ(i)n+1 + ∆γ.

Remark 2: the updated crack opening for the first iteration, ob-tained from the linearized equation, should be checked again bycomparing with the maximum equivalent separation emax.

4. Based on the updated values, the element stiffness matrix by usingimplicit-explicit integation scheme is calculated from the equations(4.49) and the internal force vector is calculated from the equation(4.51) as already described in previous chapter (4.4).

Important Remark: In case of quadrilateral element with 2*2 gauss inte-gration, the stress in the element at every gauss point is calculated andchecked with the Rankine criterion, i.e., a crack is detected, if the Rank-ine criterion is violated, in the present work we introduce the crack atevery gauss point in the element and we assume the crack is fixed once itis detected. The traction contunityv condition has to be satisfied at everygauss point in the element. The iterative solution procedure presentedabove 5.1.2 has to be called for all the gauss point where the stress ismaximum than the criterion.


5.2 Implementation of the user element in Abaqus

In this section we describe the main features related to the Abaqus userelement [1] implementation of the embedded crack model (SDA) throughthe user subroutine UEL. Abaqus provides users with an extensive arrayof user subroutines that allow them to adapt Abaqus to their particularanalysis requirements. A nonlinear finite element is implemented in usersubroutine UEL. The interface makes it possible to define any (propri-etary) element of arbitrary complexity. If coded properly, user elementscan be utilized with most analysis procedures in Abaqus/Standard. Mul-tiple user elements can be implemented in a single UEL routine and canbe utilized together.

Implementing the SDA in the commercial code Abaqus does imposes cer-tain restrictions, but it also provides access to many of the available fea-tures of such a code. We will focus on the subroutines directly related toAbaqus. We will give a general overview of the pre- and post-processingsteps used in our implementation.

5.2.1 Pre-processing and Solver

With the reference to [1], generally the user subroutines in Abaqus arewritten in Fortran programming language. User subroutine UEL will becalled for each element that is of a general user-defined element type,each time, element calculations are required and must perform all ofthe calculations for the element,appropriate to the current activity in theanalysis.

In this model, the global calculation are performend by Abaqus and thelocal calculations i.e, stiffness matrix, internal force vector and user de-fined variables(for example: seperation vecor e) of every element has tobe defined and calculated in the UEL.

According to the present embedded crack model (calculation of stiffnessmatrix, internal force vector) has to be written as UEL subroutine. Thedetailed element level iterative procedure is already presented in section(5.1) where we need the data for e.g, maximum equivalent seperatione, direction of the crack, gamma γ, e.t.c., of the every element from the

5.2. IMPLEMENTATION OF THE USER ELEMENT IN ABAQUS 69

previous load step for the present load step calculations (because thisvariables are defined by the user not the Abaqus standards), so for thispurpose we have a possibility to use Abaqus defined array called solution-dependent state variables (SVARS) which is explained in detailed below.

A user element is defined with the ∗USER ELEMENT option. This optionmust appear in the input file before the user element is invoked with the∗ELEMENT option [1].

The syntax for interfacing to UEL is as follows:

∗USER ELEMENT, TYPE=Un, NODES=, COORDINATES=,PROPERTIES=, I PROPERTIES=, VARIABLES=, UNSYMMData line(s)∗ELEMENT,TYPE=Un, ELSET=UELData line(s)∗UEL PROPERTY, ELSET=UELData line(s)∗USER SUBROUTINES, (INPUT=file name)

Parameter definition:

TYPE: (User-defined) element type of the form Un, where n is a number.NODES: Number of nodes on the element.COORDINATES: Maximum number of coordinates at any node.PROPERTIES: Number of floating point properties.I PROPERTIES: Number of integer properties.VARIABLES: Number of SDVs.UNSYMM: Flag to indicate that the Jacobian is unsymmetric.

A data line of the form dof1, dof2 where dof1 is the first degree of freedomactive at the node and dof2 is the second degree of freedom active at thenode, etc., follows the ∗USER ELEMENT option. If all nodes of the userelement have the same active degrees of freedom, no further data areneeded.

The interface to user subroutine UEL is:


SUBROUTINE UEL(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS,PROPS,NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,DTIME,KSTEP,KINC,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,PREDEF,NPREDF,LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,JPROPS,NJPROP,PERIOD)

INCLUDE ’ABA-PARAM.INC’

DIMENSION RHS(MLVARX,*), AMATRX(NDOFEL,NDOFEL), PROPS(*),SVARS(NSVARS), ENERGY(8), COORDS(MCRD,NNODE), U(NDOFEL),DU(MLVARX,*), V(NDOFEL), A(NDOFEL), TIME(2), PARAMS(*),JDLTYP(MDLOAD,*), ADLMAG(MDLOAD,*), DDLMAG(MDLOAD,*),PREDEF(2,NPREDF,NNODE), LFLAGS(*), JPROPS(*)

user coding to define RHS(nodal forces), AMATRX(stiffness matrix) and SVARS

according to this model, standard finite element procedure for calculating element stiffnessmatrix, nodal forces and embedded crack model presented in the above section (5.1.2)

RETURN

END (end of user subroutine)

UEL Variables: The following quantities are available in UEL:

• Coordinates; displacements; incremental displacements; and, for dy-namics, velocities and accelerations

• SDVs at the start of the increment

• Total and incremental values of time, temperature, and user-definedfield variables

• User element properties

• Load types as well as total and incremental load magnitudes

• Element type and user-defined element number

• Procedure type flag and, for dynamics, integration operator values


• Current step and increment numbers

The following quantities must be defined:

• Right-hand-side vector (residual nodal fluxes or forces)(RHS)

• Jacobian (stiffness) matrix(AMATRX)

• Solution-dependent state variables(SVARS)

Usually by default user subroutine is called for three times. In the firstiteration of an increment this user subroutines is called twice.During thefirst call the initial stiffness matrix is being formed using the configura-tion of the model at the start of the increment.During the second call anew stiffness, based on the updated configuration of the model, is cre-ated. In subsequent iterations the subroutines is called only once duringwhich the corrections to the model’s configuration are calculated usingthe stiffness from the end of the previous iteration.

All the available quantities and quantities that are to be defined has thedirect meaning according to the standard finite element technique exceptthe quantity called solution-dependent state variables in short form SDVs.

Solution-dependent state variables (SVARS): An array containing thevalues of the solution-dependent state variables associated with this ele-ment. The number of such variables has to be defined by the user. Userhas to define the meaning of these variables. This values can be used toevolve with the solution of an analysis.Abaqus just stores the variables forthe user subroutine. For general nonlinear steps this array is passed intoUEL containing the values of these variables at the start of the currentincrement. They should be updated to be the values at the end of theincrement, unless the procedure during which UEL is being called doesnot require such an update.

The total number of SDVs per element is set with the VARIABLES param-eter in the input file.

• If the element is integrated numerically, VARIABLES should be setequal to the number of integration points times the number of SDVsper point.


• Solution-dependent state variables can be output with the identifiersSDV1, SDV2, etc. SDVs for any element can be printed only to thedata (.dat), results (.fil), or output database (.odb) files and plottedas X–Y plots in Abaqus/Viewer.

It is clear that, user is responsible for every input variable, all the calcu-lations, output variables and variables that has to be stored for next stepcalculations related the element defined by user.

This model is imlemented and generalized for two cases:

1. Constant strain triangle element.

2. Quadrilateral element (2*2 Gauss Integration).

In the present work, variable updated in the subroutine, i.e., numberof SVARS are 1*10 for three noded element and 4*10 for four nodedelement (variables for every integration point). Solution-dependent statevariables used in the present model are as follows:

SVARS(1):Crack situation:defines whether the element is cracked (1) ornot cracked (0).SVARS(2):Direction: cos of the angle alpha from global x-axis to cracknormal, [−π/2, π/2].SVARS(3):Direction: sin of the angle alpha from global x-axis to cracknormal, [−π/2, π/2].SVARS(4):Closure:flag of crack closure (of existing crack).SVARS(5):Crack opening vector: normal direction (en).SVARS(6):Crack opening vector: tangential direction (es).SVARS(7):Maximun equivalent crack opening in history, at (n) step (en).SVARS(8):Gamma:damage state (γ).SVARS(9):Maximun equivalent crack opening in history, at (n-1) step(en−1).SVARS(10):Number of times uel is called. It is used to know the localiteration number.

The above defined variables are defined for one integration point, so therespective variables has to be stored for the respective integration points.In this work, we need the local iteration number while applying crackclosure algorithm but the iteration number is not available in UEL. The


only subroutine in Abaqus in which the iteration number is available isUINTER. But this routine is called after UEL. The dummy contact has tobe defined using UINTER in the model and to save KIT (iteration number)in a common block. In this way we make KIT (iteration number) availablefor UEL.

There are various ways to define flags and counters, with which user cancalculate the iteration number. User can define his own techniques to getthe iteration number using flags and counters saved in a common block.The difference between number of call of UEL and number of iterationhas to be taken care.

5.2.2 Post processing

For post processing currently, Abaqus does not have capabilities for user-element plotting because the code does not post-process the informationgenerated by user elements.To plot the deformed shape after an SDAanalysis, we have to use standard elements in Abaqus with very small(negligible) stiffness (called overlay elements). The overlay elementsshould have same topology as that of user elements, should share thesame nodes for retaining the same connectivity. As the overlay elementsshare the same DOFs, the deformed shape can be visualized. Only thenodal information can be overlayed by using this overlay element, butthe element information for example stresses and strains at integrationpoints cannot be visualized.

Abaqus internal procedures for post processing through the user elementare not applicable to any model, since the information generated by userelements can not be processed by Abaqus. But, the post-processing of theelement output of SDA model by using another user subroutine UVARMin Abaqus is possible.

User subroutine to generate element output (UVARM): In order togenerate the element output in Abaqus, multiple user subroutines areneeded in the analysis. The user subroutine UVARM has to be combinedwith the UEL subroutine in a single file to generate the element output.In the UEL subroutine, compute the values of the desired field outputquantities and store in the common block array. Create a UVARM sub-


routine in which the values of common block are copied to the ’UVAR’array for the standard Abaqus elements.

User subroutine UVARM:

• will be called at all material calculation points of elements for whichthe material definition includes the specification of user-defined out-put variables.

• may be called multiple times for each material point in an increment.

• will be called for each increment in a step.

• allows you to define output quantities that are functions of any ofthe available integration point quantities.

We must specify the number of user-defined output variables (NUVARM),for a given material to allocate space at each material calculation pointfor each variable. The user-defined output variables are available for bothprinted and results file output and are written to the output database andrestart files for contouring, printing and X–Y plotting in Abaqus/CAE. Anynumber of user-defined output variables can be used.

Input File Usage:USER OUTPUT VARIABLESNUVARM

The user subroutine interface is as follows:

5.3. CRACK ADAPTATION 75

SUBROUTINE UVARM(UVAR,DIRECT,T,TIME,DTIME,CMNAME,ORNAME,NUVARM,NOEL,NPT,LAYER,KSPT,KSTEP,KINC,NDI,NSHR,COORD,JMAC,JMATYP,MATLAYO,LACCFLA)

INCLUDE ’ABA-PARAM.INC’

CHARACTER*80 CMNAME,ORNAMECHARACTER*3 FLGRAY(15)DIMENSION UVAR(NUVARM),DIRECT(3,3),T(3,3),TIME(2)DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),COORD(*)

user coding to define UVAR

RETURN

END (end of user subroutine)

Parameter definition: The only variable we need for post processing isUVAR array.

UVAR(NUVARM): An array containing the user-defined output variables.These are passed in as the values at the beginning of the increment andmust be returned as the values at the end of the increment.

Limitations:

• It requires to manually update the ’nelem’ variable in both UEL andUVARM subroutines.

• The element numbers for the overlaying Abaqus elements must havea constant offset relative to their user defined counterparts.

5.3 Crack adaptation

With the foregoing equations, the model is complete and can be insertedin a finite element code to make computations. However, in most ofthe two- and three-dimensional simulations performed with the model,


the crack initially propagates correctly but after a certain crack growth,the cracking spreads over various elements simultaneously avoiding thecorrect localization as shown in Figure (5.1). Such locking seems to bedue to a bad prediction of the cracking direction in the element ahead ofthe pre-existing crack [15].

In order to solve crack locking problem, for example [10] used a smearedrotating crack model combined with a embedded crack. In that modelthe initial stage of cracking is modelled by a smeared crack model, andwhen the strain in the element reaches a critical value a displacementdiscontinuity is introduced. Other authors [23] used a rotating crackmodel to avoid locking.

To overcome this problem without introducing global algorithms (cracktracking and exclusion zones), we just introduce a certain amount ofcrack adaptability within each element [15].

Figure 5.1: Sketch of crack locking: the prediction of cracking direction in the shaded element iswrong [15].

The rationale behind the method is that the estimation of the principaldirections in a constant strain element is specially bad at crack initia-tion due to the high stress gradients in the crack tip zone where the newcracked element is usually located; after the crack grows further, the es-timation of the principal stress directions usually improves substantially.

5.3. CRACK ADAPTATION 77

Therefore, we allow the crack to adapt itself to the later variations inprincipal stress direction while its opening is small.

This crack adaptation is implemented very easily by stating that while theequivalent crack opening e is small, the crack direction is recomputed ateach step as if the crack were freshly created. After e reaches a thresh-old value eth no further adaptation is allowed and the crack directionbecomes fixed. Threshold values must be related to the softening proper-ties of the material, and values of the order of 0.1–0.2Gf/ft are usuallysatisfactory. Here Gf is the fracture energy and ft is the tensile strength.This simple expedient has proved to be extremely effective to avoid cracklocking.In our approach the limited crack adaptation is introduced to cir-cumvent the numerical deficiency in predicting accurately the principalstress directions, and has not to be taken as a material property.

Chapter 6

Validation and Application

The numerical examples in this chapter are intended to illustrate the per-formance of the proposed model in view of the simulation of discretecrack propagation by strong discontinuity approach. The analysis of thesingle element test is done for the following cases: loading case (alsovalidating Matlab and Abaqus results in loading case), loading/unloadingtest, comparison between CST element and quadrilateral element in bothcases and crack closure test.

After the analysis and validation by single element tests, different exam-ples of structural benchmarks, namely the tests of single edge notchedbeam and L-shape panel made of quasi-brittle materials, are used for thevalidation and application of the present model.

Initially the results of numerical examples, i.e., the crack pattern andload-displacement curve without the crack adaptation is applied for sin-gle edge notched beam which is discretized by triangular elements wherecrack locking phenomenon is clearly seen; when the crack adaptation isintroduced the results drastically change, with better traced crack pathand load-displacement curve. Finally this model is applied to the L-shapepanel which is discretized by triangular elements and quadrilateral ele-ments and the results are compared with expermental data reported in[4].

The computations are under control of the displacement (displacementcontrol method). Implementation is done in Abaqus and Matlab soft-wares. The crack pattern and load-displacements curve for both elementsare simulated and compared.

In the following examples, crack is assumed to be fixed with constant

79

80 CHAPTER 6. VALIDATION AND APPLICATION

opening. Only one crack is allowed during calculation for CST elementwhile for quadrilateral element one crack at every gauss point is allowed;no crack branching or intersecting is permitted.

6.1. SINGLE ELEMENT TEST 81

6.1 Single element test

The element behavior is illustrated by means of a simple but meaning-ful single element test. The proposed model is illustrated by means of asimple example. An element with length l = 5mm, height h = 1mm andthinckness t = 1mm with a prescribed crack location at centroid of trian-gular element and at gauss point for quadrilateral element is analysed,which is discretized with two similar CST elements and one quadrilateralelement as shown in the Figures (6.1 a-b ) which also shows the bound-ary conditions. The position of the crack for both considered examplesare shown in Figures (6.1 c-d). The material parameter are adoptedas: Young’s modulus E = 10000 N/mm2, Poisson’s ratio ν = 0.2, tensilestrength ft = 1.4 N/mm2 and fracture energy Gf = 0.014 N/mm.

c

c

a)

1 2

4 3

b)

c) d)

Figure 6.1: a)CST element, b)Quadrilateral element, c)-d) crack direction and position. (c:centroidof the element, 1-2-3-4:gauss points)

The element is subjected to tension by imposing the right top and rightbottom horizontal displacements. In the displacement-controlled com-putation the resulting reaction forces are compared. For the two dif-ferent cases individually, i.e., (CST element and quadrilateral element),initially the results from Matlab and Abaqus are compared for loadingcase and then the element behaviour of loading and unloading case aretested seperately. At the last stage of this section both the elements re-sults for different loading cases are compared and at the end of the sec-tion the phenomenon of closed crack is tested. As shown in the Fig-ures (6.2/6.3/6.5/6.6) the element behaves elastically until it reachesthe maximum stress. As already presented in chapter (5), the displace-ment discontinuity is embedded, if the element violates the crack initia-


tion condition. Once the strong distontinuity is introduced, softening be-haviour in the element begins which is seen in Figures (6.2/6.3/6.5/6.6)that the load displacement curve starts declining.

In Figures (6.2/6.3/6.5/6.6), the load–displacement curves obtained withthe embedded model for both the cases by using Matlab and Abaqus soft-wares are presented. When compared the curves which are obtainedfrom two softwares i.e, from Figures (6.2) and (6.3) which refer to CSTelement case and Figures (6.5) and (6.6) for four noded element caseindividually, which show complete same in both the softwares. Here wevalidated the Matlab implementation with the commercial FE softwareAbaqus. The perfect match between the load–displacement curves ob-tained with the two softwares are presented in Figures (6.4) and (6.7).

CST element:

Quadrilateral element:


0 0.005 0.01 0.0150

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

Matlab

CST element

Figure 6.2: Loading case: computed load-displacement curve, CST element, Matlab result .

0 0.005 0.01 0.0150

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

Abaqus

CST element

Figure 6.3: Loading case: computed load-displacement curve, CST element, Abaqus result.

0 0.005 0.01 0.0150

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

MatlabAbaqus

CST element

Figure 6.4: Loading case:computed load-displacement curve, CST element, comparison betweenMatlab and Abaqus results.

Loading and Unloading test:

In the previous chapter under the section (5.1), the loading and un-


0 0.005 0.01 0.0150

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

Matlab

Quadrilateral element

Figure 6.5: Loading case: computed load-displacement curve, quadrilateral element, Matlab result.

0 0.005 0.01 0.0150

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

Abaqus


Figure 6.6: Loading case: computed load-displacement curve, quadrilateral element, Abaqus result.

0 0.005 0.01 0.0150

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

MatlabAbaqus


Figure 6.7: Loading case: computed load-displacement curve, quadrilateral element, comparisonbetween Matlab and Abaqus results.

loading situation is clearly explained. Figures (6.8) and (6.9) show theloading and unloading test for CST element and quadrilateral element,


from the figures it can be observed that, the unloading step is at the pre-scribed horizontal displacement 0.0045 and 0.007. The element is underunloading step until the right top and right bottom horizontal reactionforces reaches to minimum possible positive value or zero, after then theelement starts reloading. According to the present model, during un-loading and reloading process, (i.e., form the Figures (6.8) and (6.9)at the displacement 0.0045 and 0.007), there is no further increase inthe damamge parameter γ and seperation vector untill the system startsloading, it means no more softening behaviour in the element (This canbe observed in the figures that, only after the displacement greater thanthe 0.0045 and 0.007 the system starts softening again).

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

loading & unloading

CST element

Figure 6.8: Loading/Unloading case: computed load-displacement curve, CST element.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

loading & unloading


Figure 6.9: Loading/Unloading case: computed load-displacement curve, quadrilateral element.


Comparison between CST element and Quadrilateral element:

In this model, according to assumption that in case of CST element onlyone crack passes through the element which give more accurate results,so even in case of quadrilateral element initially we started with an as-sumption that only one crack passes through the element where there isa maximum stress. The crack is initiated at one of the gauss point whichis weakest among the four. With this assumption, when compared thequadrilateral element results with CST element results, numercially it isno way closer and not even acceptable result. To know the detailed be-haviour of the model for quadilateral element case, next we allowed twocrack to pass through one element and then three cracks where gausspoints which are weaker among the four. But according to this three as-sumption when compared to three noded element, the results are veryfar and which are not even comparable. Finally we end up with the as-sumption to introduce the crack at all the gausspoint where the he stressin the element at every gauss point is calculated and checked with theRankine criterion,i.e a crack is detected, if the Rankine criterion is vio-lated. With this assumption when compared with CST element results,the perfect and exact match between the load–displacement curves isobtained which is shown in the figure 6.10 for loading and even for un-loading test shown in figure 6.11.


0 0.005 0.01 0.0150

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

CST elementQuadrilateral element

Figure 6.10: Loading case: computed load-displacement curve, comparison between CST elementand quadrilateral element.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.05

0.1

0.15

0.2

Displacement [mm]

For

ce [k

N]

CST elementQuadrilateral element

loading & unloading

Figure 6.11: Loading/Unloading case: computed load-displacement curve, comparison between CSTelement and quadrilateral element.

Crack closure:

The physical behaviour of the any body or any structure with micro cracksis, as the crack in the body starts propagating many small or tiny cracksbeside the main crack pattern (lets say) gets open and after some timethey get closed. With some restrictions the same physical phenomenonis introduced in the present model which is physically reasonable for aboundary value problem. The crack closure concept and implementationtechnique is presented in the corresponding section (4.3) and (5.1).

Figure (6.12) describes the phenomenon of closed crack from the com-puted load-displacement curve, in which the element is in elastic rangebetween the points 1−2 (elastic stiffness), 2−3 is the softening behaviour(softened stiffness), unloading of crack is between the points 3−1 and thecurve between 1 − 4 with the (arrow titled closed crack) describes thecrack closure. According to this model, if the normal componenent of


crack opening vector is negative, we say that the crack is closed, numeri-cally we make normal component as zero and instead (softened stiffness)we use (elastic stiffness) which is shown clearly in the figure 6.12. If weobserve the figure, the curve between the points 1− 2 and 1− 4 have thesame slope which is elastic.

−1 0 1 2 3 4 5

x 10−3

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Displacement [mm]

For

ce [k

N]

1

2

3

4

crack closed crack closure

Figure 6.12: Closed crack: computed load-displacement curve.

6.2. THREE-POINT BENDING TEST OF A NOTCHED BEAM 89

6.2 Three-point bending test of a notched beam

The three-point-bending beam test was analyzed using unstructured mesh.The important aspects of interest is to check the ability of the proposedmodel to trace a straight vertical crack without crack tracking. The beamdimensions were as follows: length = 2000 mm, thickness = 100 mm,and depth = 500 mm. A single notch 200 mm in depth and 5 mm in widthis introduced as shown in Figure (6.13). The material parameters weretaken to be the following: tensile strength ft = 2.5 MPa, Young modulusE = 20 GPa, Poisson’s ratio ν = 0.15, fracture energy Gf = 0.1 N/mm.

Figure 6.13: Three-point-bending test.

The computations were run under control of the displacement at the up-per midpoint. The coarse unstructured which consist of 1499 linear tri-angular elements (CST-elements) were used as shown in Figure (6.14).

Figure 6.14: Three-point-bending test: FE mesh.

A closer examination of the ability to trace the crack path is given inFigure (6.15), which compares the crack path computed with and with-out crack adaptation. Figure (6.15(a)) shows the crack path when crackadaptation is prevented. The configuration corresponds to the crack lock-ing in Figure (6.16) without crack adaptation. Initially, the crack prop-agates correctly, but along the last half part of the crack, approximately,


cracked elements can be seen that lie outside the main crack part and atthe locked crack tip cracking seems to have “delocalized”. By contrast,when crack adaptation is enabled (6.15(b)) the crack path is nearly con-tinuous and generally straight. The threshold value eth used here is 9.0 e4

mm.

Figure (6.16) without crack adaptation shows the load-displacement curvescomputed without crack adaptation together with a reference curve com-puted using the smeared-tip superposition method [15]. Obviously, theresponse of the finite element computations is too stiff and the computa-tion stops prematurely because of crack locking and lack of convergence.When crack adaption is introduced (Figure (6.17) with crack adaptation)the results drastically change: crack locking disappears and the responseof the notch beam is better captured in the tail of the curve. The notchedbeam geometric properties, material properties and the reference curveare taken from [15]. From Figure 6.17, even though the response of thenotch beam is better traced in the tail of the curve, we can see some partsof the curve exhibit snapback and jumps, because of crack adaptation(linear behaviour), i.e., the crack were freshly created until equivalentcrack opening reaches a threshold value.

Parametric study on the threshold value eth:

According to crack adaptation concept, we allow the crack to adapt itselfand is implemented by stating that while the equivalent crack openinge is small, the crack direction is recomputed at each step as if the crackwere freshly created. It means we do not use any previous load step data.After e reaches a threshold value eth no further adaptation is allowed andthe crack direction becomes fixed, then we start using the previous loadstep data.


a) b)

Figure 6.15: Three-point-bending test: Computed crack path: a) without crack adaptation and b)with crack adaptation.

0 0.2 0.4 0.6 0.7 0.80

2

4

6

8

10

Displacement [mm]

Forc

e [kN

]

unstructured mesh

smeared−tip

without crack adaptation

lock

0 0.2 0.4 0.6 0.7 0.80

2

4

6

8

10

Displacement [mm]

Forc

e [kN

]

unstructured mesh

smeared−tip

with crack adaptation

Figure 6.16: Three-point-bending test: computed load-displacement curve, without crack adapta-tion. Result compared with smeared-tip [15]

0 0.2 0.4 0.6 0.7 0.80

2

4

6

8

10

Displacement [mm]

Fo

rce

[kN

]

unstructured mesh

smeared−tip

without crack adaptation

lock

0 0.2 0.4 0.6 0.7 0.80

2

4

6

8

10

Displacement [mm]

Fo

rce

[kN

]

unstructured mesh

smeared−tip

with crack adaptation

Figure 6.17: Three-point-bending test: computed load-displacement curve, with crack adaptation.Result compared with smeared-tip [15]

Generally threshold value is taken of the order of 0.1–0.2Gf/ft (Gf is thefracture energy and ft is the tensile strength). If the high threshold value


0 0.2 0.4 0.6 0.7 0.80

2

4

6

8

10

12

Displacement [mm]F

orc

e [kN

]

unstructured mesh

smeared−tip

a) b)

Figure 6.18: Parameter study: high threshold value, three-point-bending test: a) crack path, b)load-displacement curve.

a) b)0 0.2 0.4 0.6 0.8

0

2

4

6

8

10

Displacement [mm]

Forc

e [kN

]

unstructured mesh

smeared−tip

still lock

Figure 6.19: Parameter study: small threshold value, three-point-bending test: a) crack path, b)load-displacement curve.

for example 2.0 e3 mm is used, large number of load steps are requiredto reach the given threshold value, which results in reasonable crack pathFigure (6.18(a)), but the load-displacement curve in Figure (6.18(b))exhibit snapback and big jumps, this corresponds to those almost verti-cal segments of the curve which is not realistic and acceptable. If thesmall threshold value for example 2.0 e4 mm is used, it is almost similaras standard algorithm, i.e., without crack adaptation, a little change incrack path see Figures (6.19(a)) and (6.15(a) but the computation stopsprematurely because of crack locking and lack of convergence see Figures(6.19(b)) and (6.16). Here we conclude that, it is very important to use


the appropriate threshold value.


6.3 L-shape panel test

The L-shaped panel has become a popular benchmark test for the vali-dation of computational models for the numerical simulation of crackingof plain concrete. The experimental data is from the refernece [4]. Thetest setup with the geometric properties and the boundary conditions isshown in figure (6.20(a)).

The long and the short edges of the L-shaped panel are given as 500 and250 mm,respectively; its thickness is 100 mm. The lower horizontal edgeof the vertical leg is fixed. A vertical load F,v acting uniformly across thethickness to the direction of gravity, is applied at the lower horizontalsurface of the horizontal leg at a distance of 30 mm from the verticalend face. Shortly before reaching the maximum load the experiment isswitched from load-control to displacement control. The material param-eters are: Young’s modulus, Poisson’s ratio, the uniaxial tensile strengthand the specific fracture energy are given as E = 25850N/mm2, ν = 0.18,ft = 2.70 N/mm2 and Gf = 0.09 N/mm.

500

500

250250

250

250

30

F,v

all lengths in [mm]

a) b)

Figure 6.20: Illustration of L-shaped panel test: a) Test setup, b) scatter of observed crack paths.

The grey shaded area in Figure (6.20(b)) shows the crack path of scatterof the experimental results from three tests on identical specimens andin the corresponding Figure (6.22) or (6.24) depicts the relationship be-tween the applied load F,v and the vertical displacement v at the pointof load application which is marked as red curves. The mesh data of the

6.3. L-SHAPE PANEL TEST 95

employed FE-meshes are given in Figure (6.21) consist of 1450 linear tri-angular elements (CST-elements), whereas the FE-mesh (6.23) contains620 bilinear quadrilateral elements (CPS4-elements). This model, whichis based on the strong discontinuity approach, is implemented for CST-elements and quadrilateral elements, numerical results are presented forthe meshes with CST-elements and quadrilateral elements.

Apart from the FE-meshes, Figure (6.21) and (6.23) also shows the com-puted crack paths by marking those elements with magnitude of crackopening, which are crossed by the crack, by color shading. The com-puted crack paths can be compared with the respective crack paths ob-served in the tests, which is shown in Figure (6.20(b)). The area betweenred color curves in figures 6.22 or 6.24 shows the scatter of the experi-mental results, whereas the load-displacement curves in the same figures(blue in color) refer to the numerically predicted behaviour employingthe present crack model for CST element and quadrilateral element.

According to the present model, From the section (6.1), we observed thatthe, when we compare the results between CST element and quadrilat-eral element, the perfect and exact match between the load–displacementcurves is obtained which is shown in the figure 6.10 for loading and forunloading test shown in figure 6.11. But when we apply for real struc-tures, crack pattern and load-displacement curve are quite different asshown in Figures (6.21, 6.22) and (6.23, 6.24).

From Figure (6.21), in case of CST element case, we can see that crackpath as well as load-displacement curve shown in Figure (6.22) is veryclose to realistic compared to expermental results. If we observe fromFigures (6.23, 6.24), in case of quardilateral element case, we can seethat crack path is not that realistic compared to CST element case oreven expermental results. The crack path seems to be diverting at thelast stages. We have some kind of stress locking problem. It can also beobserved that some unnecessary elements beside the main crack path getcracked, the reason can be that we started with an assumption of constantcrack opening in the element, according to the author’s knowledge.

Remark:Regarding quadrilaterals with constant approximation of the displace-ment jump, locking can be explained as the following: imagine two


Magnitude of crack opening

0

0.1

0.2

0.3

Figure 6.21: L-shaped panel test: FE-mesh (CST element) and computed crack pattern (plot ofmagnitude of crack opening).

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Displacement [mm]

For

ce [k

N]

unstructured meshexpermental spectrumexpermental spectrum

CST element

Figure 6.22: L-shaped panel test: computed load-displacement curve (CST element).

blocks separated by a fully formed crack and moving like rigid bodies,(e.g., a beam under bending after full failure of the middle cross section);if the rigid bodies rotate in opposite directions, the elements crossed bythe crack are subjected to a bending-type deformation but this cannot berelaxed by the constant displacement jump. Compressive strains appearin the top part of the element and tensile strains in the bottom part. Sincethe element itself is elastic, the corresponding stresses are generated and

6.3. L-SHAPE PANEL TEST 97

Magnitude of crack opening

0

0.1

0.2

0.3

Figure 6.23: L-shaped panel test: FE-mesh (quadrilateral element) and computed crack pattern(plot of magnitude of crack opening).

0 0.2 0.4 0.6 0.7 0.80

2

4

6

8

10

Displacement [mm]

For

ce [k

N]

unstructured meshexpermental spectrumexpermental spectrum


Figure 6.24: L-shaped panel test: computed load-displacement curve (quadrilateral element).

they lead to nodal forces that are equivalent to a bending moment. Thismoment grows proportionally to the rotation, which is typically observedat very late stages of structural response.

Chapter 7

Conclusion and outlook

7.1 Conclusion

In this thesis, a numerical model based on the finite element method forthe simulation of fracture processes in quasi-brittle structures was pre-sented. Particularly embedded crack model was formulated within theframework of the strong discontinuity approach for the nonlinear (soften-ing) behavior of the quasi-brittle material was presented.

The softening behavior of quasi-brittle material was modeled with thestatically and kinematically optimal nonsymmetric (SKON) formulationwhich improved numerical performance by optimal static and kinematicequations. A damage-based traction-separation law was included in themodel, which links the traction transmitted by the discontinuity to thedisplacement jump. The model was designed under the assumption ofsmall displacements and small strains.

In order to model physical phenomenon, numerical treatment of closedcrack was introduced in the present model which was physically reason-able for a boundary value problem. As this is a non-linear solid mechanicsproblem, to improve the robustness (computability) of the computationof softening structures, an implicit/explict integration scheme was used.In order to solve crack locking problem in the standard algorithm, crackadaptation was adopted which means allowing the embedded crack inthe finite element to adapt itself to the stress field while the crack open-ing does not exceed a small threshold value.

In the present model, Rankine type criterion for tension was applied,where the model was implemented for a constant crack opening with

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100 CHAPTER 7. CONCLUSION AND OUTLOOK

one crack was allowed to pass through the constant strain triangle ele-ment and one cracks (at every gauss point) was allowed in the quadri-lateral element. The detailed numerical implementation strategy of fi-nite elements with a localization line (embedded crack) was presented inthis thesis and a brief overview of implementing the user subroutine inAbaqus was presented.

The proposed crack model was first tested with one-element examplesfor different test cases (loading, loading/unloading, closed crack tests) forconstant strain triangle element and quadrilateral element which are im-plemented in standard codes (Matlab and Abaqus softwares). The com-parison of the results indicates that both elements behave consistently.

Next the standard model was applied for a structure without and withcrack adaptation. It has been shown that the local crack adaptability ispreferable to prevent crack locking. The model was then applied into realstructures; the results are analyzed and compared with the experimentaldata. The simulation shows good agreement to the reality.

In this way, the embedded crack approach turns out to be yet an effec-tive and robust alternative to other more sophisticated methods for thesimulation of quasi-brittle damage and fracture.

7.2 Outlook

In order to improve the current model, considering the shortcomings inthe work, the following aspects are proposed to be meaningful researchin the future:

• Improvement of the statically and kinematically optimal nonsymmet-ric formulation to linear crack opening:The present formulation is flexible only for constant crack openingwhich is acceptable for CST elements but it is not quite acceptablefor any other elements. If we observe from chapter (6) under thesection (6.3), in case of quardilateral element case, we can see thatcrack path is not realistic compared to CST element case or even ex-permental results. The crack path seems to be diverting at the laststages. we have some kind of stress locking problem. It is observed

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that some unnecessary elements beside the main crack pattern getscracked, the reason is that because we started with an assumptionof constant crack opening in the element. The present model has tobe developed in more physical form for the evolution of the crackopening.

• Improvement on the implementation:The implementation can be improved by introducing the crack track-ing algorithm for continuous crack pattern and it can be extendedfrom 2D case to 3D case. Much more flexible numerical treatment ofclosed crack concept can be adopted.

• Optimization of the programs:It is meaningful to optimize the code both the Matlab and Abaqususer subroutine , making the code work more efficiently with lesstime-consuming.

• Further investigation on the implementation of quadrilateral ele-ment:During implementation, it has been clearly mentioned in chapters(5) and (6) that, we allowed four cracks (at every gauss point) topass through the quadrilateral element (it means the element is soft-ening at every gauss points), which can be tested (improved) by al-lowing only one crack per element but softening of stiffness at thisgauss point should be multiplied by the number of gauss point whichviolated the crack initiation condition based on the current load step.The improvement is based on the success of the above menctionedpoint.

102 CHAPTER 7. CONCLUSION AND OUTLOOK

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