Comparison between thick level set (TLS) and cohesive zone models
 Andrés Parrilla Gómez^{1},
 Nicolas Moës^{1}Email author and
 Claude Stolz^{1, 2}
DOI: 10.1186/s4032301500419
© Parrilla Gómez et al. 2015
Received: 30 April 2015
Accepted: 23 July 2015
Published: 31 July 2015
Abstract
Background
Two main families of methods exist to model failure of quasibrittle structures. The first one consists on crack based models, like cohesive zone models. The second one is a continuum damage approach that leads to a local loss of stiffness. Local damage models need some regularization in order to avoid spurious localization. A recent one, the thick level set damage model, bridges both families by using levelsets. Cohesive and TLS models are presented. The cohesive one represents quasibrittle behaviors with good accuracy but requires extra equations to determine the crack path. The TLS has proved its capability to model complex crack paths while easily representing cracks (i.e. displacement jumps); contrary to most damage models.
Methods
A onedimensional analytical relationship is exhibited between TLS and cohesive models. The local damage behavior needed to obtain the same global behavior of a bar than with cohesive model is derived. It depends on the choice of some TLS parameters, notably the characteristic length \(\ell _c\). This local behavior is applied to bidimensional simulations of three point bending as well as mixedmode single edge notched specimens are performed. Results are compared to cohesive simulations, regarding both crack paths and forceCMOD curves.
Results
ForceCMOD curves obtained are very similar with both models. Theoretical analysis in 1D and numerical results in 2D indicates that, as \(\ell _c\) goes to zero, TLS results tend to CZM ones.
Conclusions
The TLS model yields very similar results to the cohesive one, without the need for extra equations to determine the crack path.
Keywords
Damage mechanics Thick level set Cohesive zone XFEM Levelset Three point bending Single edge notchBackground
Modeling the failure of quasibrittle structures is an important aim of numerical simulations. Two main kind of models have been developed [1]: crack based ones, as the cohesive zone model, and damage based ones. The first ones deals with crack evolution in elastic materials whereas damage models consider a continuous transition from sound to totally damaged materials.
Quasibrittle structures are characterized by the existence of a nonnegligible fracture process zone. Linear elasticity fracture mechanics (LEFM) does not apply as it requires this zone to be negligible. In cohesive zone models (CZM), it is assumed that the process zone is concentrated over a line (2D problems) or a surface (3D), that defines the crack path. Damage models make no strong assumption on its size and shape; damaged zone length and width are a priori not negligible compared to the size of the material domain.
The CZM definition needs to know the crack path. If it is unknown, LEFM based methods exist to predict it. Even if most cohesive models cannot handle complex crack paths, such as branching, some adaptations [2, 3] allow to do so. Regarding damage models, there exist different families. Indeed, purely local damage presents spurious localization and different regularization methods have been proposed: strain or damage averaging, higherorder gradient, phasefield, variational models... These models are able to deal with complex damaged zone shape evolution but present important difficulties to deal with jump in displacement. Moreover, as damage evolution law has to be enforced over the whole material domain, computational cost is more important than for cohesive models.
The crack band model [4] is an intermediate between the cohesive model and classical damage ones, as damage is represented as a nonlinear behavior of some elements that defines the crack path. However, damage band width is only dependent on the chosen elements that have a nonlinear behavior and the crack path has to be previously determined. More recently, a levelset based damage model, the thick level set (TLS) model, has been presented [5]. This damage model bridges both families of models as it is able to recover complex crack paths while introducing jumps in displacement in a natural way. Contrary to the crack band model, the width of the damage band is here a parameter that is not related to the mesh.
The goal of this paper is to establish a onedimensional equivalence between the cohesive zone model and a damage model with TLS regularization. Note that similar analysis concerning cohesive and gradient damage models has been performed by Lorentz in [6]. Thus, based on this equivalence, a method to derive a local damage behavior to use in TLS from any cohesive behavior will be exhibited. Later, it will be used in twodimensional simulations and results will be compared to classical cohesive simulations, that have proved their capability to reproduce experimental results.
We consider structures under quasistatic load. We restrict the study to small strain and displacement and no particular shear behavior is considered, unlike in [7]. The relationship between cohesive and TLS models is analyzed. A onedimensional equivalence is analytically first established. Then, these results are used to compare twodimensional behaviors.
Let us start with the onedimensional comparison between the cohesive and the TLS models. We consider an elastic bar of length 2L: \(x\in [L,L]\) and Young modulus E. Degradation in this bar will be modeled either with a cohesive zone located at \(x=0\) or with an evolving damage layer centered at \(x=0\). Due to symmetry, only half of the bar needs to be considered: \(x\in [0,L]\). Let u(x) denote the displacement along the bar and in particular \(u(L)=u_L\) the displacement of the extremity. The stress \(\sigma\) is here a scalar variable.
Cohesive zone model
Thick level set damage model
Summary of cohesive and damage functions
Cohesive model  Damage model 

\(g_\text {CZM}(\alpha )\)  \(g_\text {dam}(d)\) 
\(A_c(\alpha )\)  h(d) 
TLS regularization  
\(d(\phi /\ell _c)\), \(\ell _c\) 
Methods
In this section, we establish relations that must be satisfied in order to have equivalent models for the particular problem of the bar. Then, the relations will be specialized for a particular choice of \(g_\text {dam}(d)\), \(d(\phi /\ell _c)\) and finally two examples of cohesive tensionopening functions f.
Onedimensional equivalence
Imagine that, in a quasistatic analysis, some displacement \(u_L\) is imposed. We say that a CZM and a TLS models are equivalent if, for a given displacement \(u_L\), the same stress is applied and they have the same energy and dissipation [16–19].
The displacement, energy and dissipation of the bar are computed in the following sections, first for cohesive model and secondly for TLS one. The stress can be directly calculated from the constitutive models (2) and (13)
Cohesive zone model
Thick level set model
General relationships
Meaning of \(\varvec{\lambda }\)
From general to particular relations
These general relations are now derived in the case of a particular choice of \(g_\text {dam}(d)\). Later, a particular form of \(d(\hat{\phi })\) is assumed. Finally, the equations are particularized for two cohesive softening functions F: the linear and the bilinear ones.
A choice of damage function \(\varvec{g_\text {dam}}\)
A choice of TLS damage profile \(\varvec{d(\hat{\phi })}\)
Two examples of cohesive laws
Two particular cases of cohesive stressopening functions are derived from previous relationships: the linear and the bilinear ones.
Linear cohesive law
Bilinear cohesive law
The bilinear cohesive law is considered as it is one of the most popular laws to describe concrete [27–32]. It is presented in Fig. 3a. The method to obtain h is the same as previously. The result is a discontinuous and increasing h function. Corresponding strainstress curves are shown in Fig. 3b. All of them present a discontinuity that appears as a linear zone where strain and stress increase while damage value remains constant. This plateau of damage is caused by the discontinuity of slope of the cohesive behavior. Details of the calculations are given in the "Appendix 1". Some conditions on the choice of the cohesive and TLS parameters are analyzed in "Appendix 2".
Results and discussion

the necessity of representing jumps in displacement, for which classical XFEM enrichment is used [33];

the calculation of nonlocal energy release rate \(\overline{Y}\), performed by the resolution of a variational problem described in [13];

the propagation of the damage front, that is performed by an explicit algorithm with prediction described in "Appendix 3".
Simulations and results—influence of \(\varvec{\ell _c}\)
Damage zone shape at maximum load It is interesting to analyze the shape of the damaged zone at the maximum load. For different \(\ell _c\) values, the position of the damage front is drawn in Fig. 12. The width of that zone is wider as \(\ell _c\) is bigger. It is consistent with the fact that in TLS, the width of the fracture process zone is driven by \(\ell _c\). For example, the width of the damaged zone corresponding to a propagating crack is \(2\ell _c\). The length of the damaged zone ahead of the crack tip is significantly the same for all \(\ell _c\). The depth is in fact driven by the equivalent cohesive behavior. We can conclude in which concerns the fracture process zone that its length is probably the same that in cohesive simulations and its width, neglected in CZM, is now driven by TLS parameter \(\ell _c\).
Simulations and results—comparison under mixedmode load
The cohesive model used in [35] forces the crack to be in the continuity of the notch. In the TLS model, no particular assumption is done on the damage zone path. This could explain the difference in the results presented in Fig. 16. To check this assumption, another TLS simulation is presented in Fig. 17, where initial damage is bigger than previously. Here, its radius is \(0.75 \ell _c\). The CZM crack path is here overlapped with the TLS crack lips. Furthermore, loadCMOD curves are coincident until about the first half of the postpeak curve. Below that point, the TLS curve remains in the experimental envelope whereas cohesive one does not. It can be concluded that TLS simulation without an important initial damage give results slightly different from the CZM ones, but whose loadCMOD curve is closer to experimental one. Forcing similar crack paths by introducing a more important initial damage leads to very close global results.
Conclusions
It has been shown that it is possible within the TLS framework to derive an equivalent onedimensional damage behavior from any cohesive model. The damage model depends on the characteristic length \(\ell _c\). The damage behavior tends to the cohesive one when \(\ell _c\) is close to zero. Furthermore, global response of structures is independent on \(\ell _c\) for a given onedimensional equivalent cohesive law. CMODforce curves and crack path are the same for both models.
The TLS model is able to provide results of the same quality as the cohesive zone model, that is wellknown for its accuracy. Besides, it has already proven its capability to determine accurate crack paths without supplementary hypothesis or models. Moreover, it is able to represent branching, coalescence and initiation of damaged zones [5, 13]. Even if this paper only deals with quasibrittle materials, as damaged zone is concentrated over a damage band, recent developments couple local and nonlocal damage and allows diffuse damage prior to localization [11].
Abbreviations
 TLS:

thick level set
 CZM:

cohesive zone model
 LEFM:

linear elasticity fracture mechanics
 FPZ:

fracture process zone
 MTS:

maximum Tangential Stress
 TPB:

three point bending
 CMOD:

crack mouth opening displacement
Declarations
Authors’ contributions
NM and CS provided the main idea for a comparison between TLS and CZM models and performed onedimensional analysis. APG carried out the bidimensional simulations and analysis. All authors read and approved the final manuscript.
Acknowledgements
We would like to thank Alexis Salzman and Nicolas Chevaugeon for their contributions in the implementation of the TLS.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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