A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk material

A bilinear cohesive zone model (CZM) is employed in conjunction with a viscoelastic bulk (background) material to investigate fracture behavior of asphalt concrete. An attractive feature of the bilinear CZM is a potential reduction of artificial compliance inherent in the intrinsic CZM. In this study, finite material strength and cohesive fracture energy, which are cohesive parameters, are obtained from laboratory experiments. Finite element implementation of the CZM is accomplished by means of a user-subroutine which is employed in a commercial finite element code (e.g., UEL in ABAQUS). The cohesive parameters are calibrated by simulation of mode I disk-shaped compact tension results. The ability to simulate mixed-mode fracture is demonstrated. The single-edge notched beam test is simulated where cohesive elements are inserted over an area to allow cracks to propagate in any general direction. The predicted mixed-mode crack trajectory is found to be in close agreement with experimental results. Furthermore, various aspects of CZMs and fracture behavior in asphalt concrete are discussed including: compliance, convergence, and energy balance.     

Potential-based constitutive models for cohesive interfaces: Theory,implementation and examples

Cohesive interfaces appear in many materials or structures, e.g. composites or adhesive bonds. Originally introduced to model crack tips in fracture mechanics, cohesive zone models are used to describe the constitutive behavior of cohesive interfaces since the early 1990s. In the present contribution, the concept of generalized standard materials (GSM) is transferred from the modeling of bulk behavior to cohesive zones. The potential-based framework obtained hereby is referred to as standard dissipative cohesive zones (SD-CZ). Within this framework, an irreversible interface model is developed for the one-dimensional case and subsequently extended to three-dimensional transverse isotropy. While the potential structure of the constitutive law may be required for certain applications, it also brings benefits with regard to the numerical implementation. To the best knowledge of the authors, this is the first approach to interface modeling in a GSM-like framework, where dissipative effects like damage and softening are considered as well as normal-tangential coupling for mixed-mode decohesion. Comparisons to experimental data and existing cohesive zone models outline the capabilities of the approach.     

NURBS- and T-spline-based isogeometric cohesive zone modeling of interface debonding

Cohesive zone (CZ) models have long been used by the scientific community to analyze the progressive damage of materials and interfaces. In these models, non-linear relationships between tractions and relative displacements are assumed, which dictate both the work of separation per unit fracture surface and the peak stress that has to be reached for the crack formation. This contribution deals with isogeometric CZ modeling of interface debonding. The interface is discretized with generalized contact elements which account for both contact and cohesive debonding within a unified framework. The formulation is suitable for non-matching discretizations of the interacting surfaces in presence of large deformations and large relative displacements. The isogeometric discretizations are based on non uniform rational B-splines as well as analysis-suitable T-splines enabling local refinement. Conventional Lagrange polynomial discretizations are also used for comparison purposes. Some numerical examples demonstrate that the proposed formulation based on isogeometric analysis is a computationally accurate and efficient technology to solve challenging interface debonding problems in 2D and 3D.     

Reduction in mesh bias for dynamic fracture using adaptive splitting of polygonal finite elements

We present a method to reduce mesh bias in dynamic fracture simulations using the finite element method with adaptive insertion of extrinsic cohesive zone elements along element boundaries. The geometry of the domain discretization is important in this setting because cracks are only allowed to propagate along element facets and can potentially bias the crack paths. To reduce mesh bias, we consider unstructured polygonal finite elements in this work. The meshes are generated with centroidal Voronoi tessellations to ensure element quality. However, the possible crack directions at each node are limited, making this discretization a poor candidate for dynamic fracture simulation. To overcome this problem, and significantly improve crack patterns, we propose adaptive element splitting, whereby the number of potential crack directions is increased at each crack tip. Thus, the crack is allowed to propagate through the polygonal element. Geometric studies illustrate the benefits of polygonal element discretizations employed with element splitting over other structured and unstructured discretizations for crack propagation applications. Numerical examples are performed and demonstrate good agreement with previous experimental and numerical results in the literature. Copyright © 2014 John Wiley & Sons, Ltd.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

Cohesive Crack with Rate-Dependent Opening and Viscoelasticity: I. Mathematical Model and Scaling

The time dependence of fracture has two sources: (1) the viscoelasticity of material behavior in the bulk of the structure, and (2) the rate process of the breakage of bonds in the fracture process zone which causes the softening law for the crack opening to be rate-dependent. The objective of this study is to clarify the differences between these two influences and their role in the size effect on the nominal strength of stucture. Previously developed theories of time-dependent cohesive crack growth in a viscoelastic material with or without aging are extended to a general compliance formulation of the cohesive crack model applicable to structures such as concrete structures, in which the fracture process zone (cohesive zone) is large, i.e., cannot be neglected in comparison to the structure dimensions. To deal with a large process zone interacting with the structure boundaries, a boundary integral formulation of the cohesive crack model in terms of the compliance functions for loads applied anywhere on the crack surfaces is introduced. Since an unopened cohesive crack (crack of zero width) transmits stresses and is equivalent to no crack at all, it is assumed that at the outset there exists such a crack, extending along the entire future crack path (which must be known). Thus it is unnecessary to deal mathematically with a moving crack tip, which keeps the formulation simple because the compliance functions for the surface points of such an imagined preexisting unopened crack do not change as the actual front of the opened part of the cohesive crack advances. First the compliance formulation of the cohesive crack model is generalized for aging viscoelastic material behavior, using the elastic-viscoelastic analog (correspondence principle). The formulation is then enriched by a rate-dependent softening law based on the activation energy theory for the rate process of bond ruptures on the atomic level, which was recently proposed and validated for concrete but is also applicable to polymers, rocks and ceramics, and can be applied to ice if the nonlinear creep of ice is approximated by linear viscoelasticity. Some implications for the characteristic length, scaling and size effect are also discussed. The problems of numerical algorithm, size effect, roles of the different sources of time dependence and rate effect, and experimental verification are left for a subsequent companion paper. This revised version was published online in July 2006 with corrections to the Cover Date.     

Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture

Adaptive mesh refinement and coarsening schemes are proposed for efficient computational simulation of dynamic cohesive fracture. The adaptive mesh refinement consists of a sequence of edge‐split operators, whereas the adaptive mesh coarsening is based on a sequence of vertex‐removal (or edge‐collapse) operators. Nodal perturbation and edge‐swap operators are also employed around the crack tip region to improve crack geometry representation, and cohesive surface elements are adaptively inserted whenever and wherever they are needed by means of an extrinsic cohesive zone model approach. Such adaptive mesh modification events are maintained in conjunction with a topological data structure (TopS). The so‐called PPR potential‐based cohesive model (J. Mech. Phys. Solids 2009; 57 :891–908) is utilized for the constitutive relationship of the cohesive zone model. The examples investigated include mode I fracture, mixed‐mode fracture and crack branching problems. The computational results using mesh adaptivity (refinement and coarsening) are consistent with the results using uniform mesh refinement. The present approach significantly reduces computational cost while exhibiting a multiscale effect that captures both global macro‐crack and local micro‐cracks. Copyright © 2012 John Wiley & Sons, Ltd.     

General remarks on cyclic cohesive zone models

Cyclic cohesive zone models represent powerful numerical tools to describe and predict different phenomena of fatigue. In the present paper a review is given about the state of the art of cyclic cohesive zone modelling. Firstly, a commented survey of the history of cyclic cohesive zone models is presented starting from the strip-yield models of Dugdale and Barenblatt and ending with the modern creep-fatigue models. Next, the paper focuses on the thermodynamically consistent formulation of cyclic cohesive zone models. In particular, the formulation of cohesive zone potentials, the choice of appropriate deformation and damage measures as well as the assembly of damage evolution equations are discussed in detail. In this context, established models are critically compared with a model recently proposed by the authors. Finally, some numerical examples are presented for illustration, dealing with fatigue crack growth, uniaxial fatigue life prediction, and intergranular fatigue damage. These simulations demonstrate the wide range of applications of cyclic cohesive zone models.     

On the constitutive relation of materials with microstructure using a potential-based cohesive model for interface interaction

Macroscopic constitutive relationship is estimated by considering the microscopic particle/matrix interfacial debonding. For the interfacial debonding, the PPR potential-based cohesive model is utilized. The extended Mori-Tanaka model is employed for micromechanics, while a finite element-based cohesive zone model is used for the computational model. Both models (theoretical and computational) agree well each other in representing the macroscopic constitutive relationship on the basis of the PPR model. The microscopic interfacial cohesive parameters of the PPR model are estimated from macroscopic composite material behavior. In addition, different microscopic debonding processes are observed with respect to different macroscopic constitutive relationships (e.g. hardening, softening, and snap-back).     

A stable X‐FEM in cohesive transition from closed to open crack

Ignoring crack tip effects, the stability of the X‐FEM discretizations is trivial for open cracks but remains a challenge if we constrain the crack to be closed (i.e., the bi‐material problem). Here, we develop a formulation for general cohesive interactions between crack faces within the X‐FEM framework. The stability of the new formulation is demonstrated for any cohesive crack stiffness (including the closed crack) and illustrated for a nonlinear cohesive softening law. A benchmark of the new model is carried out with simpler approaches for a closed crack (i.e., Lagrange multipliers) and for a cohesive crack (i.e., penalty approach). Due to the analogies between stable cohesive X‐FEM and Nitsche's methods, the new method simplifies the implementation and is attractive in dynamic explicit codes. Copyright © 2014 John Wiley & Sons, Ltd.     

Mode I fracture of adhesive joints using tailored cohesive zone models

Cohesive zone models are explored in order to study cleavage fracture in adhesive bonded joints. A mode I cohesive model is defined which correlates the tensile traction and the displacement jump (crack faces opening) along the fracture process zone. In order to determine the traction-separation relation, the main fracture parameters, namely the cohesive strength and the fracture energy, as well as its shape, must be specified. However, owing to the difficulties associated to the direct measurement of the fracture parameters, very often they are obtained by comparing a measured fracture property with numerical predictions based on an idealized traction separation relation. Likewise in this paper the cohesive strength of an adhesive layer sandwiched between elastic substrates is adjusted to achieve a match between simulations and experiments. To this aim, the fracture energy and the load-displacement curve are adopted as input in the simulations. In order to assess whether or not the shape of the cohesive model may have an influence on the results, three representative cohesive zone models have been investigated, i.e. exponential, bilinear and trapezoidal. A good agreement between experiments and simulations has been generally observed. However, a slight difference in predicting the loads for damage onset has been found using the different cohesive models.     

Formulation of a cohesive zone model for a Mode III crack

Cohesive zone models have been proven effective in modeling crack initiation and propagation phenomena. In this work, a possible form for a Mode III cohesive zone model is formulated from elastic stress and displacement fields around a crack with a cohesive zone ahead of the crack tip. A traction-separation relation for the model is derived as a direct consequence of the formulation, which establishes some intrinsic connections between properties of the cohesive zone and those of the bulk material. Interestingly, this model states that the von Mises effective stress in the cohesive zone is constant, which may be related to the bulk material’s yield stress and is consistent with the assumption made in conventional strip-yield elastic-plastic solutions.     

Ductile Fracture Study of Stainless Steel AISI 304L Thin Sheets Using the EWF Method and Cohesive Zone Modeling

The purpose of this study is to model the ductile fracture phenomenon using experimental and numerical methods. The stainless steel AISI 304L thin sheets are studied including two thicknesses (0.8 and 1.5 mm). A mechanical characterization is firstly done in order to obtain the main mechanical properties useful for the numerical modeling. In order to determine the essential work of fracture (EWF), DENT specimens are used involving the two thicknesses. The results obtained in terms of EWF for the two thicknesses are close; we find that the essential work of fracture can be considered as an intrinsic criterion for thin sheets. A cohesive zone modeling (CZM) is used in the present study; the model is represented by a traction–separation law (TS). The cohesive elements are implemented in the finite element model, and the material parameters of the model are determined by the mechanical and fracture characterizations. A satisfactory reproduction of the experimental tests is obtained. A good correlation is also obtained between the essential work of fracture determined experimentally and the work of separation used as cohesive zone model parameter.     

A two-field modified Lagrangian formulation for robust simulations of extrinsic cohesive zone models

This paper presents the robust implementation of a cohesive zone model based on extrinsic cohesive laws (i.e. laws involving an infinite initial stiffness). To this end, a two-field Lagrangian weak formulation in which cohesive tractions are chosen as the field variables along the crack’s path is presented. Unfortunately, this formulation cannot model the infinite compliance of the broken elements accurately, and no simple criterion can be defined to determine the loading–unloading change of state at the integration points of the cohesive elements. Therefore, a modified Lagrangian formulation using a fictitious cohesive traction instead of the classical cohesive traction as the field variable is proposed. Thanks to this change of variable, the cohesive law becomes an increasing function of the equivalent displacement jump, which eliminates the problems mentioned previously. The ability of the proposed formulations to simulate fracture accurately and without field oscillations is investigated through three numerical test examples.     

Mesoscale models of interface mechanics in crystalline solids: a review

Theoretical and computational methods for representing mechanical behaviors of crystalline materials in the vicinity of planar interfaces are examined and compared. Emphasis is on continuum-type resolutions of microstructures at the nanometer and micrometer levels, i.e., mesoscale models. Grain boundary interfaces are considered first, with classes of models encompassing sharp interface, continuum defect (i.e., dislocation and disclination), and diffuse interface types. Twin boundaries are reviewed next, considering sharp interface and diffuse interface (e.g., phase field) models as well as pseudo-slip crystal plasticity approaches to deformation twinning. Several classes of models for evolving failure interfaces, i.e., fracture surfaces, in single crystals and polycrystals are then critically summarized, including cohesive zone approaches, continuum damage theories, and diffuse interface models. Important characteristics of compared classes of models for a given physical behavior include complexity, generality/flexibility, and predictive capability versus number of free or calibrated parameters.     

Finite strain fracture of plates and shells with configurational forces and edge rotations

We propose a simple and efficient algorithm for FEM‐based computational fracture of plates and shells with both brittle and ductile materials on the basis of edge rotation and load control. Rotation axes are the crack front nodes, and each crack front edge in surface discretizations affects the position of only one or two nodes. Modified positions of the entities maximize the modified mesh quality complying with the predicted crack path (which depends on the specific propagation theory in use). Compared with extended FEM or with classical tip remeshing, the proposed solution has algorithmic and generality advantages. The propagation algorithm is simpler than the aforementioned alternatives, and the approach is independent of the underlying element used for discretization. For history‐dependent materials, there are still some transfer of relevant quantities between elements. However, diffusion of results is more limited than with tip or full remeshing. To illustrate the advantages of our approach, three prototype models are used: tip energy dissipation linear elastic fracture mechanics (LEFM), cohesive‐zone approaches, and ductile fracture. Both the Sutton crack path criterion and the path estimated by the Eshelby tensor are employed. Traditional fracture benchmarks, including one with plastic hinges, and newly proposed verification tests are solved. These were found to be very good in terms of crack path and load ∕ deflection accuracy. Copyright © 2013 John Wiley & Sons, Ltd.     

Continuum coupled cohesive zone elements for analysis of fracture in solid bodies

Embedding cohesive surfaces into finite element models is a widely used technique for the numerical simulation of material separation (i.e. crack propagation). Typically, a traction-separation law is specified that relates the magnitude of the cohesive traction to the distance between the separating surfaces. Thus the characterization of fracture in such models is not directly coupled to the bulk constitutive response, in the sense that the cohesive traction does not explicitly depend on material stretching in the plane of the fracture surface. In this work, an initially-rigid cohesive-traction formulation that is coupled to the surrounding continuum is introduced as a further development of the cohesive zone idea. In this model, the traction-separation law - and therefore the fracture phenomenology - derives directly from the bulk constitutive law. The immediate goal is an improved cohesive zone framework that naturally and logically initiates cohesive separation behavior, and couples its evolution to the material state in the region of the crack tip. A cohesive element based on this model is implemented in an explicit three-dimensional finite element code. Proof-of-concept analyses using both linear elastic and Gurson void growth constitutive relations are presented. A three-point bend simulation is found to give good agreement with experimental results.     

Mapping Cohesive Fracture and Fragmentation Simulations to Graphics Processor Units

A graphics processor units(GPU)‐based computational framework is presented to deal with dynamic failure events simulated by means of cohesive zone elements. The work is divided into two parts. In the first part, we deal with pre‐processing of the information and verify the effectiveness of dynamic insertion of cohesive elements in large meshes in parallel. To this effect, we employ a novel and simplified topological data structure specialized for meshes with triangles, designed to run efficiently and minimize memory occupancy on the GPU. In the second part, we present a parallel explicit dynamics code that implements an extrinsic cohesive zone formulation where the elements are inserted ‘on‐the‐fly’, when needed and where needed. The main challenge for implementing a GPU‐based computational framework using an extrinsic cohesive zone formulation resides on being able to dynamically adapt the mesh, in a consistent way, by inserting cohesive elements on fractured facets. In order to handle that, we extend the conventional data structure used in the finite element method (based on element incidence) and store, for each element, references to the adjacent elements. This additional information suffices to consistently insert cohesive elements by duplicating nodes when needed. Currently, our data structure is specialized for triangular meshes, but an extension to tetrahedral meshes is feasible. The data structure is effective when used in conjunction with algorithms to traverse nodes and elements. Results from parallel simulations show an increase in performance when adopting strategies such as distributing different jobs among threads for the same element and launching many threads per element. To avoid concurrency on accessing shared entities, we employ graph coloring. In a pre‐processing phase, each node of the dual graph (bulk elements of the mesh as graph nodes) is assigned a color different from the colors assigned to adjacent nodes. In that fashion, elements of the same color can be processed in parallel without concurrency. All the procedures needed for the insertion of cohesive elements along fracture facets and for computing nodal properties are performed by threads assigned to triangles, invoking one kernel per color. Computations on existing cohesive elements are also performed based on adjacent bulk elements. Experiments show that GPU speedup increases with the number of nodes and bulk elements. Copyright © 2015 John Wiley & Sons, Ltd.