Cohesive-zone-model formulation and implementation using the symmetric Galerkin boundary element method for homogeneous solids

A new symmetric boundary integral formulation for cohesive cracks growing in the interior of homogeneous linear elastic isotropic media with a known crack path is developed and implemented in a numerical code. A crack path can be known due to some symmetry implications or the presence of a weak or bonded surface between two solids. The use of a two-dimensional exponential cohesive law and of a special technique for its inclusion in the symmetric Galerkin boundary element method allows us to develop a simple and efficient formulation and implementation of a cohesive zone model. This formulation is dependent on only one variable in the cohesive zone (relative displacement). The corresponding constitutive cohesive equations present a softening branch which induces to the problem a potential instability. The development and implementation of a suitable solution algorithm capable of following the growth of the cohesive zone and subsequent crack growth becomes an important issue. An arc-length control combined with a Newton–Raphson algorithm for iterative solution of nonlinear equations is developed. The boundary element method is very attractive for modeling cohesive crack problems as all nonlinearities are located along the boundaries (including the crack boundaries) of linear elastic domains. A Galerkin approximation scheme, applied to a suitable symmetric integral formulation, ensures an easy treatment of cracks in homogeneous media and excellent convergence behavior of the numerical solution. Numerical results for the wedge split and mixed-mode flexure tests are presented.     

A contact algorithm for cohesive cracks in the extended finite element method

Cracks with quasibrittle behavior are extremely common in engineering structures. The modeling of cohesive cracks involves strong nonlinearity in the contact, material, and complex transition between contact and cohesive forces. In this article, we propose a novel contact algorithm for cohesive cracks in the framework of the extended finite element method. A cohesive-contact constitutive model is introduced to characterize the complex mechanical behavior of the fracture process zone. To avoid the stress oscillations and ill-conditioned system matrix that often occur in the conventional contact approach, the proposed algorithm employs a special dual Lagrange multiplier to impose the contact constraint. This Lagrange multiplier is constructed by means of the area-weighted average and biorthogonality conditions at the element level. The system matrix can be condensed into a positive definite matrix with an unchanged size at a very low computational cost. In addition, we illustrate solving the cohesive crack contact problem using a novel iteration strategy. Several numerical experiments are performed to illustrate the efficiency and high-quality results of our method in contact analysis of cohesive cracks.     

The cohesive band model: a cohesive surface formulation with stress triaxiality

In the cohesive surface model cohesive tractions are transmitted across a two-dimensional surface, which is embedded in a three-dimensional continuum. The relevant kinematic quantities are the local crack opening displacement and the crack sliding displacement, but there is no kinematic quantity that represents the stretching of the fracture plane. As a consequence, in-plane stresses are absent, and fracture phenomena as splitting cracks in concrete and masonry, or crazing in polymers, which are governed by stress triaxiality, cannot be represented properly. In this paper we extend the cohesive surface model to include in-plane kinematic quantities. Since the full strain tensor is now available, a three-dimensional stress state can be computed in a straightforward manner. The cohesive band model is regarded as a subgrid scale fracture model, which has a small, yet finite thickness at the subgrid scale, but can be considered as having a zero thickness in the discretisation method that is used at the macroscopic scale. The standard cohesive surface formulation is obtained when the cohesive band width goes to zero. In principle, any discretisation method that can capture a discontinuity can be used, but partition-of-unity based finite element methods and isogeometric finite element analysis seem to have an advantage since they can naturally incorporate the continuum mechanics. When using interface finite elements, traction oscillations that can occur prior to the opening of a cohesive crack, persist for the cohesive band model. Example calculations show that Poisson contraction influences the results, since there is a coupling between the crack opening and the in-plane normal strain in the cohesive band. This coupling holds promise for capturing a variety of fracture phenomena, such as delamination buckling and splitting cracks, that are difficult, if not impossible, to describe within a conventional cohesive surface model.     

Instability during cohesive zone growth

Tensile microcracking of quasi-brittle materials is studied by means of micromechanics, based on (i) an elasto-damaging cohesive zone model accounting for cohesive softening and (ii) a dilute distribution of non-interacting microcracks of uniform orientation and size. Considering virgin microcracks (initially without cohesive zones), macroscopic tensile load increase results in growth of cohesive zones ahead of stationary (non-propagating) cracks and, subsequently, in crack propagation which, notably, will be encountered before the cohesive zones are fully developed i.e. onset of instable cohesive zone growth will be encountered at a load level (i) at which tractions are still transmitted across the inner edges of the cohesive zones and (ii) at which the separation at the inner edges of the cohesive zones is smaller than its critical value. Focusing on onset of instable cohesive zone growth, the chosen approach allows for accessing quantities characterizing the stability limit (e.g., the intensity of the macroscopic loading and the opening at the inner edges of the cohesive zones), without raising the need for non-linear Finite Element analyses. It is shown that the tensile macrostrength of materials containing virgin microcracks is larger than the one related to cracks with already initially fully developed cohesive zones, and related strength differences are quantified for a wide class of cohesive softening behavior. The proposed model is validated by comparing model predictions with an exact solution (available for the special case of constant cohesive tractions) and with results from reliable Finite Element analyses. The paper will be of interest for engineers involved in testing and/or in modeling of quasi-brittle media including cementitious materials and rock.     

等离子喷涂HA/TiO2复合涂层

采用大气等离子喷涂方法,成功地制备了ＨＡ／ＴｉＯ２复合涂层,对复合涂层的结合强度、微观结构、水浸渍下的表面形貌进行了较为深入的研究,结果表明,由于ＴｉＯ２的加入,ＨＡ／ＴｉＯ２涂层的结合强度明显高于纯ＨＡ涂层,而且导致涂层破坏机理由粘合破坏向内聚破坏转化,这是由于ＨＡ／ＴｉＯ２的复合缓和了涂层与基体间的膨胀系数失配现象,改善了涂层与基体之间的结合,ＳＥＭ观察显示,ＨＡ／ＴｉＯ２涂层表面有一些细小的     

A cohesive edge crack

The nucleation and growth of a cohesive edge crack is studied. This topic is germane to the initiation and early stage of crack growth during unnotched beam testing, the growth of short edge cracks in finite test pieces, and the formation of tension cracks of geological origin. This paper focuses on an edge crack in a semi-infinite plane, under a uniform far-field tensile stress acting parallel to the plane boundary. Expressions for the Mode I stress-intensity-factor and crack-opening-displacement for an edge crack subjected to arbitrary crack face loading are determined via the weight function method. All of the constants needed to define the weight function and associated integrals are themselves explicit functions of just two constants: f_r and ψ. Two types of softening behavior in the cohesive zone are examined: rectangular softening, and linear softening. In each case the process zone size, energy-release-rate, crack-opening displacement and load-ratio are examined. The different test behavior exhibited under load-control versus fixed-grip displacement control is explored. The test control conditions alter the fracture behavior significantly. For a linear softening cohesive edge crack, it is found out that under fixed-grip control (load-control), the process zone size decreases (increases) steadily with increasing traction-free crack length, approaching the semi-infinite crack asymptote from above (below). The differences between load-control versus fixed-grip control decrease rapidly with increasing traction-free crack length.     

Achieving pervasive fracture and fragmentation in three-dimensions: an unstructuring-based approach

There are few methods capable of capturing the full spectrum of pervasive fracture behavior in three-dimensions. Throughout pervasive fracture simulations, many cracks initiate, propagate, branch and coalesce simultaneously. Because of the cohesive element method framework, this behavior can be captured in a regularized manner. However, since the cohesive element method is only able to propagate cracks along element facets, a poorly designed discretization of the problem domain may introduce artifacts into the simulated results. To reduce the influence of the discretization, geometrically and constitutively unstructured means can be used. In this paper, we present and investigate the use of three-dimensional nodal perturbation to introduce geometric randomness into a finite element mesh. We also discuss the use of statistical methods for introducing randomness in heterogeneous constitutive relations. The geometrically unstructured method of nodal perturbation is then combined with a random heterogeneous constitutive relation in three numerical examples. The examples are chosen in order to represent some of the significant influencing factors on pervasive fracture and fragmentation; including surface features, loading conditions, and material gradation. Finally, some concluding remarks and potential extensions are discussed.     

Dynamic crack propagation with cohesive elements: a methodology to address mesh dependency

In this paper, two brittle fracture problems are numerically simulated: the failure of a ceramic ring under centrifugal loading and crack branching in a PMMA strip. A three‐dimensional finite element package in which cohesive elements are dynamically inserted has been developed. The cohesive elements' strength is chosen to follow a modified weakest link Weibull distribution. The probability of introducing a weak cohesive element is set to increase with the cohesive element size. This reflects the physically based effect according to which larger elements are more likely to contain defects. The calculations illustrate how the area dependence of the Weibull model can be used to effectively address mesh dependency. On the other hand, regular Weibull distributions have failed to reduce mesh dependency for the examples shown in this paper. The ceramic ring calculations revealed that two distinct phenomena appear depending on the magnitude of the Weibull modulus. For low Weibull modulus, the fragmentation of the ring is dominated by heterogeneities. Whereas many cracks were generated, few of them could propagate to the outer surface. Monte Carlo simulations revealed that for highly heterogeneous rings, the number of small fragments was large and that few large fragments were generated. For high Weibull modulus, signifying that the ring is close to being homogeneous, the fragmentation process was very different. Monte Carlo simulations highlighted that a larger number of large fragments are generated due to crack branching. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Prediction of the Work of Separation and Implications to Modeling

Following Barenblatt's idea about the modeling of cracks by the use of a cohesive zone has attracted considerable attention. Recently, this model has also been applied to the prediction of ductile crack growth. For this case the present investigation aims to compare the predictions of a cohesive zone model with the predictions of the more physically based modified Gurson relation. The results demonstrate that in case of ductile fracture the parameters cohesive strength and energy may only be regarded as material properties within a small range of stress triaxialities. This finally leads to the conclusion that special care has to be taken if a cohesive zone model is used for the analysis of ductile fracture. The use of the modified Gurson relation predicts that the cohesive energy and strength do not remain constant throughout a crack growth analysis and their change is not known a priori. Improved cohesive zone models that take a coupling to the surrounding material into account may overcome this problem. This revised version was published online in July 2006 with corrections to the Cover Date.     

Fatigue crack growth acceleration due to intermittent overstressing in adhesively bonded CFRP joints

Double cantilever beam joints were used to investigate cohesive and interlaminar crack growth in bonded composite joints under constant and variable amplitude (VA) loading. Numerical crack growth integration was used to predict the VA fatigue life using constant amplitude data. This underestimated the fatigue crack growth rate for interlaminar cracks, indicating crack growth acceleration due to load interactions. This was also the case for cohesive cracks subjected to a moderate initial strain energy release rate (G_max). An unstable crack growth regime was also identified for the case of high initial G_max cohesive crack propagation. This behaviour is attributed to the development of a damage zone ahead of the crack tip.     

等离子喷涂HA/TiO2复合涂层

采用大气等离子喷涂方法,成功地制备了HA／TiO₂复合涂层,对复合涂层的结合强度、微观结构、水浸渍下的表面形貌进行了较为深入的研究．结果表明,由于TiO₂的加入,HA／TiO₂涂层的结合强度明显高于纯HA徐层,而且导致涂层破坏机理由粘合破坏向内聚破坏转化．这是由于HA／TiO₂的复合缓和了涂层与基体间的膨胀系数失配现象,改善了涂层与基体之间的结合．SEM观察显示,HA／TiO₂涂层表面有一些细小的裂纹,但在去离子水中浸泡后就会消失,而且不容易产生新的裂纹,这说明TiO₂的加入不但改善了涂层与基体之间的结合,同时增强了涂层内部颗粒的结合．     

Fracture behavior in CFRP cross-ply laminates with initially cut fibers

This study qualitatively investigates the effects of initially cut fibers (slits) on fracture behavior in carbon fiber reinforced plastic (CFRP) cross-ply laminates, which had alternate or identical slit angle ±θ in the 0° plies. Damage progress during tensile tests was observed for several geometries of cutting. We also numerically evaluated fracture behavior in laminates with slits by a layer-wise finite-element model with cohesive elements. The simulated damage patterns included matrix cracks along the slits, splits in the 0° layer from the slit tips, and transverse cracks in the 90° layer. Delamination was also generated at the crossing point of ply cracks due to the large shear stress, and then extended to form the triangular region bounded by the slits and splits. The predicted damage extension to the final failure agreed with the observations. A numerical study demonstrated that the damage near the slits produced a stress field similar to that of a penetrating notch.     

A note on pseudo-cohesive behavior in quasi-bidimensional brittle fracture

This paper addresses the study of a planar crack in a plate with a slightly curved crack front, as found in most nominally plane strain tests. Rather than trying to make a full three dimensional analysis of the elastic fields, we look for a systematic procedure to project the three-dimensional features on the plane of the plate. This is achieved by though-the-thickness averaging of the stress and displacement fields. It is shown that the resulting fields are pseudo-cohesive, which means that their expressions are formally identical to those corresponding to a traditional cohesive zone. The results for two simple cases are given to illustrate the pseudo-cohesive behavior, although general results are derived as obvious consequences of the averaging process. The general procedure to carry out the analysis for any kind of crack front shape is indicated, and the application to special cases such as fatigue-grown cracks in metals and stably growing cracks in brittle polymers is shortly discussed.     

Extended Voronoi cell finite element model for multiple cohesive crack propagation in brittle materials

This paper introduces an extended Voronoi cell finite‐element model (X‐VCFEM) for modelling cohesive crack propagation in brittle materials with multiple cracks. The cracks are modelled by a cohesive zone model and their incremental directions and growth lengths are determined in terms of the cohesive energy near the crack tip. Extension to VCFEM is achieved through enhancements in stress functions in the assumed stress hybrid formulation. In addition to polynomial terms, the stress functions include branch functions in conjunction with level set methods, and multi‐resolution wavelet functions in the vicinity of crack tips. The wavelet basis functions are adaptively enriched to accurately capture crack‐tip stress concentrations. Conditions and methods of stability are enforced in X‐VCFEM for improved convergence with propagating cracks. Two classes of problems are solved and compared with existing solutions in the literature for validation of the X‐VCFEM algorithms. The first set corresponds to results for static cracks, while in the latter set, the propagation of cohesive cracks are considered. Comparison of X‐VCFEM simulation results with results in literature for several fracture mechanics problems validates the effectiveness of X‐VCFEM. Copyright © 2005 John Wiley & Sons, Ltd.     

Cohesive Fracture Model Based on Necking

A linear hardening model together with a linear elastic background material is first used to discuss some aspects of the mathematical and physical limitations and constraints on cohesive laws. Using an integral equation approach together with the cohesive crack assumption, it is found that in order to remove the stress singularity at the tip of the cohesive zone, the cohesive law must have a nonzero traction at the initial zero opening displacement. A cohesive zone model for ductile metals is then derived based on necking in thin cracked sheets. With this model, the cohesive behavior including peak cohesive traction, cohesive energy density and shape of the cohesive traction–separation curve is discussed. The peak cohesive traction is found to vary from 1.15 times the yield stress for perfectly plastic materials to about 2.5 times the yield stress for modest hardening materials (power hardening exponent of 0.2). The cohesive energy density depends on the critical relative plate thickness reduction at the root of the neck at crack initiation, which needs to be determined by experiments. Finally, an elastic background medium with a center crack is employed to re-examine the shape effect of cohesive traction–separation curve, and the relation between the linear elastic fracture mechanics (LEFM) and cohesive zone models by considering the cohesive zone development and crack growth in the infinite elastic medium. It is shown that the shape of the cohesive curve does affect the cohesive zone size and the apparent energy release rate of LEFM for the crack growth in the elastic background material. The apparent energy release rate of LEFM approaches the cohesive energy density when the crack extends significantly longer than the characteristic length of the cohesive zone.     

The effects of inter-surface cohesive tractions on linear and penny-shaped cracks

A mesoscopic fracture model of equilibrium slit cracks in brittle solids, including inter-surface cohesive tractions acting near the crack tip, is analyzed and the effects of the cohesive tractions on the in-plane stress fields, crack-opening displacement profiles, and crack driving forces examined quantitatively for linear and penny-shaped cracks. The (numerical) analysis method is described in detail, along with results for four different cohesive forces. The assumed distribution of cohesive tractions were found to suppress the in-plane stress field adjacent to cracks in a homogeneous, isotropic medium when uniformly loaded in mode-I, and the suppression was a function of crack length. The crack-opening displacement profile was also perturbed and a new regime identified between the near-field Barenblatt zone and the far-field continuum zone. The extent of this `cohesive zone' was quantified by use of an interpolating function fit to the calculated profiles and found to be independent of crack size for a given cohesive tractions distribution. Furthermore, the crack-opening displacement at the edge of the cohesive zone was found to be independent of crack size, implying that despite significant perturbations to the stress field, the crack driving force at unstable equilibrium remains unchanged with crack size.     

Computational model for predicting nonlinear viscoelastic damage evolution in materials subjected to dynamic loading

Many inelastic solids accumulate numerous cracks before failure due to impact loading, thus rendering any exact solution of the IBVP untenable. It is therefore useful to construct computational models that can accurately predict the evolution of damage during actual impact/dynamic events in order to develop design tools for assessing performance characteristics. This paper presents a computational model for predicting the evolution of cracking in structures subjected to dynamic loading. Fracture is modeled via a nonlinear viscoelastic cohesive zone model. Two example problems are shown: one for model validation through comparison with a one-dimensional analytical solution for dynamic viscoelastic debonding, and the other demonstrates the applicability of the approach to model dynamic fracture propagation in the double cantilever beam test with a viscoelastic cohesive zone.