Viscoelastic-damage interface model formulation with friction to simulate the delamination growth in mode II shear

In this paper a formulation of a viscoelastic-damage interface model with friction in mode-II is presented. The cohesive constitutive law contains elastic and damage regimes. It has been assumed that the shear stress in the elastic regime follows the viscoelastic properties of the matrix material. The three element Voigt model has been used for the formulation of relaxation modulus of the material. Damage evolution proceeds according to the bilinear cohesive constitutive law combined with friction stress consideration. Combination of damage and friction is based on the presumption that the damaged area, related to an integration point, can be dismembered into the un-cracked area with the cohesive damage and cracked area with friction. Samples of a one element model have been presented to see the effect of parameters on the cohesive constitutive law. A comparison between the predicted results with available results of end-notched flexure specimens in the literature is also presented to verify the model. Transverse crack tension specimens are also simulated for different applied displacement velocities.     

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.     

Numerical analysis of dynamic crack propagation in rubber

In the present paper, dynamic crack propagation in rubber is analyzed numerically using the finite element method. The problem of a suddenly initiated crack at the center of stretched sheet is studied under plane stress conditions. A nonlinear finite element analysis using implicit time integration scheme is used. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using a cohesive zone model with a bilinear traction-separation law. Hence, the numerical model is able to model and predict the different contributions to the fracture toughness, i.e. the surface energy, viscoelastic dissipation, and inertia effects. The separation work per unit area and the strength of the cohesive zone have been parameterized, and their influence on the separation process has been investigated. A steadily propagating crack is obtained and the corresponding crack tip position and velocity history as well as the steady crack propagation velocity are evaluated and compared with experimental data. A minimum threshold stretch of 3.0 is required for crack propagation. The numerical model is able to predict the dynamic crack growth. It appears that the strength and the surface energy vary with the crack speed. Finally, the maximum principal stretch and stress distribution around steadily propagation crack tip suggest that crystallization and cavity formation may take place.     

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.     

A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials

Numerical investigations are conducted to simulate high-speed crack propagation in pre-strained PMMA plates. In the simulations, the dynamic material separation is explicitly modeled by cohesive elements incorporating an initially rigid, linear-decaying cohesive law. Initial attempts using a rate-independent cohesive law failed to reproduce available experimental results as numerical crack velocities consistently overestimate experimental observations. As proof of concept, a phenomenological rate-dependent cohesive law, which bases itself on the physics of microcracking, is introduced to modulate the cohesive law with the macroscopic crack velocity. We then generalize this phenomenological approach by establishing a rate-dependent cohesive law, which relates the traction to the effective displacement and rate of change of effective displacement. It is shown that this new model produces numerical results in good agreement with experimental data. The analysis demonstrates that the simulation of high-speed crack propagation in brittle structures necessitates the use of rate-dependent cohesive models, which account for the complicated rate-process of dynamic fracture at the propagating crack tip.     

A boundary integral method for a dynamic, transient mode I crack problem with viscoelastic cohesive zone

We consider the problem of the dynamic, transient propagation of a semi-infinite, mode I crack in an infinite elastic body with a nonlinear, viscoelastic cohesize zone. Our problem formulation includes boundary conditions that preclude crack face interpenetration, in contrast to the usual mode I boundary conditions that assume all unloaded crack faces are stress-free. The nonlinear viscoelastic cohesive zone behavior is motivated by dynamic fracture in brittle polymers in which crack propagation is preceeded by significant crazing in a thin region surrounding the crack tip. We present a combined analytical/numerical solution method that involves reducing the problem to a Dirichlet-to-Neumann map along the crack face plane, resulting in a differo-integral equation relating the displacement and stress along the crack faces and within the cohesive zone.     

Dynamic response of interfacial finite cracks in orthotropic materials subjected to concentrated loads

A cohesive zone model has been proposed to model crack growth with a part-through process zone in a thin solid. With the solid being modeled in Kirchhoff’s plate theory, the crack with a relatively long, inclined front is modeled as a line discontinuity with a finite cohesive zone within the plate. A cohesive force law is adopted to capture the effect of residual strength and residual rigidity of a plate cross-section gradually cracking through the thickness. It is derived by a plane-strain elasticity analysis of a cross section normal to the part-through crack. It is then applied in the plate formulation of a line crack to simulate its propagation within the plate plane. This model essentially resolves the originally three-dimensional crack problem in two hierarchical steps, i.e., in the thickness and in the in-plane directions. In the present study, the bending case is considered. A boundary element method is applied to numerically derive the cohesive force law and simulate the crack growth in a thin titanium-alloy plate. The computational efficiency of the model is demonstrated. The plate is shown to fracture in a nominally brittle or ductile manner depending on its thickness.     

A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks

A cracked elastostatic structure is artificially divided into subdomains of simpler topology such that the well-developed classic dual integral equations can be applied appropriately to each domain. Applying the continuity and equilibrium conditions along artificial boundaries and properties of the integral kernels a single-domain dual-boundary-integral equation formulation is derived for a cracked elastic structure. A cohesive zone model is used to model the crack tip processes and is coupled with the single-domain dual-boundary-integral equation formulation; the resulting nonlinear equations are solved using the iterative method of successive-over-relaxation. The constitutive law used for a crack includes three parts: a law relating cohesive force to crack displacement difference when a crack is opening, a characterization of tangential interaction between crack surfaces when the crack surfaces are in contact, and a maximum principal stress criterion of crack advance. Incorporation of local unloading effect of the cohesive zone material has enabled a simulation of fracture with initial damage, partial development of the failure process zone at structural instability and multiple crack interaction. Some of the features of the method are demonstrated by considering three examples. The first problem is a single-edge-cracked specimen that exhibits a snap-back instability. The second example is the development of wing cracks from an angled crack under compression. The last example demonstrates the capability to consider mixed-mode crack growth and interaction of cracks. Thus, the problem of crack growth has been reduced to the determination of the cohesive model for the fracture process. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical study of geometric constraint and cohesive parameters in steady-state viscoelastic crack growth

This paper presents a finite element study of cohesive crack growth in a thin infinite viscoelastic strip to investigate the effects of viscoelastic properties, strip height, and cohesive model parameters on the crack growth resistance. The results of the study show that the dependence of the fracture energy on the viscoelastic properties for the strip problem is similar to that obtained for the infinite body problem even when the cohesive zone length is large compared to the height of the strip. The fracture energy also depends on the crack speed v through the dimensionless parameter v τ/L _∞ where L _∞ is the characteristic length of the cohesive zone and τ is the characteristic relaxation time of the bulk material. This relationship confirms that at least two properties of the fracture process must be prescribed accurately to model viscoelastic crack growth. In contrast, the fracture energy and crack speed are insensitive to the strip height even in situations where the growth of the dissipation zone is severely constrained by the strip boundaries. We observe that at high speeds, where the fracture energy asymptotically approaches the maximum value, the material surrounding the cohesive zone is in the rubbery (equilibrium) state and not the glassy state.     

Modeling of cohesive crack growth using an adaptive mesh refinement via the modified-SPR technique

In this paper, an adaptive finite element procedure is presented in modeling of mixed-mode cohesive crack propagation via the modified superconvergent path recovery technique. The adaptive mesh refinement is performed based on the Zienkiewicz–Zhu error estimator. The weighted-SPR recovery technique is employed to improve the accuracy of error estimation. The Espinosa–Zavattieri bilinear cohesive zone model is applied to implement the traction-separation law. It is worth mentioning that no previous information is necessary for the path of crack growth and no region of the domain is necessary to be filled by the cohesive elements. The maximum principal stress criterion is employed for predicting the direction of extension of the cohesive crack in order to implement the cohesive elements. Several numerical examples are analyzed numerically to demonstrate the capability and efficiency of proposed computational algorithm.     

On damage accumulations in the cyclic cohesive zone model for XFEM analysis of mixed-mode fatigue crack growth

Predicting mixed-mode fatigue crack propagation is an important and troublesome issue in structure assessment for decades. In the present paper an extended finite element method (XFEM) combined with a new cyclic cohesive zone model (CCZM) is introduced for simulating fatigue crack propagation under mixed-mode loading conditions, which has been implemented in the commercial general purpose software ABAQUS. The algorithm allows introducing a new crack surface at arbitrary locations and directions in a finite element mesh, without re-meshing. The cyclic cohesive zone model is based on the known S–N curves and Goodman diagram for metallic materials and validated by uniaxial tension results. Furthermore, the sensitivity of the model parameter is investigated for mixed-mode fatigue. The virtual crack closure technique has been extended to the cohesive zone model and proposed to calculate the energy release rate for the generalized Paris’ law. Finally, the crack propagation rate and direction under mixed-mode fatigue loading conditions are studied.     

Quasi‐static crack propagation in plane and plate structures using set‐valued traction‐separation laws

We introduce a numerical technique to model set‐valued traction‐separation laws in plate bending and also plane crack propagation problems. By using of recent developments in thin (Kirchhoff–Love) shell models and the extended finite element method, a complete and accurate algorithm for the cohesive law is presented and is used to determine the crack path. The cohesive law includes softening and unloading to origin, adhesion and contact. Pure debonding and contact are obtained as particular (degenerate) cases. A smooth root‐finding algorithm (based on the trust‐region method) is adopted. A step‐driven algorithm is described with a smoothed law which can be made arbitrarily close to the exact non‐smooth law. In the examples shown the results were found to be step‐size insensitive and accurate. In addition, the method provides the crack advance law, extracted from the cohesive law and the absence of stress singularity at the tip. Copyright © 2007 John Wiley & Sons, Ltd.     

A new nonlocal cohesive stress law and its applicable range

In this paper, we attempt to provide a new analytical method to determine the cohesive law in the framework of nonlocal continuum mechanics. Firstly, the equivalence between the cohesive stress and the surface-induced traction (nonlocal surface residual) is established on the basis of the nonlocal stress boundary condition. Then a new cohesive stress law is derived logically from the perspective of rational mechanics, which characterizes the dependence of the cohesive stress on the crack opening displacement (COD) within the cohesive zone. Finally, we apply this new cohesive crack model to two fracture examples with different cohesive zone sizes, and investigate the stress field ahead of the crack tip in detail. The results show that the stress singularity at the crack tip is removed, and the maximum stress occurs within the cohesive zone away from the crack tip. Moreover, the stress in the large-scale cohesive zone drops rapidly to a constant approaching zero, exhibiting a stronger softening behavior.     

Crack growth predictions in polyethylene using measured traction–separation curves

The accuracy of predicting the crack growth in any cohesive zone model calculation depends critically on the choice of cohesive law. A novel experimental method was used to measure directly such a cohesive law or ‘traction–separation curve’ in polyethylene. Deep notched tensile specimens were tested under constant displacement rate conditions, which facilitated a localisation of the damage mechanisms thought to precede crack growth and allowed a quantification of these processes independent of bulk deformation. Results showed that both the fracture energy and cohesive strength measured in this manner are a function of the applied rate and specimen geometry.Here we present a cohesive zone model within the finite-volume method to predict crack initiation and propagation history in three grades of polyethylene of different toughness, using the experimental measurements described above. The choice of cohesive law is crucial as it has a fundamental bearing on the predicted crack growth rates, particularly in tough polymers, where changes in the prevailing rate, constraint and temperature may affect the magnitude of the holding tractions within the damage zone. Initially a single experimentally measured, fixed rate traction–separation curve was used in the model as the fracture criterion but was unable to provide satisfactory crack growth predictions. By contrast, use of a more physically realistic family of curves measured at different rates provided better agreement of the prediction with experiment for the tough polyethylenes and very good agreement for the more brittle polyethylene. It was concluded that along with a rate dependent cohesive law, an accurate prediction of the crack growth history of tough polyethylenes would also require an incorporation of the effects of variations in constraint and perhaps also temperature. The ultimate goal may therefore be the development of a physical material model, sufficiently calibrated by experimental data, which would be able to accurately describe the local fracture process via a rate, constraint and temperature dependent traction–separation law.     

Suggestions to the cohesive traction–separation law from atomistic simulations

The cohesive model becomes popular in crack analysis for its clear physical background and flexible implementation. The cohesive traction–separation law, however, is a critical point and will generally be empirically assumed. In the present paper the cohesive traction–separation law is investigated based on constrained three-dimensional atomistic simulations. The computations under mode I conditions show that crack growth even in the nano-scale single-crystal aluminum is in the form of void nucleation, growth and coalescence, which is similar to ductile fracture at meso-scale. The concentrations of the atomic tensile stress and the atomic hydrostatic stress at a certain distance from the crack tip characterize void nucleation and final crack growth. The Mises stress does not play a role in the material failure in the nano-scale. This implies that ductile failure under mode I loading condition is dominated by the normal traction, which agrees with the assumption of the cohesive zone model. Variations of the atomic stresses near the crack tip provide the theoretical background for the cohesive zone model and can be used to identify the cohesive traction–separation law. The traction curve is very sensitive to the distance to the crack tip, which is related with the stress triaxiality. The atomistic simulations show tendentious agreement with the known cohesive traction–separation laws, whereas the scattering of the atomic stress versus separation implies effects of the hydrostatic stress in the traction–separation law. The computation provides important information for constructing the cohesive zone model.     

Prediction of 3D small fatigue crack propagation in shot-peened specimens

Shot peening has been widely applied in industrial design to improve fatigue durability of high loaded machine components. The compressive residual stress induced by shot peening is in general assumed to be responsible for the improvement of material fatigue strength. In the present work a cyclic cohesive zone model is extended to analyze three-dimensional fatigue crack growth in shot-peened specimens. Fatigue crack growth behaviors in both unpeened and peened specimens are investigated using 3D finite element analysis. The parameters of the cohesive zone model have been identified in 2D unpeened specimens and are applied to predict peened specimens directly. The results indicate that shot peening strongly affects crack initiation time and crack profiles, but has little effect on crack propagation rate. It implies that the shot peening will hardly change Paris’ law used for the damage tolerant design.     

A Viscoelastic Fictitious Crack Model for the Fracture of Sea Ice

The fracture of sea ice is modeled using a viscoelastic fictitious crack(cohesive zone) model. The sea ice is modeled as a linear viscoelastic material. The fictitious crack model is implemented via the weight function method. The associated stress-separation curve can be rate dependent. The impact of assuming viscoelastic behavior in the bulk as opposed to elastic behavior is studied. Results from the model are compared to the available exact results for various test cases. The model is applied to a large scale in situ sea ice fracture test. Various implications of such applications are pointed out. This viscoelastic fictitious crack model is found to be a promising tool in investigations pertaining to the fracture of sea ice. This revised version was published online in July 2006 with corrections to the Cover Date.     

A cohesive model for fatigue failure of polymers

A cohesive failure model is proposed to simulate fatigue crack propagation in polymeric materials. The model relies on the combination of a bi-linear cohesive failure law used for fracture simulations under monotonic loading and an evolution law relating the cohesive stiffness, the rate of crack opening displacement and the number of cycles since the onset of failure. The fatigue component of the cohesive model involves two parameters that can be readily calibrated based on the classical log-log Paris failure curve between the crack advance per cycle and the range of applied stress intensity factor. The paper also summarizes a semi-implicit implementation of the cohesive model into a cohesive-volumetric finite element framework, allowing for the simulation of a wide range of fatigue fracture problems.     

Damage Dependent Constitutive Behavior and Energy Release Rate for a Cohesive Zone in a Thermoviscoelastic Solid

In this paper a cohesive zone is introduced ahead of a crack tip in order to avoid the singularity at the crack tip. By applying thermodynamics to the cohesive zone and the surrounding body, a fracture criterion will be established so that the inelastic energy dissipation both in the cohesive zone and the surrounding bulk material can be distinguished from the energy released by fracture, and the propagation of crack can be predicted. In addition, the cohesive zone constitutive equation is constructed utilizing the Helmholtz free energy in the form of a single hereditary integral for a nonlinear viscoelastic material. The resulting constitutive model for the cohesive zone contains an internal state variable which represents the damage state within the cohesive zone. When the cohesive zone opening displacement is known, the energy release rate is thus history dependent, which is expressed in terms of the damage state, the length of separation in the cohesive zone and the geometric configuration of the cohesive zone opening displacement. Example results contained herein demonstrate this effect. This revised version was published online in July 2006 with corrections to the Cover Date.