Assessment of energy dissipation during mixed-mode delamination growth using cohesive zone models

The use of cohesive elements to simulate delamination growth involves modeling the inelastic region existing ahead of the crack tip. Recent numerical and experimental findings indicate that the mixed-mode ratio varies at each material point within the inelastic region ahead of the crack tip during crack propagation, even for those specimens whose mixed-mode ratio is expected to be constant. Although the local variation of the mode mixity may adversely affect the predicted numerical results, most existing formulations do not take it into account. In this work, the mode-decomposed J-integral is implemented as a finite element post-processing tool to obtain the strain energy release rates and the mixed-mode ratio of the inelastic region as a whole, allowing the assessment of crack propagation in terms of energy dissipation and mixed-mode ratio computation. Different cohesive elements are assessed with this method.     

Simulating damage and delamination in fibre metal laminate joints using a three-dimensional damage model with cohesive elements and damage regularisation

This paper investigates the capability of a three-dimensional finite element model with damaging material behaviour, cohesive elements and damage regularisation to simulate complex damage patterns in fibre metal laminate (FML) joints. The model incorporates a three-dimensional continuum damage mechanics approach for the composite plies, a plasticity model for the aluminium layers, and a delamination model between layers. A nonlocal averaging scheme is implemented to mitigate the mesh sensitivity that occurs with strain-softening material models. Bearing stress-strain responses and variations in stiffness are calculated, and damage progression is described in detail for all plies and interfaces. Microscopy and stress-strain data from a parallel series of experimental tests are presented, and damage and failure phenomena observed in the tests are compared with the model. Generally, good agreement between model and tests was achieved but certain limitations of the numerical model were observed and are discussed. The combined numerical and experimental information provide a detailed understanding of the failure sequence of FML joints.     

The proper generalized decomposition for the simulation of delamination using cohesive zone model

The use of cohesive zone models is an efficient way to treat the damage especially when the crack path is known a priori. It is the case in the modeling of delamination in composite laminates. However, the simulations using cohesive zone models are expensive in a computational point of view. When using implicit time integration or when solving static problems, the non‐linearity related to the cohesive model requires many iteration before reaching convergence. In explicit approaches, an important number of iterations are also needed because of the time step stability condition. In this article, a new approach based on a separated representation of the solution is proposed. The proper generalized decomposition is used to build the solution. This technique coupled with a cohesive zone model allows a significant reduction of the computational cost. The results approximated with the proper generalized decomposition are very close the ones obtained using the classical finite element approach. Copyright © 2014 John Wiley & Sons, Ltd.     

A self-adaptive finite element approach for simulation of mixed-mode delamination using cohesive zone models

Oscillations observed in the load–displacement response of brittle interfaces modeled by cohesive zone elements in a quasi-static finite element framework are artifacts of the discretization. The typical limit points in this oscillatory path can be traced by application of path-following techniques, or avoided altogether by adequately refining the mesh until the standard iterative Newton–Raphson method becomes applicable. Both strategies however lead to an unacceptably high computational cost and a low efficiency, justifying the development of a process driven hierarchical extension of the discretization used in the process zone of a cohesive crack. A self-adaptive enrichment scheme within individual cohesive zone elements driven by the physics governing the problem, is an efficient solution that does not require further mesh refinements. A two-dimensional mixed-mode example in a general framework with an irreversible cohesive zone law shows that an enriched formulation restores the smoothness of the solution in structures that are discretized in a relatively coarse manner.     

An enriched cohesive zone model for delamination in brittle interfaces

Application of standard cohesive zone models in a finite element framework to simulate delamination in brittle interfaces may trigger non‐smooth load–displacement responses that lead to the failure of iterative solution procedures. This non‐smoothness is an artifact of the discretization; and hence it can be avoided by sufficiently refining the mesh leading to unacceptably high computational costs and a low efficiency and robustness. In this paper, a process‐driven hierarchical extension is proposed to enrich the separation approximation in the process zone of a cohesive crack. Some numerical examples show that instead of mesh refinement, a more efficient enriched formulation can be used to prevent a non‐smooth solution. Copyright © 2009 John Wiley & Sons, Ltd.     

A cohesive finite element formulation for modelling fracture and delamination in solids

In recent years, cohesive zone models have been employed to simulate fracture and delamination in solids. This paper presents in detail the formulation for incorporating cohesive zone models within the framework of a large deformation finite element procedure. A special Ritz-finite element technique is employed to control nodal instabilities that may arise when the cohesive elements experience material softening and lose their stress carrying capacity. A few simple problems are presented to validate the implementation of the cohesive element formulation and to demonstrate the robustness of the Ritz solution method. Finally, quasi-static crack growth along the interface in an adhesively bonded system is simulated employing the cohesive zone model. The crack growth resistance curves obtained from the simulations show trends similar to those observed in experimental studies     

An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models

A methodology to determine the constitutive parameters for the simulation of progressive delamination is proposed. The procedure accounts for the size of a cohesive finite element and the length of the cohesive zone to ensure the correct dissipation of energy. In addition, a closed-form expression for estimating the minimum penalty stiffness necessary for the constitutive equation of a cohesive finite element is presented. It is shown that the resulting constitutive law allows the use of coarser finite element meshes than is usually admissible, which renders the analysis of large-scale progressive delamination problems computationally tractable.     

A cohesive zone model for predicting delamination suppression in z-pinned laminates

This paper presents a cohesive zone model based finite element analysis of delamination resistance of z-pin reinforced double cantilever beam (DCB). The main difference between this and existing cohesive zone models is that each z-pin bridging force is governed by a traction-separation law derived from a meso-mechanical model of the pin pullout process, which is independent of the fracture toughness of unreinforced laminate. Therefore, two different traction-separation laws are used: one representing the toughness of unreinforced laminate and the other the enhanced delamination toughness owing to the pin bridging action. This approach can account for the large scale bridging effect and avoid using concentrated pin forces, thus removing the mesh dependency and permitting more accurate analysis solution. Computations were performed using a simplified unit strip model. Predicted delamination growth and load vs. displacement relation are in excellent agreement with the prediction by a complete model, and both models are in good agreement with test measured load vs. displacement relation. For a pinned DCB specimen, the unit strip model can reduce the computing time by 85%.     

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.     

复合材料分层问题中界面层方法的数值研究

通过对双悬臂梁(DCB) 开裂问题的数值模拟, 研究了使用界面层方法时所遇到的数值问题。重点讨论了静态方法和准静态方法对计算效率的影响以及界面层的应力应变函数对计算精度的影响。对模拟过程中所遇到的精度及收敛性问题进行了研究。研究表明, 准静态方法可有效模拟分层问题, 但界面层的本构关系应具有一个缓慢的退化过程。      

Fatigue damage modelling of adhesively bonded joints under variable amplitude loading using a cohesive zone model

In this paper, the fatigue response of adhesively bonded joints under variable amplitude (VA) cyclic loading was predicted using a numerical model. The adhesive layer was modelled using the cohesive zone model with a bi-linear traction-separation response. A damage model, incorporating fatigue load ratio effects, was utilised in conjunction with the cohesive zone model to simulate the detrimental influence of VA fatigue loading. This model was validated against published experimental results obtained from fatigue tests of adhesively bonded single lap joints subjected to various types of VA fatigue loading spectra. This model successfully predicted the damaging effect of VA fatigue loading on the adhesively bonded joints and was generally found to be a significant improvement on the other damage models considered.     

Modeling the progressive damage of the microdroplet test using contact surfaces with cohesive behavior

A quarter-symmetric, three-dimensional finite element model is used to determine the stress state and progressive failure of the fiber–matrix interface during a microdroplet pull-out test of a glass fiber/epoxy matrix system. The microdroplet system interphase region is modeled using contact with cohesive behavior and is able to undergo progressive damage. The cohesive model strength and toughness are varied to determine their influence on the macro-behavior of the system and the simulations are compared to experimental results from the literature. Additionally, geometric test parameters including the blade opening, fiber free length, and fiber diameter are varied to assess their relative influence on the macro-response of the system.     

Adaptive hierarchical enrichment for delamination fracture using a decohesive zone model

The paper describes a method for modelling delamination in fibre‐reinforced composite structures with the aid of a decohesive zone model and interface elements. Unless a fine mesh is provided, the resulting load/deflection responses are very non‐smooth and the iterative non‐linear solution procedure may fail. To overcome this problem, the elements around the softening process zone are enriched with hierarchical polynomial functions. The enriched zones change as the analysis proceeds and the cracks propagate. This procedure is implemented using a technique which continually modifies the boundary conditions. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Modelling the quasi-static behaviour of bituminous material using a cohesive zone model

This paper investigates the applicability of a cohesive zone model for simulating the performance of bituminous material subjected to quasi-static loading. The Dugdale traction law was implemented within a finite volume code in order to simulate the binder course mortar material response when subjected to indirect tensile loading. A uniaxial tensile test and a three-point bend test were employed to determine initial stress-strain curves at different test rates and the cohesive zone parameters (specifically, fracture energy and cohesive strength). Numerical results agree well with the experimental data up to the peak load and onset of fracture, demonstrating the value of the cohesive zone modelling technique in successfully predicting fracture initiation and maximum material strength.     

Homogenization of glass/alumina two-phase materials using a cohesive zone model

This paper is devoted to glass/alumina materials which can exhibit important damages at the interface in the case of a difference of the coefficients of thermal expansion between the matrix and the inclusions. One of the key-issue for this type of materials use is to determine and quantify this damage. This paper presents numerical and experimental approaches for this purpose. A numerical model is used and coupled to a homogenization procedure to evaluate damages provoked by the difference of the coefficients of thermal expansion. Cohesive elements have been introduced in the FE model so as to simulate the interface damage. Some thermal tests have been done on two-phase specimens. Experimental and numerical results are compared and analyzed.     

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.