Removing mesh bias in mixed-mode cohesive fracture simulation with stress recovery and domain integral

To remove mesh bias and provide an accurate crack path representation in mixed-mode investigation, a novel stress recovery technique is proposed in conjunction with a domain integral and element splits. Based on a domain integral and stress recovery technique, a maximum strain energy release rate is estimated to determine a crack path direction. Then, for a given crack path direction, continuum elements are split, and a cohesive surface element is adaptively inserted. One notes that the proposed stress recovery technique provides a more accurate stress field than a standard stress evaluation procedure. The proposed computational framework is verified and validated by solving mode-I and mixed-mode examples. Computational results demonstrate that the domain integral with the stress recovery accurately evaluates a crack path, even with a lower-quality mesh and under a biaxial stress state. Furthermore, the cohesive surface element approach, with the element split in conjunction with the stress recovery and the domain integral, predicts mixed-mode fracture behaviors while removing mesh bias in the crack path representation. Additionally, the condition numbers of stiffness matrices are within the same order of magnitude during cohesive fracture simulation.     

Modeling of crack propagation via an automatic adaptive mesh refinement based on modified superconvergent patch recovery technique

In this paper, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in a 2D finite element mesh. The Zienkiewicz-Zhu error estimator is employed in conjunction with a modified SPR technique based on the recovery of gradients using analytical crack-tip fields in order to obtain more accurate estimation of errors. The optimization of crack-tip singular finite element size is achieved through the adaptive mesh strategy. Finally, several numerical examples are illustrated to demonstrate the effectiveness, robustness and accuracy of computational algorithm in calculation of fracture parameters and prediction of crack path pattern.     

Crack Extension at an Interface: Prediction of Fracture Toughness and Simulation of Crack Path Deviation

Ductile tearing of laser welded Al sheets is studied both experimentally and numerically. The mechanical behaviour and the microstructure of the various zones of the weld are characterised. Mechanical tests on compact tension (C(T))-specimen are carried out, with the position of the initial crack in the heat affected zone. Due to the asymmetry of the configuration, crack path deviation towards the softer fusion zone is observed. The topography of the non-planar fracture surface is measured using laser equipment. This work is focussed on the prediction of the fracture resistance and the simulation of crack path deviation for the respective configuration. The numerical simulations are based on two different models for ductile damage: the micromechanical Gurson–Tvergaard– Needleman (GTN) model and the phenomenological cohesive model. In the case of the GTN-model, the crack front may follow an arbitrary path. In contrast, the crack propagation direction for the cohesive model is prescribed by the morphology of the finite element mesh. The GTN-model is used to investigate crack path deviation and to derive limits for simplifications used together with the cohesive approach. The latter allows for a cost-efficient 2D simulation. Good agreement between experimental results and numerical simulations could be achieved in all cases     

J resistance behavior in functionally graded materials using cohesive zone and modified boundary layer models

This paper describes elastic–plastic crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) under mode I loading conditions using cohesive zone and modified boundary layer (MBL) models. For this purpose, we first explore the applicability of two existing, phenomenological cohesive zone models for FGMs. Based on these investigations, we propose a new cohesive zone model. Then, we perform crack growth simulations for TiB/Ti FGM SE(B) and SE(T) specimens using the three cohesive zone models mentioned above. The crack growth resistance of the FGM is characterized by the J-integral. These results show that the two existing cohesive zone models overestimate the actual J value, whereas the model proposed in the present study closely captures the actual fracture and crack growth behaviors of the FGM. Finally, the cohesive zone models are employed in conjunction with the MBL model. The two existing cohesive zone models fail to produce the desired K–T stress field for the MBL model. On the other hand, the proposed cohesive zone model yields the desired K–T stress field for the MBL model, and thus yields J _R curves that match the ones obtained from the SE(B) and SE(T) specimens. These results verify the application of the MBL model to simulate crack growth resistance in FGMs.     

Numerical Analysis of Stiffener Runout Sections

The recent trend of incorporating more composite material in primary aircraft structures has highlighted the vulnerability of stiffened aerostructures to through-thickness stresses, which may lead to delamination and debonding at the skin–stiffener interface, leading to collapse. Stiffener runout regions are particularly susceptible to this problem and cannot be avoided due to the necessity to terminate stiffeners at rib intersections or at cutouts, interrupting the stiffener load path. In this paper, experimental tests relating to two different stiffener runout specimens are presented and the failure modes of both specimens are discussed in detail. A thinner-skinned specimen showed sudden and unstable crack propagation, while a thicker-skinned specimen showed initially unstable but subsequent stable crack growth. Detailed finite element models of the two specimens are developed, and it is shown how such models can explain and predict the behaviour and failure mode of stiffener runouts. The models contain continuum shell elements to model the skin and stiffener, while cohesive elements using a traction-separation law are placed at the skin–stiffener interface to effectively model the debonding which promotes structural failure.     

Unstructured polygonal meshes with adaptive refinement for the numerical simulation of dynamic cohesive fracture

This paper presents a scheme for adaptive mesh refinement on unstructured polygonal meshes to better capture crack patterns in dynamic cohesive fracture simulations. A randomly seeded polygonal mesh leads to an isotropic discretization of the problem domain, which does not bias crack patterns, but restricts the number of paths that a crack may travel at each node. An adaptive refinement scheme is proposed and investigated through a detailed set of geometric studies. The refinement scheme is selectively chosen to optimize the number of paths that a crack may travel, while still maintaining a conforming domain discretization. The details of the refinement scheme are outlined, along with the criterion used to determine the region of refinement and the method of interpolating nodal attributes. Extrinsic cohesive elements are inserted when and where necessary, and follow the constitutive response of the Park–Paulino–Roesler cohesive model. The influence of bulk and cohesive material heterogeneity is investigated through the use of a statistical distribution of material properties. The adaptive mesh modifications are handled through a compact topological data structure. Numerical examples highlight the features of adaptive refinement in capturing physical fracture patterns while addressing computational cost. Thus, the present approach is a step towards obtaining accurate dynamic fracture patterns and fields with polygonal elements.     

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.     

Numerical simulation of multiple crack growth in brittle materials with adaptive remeshing

In this paper, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in multiple cracked bodies. The Zienkiewicz–Zhu error estimator is employed in conjunction with the modified superconvergent patch recovery (SPR) technique to obtain more accurate estimation of error. A stability analysis is performed to determine active cracks from a set of competitive cracks. Various crack growth criteria together with the respective crack trajectory prediction are compared. Several numerical examples are illustrated to demonstrate the efficiency, robustness and accuracy of computational algorithm in the simulation of multiple crack growth. Copyright © 2010 John Wiley & Sons, Ltd.     

Error estimation and adaptivity for discontinuous failure

Discontinuous failure is simulated via the introduction of a geometrical discontinuity. The cohesive zone is modelled via the use of the partition‐of‐unity property of the finite element interpolation. By this approach, a crack can pass through the elements without any restriction to the underlying mesh. Despite such a feature, it has been confirmed that a sufficiently fine mesh discretization still needs to be ensured in order to obtain a correct crack path and mechanical response. The p‐adaptive scheme, driven by error in an energy norm measure or a goal‐oriented measure, has been examined due to its implementational simplicity. The results have shown that, if considering only increasing the polynomial degree, the p‐approach can greatly improve the results. Copyright © 2008 John Wiley & Sons, Ltd.     

Fracture analysis in 2D magneto–electro–elastic media by the boundary element method

An integral formulation for 2D cracked infinite anisotropic magneto–electro–elastic media is presented. Based on the method proposed by Garcia-Sanchez et al. (Comput Struct 83: 804–820, 2005), the hypersingular kernels are analytically transformed into weakly singular and regular integrals with known analytical solution. Special quadratic discontinuous crack tip elements are employed to model the singular characteristics of the stresses, electric displacements and magnetic inductions. The extended stress intensity factors at the crack tips are calculated using the extended discontinuous displacements at crack tip elements based on one point extended displacement formulation. Some results for curved cracks in magneto–electro–elastic media are also presented.     

Two-dimensional analysis of anisotropic crack problems using coupled meshless and fractal finite element method

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with fractal the finite element method (FFEM) for analyzing homogeneous, anisotropic, and two dimensional linear-elastic cracked structures subjected to mixed-mode (modes I and II) loading conditions. FFEM is adopted for discretization of domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprises both the element-free Galerkin and the finite element shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM-FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary and no post-processing (employing any path independent integrals) is needed to determine fracture parameters such as stress-intensity factors (SIFs) and T − stress. The numerical results based on all four orthotropic cases show that SIFs and T − stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions.     

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     

Experimental validation of large-scale simulations of dynamic fracture along weak planes

A well-controlled and minimal experimental scheme for dynamic fracture along weak planes is specifically designed for the validation of large-scale simulations using cohesive finite elements. The role of the experiments in the integrated approach is two-fold. On the one hand, careful measurements provide accurate boundary conditions and material parameters for a complete setup of the simulations without free parameters. On the other hand, quantitative performance metrics are provided by the experiments, which are compared a posteriori with the results of the simulations. A modified Hopkinson bar setup in association with notch-face loading is used to obtain controlled loading of the fracture specimens. An inverse problem of cohesive zone modeling is performed to obtain accurate mode-I cohesive zone laws from experimentally measured deformation fields. The speckle interferometry technique is employed to obtain the experimentally measured deformation field. Dynamic photoelasticity in conjunction with high-speed photography is used to capture experimental records of crack propagation. The comparison shows that both the experiments and the numerical simulations result in very similar crack initiation times and produce crack tip velocities which differ by less than 6%. The results also confirm that the detailed shape of the non-linear cohesive zone law has no significant influence on the numerical results.     

3D adaptive finite element modeling of non-planar curved crack growth using the weighted superconvergent patch recovery method

In this paper, an adaptive finite element analysis is presented for 3D modeling of non-planar curved crack growth. The fracture mechanical evaluation is performed based on a general technique for non-planar curved cracks. The Schollmann’s crack kinking criterion is used for the process of crack propagation in 3D problems. The Zienkiewicz-Zhu error estimator is employed in conjunction with a weighted SPR technique at each patch to improve the accuracy of error estimation. Applying the proposed technique to 3D non-planar curved crack growth problems shows significant improvements particularly at the boundaries and near crack tip regions. Several numerical examples are presented to illustrate the robustness of the proposed technique.     

A Cohesive Zone Approach for Fatigue-Driven Delamination Analysis in Composite Materials

A new model for prediction of fatigue-driven delamination in laminated composites is proposed using cohesive interface elements. The presented model provides a link between cohesive elements damage evolution rate and crack growth rate of Paris law. This is beneficial since no additional material parameters are required and the well-known Paris law constants are used. The link between the cohesive zone method and fracture mechanics is achieved without use of effective length which has led to more accurate results. The problem of unknown failure path in calculation of the energy release rate is solved by imposing a condition on the damage model which leads to completely vertical failure path. A global measure of energy release rate is used for the whole cohesive zone which is computationally more efficient compared to previous similar models. The performance of the proposed model is investigated by simulation of well-known delamination tests and comparison against experimental data of the literature.     

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.     

Interaction of Matrix Cracking and Delamination in Cross-ply Laminates: Simulations with Stochastic Cohesive Zone Elements

Two main damage mechanisms of laminates—matrix cracking and inter-ply delaminationare closely linked together (Joshi and Sun 1). This paper is focussed on interaction between matrix cracking and delamination failure mechanisms in CFRP cross-ply laminates under quasi-static tensile loading. In the first part of the work, a transverse crack is introduced in 90o layers of the cross-ply laminate [01/904/01], and the stresses and strains that arise due to tensile loading are analyzed. In the second part, the cohesive zone modelling approach where the constitutive behaviour of the cohesive elements is governed by traction-displacement relationship is employed to deal with the problem of delamination initiation from the matrix crack introduced in the 90o layers of the laminate specimen. Additionally, the effect of microstructural randomness, exhibited by CFRP laminates on the damage behaviour of these laminates is also accounted for in simulations. This effect is studied in numerical finite-element simulations by introducing stochastic cohesive zone elements. The proposed damage modelling effectively simulated the interaction between the matrix crack and delamination and the variations in the stresses, damage and crack lengths of the laminate specimen due to the microstructural randomness.     

Towards optimization of crack resistance of composite materials by adjustment of fiber shapes

We investigate the evolution and propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. In addition to the classical variational formulation, where the standard potential energy is minimized over the cracked domain under physical conditions characterizing the behavior of the material close to the crack (e.g. non-penetration conditions), we include a ‘cohesive traction term’ in the energy expression. In this way we obtain a mathematically concise set of partial differential equations with non-linear boundary conditions at the crack interfaces. We perform a finite element discretization using a combination of standard continuous finite elements and so-called cohesive elements. During the simulation process cohesive elements are adaptively inserted at positions where a certain stress bound is exceeded. In our numerical studies we consider domains consisting of a matrix material with fiber inclusions. Beyond pure crack path simulation, our ultimate goal is to determine an optimal shape of the fibers resulting in a crack path that releases for a given load scenario as much energy as possible without destroying the specimen completely. We develop a corresponding optimization model and propose a solution algorithm for the same. The article is concluded by numerical results.     

Discrete cohesive zone model for mixed-mode fracture using finite element analysis

The discrete cohesive zone model (DCZM) is implemented using the finite element (FE) method to simulate fracture initiation and subsequent growth when material non-linear effects are significant. Different from the widely used continuum cohesive zone model (CCZM) where the cohesive zone model is implemented within continuum type elements and the cohesive law is applied at each integral point, DCZM uses rod type elements and applies the cohesive law as the rod internal force vs. nodal separation (or rod elongation). These rod elements have the provision of being represented as spring type elements and this is what is considered in the present paper. A series of 1D interface elements was placed between node pairs along the intended fracture path to simulate fracture initiation and growth. Dummy nodes were introduced within the interface element to extract information regarding the mesh size and the crack path orientation. To illustrate the DCZM, three popular fracture test configurations were examined. For pure mode I, the double cantilever beam configuration, using both uniform and biased meshes were analyzed and the results show that the DCZM is not sensitive to the mesh size. Results also show that DCZM is not sensitive to the loading increment, either. Next, the end notched flexure for pure mode II and, the mixed-mode bending were studied to further investigate the approach. No convergence difficulty was encountered during the crack growth analyses. Therefore, the proposed DCZM approach is a simple but promising tool in analyzing very general two-dimensional crack growth problems. This approach has been implemented in the commercial FEA software ABAQUS^® using a user defined subroutine and should be very useful in performing structural integrity analysis of cracked structures by engineers using ABAQUS^®.