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.     

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.     

Phase-field modeling of brittle fracture in a 3D polycrystalline material via an adaptive isogeometric-meshfree approach

Phase-field modeling, which introduces the regularized representation of sharp crack topologies, provides a convenient strategy for tackling 3D fracture problems. In this work, an adaptive isogeometric-meshfree approach is developed for the phase-field modeling of brittle fracture in a 3D polycrystalline material. The isogeometric-meshfree approach uses moving least-squares approximations to construct the equivalence between isogeometric basis functions and meshfree shape functions, thus inheriting the flexible local mesh refinement scheme from a meshfree method. This refinement scheme is improved by introducing an error estimator that includes both the phase field and its gradient. With the present approach, numerical implementations of the adaptive phase-field modeling that introduces the anisotropy of fracture resistance in polycrystals are proposed. In this way, propagating cracks can be dynamically tracked, and the mesh near cracks is refined in a meshfree manner without requiring a priori knowledge of crack paths. Furthermore, the intergranular and transgranular crack propagation patterns in polycrystalline materials can be simulated by the present approach. A series of numerical examples that deal with the isotropic and anisotropic fracture are investigated to demonstrate the robustness and effectiveness of the proposed approach.     

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.     

Multiscale dynamic fracture analysis of composite materials using adaptive microstructure modeling

In this study, a computational framework is proposed to investigate multiscale dynamic fracture phenomena in materials with microstructures. The micro- and macro-scales of a composite material are integrated by introducing an adaptive microstructure representation. Then, the far and local fields are simultaneously computed using the equation of motion, which satisfies the boundary conditions between the two fields. Cohesive surface elements are dynamically inserted where and when needed, and the Park-Paulino-Roesler cohesive model is employed to approximate nonlinear fracture processes in a local field. A topology-based data structure is utilized to efficiently handle adjacency information during mesh modification events. The efficiency and validity of the proposed computational framework are demonstrated by checking the energy balances and comparing the results of the proposed computation with direct computations. Furthermore, the effects of microstructural properties, such as interfacial bonding strength and unit cell arrangement, on the dynamic fracture behavior are investigated. The computational results demonstrate that local crack patterns depend on the combination of microstructural properties such as unit cell arrangement and interfacial bonding strength; therefore, the microstructure of a material should be carefully considered for dynamic cohesive fracture investigations.     

A finite element method with mesh‐separation‐based approximation technique and its application in modeling crack propagation with adaptive mesh refinement

This paper presents a FEM with mesh‐separation‐based approximation technique that separates a standard element into three geometrically independent elements. A dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation. The novel technique makes it very easy for mesh generation of problems with complex or solution‐dependent, varying geometry. It offers a flexible way to construct displacement approximations and provides a unified framework for the FEM to enjoy some of the key advantages of the Hansbo and Hansbo method, the meshfree methods, the semi‐analytical FEMs, and the smoothed FEM. For problems with evolving discontinuities, the method enables the devising of an efficient crack‐tip adaptive mesh refinement strategy to improve the accuracy of crack‐tip fields. Both the discontinuities due to intra‐element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level. The effectiveness and robustness of the present method are benchmarked with several numerical examples. The numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test, and it is essential to apply local adaptive mesh refinement for low fracture energy problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Modelling multiple cohesive crack propagation using a finite element–scaled boundary finite element coupled method

This paper presents an extension of the recently-developed finite element–scaled boundary finite element (FEM–SBFEM) coupled method to model multiple crack propagation in concrete. The concrete bulk and fracture process zones are modelled using SBFEM and nonlinear cohesive interface finite elements (CIEs), respectively. The CIEs are automatically inserted into the SBFEM mesh as the cracks propagate. The algorithm previously devised for single crack propagation is augmented to model problems with multiple cracks and to allow cracks to initiate in an un-cracked SBFEM mesh. It also addresses crack propagation from one subdomain into another, as a result of partitioning a coarse SBFEM mesh, required for some mixed–mode problems. Each crack in the SBFEM mesh propagates when the sign of the Mode-I stress intensity factor at the crack tip turns positive from negative. Its propagation angle is determined using linear elastic fracture mechanics criteria. Three concrete beams involving multiple crack propagation are modelled. The predicted crack propagation patterns and load–displacement curves are in good agreement with data reported in literature.     

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.     

Smart computational fracture of materials and structures

The use of the finite element method for complex engineering problems is now common. To ease the burden on the engineer the development of smart or adaptive computational methods is now required to model complex problems. In this paper we investigate the development of an adaptive finite element method for fracture-related problems. The adaptive method involves various stages which include the finite element analysis, error estimation/indication, mesh refinement and fracture/failure analysis in a loop. Some simple error estimators, based on stress projection, are used to investigate the adaptive finite element process. Element refinement is based on three schemes; the first and second are a simple and hierarchical refinement scheme with transitioning which avoids the need for constraint equations between element boundaries. Another scheme based on constraint equations between elements is also examined. The energy norm is used to estimate the element error. The software has the ability to introduce a discrete fracture in the structure according to standard fracture analysis practice. Crack tip parameters are calculated using a least-squares fit of the displacements into the asymptotic crack tip displacement field. Some simple examples are used to investigate the adaptive process, its behavior and some of the practical problems encountered. The convergence and equilibrium of the adaptive process, in terms of global error in the energy norm, are investigated. In the example the same problem is analyzed using both a fine computational grid and a coarse one. The coarse mesh is then adapted using the three different procedures available. The estimated error in the solution and the stress intensity are shown against the number of elements and number of iterations. Some further areas of research in adaptive finite element analysis are discussed.     

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.     

Cohesive modeling of dynamic fracture in functionally graded materials

The dynamic fracture of functionally graded materials (FGMs) is modeled using an explicit cohesive volumetric finite element scheme that incorporates spatially varying constitutive and failure properties. The cohesive element response is described by a rate-independent bilinear cohesive failure model between the cohesive traction acting along the cohesive zone and the associated crack opening displacement. A detailed convergence analysis is conducted to quantify the effect of the material gradient on the ability of the numerical scheme to capture elastodynamic wave propagation. To validate the numerical scheme, we simulate dynamic fracture experiments performed on model FGM compact tension specimens made of a polyester resin with varying amounts of plasticizer. The cohesive finite element scheme is then used in a parametric study of mode I dynamic failure of a Ti/TiB FGM, with special emphasis on the effect of the material gradient on the initiation, propagation and arrest of the crack.     

Multi‐level hp‐adaptivity for cohesive fracture modeling

Discretization‐induced oscillations in the load–displacement curve are a well‐known problem for simulations of cohesive crack growth with finite elements. The problem results from an insufficient resolution of the complex stress state within the cohesive zone ahead of the crack tip. This work demonstrates that the hp‐version of the finite element method is ideally suited to resolve this complex and localized solution characteristic with high accuracy and low computational effort. To this end, we formulate a local and hierarchic mesh refinement scheme that follows dynamically the propagating crack tip. In this way, the usually applied static a priori mesh refinement along the complete potential crack path is avoided, which significantly reduces the size of the numerical problem. Studying systematically the influence of h‐refinement, p‐refinement, and hp‐refinement, we demonstrate why the suggested hp‐formulation allows to capture accurately the complex stress state at the crack front preventing artificial snap‐through and snap‐back effects. This allows to decrease significantly the number of degrees of freedom and the simulation runtime. Furthermore, we show that by combining this idea with the finite cell method, the crack propagation within complex domains can be simulated efficiently without resolving the geometry by the mesh. Copyright © 2016 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.     

On the convergence of 3D free discontinuity models in variational fracture

Free discontinuity problems arising in the variational theory for fracture mechanics are considered. A Γ -convergence proof for an r-adaptive 3D finite element discretization is given in the case of a brittle material. The optimal displacement field, crack pattern and mesh geometry are obtained through a variational procedure that encompasses both mechanical and configurational forces. Possible extensions to cohesive fracture and quasi-static evolutions are discussed.     

Bounds for element size in a variable stiffness cohesive finite element model

The cohesive finite element method (CFEM) allows explicit modelling of fracture processes. One form of CFEM models integrates cohesive surfaces along all finite element boundaries, facilitating the explicit resolution of arbitrary fracture paths and fracture patterns. This framework also permits explicit account of arbitrary microstructures with multiple length scales, allowing the effects of material heterogeneity, phase morphology, phase size and phase distribution to be quantified. However, use of this form of CFEM with cohesive traction–separation laws with finite initial stiffness imposes two competing requirements on the finite element size. On one hand, an upper bound is needed to ensure that fields within crack‐tip cohesive zones are accurately described. On the other hand, a lower bound is also required to ensure that the discrete model closely approximates the physical problem at hand. Both issues are analysed in this paper within the context of fracture in multi‐phase composite microstructures and a variable stiffness bilinear cohesive model. The resulting criterion for solution convergence is given for meshes with uniform, cross‐triangle elements. A series of calculations is carried out to illustrate the issues discussed and to verify the criterion given. These simulations concern dynamic crack growth in an Al₂O₃ ceramic and in an Al₂O₃/TiB₂ ceramic composite whose phases are modelled as being hyperelastic in constitutive behaviour. Copyright © 2004 John Wiley & Sons, Ltd.     

New crack‐tip elements for XFEM and applications to cohesive cracks

An extended finite element method scheme for a static cohesive crack is developed with a new formulation for elements containing crack tips. This method can treat arbitrary cracks independent of the mesh and crack growth without remeshing. All cracked elements are enriched by the sign function so that no blending of the local partition of unity is required. This method is able to treat the entire crack with only one type of enrichment function, including the elements containing the crack tip. This scheme is applied to linear 3‐node triangular elements and quadratic 6‐node triangular elements. To ensure smooth crack closing of the cohesive crack, the stress projection normal to the crack tip is imposed to be equal to the material strength. The equilibrium equation and the traction condition are solved by the Newton–Raphson method to obtain the nodal displacements and the external load simultaneously. The results obtained by the new extended finite element method are compared to reference solutions and show excellent agreement. Copyright © 2003 John Wiley & Sons, Ltd.     

Modelling of crack propagation of gravity dams by scaled boundary polygons and cohesive crack model

Crack propagation in concrete gravity dams is investigated using scaled boundary polygons coupled with interface elements. The concrete bulk is assumed to be linear elastic and is modelled by the scaled boundary polygons. The interface elements model the fracture process zone between the crack faces. The cohesive tractions are modelled as side-face tractions in the scaled boundary polygons. The solution of the stress field around the crack tip is expressed semi-analytically as a power series. It reproduces the singular and higher-order terms in an asymptotic solution, such as the William’s eigenfunction expansion when the cohesive tractions vanish. Accurate results can be obtained without asymptotic enrichment or local mesh refinement. The stress intensity factors are obtained directly from their definition and provide a convenient and accurate means to assess the zero-K condition, which determines the stability of a cohesive crack. The direction of crack propagation is determined from the maximum circumferential stress criterion. To accommodate crack propagation, a local remeshing algorithm that is applicable to any polygon mesh is augmented by inserting cohesive interface elements between the crack surfaces as the cracks propagate. Three numerical benchmarks involving crack propagation in concrete gravity dams are modelled. The results are compared to the experimental and other numerical simulations reported in the literature.     

A numerical approach to the analysis of failure modes in anisotropic plates

This study is motivated by the attempt to characterize failure modes of silicon chips commonly used in electronic industries. Previous experimental investigations provided the failure probability of dies made of a single-crystal and produced a large variety of crack patterns, but were not able to elucidate the link between defect distributions and crack initiation and propagation. To get some insight in the fracture activation and propagation mechanisms, we resort to finite element analyses and adopt an explicit methodology for crack tracking, based on the self-adaptive insertion of cohesive elements into a coherent mesh of solid elements. Finite kinematics material models with anisotropic features for both bulk and cohesive surfaces are employed to describe the behavior of single-crystal silicon plates undergoing a particular bending test up to failure. The cohesive model adopted in the calculation is fully anisotropic and newly formulated to accomplish the present study. Numerical simulations considering different material properties were able to ascertain the effects of particular flaws on failure modes of brittle silicon plates.