Theory and numerics for finite deformation fracture modelling using strong discontinuities

A general finite element approach for the modelling of fracture is presented for the geometrically non‐linear case. The kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements. Thus, the deformation map is defined in terms of one continuous and one discontinuous portion, considered as mutually independent, giving rise to a weak formulation of the equilibrium consisting of two coupled equations. In addition, two different fracture criteria are considered. Firstly, a principle stress criterion in terms of the material Mandel stress in conjunction with a material cohesive zone law, relating the cohesive Mandel traction to a material displacement ‘jump’ associated with the direct discontinuity. Secondly, a criterion of Griffith type is formulated in terms of the material‐crack‐driving force (MCDF) with the crack propagation direction determined by the direction of the force, corresponding to the direction of maximum energy release. Apart from the material modelling, the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria. Copyright © 2005 John Wiley & Sons, Ltd.     

A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization

A variational formulation of quasi-static brittle fracture in elastic solids at small strains is proposed and an associated finite element implementation is presented. On the theoretical side, a consistent thermodynamic framework for brittle crack propagation is outlined. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip. On the numerical side, we first consider a standard finite element discretization in the two-dimensional space which yields a discrete formulation of the global dissipation in terms of configurational nodal forces. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity for two-dimensional problems is performed by the doubling of critical nodes and interface segments of the mesh. A crucial step for the success of this procedure is its embedding into a r-adaptive crack-segment re-orientation algorithm governed by configurational-force-based directional indicators. Here, successive crack propagation is performed by a staggered loading-release algorithm of energy minimization at frozen crack state followed by nodal releases at frozen deformation. We compare results obtained by the proposed formulation with other crack propagation criteria. The computational method proposed is extremely robust and shows an excellent performance for representative numerical simulations.     

A cohesive XFEM model for simulating fatigue crack growth under mixed-mode loading and overloading

Structures are subjected to cyclic loads that can vary in direction and magnitude, causing constant amplitude mode I simulations to be too simplistic. This study presents a new approach for fatigue crack propagation in ductile materials that can capture mixed-mode loading and overloading. The extended finite element method is used to deal with arbitrary crack paths. Furthermore, adaptive meshing is applied to minimize computation time. A fracture process zone ahead of the physical crack tip is represented by means of cohesive tractions from which the energy release rate, and thus the stress intensity factor can be extracted for an elastic-plastic material. The approach is therefore compatible with the Paris equation, which is an empirical relation to compute the fatigue crack growth rate. Two different models to compute the cohesive tractions are compared. First, a cohesive zone model with a static cohesive law is used. The second model is based on the interfacial thick level set method in which tractions follow from a given damage profile. Both models show good agreement with a mode I analytical relation and a mixed-mode experiment. Furthermore, it is shown that the presented models can capture crack growth retardation as a result of an overload.     

An implicit adaptive node‐splitting algorithm to assess the failure mechanism of inelastic elastomeric continua

This contribution presents a mesh adaptive crack propagation scheme for the evaluation of the viscoelastic fracture response of elastomers at large strains and up to high loading rates. The approach accounts for micromechanical based features of both elastic and viscoelastic bulk responses of idealized polymer networks. To this end, the Bergstörm–Boyce model is considered to introduce hyperelastic and nonlinear finite viscoelastic responses. Moreover, the crack driving force and the crack driving direction are predicted by the material force approach. A consistent thermodynamic framework for the combined configurational motion in viscoelastic continua at finite strain regime is discussed. The fracture toughness of non‐strain‐crystallizing elastomers shows strong rate dependency and the energy release rate versus the rate of tearing to be a fundamental material property. Therefore, in this contribution, a dynamic fracture criterion, which is a function of the rate of crack growth, is shown to be adequate in numerical simulations. The use of the presented method enables to study fracture behaviour of any material nonlinearity within the implicit time integration. Main feature of the proposed algorithm is restructuring the overall discrete system by duplication of crack front DOFs based on minimization of the overall energy via the Griffith criterion. Copyright © 2014 John Wiley & Sons, Ltd.     

An enriched element‐failure method (REFM) for delamination analysis of composite structures

This paper develops an enriched element‐failure method for delamination analysis of composite structures. This method combines discontinuous enrichments in the extended finite element method and element‐failure concepts in the element‐failure method within the finite element framework. An improved discontinuous enrichment function is presented to effectively model the kinked discontinuities; and, based on fracture mechanics, a general near‐tip enrichment function is also derived from the asymptotic displacement fields to represent the discontinuity and local stress intensification around the crack‐tip. The delamination is treated as a crack problem that is represented by the discontinuous enrichment functions and then the enrichments are transformed to external nodal forces applied to nodes around the crack. The crack and its propagation are modeled by the ‘failed elements’ that are applied to the external nodal forces. Delamination and crack kinking problems can be solved simultaneously without remeshing the model or re‐assembling the stiffness matrix with this method. Examples are used to demonstrate the application of the proposed method to delamination analysis. The validity of the proposed method is verified and the simulation results show that both interlaminar delamination and crack kinking (intralaminar crack) occur in the cross‐ply laminated plate, which is observed in the experiment. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical study of crack shielding and deflection under extensive plasticity

Experimentally observed crack deflection events in multi-layered material systems, occurring even under pure mode-I loading, are here simulated and explained through elasto-plastic finite element modelling. The crack tip opening displacement is adopted as the crack driving force and estimated along crack paths whose deflection is predicted using the maximum tangential strain criterion. Shielding conditions that promote deflection and bifurcation are evaluated. It is shown that, under conditions of extended plasticity, CTOD evolution as a crack approaches an interface can reveal crack shielding and amplification, and that crack deflection and growth can be assessed from the variation of tangential strains around the crack tip.     

Extended finite element method for cohesive crack growth

The extended finite element method allows one to model displacement discontinuities which do not conform to interelement surfaces. This method is applied to modeling growth of arbitrary cohesive cracks. The growth of the cohesive zone is governed by requiring the stress intensity factors at the tip of the cohesive zone to vanish. This energetic approach avoids the evaluation of stresses at the mathematical tip of the crack. The effectiveness of the proposed approach is demonstrated by simulations of cohesive crack growth in concrete.     

Application of optical deformation analysis system on wedge splitting test and its inverse analysis

An alternative approach of fracture tests evaluation based on optical measurements of displacements is investigated in this paper. The non-linear hinge model based inverse analysis outgoing from the optically measured crack mouth opening displacements is introduced for the wedge splitting test. Results of the inverse analysis are compared with traditional inverse analysis based on clip gauge data. Then the optically measured crack profile and crack tip position are compared with predictions done by the non-linear hinge model and a finite element analysis. It is shown that the inverse analysis based on the optically measured data can provide material parameters of the fictitious crack model matching favorably those obtained by classical inverse analysis based on the clip gauge data. Further advantages of using of the optical deformation analysis lie in identification of such effects as aggregates bridging and crack branching. These effects would remain hidden if the crack profile is simulated by a model based on the fictitious crack model.     

Computational modeling of strong and weak discontinuities using the extended finite element method

In this article, the extended finite element method is employed to solve problems, including weak and strong discontinuities. To this end, a level set framework is used to represent the discontinuities location, and the Heaviside and Branch function are included in the standard finite element method. The case of two arbitrary curved cracks is solved numerically and stress intensity factor (SIF) values at the crack tips are calculated based on the evaluation of the crack tip opening displacement. Afterwards, J-integral methodology is adopted to evaluate the SIFs for isotropic and anisotropic bi-material interface crack problems. Numerical results are verified with those presented in the literature.     

Direct evaluation of accurate coefficients of the linear elastic crack tip asymptotic field

An improvement to the extended finite element method (XFEM) and generalised finite element method (GFEM) is introduced. It enriches the finite element approximation of the crack tip node as well as its surrounding nodes with not only the first term but also the higher order terms of the linear elastic crack tip asymptotic field using a partition of unity method (PUM). Numerical results show that together with a reduced quadrature rule to the enriched elements, this approach predicts accurate stress intensity factors (SIFs) directly (i.e. without extra post‐processing) after constraining the enriched nodes properly. However, it does not predict accurately the coefficients of the higher order terms. For that a hybrid crack element (HCE) is introduced which is powerful and convenient not only for directly determining the SIF but also the coefficients of higher order terms in the plane linear elastic crack tip asymptotic field. Finally, the general expressions for the coefficients of the second to fifth terms of the linear elastic crack tip asymptotic field of three‐point bend single edge notched beams (TPBs) with span to depth ratios widely used in testing are extended to very deep cracks with the use of the HCE.     

基于X-SBFEM的裂纹体非网格重剖分耦合模型研究

扩展有限元法利用了非网格重剖分技术,但需要基于裂尖解析解构造复杂的插值基函数,计算精度受网格疏密和插值基函数等因素影响。比例边界有限元法则在求解无限域和裂尖奇异性问题优势明显,两者衔接于有限元法理论内,可建立一种结合二者优势的断裂耦合数值模型。该文从虚功原理出发,利用位移协调与力平衡机制,提出了一种断裂计算的新方法X-SBFEM,达到了扩展有限元模拟裂纹主体、比例边界有限元模拟裂尖的目的。在数值算例中,通过边裂纹和混合型裂纹的应力强度因子计算,并与理论解对比,验证了该方法的准确性和有效性。     

Modeling crack discontinuities without element‐partitioning in the extended finite element method

In this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment

A methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial differential equation loses hyperbolicity. The approach is limited to rate‐independent materials, so that the transition occurs on a set of measure zero. The discrete discontinuity is treated by the extended finite element method (XFEM) whereby arbitrary discontinuities can be incorporated in the model without remeshing. Loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model. A new method was developed for the case when the discontinuity ends within an element; it facilitates the modelling of crack tips that occur within an element in a dynamic setting. The method is applied to several dynamic crack growth problems including the branching of cracks. Copyright © 2003 John Wiley & Sons, Ltd.     

Simulation of failure processes with the variational multiscale method

In the present contribution a multiscale method for the modeling an simulation of failure and crack propagation is presented. Thereby, the variational multiscale method initially introduced by Hughes provides the methodological multiscale framework. The basis of this method is a decomposition of the solution into coarse scale and fine scale contributions, the latter incorporating the local behaviour. The method involves the propagation of cracks at finite strains. A twoscale approach, macro-meso, is adopted and both scales are discretized with finite elements whereby certain locality assumptions are prescribed to the mesoscopic solution. At the fine scale an evolving mesostructure induced by crack propagation is taken into account. The multiscale framework implies naturally a refined discretization in the area of the crack tip. Nevertheless, when crack propagation is modelled, the crack direction should not depend on the discretization. To prevent constant remeshing a discretization with discontinuous elements at the fine scale is applied. The applicability of the method to simulate multiscale failure processes at finite strains is demonstrated by numerical examples.     

Analytical and computational investigation into the influence of the compressive stress cycles on crack growth under variable amplitude loading using CTOD

Fatigue crack growth rate of 6061 T651 aluminum alloy centre‐cracked plates, subject to a variable amplitude loading (VAL), is established using a coupled analytical/computational approach. The method utilises the generalised Willenborg retardation model in conjunction with the Walker model, with several of the models parameters obtained through nonlinear finite element analyses. A two‐parameter approach is also used in this study to explore the influence of compressive stress cycles in a VAL scenario on the crack tip opening displacement, and in turn, on the resulting fatigue crack growth rate of the material. The work includes the evaluation of the crack tip opening displacement and residual stresses in the vicinity of a crack by nonlinear finite element analysis, and application of the Generalised Willenborg model to evaluate the fatigue crack growth; particularly, under the influence of the compressive stress cycles, tensile overloads and underloads have been investigated. The finite element analysis results are compared with the experimental results reported in earlier studies by the authors. The results further demonstrate two important phenomena, that is, (i) the influence of FCG retardation effects due to the tensile overloads and (ii) the acceleration effect due to the applied compressive underloads of a VAL stress–time history.     

Proposal of the concept of splice-type arrester for foam core sandwich panels

A new type of crack arrester concept, named the splice-type crack arrester, was invented and applied to a core–core splice in a foam core sandwich panel in order to suppress interfacial crack growth. An analytical evaluation of this crack arrester including parametric studies was carried out. It was confirmed by finite element (FE) analysis that interfacial crack propagation was suppressed by a decrease in the energy release rate at the crack tip under constant loading owing to the splice-type crack arrester as the crack tip approached the edge of the arrester. Through this study, it was revealed that the leading edge of the splice-type crack arrester, its shape and material, have strong effects on the crack suppression capability.     

Plane strain crack closure and cyclic hardening

This study is focused on the understanding of the mechanical effects of cyclic hardening on crack tip plasticity and on plasticity-induced crack closure. Various finite element analyses were conducted using abaqus. Cyclic hardening is found to affect both crack closure and the shape of the plastic zone at the crack tip. Crack growth modelling in plane strain conditions in a cyclically hardening material is discussed. An empirical formula is provided which allows the calculation of the crack tip plastic zone size under plane strain conditions in a cyclically hardening material. The effects of overloads are also examined.     

An adaptive singular finite element in nonlinear fracture mechanics

In the case of nonlinear fracture mechanics the type of singularity induced by the crack tip is commonly not known. This results in a poor approximation of the near crack tip fields in a finite element setting and induces so called spurious—or residual—discrete material forces in the vicinity of the crack tip. Thus the numerical calculation of the crack driving material force in nonlinear fracture is often not that precise as in linear elasticity where we can use special crack tip elements and/or path independency. To overcome this problem we propose an adaptive singular element, which adapts automatically to the type of singularity. The adaption is based on an optimisation procedure using a variational principle.     

An edge-based smoothed XFEM for fracture in composite materials

In this work, an edge-based smoothed extended finite element method (ES-XFEM) is extended to fracture analysis in composite materials. This method, in which the edge-based smoothing technique is married with enrichment in XFEM, shows advantages of both the extended finite element method (XFEM) and the edge-based smoothed finite element method (ES-FEM). The crack tip enrichment functions are specially derived to represent the characteristic of the displacement field around the crack tip in composite materials. Due to the strain smoothing, the necessity of integrating the singular derivatives of the crack tip enrichment functions is eliminated by transforming area integration into path integration, which is an obvious advantage compared with XFEM. Two examples are presented to testify the accuracy and convergence rate of the ES-XFEM.