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.     

A mixed finite volume formulation for determining the small strain deformation of incompressible materials

A finite volume formulation for determining small strain deformations in incompressible materials is presented in detail. The formulation includes displacement and hydrostatic pressure variables. The displacement field varies linearly along and across each cell face. The hydrostatic pressure field associated with each face is uniform. The cells that discretize the structure are geometrically unrestricted, each cell can have an arbitrary number of faces. The formulation is tested on a number of linear elastic plane strain benchmark problems. This testing reveals that when meshes of multifaceted cells are employed to represent the structure then locking behaviour is exhibited, but when triangular cells are used then accurate predictions of the displacement and stress fields are produced. Copyright © 1999 John Wiley & Sons, Ltd.     

A direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures

This paper presents a direct complementarity approach for carrying out the elastoplastic analysis of plane stress and plane strain structures. Founded on a traditional finite‐step formulation, our approach, however, avoids the typically cumbersome implementation of iterative predictor–corrector procedures associated with the ubiquitous return mapping algorithm. Instead, at each predefined step, the governing formulation—cast in its most natural mathematical programming format known as a mixed complementarity problem—is directly solved by using a complementarity solver run from within a mathematical modeling system. We have chosen the industry‐standard General Algebraic Modeling System/PATH mixed complementarity problem solver that is called from within the General Algebraic Modeling System environment. We consider both von Mises and Tresca materials, with perfect or hardening (kinematic and isotropic) behaviors. Our numerical tests, five (benchmark) examples of which are presented in this paper, have been carried out using models constructed from the mixed finite element of Capsoni and Corradi (Comput. Methods Appl. Mech. Eng. 1997; 141 :67–93), which beneficially offers a locking‐free behavior and coarse‐mesh accuracy. The results indicate, in addition to an isochoric locking‐free behavior, good accuracy and the ability to circumvent the difficult singularity problem associated with the corners of Tresca yield surfaces. Copyright © 2012 John Wiley & Sons, Ltd.     

Characteristics of three-dimensional stress fields in plates with a through-the-thickness crack

Three-dimensional finite element analyses were performed on plates with a through-the-thickness crack. Global-local finite element technique with sub-modeling was used to achieve the refinement required to obtain an accurate stress field. The existence of a weaker singularity was verified, and a model was proposed to explain the behavior of stresses in the boundary layer. This model is able to account for the competing interaction between the inverse square root singular term and the vertex singular term. The energy release rate was calculated using the modified crack closure method and energy balance. A simple technique without 3-D calculation was suggested for evaluating an approximate 3-D stress intensity factor at the mid-plane. The effect of plate thickness on the size of the three-dimensional region was studied, and the validity of two-dimensional linear elastic fracture mechanics was discussed.     

Nonlinear finite element analysis of stress and strain distributions across the adhesive thickness in composite single-lap joints

A geometrically nonlinear, two-dimensional (2D) finite element analysis has been performed to determine the stress and strain distributions across the adhesive bond thickness of composite single-lap joints. The results of simulations for 0.13 and 0.26 mm bond thickness are presented. Using 2-element and 6-element mesh schemes to analyze the thinner bond layer, good agreement is found with the experimental results of Tsai and Morton. Further mesh refinement using a 10-element analysis for the thicker bond has shown that both the tensile peel and shear stresses at the bond free edges change significantly across the adhesive thickness. Both stresses became increasingly higher with distance from the centerline and peak near but not along the adherend–adhesive interface. Moreover, the maximum shear and peel stresses occur near the overlap joint corner ends, suggesting that cohesive crack initiation is most likely to occur at the corners. The dependence of stress and corresponding strain distributions on bond thickness and adhesive elastic modulus are also presented. It is observed that the peak shear and peel stresses increase with the bond thickness and elastic modulus.     

Analysis of singular stress fields at multi‐material corners under thermal loading

The scaled boundary finite‐element method is extended to the modelling of thermal stresses. The particular solution for the non‐homogeneous term caused by thermal loading is expressed as integrals in the radial direction, which are evaluated analytically for temperature changes varying as power functions of the radial coordinate. When applied to model a multi‐material corner, only the boundary of the problem domain is discretized. The boundary conditions on the straight material interfaces and the side‐faces forming the corner are satisfied analytically without discretization. The stress field is expressed semi‐analytically as a series solution. The stress distribution along the radial direction, including both the real and complex power singularity and the power‐logarithmic singularity, is represented analytically. The stress intensity factors are determined directly from their definitions in stresses. No knowledge on asymptotic expansions is required. Numerical examples are calculated to evaluate the accuracy of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.     

An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi‐brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well‐known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.     

An adaptive finite element framework for fatigue crack propagation

A new finite element (FE) framework for fatigue crack propagation (FCP) analysis is proposed. This framework combines the simplicity of standard industrial FCP analysis with the generality and accuracy of a full FE analysis and can be implemented on a small computer by combining standard existing computational tools. In this way it constitutes an attractive alternative to existing approaches. The framework is based on linear elastic fracture mechanics and on FE mesh adaptation. Some novel features are introduced in several of its steps in order to make it efficient and at the same time reasonably accurate. Various computational aspects of the scheme are discussed. A few two‐dimensional numerical examples involving FCP in thin sheets under plane‐stress conditions are presented to demonstrate the performance of the framework. Some of the numerical results are compared to those of laboratory experiments. Copyright © 2002 John Wiley & Sons, Ltd.     

Non‐oscillatory and non‐singular asymptotic solutions to stress fields at interface cracks

This work concerns the complex oscillatory singularities revealed in Williams's asymptotic solutions to stress fields around arbitrary interface cracks, which are the foundation of phenomenological interface fracture mechanics. First, we highlight the fatal discrepancy between the asymptotic stress fields for cracks in a homogeneous material obtained by assigning an identical material on both regions embracing an interface crack, and the solutions directly derived from cracks in a single material. Next, following a brief introduction to Williams's formulation process, we adopt the method of repeatedly eliminating variables instead of solving the determinant equation for the coefficient matrix to reformulate the asymptotic analysis of stress fields at arbitrary interface cracks. The resultant stresses get rid of oscillatory character. Further, under two specific loading conditions, namely, remotely uniaxial tension or shear, non‐oscillatory and non‐singular asymptotic solutions to stress fields around interface cracks are obtained.     

Classification of stress modes in assumed stress fields of hybrid finite elements

A classification method is presented to classify stress modes in assumed stress fields of hybrid finite element based on the eigenvalue examination and the concept of natural deformation modes. It is assumed that there only exist m (=n−r) natural deformation modes in a hybrid finite element which has n degrees of freedom and r rigid-body modes. For a hybrid element, stress modes in various assumed stress fields proposed by different researchers can be classified into m stress mode groups corresponding to m natural deformation modes and a zero-energy stress mode group corresponding to rigid-body modes by the m natural deformation modes. It is proved that if the flexibility matrix [H] is a diagonal matrix, the classification of stress modes is unique. Each stress mode group, except the zero-energy stress mode group, contains many stress modes that are interchangeable in an assumed stress field and do not cause any kinematic deformation modes in the element. A necessary and sufficient condition for avoiding kinematic deformation modes in a hybrid element is also presented. By means of the m classified stress mode groups and the necessary and sufficient condition, assumed stress fields with the minimum number of stress modes can be constructed and the resulting elements are free from kinematic deformation modes. Moreover, an assumed stress field can be constructed according to the problem to be solved. As examples, 2-D, 4-node plane element and 3-D, 8-node solid element are discussed. © 1997 John Wiley & Sons, Ltd.     

A mixed finite element for the nonlinear analysis of in-plane loaded masonry walls

A mixed membrane eight-node quadrilateral finite element for the analysis of masonry walls is presented. Assuming that a nonlinear and history-dependent 2D stress-strain constitutive law is used to model masonry material, the element derivation is based on a Hu-Washizu variational statement, involving displacement, strain, and stress fields as primary variables. As the behavior of masonry structures is often characterized by strain localization phenomena, due to strain softening at material level, a discontinuous, piecewise constant interpolation of the strain field is considered at element level, to capture highly nonlinear strain spatial distributions also within finite elements. Newton's method of solution is adopted for the element state determination problem. For avoiding pathological sensitivity to the finite element mesh, a novel algorithm is proposed to perform an integral-type nonlocal regularization of the constitutive equations in the present mixed formulation. By the comparison with competing serendipity displacement-based formulation, numerical simulations prove high performances of the proposed finite element, especially when coarse meshes are adopted.     

A systematic construction of B‐bar functions for linear and non‐linear mixed‐enhanced finite elements for plane elasticity problems

In a previous paper a modified Hu–Washizu variational formulation has been used to derive an accurate four node plane strain/stress finite element denoted QE2. For the mixed element QE2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations a priori for the linear elastic case. In this paper an alternative approach is discussed. The new formulation leads to the same accuracy for linear elastic problems as the QE2 element; however it turns out to be more efficient in numerical simulations, especially for large deformation problems. Using orthogonal stress and strain functions we derive B̄ functions which avoid numerical inversion of matrices. The B̄ ‐strain matrix is sparse and has the same structure as the strain matrix B obtained from a compatible displacement field. The implementation of the derived mixed element is basically the same as the one for a compatible displacement element. The only difference is that we have to compute a B̄ ‐strain matrix instead of the standard B ‐matrix. Accordingly, existing subroutines for a compatible displacement element can be easily changed to obtain the mixed‐enhanced finite element which yields a higher accuracy than the Q4 and QM6 elements. Copyright © 1999 John Wiley & Sons, Ltd.     

An interaction integral method for computing mixed mode stress intensity factors for curved bimaterial interface cracks in non-uniform temperature fields

In finite element analysis the interaction integral has been a useful tool for computing the stress intensity factors for fracture analysis. This work extends the interaction integral to account for non-uniform temperatures in the calculation of stress intensity factors for three dimensional curvilinear cracks either in a homogeneous body or on a bimaterial interface. First, the derivation of the computational algorithm, which includes the additional terms developed by the non-zero gradient of the temperature field, is presented in detail. The algorithm is then implemented in conjunction with commercial finite element software to calculate the stress intensity factors of a crack undergoing non-uniform temperatures on both a homogeneous and a bimaterial interface. The numerical results displayed path independence and showed excellent agreement with available analytical solutions.     

Representation of singular fields without asymptotic enrichment in the extended finite element method

In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi‐analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix, and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A Heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended SBFEM. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code. Copyright © 2013 John Wiley & Sons, Ltd.     

A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems

This paper presents a novel numerical method for effectively simulating the singular stress field for mode-I fracture problems based on the edge-based smoothed finite element method (ES-FEM). Using the unique feature of the ES-FEM formulation, we need only the assumed displacement values (not the derivatives) on the boundary of the smoothing domains, and hence a new technique to construct singular shape functions is devised for the crack tip elements. Some examples have demonstrated that results of the present singular ES-FEM in terms of strain energy, displacement and J-integral are much more accurate than the finite element method using the same mesh.     

The least‐squares finite element method in elasticity—Part I: Plane stress or strain with drilling degrees of freedom

A new least‐squares finite element method (LSFEM) for plane elasticity problems is developed based on the first‐order displacement–stress–rotation formulation which includes two new first‐order compatibility constraints among the stresses and the drilling rotation. This LSFEM can accommodate all kinds of equal‐order interpolations. Numerical experiments on various examples including incompressible materials show that the method achieves an optimal rate of convergence for all variables. Copyright © 2001 John Wiley & Sons, Ltd.     

平面应变断裂韧度评定中临界载荷研究

针对新版断裂韧性测试规范扩大的裂纹长度适用范围,利用有限元精细分析用于平面应变断裂韧度KIC评定中割线法确定临界载荷PQ的相对割线斜率ΔS的合理性,提出0.45≤a/W≤0.7范围内的相对割线斜率ΔS的表达式。结果表明:现行规范推荐的相对割线斜率ΔS不再适合于确定0.55a/W≤0.7范围内的临界载荷PQ,其最大相对误差已近8%。     

Improvement of plane stress solutions using adaptive finite elements

Abstract

A finite element method is combined with an adaptive meshing technique to improve the accuracy of the finite element solution. The effectiveness of the combined method is evaluated by a plane stress problem that has an exact solution. The problem is that of a panel with a circular cutout subjected to an applied load. The result demonstrates that the adaptive meshing technique can reduce the numbers of the finite elements, the analysis computational time, and improve the accuracy of the analysis solution.