A contact algorithm for frictional crack propagation with the extended finite element method

We present an incremental quasi‐static contact algorithm for path‐dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non‐linear contact mechanics and embedded within a localized element by penalty method. Newton–Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ‘LATIN’ for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Sequential quadratic programming for non-linear elastic contact problems

The physical problem considered in this paper is that of a non-linear elastic body being indented by a rigid punch. The treatment is based on finite element discretization and sequential quadratic programming (SQP). The finite element formulation is obtained through a variational formulation, which generalizes to frictionless contact a three-field principle which involves deformation, volume strain and hydrostatic pressure as independent fields. We compare an incremental load method and a method where the indentation for the final load is sought directly. Crucial for the second method is the use of a line search with respect to a merit function which measures the infeasibility in the optimality criteria for the problem; this line search also includes a check of the orientation-preserving condition of a positive determinant of the deformation gradient. Each iteration within an SQP method requires the solution of a quadratic programming (QP) subproblem, and four different methods for the solution of these subproblems are compared. The performance of the overall procedure is also compared to that of a commercially available system. Test examples ranging from 23 to 770 displacement degrees of freedom are treated. The computational results show that the proposed solution concept is feasible and efficient. Furthermore, it can be applied to general non-linear elastic contact problems, since it does not include any ad hoc rules.     

A domain-independent interaction integral for linear elastic fracture analysis of micropolar materials

This paper establishes a domain-independent interaction integral (DII-integral) for linear elastic fracture mechanics of micropolar elastic solids. The DII-integral has three amazing features that make it effective for solving the fracture parameters of complex micropolar materials. The first one is that the DII-integral can decouple the stress intensity factors (SIFs) and couple stress intensity factors (CSIFs) both of which are the key fracture parameters charactering the crack-tip asymptotic singular fields. In details, the DII-integral is derived from the J-integral by superimposing an actual field and an auxiliary field. By assigning the fracture parameters in the auxiliary field with different values, the SIFs and CSIFs of different crack opening modes can be obtained separately through the DII-integral. The second important feature is that the DII-integral is domain-independent for material nonhomogeneity and discontinuity. Thanks to this feature, the DII-integral becomes extremely effective for the micropolar materials with arbitrary nonhomogeneous properties or complex interfaces. The third feature is that the DII-integral does not contain any derivatives of material properties, which feature facilitate the practical implementation of the DII-integral on complex micropolar materials. Finally, the DII-integral combined with the extended finite element method (XFEM) is employed to solve four representative crack problems and the results show good validity of the DII-integral for complex micropolar materials.     

Fracture mechanics analysis using the wavelet Galerkin method and extended finite element method

This paper presents fracture mechanics analysis using the wavelet Galerkin method and extended finite element method. The wavelet Galerkin method is a new methodology to solve partial differential equations where scaling/wavelet functions are used as basis functions. In solid/structural analyses, the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain. To improve accuracy, wavelet functions are superposed on the scaling functions within a region having a high stress concentration, such as near a hole or notch. Thus, the method can be considered a refinement technique in fixed‐grid approaches. However, because the basis functions are assumed to be continuous in applications of the wavelet Galerkin method, there are difficulties in treating displacement discontinuities across the crack surface. In the present research, we introduce enrichment functions in the wavelet Galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method. This paper presents the mathematical formulation and numerical implementation of the proposed technique. As numerical examples, stress intensity factor evaluations and crack propagation analyses for two‐dimensional cracks are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

A linear programming formulation for incremental contact analysis

A novel algorithm for the analysis of contact problems in elasticity has been presented in this paper. The algorithm is based on the boundary element method and a direct approximation of the contact complementarity conditions using linear programming. An incremental loading scheme has been developed to ensure an accurate approximation of the deformation path that the object experiences during the process of contact. Several numerical examples have been analysed to illustrate the validity of the proposed formulations.     

A hybrid boundary element method for three-dimensional fracture analysis

At first, a hybrid boundary element method used for three-dimensional linear elastic fracture analysis is established by introducing the relative displacement fundamental function into the first and the second kind of boundary integral equations. Then the numerical approaches are presented in detail. Finally, several numerical examples are given out to check the proposed method. The numerical results show that the hybrid boundary element method has a very high accuracy for analysis of a three-dimensional stress intensity factor.     

Mixed-mode thermoelastic fracture analysis of orthtropic composites

This study demonstrates ability to determine the in-plane stress intensity factors, K _I and K _II, simultaneously under mixed-mode conditions in orthotropic composites by the combined use of least-squares, stress representations which are valid away from the crack and distant measured temperatures. Recognizing the stresses near a crack-tip are dominated by the stress intensity factors, it has not been uncommon to neglect the higher-order stress terms when evaluating these factors. However, and among other considerations, it is typically difficult to obtain accurate temperature information very near the crack-tip. It can therefore be advantageous to employ measured data which originate away from the crack and to retain six to eight terms in the stress functions when evaluating the stress intensity factors. On the other hand, errors in K _I and/or K _II can be appreciable if only the r^–1/2 terms are employed with distant input information.     

Frictional contact analysis of composite materials

In this paper, the highly non-linear frictional contact problems of composite materials are analysed. A proportional loading, the potential contact zone method and finite element analysis are used to solve the problems. A tree-like searching method is used to obtain the solution of the parametric linear complementary problem, which may overcome the anisotropic properties of contact equations caused by composite materials. In the frictional contact analysis of composite materials, the distributions of normal contact pressures, tangential contact stresses and relative tangential displacements are presented for different contact material systems and different coefficients of friction. The results show that the solutions in the paper have good agreement with Hertzian solutions. The influence of different contact material systems and different coefficients of friction on the contact stresses and displacements is large. As a numerical example, ball-indentation tests of composite materials are modelled by the three-dimensional finite element method.     

Computing a-posteriori bounds for stress intensity factors in elastic fracture mechanics

This paper mainly focuses on computing the lower and upper bounds on stress intensity factors in elastic fracture mechanics with an efficient finite element output bound procedure on quantities of interest in engineering. The bounds procedure is obtained by minimizing the quadratic energy functional of output with constraints of equilibrium conditions of mechanics and continuity conditions of finite element space. The computation is based on solving the elemental Neumann residual problems for the bounds on energy norm of error in finite element solutions. The lower and upper bounds on the intensity factors of an open mode and a shear mode elastic fracture problems are computed in this paper.     

Interfacial steady‐state and transient thermal fracture of dissimilar media using the boundary element contact analysis

A boundary element procedure is formulated to treat frictional contact problems of thermally stressed structures. The aim is the computation of steady‐state and time‐dependent thermal stress intensity factors of interfacial cracks in two‐dimensional bimaterial structures when crack closure conditions are present. The effect of friction between the crack faces is taken into consideration. The problems considered are formulated in an incremental and iterative fashion because of their non‐linear nature. Near crack tip singularities of temperature and displacement fields are modelled through appropriate quarter‐point singular elements. Fracture parameters are evaluated from nodal displacements of singular elements utilizing proper formulas. Numerical results are compared with available solutions from the literature, where possible. Good agreement between them can be found. Copyright © 2005 John Wiley & Sons, Ltd.     

Transient dynamic fracture analysis using scaled boundary finite element method: a frequency-domain approach

In the scaled boundary finite element method (SBFEM), the analytical nature of the solution in the radial direction allows accurate stress intensity factors (SIFs) to be determined directly from the definition, and hence no special crack-tip treatment, such as refining the crack-tip mesh or using singular elements (needed in the traditional finite element and boundary element methods), is necessary. In addition, anisotropic material behaviour may be handled with ease. These advantages are used in this study, in which a newly-developed Frobenius solution procedure in the frequency domain for solving the governing differential equations of the SBFEM, is applied to model transient dynamic fracture problems. The complex frequency-response functions are first computed using the Frobenius solution procedure. The dynamic stress intensity factors (DSIFs) are then extracted directly from the response functions. This is followed by a fast Fourier transform (FFT) of the transient load and a subsequent inverse FFT to obtain the time history of DSIFs. Benchmark problems with isotropic and anisotropic material behaviour are modelled using the developed frequency-domain approach. Excellent agreement is observed between the results of this study and those in published literature. The effects of the mesh density, the material internal damping coefficient, the maximum frequency and the frequency interval determining the frequency-response functions on the resultant accuracy and the computational cost are also discussed.     

A mixed augmented Lagrangian-extended finite element method for modelling elastic–plastic fatigue crack growth with unilateral contact

The complete modelling of fatigue crack growth is still an industrial challenging issue for numerical methods. A new technique for the finite element modelling of elastic–plastic fatigue crack growth with unilateral contact on the crack faces is presented. The extended finite element method (X-FEM) is used to discretize the equations, allowing for the modelling of arbitrary cracks whose geometries are independent of the finite element mesh. This paper presents an augmented Lagrangian formulation in the X-FEM framework that is able to deal with elastic–plastic crack growth with treatment of contact. An original formulation, which takes advantages of two powerful numerical methods, is presented. Next the numerical issues such as contact treatment and numerical integration are addressed, and finally numerical examples are shown to validate the method. Copyright © 2007 John Wiley & Sons, Ltd.     

QPSchur: A dual, active-set, Schur-complement method for large-scale and structured convex quadratic programming

We describe an active-set, dual-feasible Schur-complement method for quadratic programming (QP) with positive definite Hessians. The formulation of the QP being solved is general and flexible, and is appropriate for many different application areas. Moreover, the specialized structure of the QP is abstracted away behind a fixed KKT matrix called K_o and other problem matrices, which naturally leads to an object-oriented software implementation. Updates to the working set of active inequality constraints are facilitated using a dense Schur complement, which we expect to remain small. Here, the dual Schur complement method requires the projected Hessian to be positive definite for every working set considered by the algorithm. Therefore, this method is not appropriate for all QPs. While the Schur complement approach to linear algebra is very flexible with respect to allowing exploitation of problem structure, it is not as numerically stable as approaches using a QR factorization. However, we show that the use of fixed-precision iterative refinement helps to dramatically improve the numerical stability of this Schur complement algorithm. The use of the object-oriented QP solver implementation is demonstrated on two different application areas with specializations in each area; large-scale model predictive control (MPC) and reduced-space successive quadratic programming (with several different representations for the reduced Hessian). These results demonstrate that the QP solver can exploit application-specific structure in a computationally efficient and fairly robust manner as compared to other QP solver implementations.     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.     

渐开线齿轮的接触分析

以Hertz应力表达式为基础,将单齿啮合的渐开线齿轮等效为相互挤压的两个圆柱体,推导了齿轮接触应力的理论表达式．建立一套单齿对啮合的有限元分析方法,包括引入渐开线和齿根过渡曲线方程及对应的自变量区间,建立参数化齿廓;模型轮缘厚度取3倍的模数,周向宽度取3倍的齿厚;在接触面上进行网格细化处理;在对应的主、从动齿轮内缘分别施加均匀切向力和固定约束等．计算结果显示,有限元解和理论解吻合较好,最大偏差不超过5%,该方法适用于不同的啮合轮齿参数．由有限元方法得到的渐开线轮齿的接触应力符合Hertz理论中的半椭圆分布规律,Mises应力和剪应力分布也符合接触力学理论,但应力分布的对称中心存在偏离或者偏斜现象,分析认为,是由于轮齿弯曲变形造成的．     

Dual boundary element method for anisotropic dynamic fracture mechanics

In this work, the dual boundary element method formulation is developed for effective modelling of dynamic crack problems. The static fundamental solutions are used and the domain integral, which comes from the inertial term, is transformed into boundary integrals using the dual reciprocity technique. Dynamic stress intensity factors are computed from crack opening displacements. Comparisons are made with quasi‐isotropic as well as anisotropic results, using the sub‐region technique. Several examples are presented to assess the accuracy and efficiency of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

Solving 2D transient rolling contact problems using the BEM and mathematical programming techniques

This work presents a new approach to the transient rolling contact of two‐dimensional elastic bodies. A solution will be obtained by minimizing a general B‐differentiable function representing the equilibrium equations and the contact conditions at each time step. Inertial effects are not taken into account and the boundary element method is used to compute the elastic influence coefficients of the surface points involved in contact (equilibrium equations). The contact conditions are represented with the help of variational inequalities and projection functions. Finally, the minimization problem is solved using the Generalized Newton's Method with line search. The results are compared with some example problems and the influence of discretization and integration time step on the results is discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Three dimensional boundary element analysis of internal cracks under sliding contact load with a spherical indenter

Using boundary element based three dimensional modelling for linear fracture mechanics, we present an analysis of cracking in a homogeneous medium subject to contact load. The proposed iterative solution procedure allows a simultaneous treatment of a reasonable number of partially closed cracks. It is shown that the most probable direction of propagation of a vertical internal crack is strongly dependent on its size compared to the contact radius and its location with respect to the axis of maximum normal load.     

A phase-field method for modeling cracks with frictional contact

We introduce a phase-field method for continuous modeling of cracks with frictional contacts. Compared with standard discrete methods for frictional contacts, the phase-field method has two attractive features: (i) it can represent arbitrary crack geometry without an explicit function or basis enrichment, and (ii) it does not require an algorithm for imposing contact constraints. The first feature, which is common in phase-field models of fracture, is attained by regularizing a sharp interface geometry using a surface density functional. The second feature, which is a unique advantage for contact problems, is achieved by a new approach that calculates the stress tensor in the regularized interface region depending on the contact condition of the interface. Particularly, under a slip condition, this approach updates stress components in the slip direction using a standard contact constitutive law, while making other stress components compatible with stress in the bulk region to ensure nonpenetrating deformation in other directions. We verify the proposed phase-field method using stationary interface problems simulated by discrete methods in the literature. Subsequently, by allowing the phase field to evolve according to brittle fracture theory, we demonstrate the proposed method's capability for modeling crack growth with frictional contact.