Mixed-mode Crack Problems Research Articles

Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity

A boundary element method is developed for fracture analysis of gradient elastic planar solids under static loading. A simple version of Mindlin’s general theory of gradient elastic materials is employed and the two required boundary integral equations, one for the displacement vector and the other for its normal derivative are presented. Use is made of the fundamental solution of the problem and this leads to a formulation that requires only a boundary discretization. Two representative numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The first deals with a mode I crack, while the second with a mixed mode (I & II) crack. For the second case the proposed method is used in conjunction with the method of subregions. The method is employed with regular and special (near the crack tip) boundary elements. The gradient effect consists of modifying both the displacement and the stress field around the crack tip and resulting in a response which is more physically acceptable than the one coming from the classical theory of elasticity.

Functionally graded piezoelectric strip with a penny-shaped crack under electromechanical loadings

In this paper, the mixed-mode penny-shaped crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electromechanical loadings. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.

On the full set of elastic T-stress terms of internal circular cracks under mixed-mode loading conditions

Analytical expressions for all non-singular stress terms of the asymptotic crack tip field, the so-called T-stresses of internal mixed mode circular (penny shaped) cracks in infinite homogeneous and isotropic elastic solids are addressed. To solve the problem the mixed mode crack problem is divided into sub-problems using the superposition method, and each sub-problem is then solved for the asymptotic stress field. Considering the expansion of the local stress field at the crack front, the elastic T-stress terms are derived for each sub-problem. The results are superimposed to give the analytical expressions of the so far missing elastic T-stresses of internal mixed mode penny shaped cracks. The effect of the T-stresses on the size and shape of the plastic zone at the crack tip is discussed, and analytical results are compared to the ones from finite element analyses, both for the T-stress components and the size of the plastic zone. For an accurate prediction of the plastic zone all the singular terms and the constant terms ( T-stresses) in the stress expansion formulae should be considered. It is observed that negative T-stresses increase the size of the plastic zone, while positive ones reduce it.

Electroelastic Response of a Parallel Crack in a Functionally Graded Piezoelectric Strip

In this paper, the mixed-mode crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in plane electric loading. The problem is formulated in terms of a system of singular integral equations. The stress and the electric displacement intensity factors, and the energy density factor are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.

3652 平行き裂を有する傾斜機能圧電セラミックス厚板の電気弾性相互干渉(S24-4 特異応力・刃状転位,S24 数理・数値解析による先端材料評価)

In this paper, the mixed-mode crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electric loading. The problem is formulated in terms of a system of singular integral equations. The stress and the electric displacement intensity factors, and the energy density factor are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.

A finite crack in a semi-infinite strip of a graded piezoelectric material under electric loading

In this paper, the mixed-mode crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electric loading. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.

The mixed mode crack problem in an FGM layer bonded to a homogeneous half-plane

In this paper, the plane elasticity problem of an arbitrarily oriented crack in a FGM layer bonded to a homogeneous half-plane is considered. The problem is modeled by assuming that the elastic properties of the FGM layer are exponential functions of the thickness coordinate and are continuous at the interface of the FGM layer and the half-plane. The Fourier transform technique is used to reduce the problem to the solution of a system of Cauchy-type singular integral equations, which are solved numerically. The stress intensity factors are computed for various crack orientations, crack locations and material parameters. The results show that crack length, crack orientation and the non-homogeneity parameter of the strip material have significant effect on the fracture of the FGM layer.

T-stress in orthotropic functionally graded materials: Lekhnitskii and Stroh formalisms

A new interaction integral formulation is developed for evaluating the elastic T-stress for mixed-mode crack problems with arbitrarily oriented straight or curved cracks in orthotropic nonhomogeneous materials. The development includes both the Lekhnitskii and Stroh formalisms. The former is physical and relatively simple, and the latter is mathematically elegant. The gradation of orthotropic material properties is integrated into the element stiffness matrix using a “generalized isoparametric formulation” and (special) graded elements. The specific types of material gradation considered include exponential and hyperbolic-tangent functions, but micromechanics models can also be considered within the scope of the present formulation. This paper investigates several fracture problems to validate the proposed method and also provides numerical solutions, which can be used as benchmark results (e.g. investigation of fracture specimens). The accuracy of results is verified by comparison with analytical solutions.

Fracture mechanics analysis models for functionally graded materials with arbitrarily distributed properties (Modes II and III problems)

In Wang et al. (2002), the mode I crack problem has been analyzed for a functionally graded material strip with arbitrarily distributed properties. For the completeness of the research, this paper provides an analysis method for the mode II in plane and mode III anti-plane elastic crack problems for a functionally graded material (FGM) strip containing a crack along the gradient direction. In the analysis, the gradient region is treated as a number of homogeneous layers stacked along the gradient direction. The analytical models developed in this paper for the modes II and III problems, together with the model developed in Wang et al. (2002) for the mode I crack problem complete the analysis of the mixed-mode crack problem for functionally graded materials with a crack parallel the gradient direction. Those analytical models allow the material properties be any continuous functions along the thickness direction of the strip.

Mixed-mode J-integral formulation and implementation using graded elements for fracture analysis of nonhomogeneous orthotropic materials

The path-independent J k *-integral, in conjunction with the finite element method (FEM), is presented for mode I and mixed-mode crack problems in orthotropic functionally graded materials (FGMs) considering plane elasticity. A general procedure is presented where the crack is arbitrarily oriented, i.e. it does not need to be aligned with the principal orthotropy directions. Smooth spatial variations of the independent engineering material properties are incorporated into the element stiffness matrix using a “generalized isoparametric formulation”, which is natural to the FEM. Both exponential and linear variations of the material properties are considered. Stress intensity factors and energy release rates for pure mode I and mixed-mode boundary value problems are numerically evaluated by means of the equivalent domain integral especially tailored for orthotropic FGMs. Numerical results are discussed and validated against available theoretical and numerical solutions.

On the dynamic crack propagation in an interphase with spatially varying elastic properties under inplane loading

This article provides a comprehensive theoretical investigation on a finite crack with constant length (Yoffe type crack) propagating in an interfacial layer with spatially varying elastic properties under inplane loading. The analytical formulations are developed using Fourier transforms and solving the resulting singular integral equations in terms of the opening and sliding displacements of the crack. The dynamic stress intensity factors and energy release rate are analyzed to study the dynamic fracture property of this inherent mixed mode crack problem. Numerical examples are provided to show the effects of the material properties, the thickness of the interfacial layer, the crack position and speed upon the dynamic fracture behaviour, and the singularity transition between the current crack and the corresponding interfacial crack for thin interphase.

Unification of stress intensity factor (SIF) extraction methods with an h‐adaptive finite element scheme

AbstractAn h‐adaptive finite element procedure consisting of recently developed mesh generator (advancing front type and exclusively suitable for two‐dimensional domains containing cracks), ZZ‐error estimator and a refinement strategy based on strain energy concentration index have been used to refine the meshes of domains containing cracks adaptively to a target error of 3 per cent. Seven mixed‐mode SIF extraction methods have been coupled with the present h‐adaptive finite element analysis in order to examine the performance of each method in an environment of randomly changing meshes. Based on the performance of each method the SIF values of a mixed‐mode crack problem that does not have analytical solution have been suggested. Copyright © 2001 John Wiley & Sons, Ltd.

Adaptive finite element analysis of mixed-mode fracture problems containing multiple crack-tips with an automatic mesh generator

A fully automatic advancing front type mesh generator to take care of crack problems has been presented. It is coupled with the Zienkiewicz and Zhu error estimator and the refinement methodology depends on the concept of strain energy concentration for completely automatic adaptive analysis of mixed-mode crack problems. For the first time energy based path independent M1-integral has been used to extract mixed-mode stress intensity factors in randomly changing quadratic triangular meshes. To fulfill the objective of automatic adaptive procedures, an approach has been suggested and validated for generation of integration paths automatically without user intervention. Stress intensity factors have been obtained within engineering accuracy.

On the solution of three-dimensional thermoelastic mixed-mode edge crack problems by the dual boundary element method

This paper describes the formulation and numerical implementation of the three-dimensional dual boundary element method (DBEM) for the thermoelastic analysis of mixed-mode crack problems. The DBEM incorporates two pairs of independent boundary integral equations: namely, the temperature and displacement, and the flux and traction equations. In this technique, one pair is applied on one of the crack faces and the other on the opposite one. On non-crack boundaries, the temperature and displacement equations are applied. Special shape functions are implemented to model the r behaviour of crack tip fields. Stress intensity factors were calculated with the crack opening displacement formulae and by using standard and special elements at the crack front. To demonstrate the accuracy and efficiency of the proposed method, several examples were solved and the results compared, where possible, with existing solutions.

Variable singularity boundary element and its applications in computation of SIFs

Two boundary elements have been proposed for simulation of variable order singularities in two dimensions. The first can model the variable order strain singularity near a crack tip and the second can simulate both the strain and traction. These elements can be easily incorporated in a standard boundary element computer programme. The elements are useful for computation of stress intensity factors (SIFs) in fracture mechanics. To improve the accuracy of such computations further, a modified crack closure integral (MCCI) based method for mechanical loading are presented. Examples of mode I and mixed mode crack problems are examined to illustrate the performance of the proposed elements and the MCCI based calculations. The effects of order of Gauss quadrature associated with such elements on the accuracy of the SIFs are also reported.

Crack analysis under fretting condition by rounded punch

Surface edge crack subjected to contact stresses is analysed. A punch with corner radii is considered to press the semi-infinite plane. Partial slip problem is solved when a shear force is applied to the punch. Dislocation density function method is used to solve the present mixed mode crack problem. The crack length of positive K1 is examined, which is affected by the ratio of the flat portion to the total width of the punch. Surface traction during one cycle of the shear force is evaluated to simulate the fretting condition. The compliance change of the contact surface is also investigated during the shear cycle. It is found that the crack grows during only a part of the cycle, which may be termed as effective period of crack growing. A design method for restraining the fretting failure is discussed, from which recommendable geometry of the punch is suggested.

Adaptive finite element analysis of mixed-mode crack problems with automatic mesh generator

Adaptive finite element analysis of mixed-mode crack problems with automatic mesh generator

Dual Boundary Element Methods for Three-Dimensional Dynamic Crack Problems

In this paper, three formulations of a single-region dual boundary element method for dynamic crack problems are presented. The dynamic stress intensity factors are calculated by crack opening displacement. These methods are developed to study the dynamic behaviour of stationary cracks in threedimensional analysis. Formulations of the methods and computational efficiency characterized by memory and central processing unit time are discussed. In order to compare these three approaches, they are applied to pure mode and mixed-mode crack problems in infinite and finite domains.

A dislocation and point force approach to the boundary element method for mixed mode crack analysis of plane anisotropic solids

In this paper we formulate a direct boundary element method (BEM) for plane anisotropic elasticity (i.e., the in‐plane deformation decoupled from the out‐of‐plane deformation) based on distributions of point forces and dislocation dipoles. According to a physical interpretation of Somigliana's identity the displacement field in a finite body R is represented by the continuous distributions of point forces and dislocation dipoles along the imaginary boundary ?R of the finite domain R embedded in an infinite body. We adopt Stroh's complex variable formalism for anisotropic elasticity and represent the point force and the dislocation, their dipoles, and continuous distributions systematically exploiting the duality relations between the point force and the dislocation solutions. Explicit formulas for the displacement and the traction formulations, obtained by analytical integration of the boundary integrals, are given. We apply these formulas to mixed mode crack problems for multiply cracked anisotropic bodies by extending the physical interpretation of Somigliana's identity to cracked bodies and representing the crack by the continuous distribution of dislocation dipoles. With the help of the conservation integrals of anisotropic elasticity, we will demonstrate the capability of the method to determine the mixed mode stress intensity factors (KI and KII ) accurately.

Dual boundary element methods for three-dimensional dynamic crack problems

In this paper, three formulations of a single-region dual boundary element method for dynamic crack problems are presented. The dynamic stress intensity factors are calculated by crack opening displacement. These methods are developed to study the dynamic behaviour of stationary cracks in threedimensional analysis. Formulations of the methods and computational efficiency characterized by memory and central processing unit time are discussed. In order to compare these three approaches, they are applied to pure mode and mixed-mode crack problems in infinite and finite domains.