A generalized comninou contact model for interface cracks in anisotropic elastic solids

An open question of interest to the mechanics of interface fracture is how to generalize the Comninou contact model for interface cracks in isotropic solids to the general anisotropic case. Part of the difficulty lies in that the peculiar oscillatory behavior can not be fully eliminated by Comninou's original assumption of pure pressure contact between the crack surfaces. In this paper, we propose a model that strictly enforces the non-oscillatory condition by allowing the crack face contact force to have a shear component normal to the direction of slip, which is somewhat reminiscent of frictionless slip between a pair of grooved surfaces. Based on that model, complex variable representations are adopted to determine the complete series expansion for the crack-tip field. The solutions are incorporated into a hybrid finite element procedure to develop a special element for closed interfacial crack tips obeying the generalized contact model. Numerical examples involving a partially closed crack between a pair of misoriented cubic crystals are given to illustrate how the special crack-tip element helps in determining the stress intensity factors as well as the contact zone geometry.     

Hertz contact problems for an anisotropic physically nonlinear elastic medium

Conditions under which Hertz contact problems exhibit the property of self-similarity are determined. Qualitative conclusions concerning the character of selfsimilitude solutions from which, among other things, an equation similar to the familiar Mayer equation follows, are drawn. The problem of the collision between nonlinearly elastic bodies is also examined.Moscow Institute of Radio Engineering, Electronics, and-Automated Equipment. Translated from Problemy Prochnosti, No. 12, pp. 47–53, December, 1989.     

On a generalised plane strain crack problem for inhomogeneous anisotropic elastic materials

A generalised plane strain crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material are considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

BEM for crack-hole problems in thermopiezoelectric materials

The boundary element formulation for analysing interaction between a hole and multiple cracks in piezoelectric materials is presented. Using Green's function for hole problems and variational principle, a boundary element model (BEM) for a 2-D thermopiezoelectric solid with cracks and holes has been developed and used to calculate stress intensity factors of the crack-hole problem. In BEM, the boundary condition on the hole circumference is satisfied a priori by Green's function, and is not involved in the boundary element equations. The method is applicable to multiple-crack problems in both finite and infinite solids. Numerical results for stress and electric displacement intensity factors at a particular crack tip in a crack-hole system of piezoelectric materials are presented to illustrate the application of the proposed formulation.     

Numerical treatment of generalized plasticity with anisotropic elastic range in the case of plane stress

 The generalized plasticity model as presented by Lubliner (1991) and applied by Taylor and Auricchio (1995) is an extension of classical rate independent plasticity with a yield surface. For the generalized plasticity concept in Huettel and Matzenmiller (1999a) an integration algorithm under the restrictions of plane stress is presented that reduces the discretized tensor equations in the case of isotropy even for nonlinear hardening to the solution of a single scalar equation, as it is typical for the classical J₂-plasticity model in the complete three-dimensional case. Received 27 October 1999     

An alternative multi-region BEM technique for three-dimensional elastic problems

The main objective of this work is to present an alternative boundary element method (BEM) formulation for the static analysis of three-dimensional non-homogeneous isotropic solids. These problems can be solved using the classical boundary element formulation, analyzing each subregion separately and then joining them together by introducing equilibrium and displacements compatibility. Establishing relations between the displacement fundamental solutions of the different domains, the alternative technique proposed in this paper allows analyzing all the domains as one unique solid, not requiring equilibrium or compatibility equations. This formulation also leads to a smaller system of equations when compared to the usual subregion technique, and the results obtained are even more accurate.     

Time-domain BEM for transient interfacial crack problems in anisotropic piezoelectric bi-materials

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study transient response of interfacial cracks in piecewise homogeneous, anisotropic and linear piezoelectric bi-materials under electrical and mechanical impacts. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. To determine the real or complex dynamic stress intensity factors and the dynamic electrical displacement intensity factor of the interfacial cracks, an explicit extrapolating formula in a typical state of the crack plane perpendicular to the poling direction is presented in this paper. Numerical examples are presented; and the effects of the load combination and material combination on dynamic intensity factors and dynamic energy release rate are discussed.     

Kinked cracks in anisotropic elastic materials

The problem of a kinked crack is analysed for the most general case of elastic anisotropy. The kinked crack is modelled by means of continuous distributions of dislocations which are assumed to be singular both at the crack tips and at the kink vertex. The resulting system of singular integral equations is solved numerically using Chebyshev polynomials and the reciprocal theorem. The stress intensity factors for modes I, II and III and the generalised stress intensity factor at the vertex are obtained directly from the dislocation densities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stochastic identification of elastic constants for anisotropic materials

This paper presents an energy‐based characterization technique that stochastically identifies the elastic constants of anisotropic materials by modeling the measurement noise and removing its effect unlike conventional deterministic techniques, which deterministically identify the elastic constants directly from noisy measurements. The technique recursively estimates the elastic constants at every acquisition of measurements using Kalman Filter. Owing to the non‐linear expression of the measurement model, a Kalman gain has been newly derived and achieves optimal estimation. Since the variances in addition to the means are computed, the proposed technique can not only identify the elastic constants but also describe their certainty as an additional advantage. The validity of the proposed technique and its superiority to the conventional technique were first demonstrated via parametric studies of low‐dimensional problems. The proposed technique was then successfully applied to the identification of elastic constants of an anisotropic material. Copyright © 2009 John Wiley & Sons, Ltd.     

A BEM solution for plates on elastic half-space with unilateral contact

The method developed to analyze the stress and deflection of plates on unilateral foundation is different from the traditional method. The behavior of plates on unilateral foundation belongs to that of free-boundary problems. It has been proved effective to solve the problems of free-boundary using the linear complementary equation method. In this paper, a boundary element–linear complementary equation is derived according to contact theory. This equation is used to analyze plate-bending on elastic half-space foundation, especially considering the impact on the internal force and displacement of plate, which is caused by the neighboring loads acting on the foundation around the plate. The effectiveness of the method is illustrated by numerical results.     

Three‐dimensional BEM for scattering of elastic waves in general anisotropic media

Based on the full‐space Green's functions, a three‐dimensional time‐harmonic boundary element method is presented for the scattering of elastic waves in a triclinic full space. The boundary integral equations for incident, scattered and total wave fields are given. An efficient numerical method is proposed to calculate the free terms for any geometry. The discretization of the boundary integral equation is achieved by using a linear triangular element. Applications are discussed for scattering of elastic waves by a spherical cavity in a 3D triclinic medium. The method has been tested by comparing the numerical results with the existing analytical solutions for an isotropic problem. The results show that, in addition to the frequency of the incident waves, the scattered waves strongly depend on the anisotropy of the media. Copyright © 2003 John Wiley & Sons, Ltd.     

BEM for crack‐inclusion problems of plane thermopiezoelectric solids

The problem of interactions between an inclusion and multiple cracks in a thermopiezoelectric solid is considered by boundary element method (BEM) in this paper. First of all, a BEM for the crack–inclusion problem is developed by way of potential variational principle, the concept of dislocation, and Green's function. In the BE model, the continuity condition of the interface between inclusion and matrix is satisfied, a priori, by the Green's function, and not involved in the boundary element equations. This is then followed by expressing the stress and electric displacement (SED) and elastic displacements and electric potential (EDEP) in terms of polynomials of complex variables ξ_t and ξ_k in the transformed ξ‐plane in order to simulate SED intensity factors by the BEM. The least‐squares method incorporating the BE formulation can, then, be used to calculate SED intensity factors directly. Numerical results for a piezoelectric plate with one inclusion and a crack are presented to illustrate the application of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

An arbitrarily-oriented plane crack in an anisotropic elastic slab

The problem of an arbitrarily-oriented plane crack in an anisotropic elastic slab is considered. Through the use of a Fourier transform technique, the problem is reduced to a system of simultaneous Fredholm integral equations of the second kind. Once these integral equations are solved, relevant quantities such as the crack energy can be readily computed. Numerical results pertaining to the stability of a plane crack in a particular elastic slab are given.     

A plane contact problem for an elastic orthotropic strip

The contact of a punch with an elastic orthotropic strip is considered. A singular integral equation is derived for the contact pressure. The analytic expression of the associated kernel is unique for all types of orthotropy. An iterative solution method is developed to investigate a thick strip. A direct asymptotic procedure proposed for a thin strip leads to simple explicit formulae. Numerical examples are presented for various values of the relative strip thickness.     

2D analysis for geometrically non-linear elastic problems using the BEM

A boundary element method (BEM) approach for the solution of the elastic problem with geometrical non-linearities is proposed. The geometrical non-linearities that are considered are both finite strains and large displacements. Material non-linearities are not considered in this paper, so the constitutive law employed is Hooke's elastic one and the fundamental solution introduced in the integral equations is the usual one for isotropic linear elasticity. In order to deal with the intricate non-linear equations that govern the problem, an incremental–iterative method is proposed. The equations are linearized and a Total Lagrangian Formulation is adopted. The integral equations of the BEM are developed in an incremental form. The iterative process is necessary in order to achieve a good approximation to the governing equations. The problem of a slab under homogeneous deformation is solved and the results obtained agree with the analytical solution. The problem of a hollow cylinder under internal pressure is also solved and its solution compared with that obtained by a standardized finite element method code.     

A hypersingular boundary integral equation for antiplane crack problems for a class of inhomogeneous anisotropic elastic materials

An antiplane multiple crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material is considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

3D Frequency domain BEM for solving dipolar gradient elastic problems

A three dimensional (3D) boundary element method (BEM) for treating time harmonic problems in linear elastic structures exhibiting microstructure effects is presented. These microstructural effects are taken into account with the aid of the dipolar gradient elastic theory, which is the simplest dynamic version of Mindlins generalized elastic theory. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The dipolar gradient frequency domain elastodynamic fundamental solution is explicitly derived and used to construct the boundary integral representation of the solution with the aid of a reciprocal integral identity. In addition to a boundary integral representation for the displacement, a boundary integral representation for its normal derivative is also necessary for the complete formulation of a well posed problem. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. The solution procedure is described in detail. A numerical example serves to illustrate the method and demonstrate its accuracy     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.