An integral formulation for steady-state elastoplastic contact over a coated half-plane

 A boundary-domain integral equation for a coated half-space (elastically isotropic homogeneous substratum, possibly anisotropic coating layer) is developed. The half-space fundamental solution is used, so that the discretization is limited to the potential contact zone (boundary elements), the potentially plastic part of the substratum and the coating layer (domain integration cells). Steady-state elastoplastic analysis is implemented within this framework, for plane-strain conditions, for solving rolling and/or sliding contact problems, where at the moment the contact load comes from either a purely elastic contact analysis or is of Hertz type. The constitutive integration is of implicit type. In order to improve accuracy and computational efficiency, infinite elements are used. Comparison of numerical results with other sources, when available, is satisfactory. The present formulation is also used to compute the contact pressure for an isotropic (or anisotropic) coating on an isotropic homogeneous half-space indented by an elastic punch. Received 29 May 2001     

Localized direct boundary-domain integro-differential formulations for incremental elasto-plasticity of inhomogeneous body

A quasi-static mixed boundary value problem of incremental elasto-plasticity for a continuously inhomogeneous body is considered. Using the two-operator Green–Betti formula and the fundamental solution of a reference homogeneous linear elasticity problem, with frozen initial or tangent elastic coefficients, a boundary-domain integro-differential formulation of the elasto-plastic problem is presented, with respect to the displacement rates and their gradients. Using a cut-off function approach, the corresponding localized parametrix of the reference problem is constructed to reduce the elasto-plastic problem to a nonlinear localized boundary-domain integro-differential equation. Algorithms of mesh-based and mesh-less discretizations are presented resulting in sparsely populated systems of nonlinear algebraic equations for the displacement increments.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Direct localized boundary-domain integro-differential formulations for physically nonlinear elasticity of inhomogeneous body

A static mixed boundary value problem (BVP) of physically nonlinear elasticity for a continuously inhomogeneous body is considered. Using the two-operator Green-Betti formula and the fundamental solution of an auxiliary linear operator, a non-standard boundary-domain integro-differential formulation of the problem is presented, with respect to the displacements and their gradients. Using a cut-off function approach, the corresponding localized parametrix is constructed to reduce the nonlinear BVP to a nonlinear localized boundary-domain integro-differential equation. Algorithms of mesh-based and mesh-less discretizations are presented resulting in sparsely populated systems of nonlinear algebraic equations.     

Doubly periodic volume?surface integral equation formulation for modelling metamaterials

A generalised volume-surface integral equation is extended by way of the periodic Green's function to model arbitrarily complex designs of metamaterials consisting of high-contrast inhomogeneous anisotropic material regions as well as metallic inclusions. The unique aspect of the formulation is the integration of boundary and volume integral equations to increase modelling efficiency and capability. Specifically, the boundary integral approach with equivalent surface currents is adopted over regions consisting of piecewise homogeneous materials as well as metallic perfect electric/magnetic conductor inclusions, whereas the volume integral equation is employed only in inhomogeneous and/or anisotropic material regions. Because the periodic Green's function only needs to be evaluated for the equivalent surface currents enclosing an inhomogeneous and/or anisotropic region, matrix fill time is much less as compared to using a volume formulation. Furthermore, the incorporation of curvilinear finite elements allows for greater geometrical modelling flexibility for arbitrarily shaped high-contrast regions found in typical designs of engineered metamaterials     

The problem of two plastic and heterogeneous inclusions in an anisotropic medium

We demonstrate an integral equation for the total local strain ε^T in an anisotropic heterogeneous medium with incompatible strain ε^p and which is at the same time submitted to an exterior field. The integral equation is solved in the case of an heterogeneous and plastic pair of inclusions, for which we calculate the average fields in each inclusion as well as the different parts of the elastic energy stocked in the medium.The solution is applied to the case of two isotropic and spherical inclusions in an isotropic matrix loaded in shear. The results are compared with those deduced from a more approximate method based on Horn's approximation of the integral equation. In appendix we give a numerical method for calculating the interaction tensors between anisotropic inclusions in an anisotropic medium as well as the analytic solution in the case of two spherical inclusions located in an isotropic medium.     

An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials

A new meshless method for computing the dynamic stress intensity factors (SIFs) in continuously non-homogeneous solids under a transient dynamic load is presented. The method is based on the local boundary integral equation (LBIE) formulation and the moving least squares (MLS) approximation. The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by a circle centered at the collocation point. The boundary-domain integral formulation with elastostatic fundamental solutions for homogeneous solids in Laplace-transformed domain is used to obtain the weak solution for subdomains. On the boundary of the subdomains, both the displacement and the traction vectors are unknown generally. If modified elastostatic fundamental solutions vanishing on the boundary of the subdomain are employed, the traction vector is eliminated from the local boundary integral equations for all interior nodal points. The spatial variation of the displacements is approximated by the MLS scheme.     

Boundary-domain integral method using a velocity-vorticity formulation

The paper deals with the numerical solution of fluid dynamics (transport phenomena in incompressible fluid flow) using a boundary-domain integral method. A velocity-vorticity formulation of the Navier-Stokes equations is adopted, where the kinematic equation is written in its parabolic version.     

Mesh-based numerical implementation of the localized boundary-domain integral-equation method to a variable-coefficient Neumann problem

An implementation of the localized boundary-domain integral-equation (LBDIE) method for the numerical solution of the Neumann boundary-value problem for a second-order linear elliptic PDE with variable coefficient is discussed. The LBDIE method uses a specially constructed localized parametrix (Levi function) to reduce the BVP to a LBDIE. After employing a mesh-based discretization, the integral equation is reduced to a sparse system of linear algebraic equations that is solved numerically. Since the Neumann BVP is not unconditionally and uniquely solvable, neither is the LBDIE. Numerical implementation of the finite-dimensional perturbation approach that reduces the integral equation to an unconditionally and uniquely solvable equation, is also discussed.     

Interactions between inclusions and various types of cracks

The problems of a crack inside, outside, penetrating or lying along the interface of an anisotropic elliptical inclusion are considered in this paper. Because the crack may be represented by a distribution of dislocation, integrating the analytical solutions of dislocation problems along the crack and applying the technique of numerical solution on the singular integral equation, we can obtain the general solutions to the problems of interactions between cracks and anisotropic elliptical inclusions. Since there are no analytical solutions existing for the general cases of interactions between cracks and inclusions, the comparison is made with the numerical results obtained by other methods or with the analytical results for the special cases which can be reduced from the present problems. These results show that our solutions are correct and universal     

Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods

This paper presents a formulation of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of two-dimensional mixed boundary-value problems (BVP) for a second-order linear elliptic partial differential equation (PDE) with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the BVP to a BDIE or BDIDE. The numerical formulation of the BDIDE employs an approximation for the boundary fluxes in terms of the potential function within the domain cells; therefore, the solution is fully described in terms of the variation of the potential function along the boundary and domain. Linear basis functions localised on triangular elements and standard quadrature rules are used for the calculation of boundary integrals. For the domain integrals, we have implemented Gaussian quadrature rules for two dimensions with Duffy transformation, by mapping the triangles into squares and eliminating the weak singularity in the process. Numerical examples are presented for several simple problems with square and circular domains, for which exact solutions are available. It is shown that the present method produces accurate results even with coarse meshes. The numerical results also show that high rates of convergence are obtained with mesh refinement.     

A boundary element method solution for anisotropic nonhomogeneous elasticity

A new BEM approach is presented for the plane elastostatic problem for nonhomogeneous anisotropic bodies. In this case the response of the body is described by two coupled linear second order partial differential equations in terms of displacement with variable coefficient. The incapability of establishing the fundamental solution of the governing equations is overcome by uncoupling them using the concept of analog equation, which converts them to two Poisson’s equations, whose fundamental solution is known and the necessary boundary integral equations are readily obtained. This formulation introduces two additional unknown field functions, which physically represent the two components of a fictitious source. Subsequently, they are determined by approximating them globally with radial basis functions series. The displacements and the stresses are evaluated from the integral representation of the solution of the substitutes equations. The presented method maintains the pure boundary character of the BEM. The obtained numerical results demonstrate the effectiveness and accuracy of the method.     

Effects of a viscoelastic substrate on a cracked body under in-plane concentrated loading

Summary The effect of a viscoelastic substrate on an anisotropic elastic cracked body under in-plane concentrated loading is studied in this paper. Based on the correspondence principle, the viscoelastic solution is directly obtained from the corresponding elastic one. The fundamental elastic solution is solved as three complex potentials via the property of analytical continuation to satisfy the continuity condition along the interface between dissimilar media. A singular integral technique in association with the dual coordinate transformation is applied to obtain the stress intensity factors for various crack orientations. Using the standard solid model to formulate the viscoelastic constitutive equation, some numerical examples are considered to demonstrate the use of the present approach.     

A general integral equation formulatin for homogeneous orthotropic potential problems

In this paper an integral equation formulation is proposed for the analysis of orthotropic potential problems. The two primary integral equations of the method are derived from the original governing differential equation firstly by rewriting it in a slightly different form and then applying the direct boundary element method formulation. The solution procedure is based on the use of the fundamental solutions for the isotropic potential case and special attention is given to the differentiation of a singular integral which yields an additional term as well as to the evaluation of the resulting Cauchy principal value integral. A simple discretization for the boundary and its interior domain is adopted in order to express the primary integral equations of the method in matrix form. Three examples are presented, the results of which illustrate the satisfactory accuracy of the method. The main feature of the proposed formulation is its generality, which makes possible its direct extension to solve such as heat conduction or subsurface flow in anisotropic media and, foremost, to orthotropic and anisotropic elasticity or elastoplasticity.     

BIEM for 2D steady-state problems in cracked anisotropic materials

The two-dimensional ‘in-plane’ time-harmonic elasto-dynamic problem for anisotropic cracked solid is studied. The non-hypersingular traction boundary integral equation method (BIEM) is used in conjunction with closed form frequency dependent fundamental solution, obtained by Radon transform. Accuracy and convergence of the numerical solution for stress intensity factor (SIF) is studied by comparison with existing solutions in isotropic, transversely-isotropic and orthotropic cases. In addition a parametric study for the wave field sensitivity on wave, crack and anisotropic material parameters is presented.     

Elastic analysis of a half-plane containing an inclusion and a void using a mixed volume and boundary integral equation method

A mixed volume and boundary integral equation method is used to calculate the plane elastostatic field in an isotropic elastic half-plane containing an isotropic or anisotropic inclusion and a void subject to remote loading parallel to a traction-free boundary. A detailed analysis of the stress field is carried out for three different geometries of the problem. It is demonstrated that the method is very accurate and effective for investigating local stresses in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions and multiple voids.     

Quasistationary Model of Electromagnetic Fields for a Piecewise-Homogeneous Body

We consider a two- or three-dimensional body of arbitrary shape containing foreign inclusions. These inclusions are in perfect electromagnetic contact with the matrix. A boundary condition of the third kind is imposed on the surface of the body. We propose a method of combined application of the fundamental solution of the nonstationary of heat-conduction equation, near-boundary and contact elements, and a step-by-step time scheme of common initial condition for the construction of integral transformations of the components of the vectors of electromagnetic-field intensity at an arbitrary point of space and time. In the case of perfect electromagnetic contact between the zones, the application of contact elements on the interface of the media (instead of boundary or near-boundary elements) enables one to automatically satisfy the first condition of contact (the equality for the components of the vector of electric-field intensity). As a result, the number of unknown fictitious current sources decreases as compared with the case of application of traditional indirect methods of boundary and near-boundary elements. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 2, pp. 89–96, March–April, 2005.     

横观各向同性结构边界元分析

本文利用边界单元法分析三维横观各向同性结构,引用三个位势函数并利用叠加原理导出了基本解,并且利用基本解的新型结构形式避免了在分界单元计算中所遇到的所谓“奇异积分”的问题。     

Multiple elliptical inclusions of arbitrary orientation in composites

A volume integral equation method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing multiple elliptical inclusions of arbitrary orientation subjected to uniform tensile stress at infinity. The inclusions are assumed to be long parallel elliptical cylinders composed of isotropic and anisotropic elastic material perfectly bonded to the isotropic matrix. The solid is assumed to be under plane strain on the plane normal to the cylinders. A detailed analysis of the stress field at the matrix–inclusion interface for square and hexagonal packing arrays is carried out, taking into account different values for the number, orientation angles and concentration of the elliptical inclusions. The accuracy and efficiency of the method are examined in comparison with results available in the literature.     

A novel method for solving the displacement and stress fields of an infinite domain with circular holes and/or inclusions subject to a screw dislocation

In this paper, the degenerate kernel and superposition technique are employed to solve the screw dislocation problems with circular holes or inclusions. The problem is decomposed into the screw dislocation problem with several holes and the interior Laplace problems for several circular inclusions. Following the success of the null-field integral equation approach, the typical boundary value problems can be solved easily. The kernel functions and unknown boundary densities are expanded by using the degenerate kernel and Fourier series, respectively. To the authors?? best knowledge, the angle-type fundamental solution is first derived in terms of degenerate kernel in this paper. Finally, four examples are demonstrated to verify the validity of the present approach.