Equations for small fields superposed on finite biasing fields in a thermoelectroelastic body

Equations for infinitesimal incremental fields superposed on finite biasing fields in a thermoelectroelastic body are derived from the nonlinear equations of thermoelectroelasticity. The equations are general in the sense that no assumptions are made on the biasing fields. The general equations obtained are reduced to various special cases under different approximations.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Stroh formalism based boundary integral equations for 2D magnetoelectroelasticity

This paper presents a novel approach for obtaining boundary integral equations of 2D anisotropic magnetoelectroelasticity. This approach is based on the holomorphy theorems and the Stroh formalism and allows developing of the integral equations for the aperiodic, singly and doubly periodic problems of magnetoelectroelasticity. Obtained equations contain the unknown discontinuities of displacement, electric and magnetic potentials and also traction, electric displacement and magnetic induction that allow adopting the existing boundary element procedures for their solution. Analytical solutions for systems of collinear permeable or impermeable cracks are obtained. Numerical boundary element solutions are obtained for the singly and doubly periodic sets of permeable and impermeable cracks in the magnetoelectroelastic medium and a half-plane. Comparison with analytical solutions and other available results validate the present formulations and numerical computation.     

Direct boundary integral equations for elastodynamics in 3-D half-spaces

Vector boundary integral equations (BIE's) based on Somigliana's integral formula are presented with Stokes' (full-space) and Lamb's (half-space) fundamental tensors (Green's functions), for radiation and scattering of time-harmonic elastic waves by bodies embedded in or laying on the surface of a three-dimensional homogeneous, isotropic, linear-elastic half-space. Numerical work is based on BIEs free from principal-value integrals. Whereas the Stokes'-tensor BIE requires discretization of the infinite half-space surface, all discretization is confined to (finite) surfaces of the body when Lamb's tensors are used. The nonuniqueness of the integral equation solution at fictitious eigenfrequencies is addressed. Numerical results are presented for a rigid circular footing, a rigid hemispherical foundation and a fully embedded spherical cavity.     

Identities for the fundamental solution of thin plate bending problems and the nonuniqueness of the hypersingular BIE solution for multi-connected domains

Four integral identities for the fundamental solution of thin plate bending problems are presented in this paper. These identities can be derived by imposing rigid-body translation and rotation solutions to the two direct boundary integral equations (BIEs) for plate bending problems, or by integrating directly the governing equation for the fundamental solution. These integral identities can be used to develop weakly-singular and nonsingular forms of the BIEs for plate bending problems. They can also be employed to show the nonuniqueness of the solution of the hypersingular BIE for plates on multi-connected (or multiply-connected) domains. This nonuniqueness is shown for the first time in this paper. It is shown that the solution of the singular (deflection) BIE is unique, while the hypersingular (rotation) BIE can admit an arbitrary rigid-body translation term in the deflection solution, on the edge of a hole. However, since both the singular and hypersingular BIEs are required in solving a plate bending problem using the boundary element method (BEM), the BEM solution is always unique on edges of holes in plates on multi-connected domains. Numerical examples of plates with holes are presented to show the correctness and effectiveness of the BEM for multi-connected domain problems.     

The use of Stokes' fundamental solution for the boundary only element formulation of the three-dimensional Navier–Stokes equations for moderate Reynolds numbers

This paper presents a boundary element formulation for the permanent Navier–Stokes equations in which the well-known closed-form fundamental solution for the steady Stokes equations is employed. In this way, from the integral representation formulae for the Stokes' equations, an integral equation is found in which the original non-linear convective terms of the Navier–Stokes equations appear as a domain integral. Additionally, the method of dual reciprocity is used to transform the domain integral to boundary integrals (this method is closely related to the method of particular integrals also used in the literature to transform domain integrals to boundary integrals). Numerical results are presented for the three-dimensional internal flow in a cylindrical container with a rotating cover, in which the accuracy of the method is shown.     

弹性地基板广义边值问题的边界元法

本文利用Ｈａｎｋｅｌ变换导出了弹性地基板弯曲问题的基本解，该基本解对于Ｗｉｎｋｌｅｒ地基、Ｐａｓｔｅｒｎａｋ地基和弹性半空间地基模型具有统一的表达形式。在此基础上，建立了适用于弹性地基板广义边值问题的边界积分方程组，最后文中给出了若干数值算例。     

A comprehensive study on the 2D boundary element method for anisotropic thermoelectroelastic solids with cracks and thin inhomogeneities

This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displacement in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results.     

A boundary integral method for multiple circular holes in an elastic half-plane

This paper presents a semi-analytical method for solving the problem of an isotropic elastic half-plane containing a large number of randomly distributed, non-overlapping, circular holes of arbitrary sizes. The boundary of the half-plane is assumed to be traction-free and a uniform far-field stress acts parallel to that boundary. The boundaries of the holes are assumed to be either traction-free or subjected to constant normal pressure. The analysis is based on solution of complex hypersingular integral equation with the unknown displacements at each circular boundary approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-plane. Several examples available in the literature are re-examined and corrected, and new benchmark examples with multiple holes are included to demonstrate the effectiveness of the approach.     

Null-field approach for the antiplane problem with elliptical holes and/or inclusions

In this paper, we extend the successful experience of solving an infinite medium containing circular holes and/or inclusions subject to remote shears to deal with the problem containing elliptical holes and/or inclusions. Arbitrary location, different orientation, various size and any number of elliptical holes and/or inclusions can be considered. By fully employing the elliptical geometry, fundamental solutions were expanded into the degenerate kernel by using an addition theorem in terms of the elliptic coordinates and boundary densities are described by using the eigenfunction expansion. The difference between the proposed method and the conventional boundary integral equation method is that the location point can be exactly distributed on the real boundary without facing the singular integral and calculating principal value. Besides, the boundary stress can be easily calculated free of the Hadamard principal values. It is worthy of noting that the Jacobian terms exist in the degenerate kernel, boundary density and contour integral; however, these Jacobian terms would cancel each other out and the orthogonal property is preserved in the process of contour integral. This method belongs to one kind of meshless methods since only collocation points on the real boundary are required. In addition, the solution is regarded as semi-analytical form because error purely attributes to the number of truncation term of eigenfunction. An exact solution for a single elliptical inclusion is also derived by using the proposed approach and the results agree well with Smith’s solutions by using the method of complex variables. Several examples are revisited to demonstrate the validity of our method.     

Thermoelectroelastic analysis of cracks in piezoelectric half-plane by BEM

The Green's function and the boundary element method for analysing fracture behaviour of cracks in piezoelectric half-plane are presented in this paper. By combining Stroh formalism and the concept of perturbation, a general thermoelectroelastic solution for half-plane solid subjected to point heat source and/or temperature discontinuity has been derived. Using the proposed solution and the potential variational principle, a boundary element model (BEM) for 2-D half-plane solid with multiple cracks has been developed and used to calculate the stress intensity factors of the multiple crack problem. The method is available for multiple crack problems in both finite and infinite solids. Numerical results for a two-crack system are presented and compared with those from finite element method (FEM).     

Mathematical boundary integral equation analysis of an embedded shell under dynamic excitations

A boundary integral equation method is presented for the analysis of a thin cylindrical shell embedded in an elastic half-space under axisymmetric excitations. By virtue of a set of ring-load Green's functions for the shell and a group of dynamic fundamental solutions for the semi-infinite medium, the structure–medium interaction problem of wave propagation is shown to be reducible to a set of coupled boundary integral equations. Through the analysis of an auxiliary pair of Cauchy integral equations, the singularities of the contact stress distributions arc rendered explicit. With a direct incorporation of such analytical features into the formulation, an effective computational procedure is developed which involves an interpolation of regular functions only. Typical results for the dynamic contact load distributions, displacements, and complex compliance functions are included as illustrations. In addition to furnishing quantities of direct engineering interest, this treatment is apt to be useful as a foundation for further rigorous as well as approximate developments for various related physical problems and boundary integral methods.     

A complex variable boundary element method: Development

A generalized boundary integral equation method for the solution of the Laplace equation is developed based on the Cauchy integral theorem for analytical complex variable functions. Although the approach is complicated by the utilization of complex variable theory, the resulting model involves direct integration along straight-line boundary segments (elements) rather than using quadrature formulae that are required in current real variable boundary element formulations. Previously published boundary integral equation methods based on the Cauchy integral theorem are shown to be a subset of the generalized model theory developed in this paper.     

Boundary element analysis of dynamic coupled thermoelasticity problems

Some possibility of numerical analysis of coupled dynamic problems of linear elastic heat conductors on classical thermoelasticity theory by using the boundary element method is shown in this paper. The boundary integral equation formulation and its numerical implementation of the two-dimensional problem are developed in the manner by the newly derived fundamental solution for the coupled equations of elliptic type in the transformed space and the numerical inversion of Laplace transformation. The boundary element unsteady solutions of the first and second Danilovskaya problems and the Sternberg and Chakravorty problem in the half-space are demonstrated through comparison with the existing solutions.     

Transient behavior of a few dies on an elastic half-space

Summary An asymptotic approach to dynamic interaction between a few distant dies and an elastic half-space is proposed. The transient motion of the dies under low-frequency vertical load is under consideration. The explicit expression for the fundamental singular solution of Lamb's problem is used to derive the boundary integral equation of contact. Then this equation is asymptotically simplified and solved numerically in combination with equations of motion of the dies.Equations obtained in the asymptotic limit describe both the die-medium dynamic interaction and the interaction between dies through the elastic medium. These equations take into account the energy dissipation phenomenon associated with energy transfer deep into the medium by outgoing elastic waves, of so called geometrical damping.Equations proposed are asymptotically correct within the corresponding range of parameters, as such improving the state-of-the-art.     

Boundary integral equations for notch problems in an anisotropic half-space

A crack or a hole embedded in an anisotropic half-plane space subjected to a concentrated force at its surface is analyzed. Based on the Stroh formalism and the fundamental solutions to the half-plane solid due to point dislocations, the problem can be formulated by a system of boundary integral equations for the unknown dislocation densities defined on the crack or hole border. These integral equations are then reduced to algebraic equations by using the properties of the Chebyshev polynomials in conjunction with the appropriate transformations. Numerical results have been carried out for both crack problems and hole problems to elucidate the effect of geometric configurations on the stress intensity factors and the stress concentration.     

Singular integral equation method for multiple Zener–Stroh crack problems in antiplane elasticity

In this paper, the multiple Zener–Stroh crack problems in anti-plane elasticity are studied. The crack faces are assumed to be traction free, and dislocation distributions on the cracks are chosen as the unknown functions in the solution. The singular integral equations for the problem are obtained. The constraint equations are also derived from the condition of the accumulation of dislocation on the cracks. After solving the integral equations, the stress intensity factors at crack tips can be evaluated immediately. Numerical examples are given. It is found that interactions between the Zener–Stroh cracks are quite different from those for the Griffith cracks, in qualitative and quantitative aspects.     

Green's function for two-and-a-half dimensional elastodynamic problems in a half-space

 This paper presents an analytical solution, together with explicit expressions, for the steady state response of a homogeneous three-dimensional half-space subjected to a spatially sinusoidal, harmonic line load. These equations are of great importance in the formulation of three-dimensional elastodynamic problems in a half-space by means of integral transform methods and/or boundary elements. The final expressions are validated here by comparing the results with those obtained with the boundary element method (BEM) solution, for which the free surface of the ground is discretized with boundary elements.     

Perturbation method for the solution of a Zener–Stroh crack with a slightly curved configuration

This paper investigates the Zener–Stroh crack with curved configuration in plane elasticity. A singular integral equation is suggested to solve the problem. Formulae for evaluating the SIFs and T-stress at the crack tip are suggested. If the curve configuration is a product of a small parameter and a quadratic function, a perturbation method based on the singular integral equation is suggested. In the method, the singular integral equation can be expanded into a series with respect to the small parameter. Therefore, many singular integral equations can be separated from the same power order for the small parameter. These singular integral equations can be solved successively. The solution of the successive singular integral equations will provide results for stress intensity factors and T-stress at the crack tip. It is found that the behaviors for the solution of SIFs and T-stress in the Zener–Stroh crack and the Griffith crack are quite different. This can be seen from the presented comparison results.     

Interaction between a crack and a free or welded boundary in media with arbitrary anisotropy

The problem of an arbitrarily oriented straight crack near a straight free or welded boundary, in media with the most general anisotropy, is considered. Using the solution given by Stroh [2] for a dislocation in an infinite medium, we choose appropriate stress functions to obtain the solution for a dislocation near the boundary. A system of singular integral equations is then set up for the crack and its numerical solution gives the dislocation density of it. Results related to a wear application are presented and the variation of the stress intensity factors with the material anisotropy and the position of the crack is shown.