Green's functions for Kirchhoff plates of irregular shape

A method is proposed for the construction of Green's functions for the Sophie Germain equation in regions of irregular shape with mixed boundary conditions imposed. The method is based on the boundary integral equation approach where a kernel vector function B satisfies the biharmonic equation inside the region. This leads to a regular boundary integral equation where the compensating loads and moments are applied to the boundary. Green's function is consequently expressed in terms of the kernel vector function B, the fundamental solution function of the biharmonic equation, and kernel functions of the inverse regular integral operators. To compute moments and forces, the kernel functions are differentiated under the integral sign. The proposed method appears highly effective in computing both displacements and stress components.     

Analysis of elliptical cracks perpendicular to the interface of two joined transversely isotropic solids

This paper presents a boundary element analysis of elliptical cracks in two joined transversely isotropic solids. The boundary element method is developed by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of stress intensity factors (SIFs) are obtained by using crack opening displacements. The results of the proposed method compare well with the existing exact solutions for an elliptical crack parallel to the isotropic plane of a transversely isotropic solid of infinite extent. Elliptical cracks perpendicular to the interface of transversely isotropic bi-material solids of either infinite extent or occupying a cubic region are further examined in detail. The crack surfaces are subject to the uniform normal tractions. The stress intensity factor values of the elliptical cracks of the two types are analyzed and compared. Numerical results have shown that the stress intensity factors are strongly affected by the anisotropy and the combination of the two joined solids.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).     

3D analytical solution for a functionally graded transversely isotropic piezoelectric circular plate under tension and bending

A three-dimensional (3D) analysis of a functionally graded piezoelectric circular plate under tension and bending is carried out. A direct displacement method is developed, with analytical solutions obtained for plate with either free or simply-supported edge conditions. The material properties of the plate can vary arbitrarily along the thickness except that the strain-energy function should be positive definite as required for stable materials and certain integrable conditions are assumed valid during the derivation. The validity of the present solutions is discussed both analytically and numerically. Numerical analyses are made for a specific functionally graded material to show the influence of material heterogeneity on the piezoelastic field.     

Green’s functions for transversely isotropic magnetoelectroelastic media

Based on the governing equations of transversely isotropic magnetoelectroelastic media, four general solutions on the cases of distinct eigenvalues and multiple eigenvalues are given and expressed in five mono-harmonic displacement functions. Then, based on these general solutions, employing the trial-and-error method, the three-dimensional Green’s functions of infinite, two-phase and semi-infinite magnetoelectroelastic media under point forces, point charge and magnetic monopole are all presented in terms of elementary functions for all cases of distinct eigenvalues and multiple eigenvalues. Numerical results are also presented.     

Distributed-elementary-source self-regularized dyadic Green's functions for modeling the massloading effect in acoustic devices

A concept for the simulation of two-dimensional models of massive electrodes in micro-acoustic devices has been presented. The method is based on a mesh-less analysis of the underlying boundary value problem. An efficient procedure for the calculation of the involved dyadic Green's functions has been introduced. Major advantage of the proposed method is in the ability of pre-calculating and storing relevant data for the characterization of individual substructures. The latter property is by construction amenable to parallel computing. Glimpse of the numerical results and figures facilitate the discussion of the underlying ideas.     

The boundary contour method for piezoelectric media with linear boundary elements

This paper presents a development of the boundary contour method (BCM) for piezoelectric media. First, the divergence‐free property of the integrand of the piezoelectric boundary element is proved. Secondly, the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and Green's functions (Computer Methods in Applied Mechanics and Engineering 1998; 158 : 65) for piezoelectric media. The BCM is applied to the problem of piezoelectric media. Finally, numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method (BEM). The numerical results of the BCM coincide very well with the exact solution, and the feasibility and efficiency of the method are verified. Copyright © 2003 John Wiley & Sons, Ltd.     

The time-dependent Green's function of the transverse vibration of a composite rectangular membrane

A new method for the approximate computation of the time-dependent Green's function for the equations of the transverse vibration of a multi stepped membrane is suggested. This method is based on generalization of the Fourier series expansion method and consists of the following steps. The first step is finding eigenvalues and an orthogonal set of eigenfunctions corresponding to an ordinary differential operator with boundary and matching conditions. The second step is a regularization (approximation) of the Dirac delta function in the form of the Fourier series with a finite number of terms, using the orthogonal set of eigenfunctions. The third step is an approximate computation of the Green's function in the form of the Fourier series with a finite number of terms relative to the orthogonal set of eigenfunctions. The computational experiment confirms the robustness of the method.     

3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium

This paper considers the non-axisymmetric three-dimensional problem of a penny-shaped crack with permeable electric conditions imposed on the crack surfaces, subjected to a pair of point normal forces applied symmetrically with respect to the crack plane. The crack is embedded in an infinite transversely isotropic piezoelectric body with the crack face perpendicular to the axis of material symmetry. Applying the symmetry of the problem under consideration then leads to a mixed–mixed boundary value problem of a half-space, for which potential theory method is employed for the purpose of analysis. The cases of equal eigenvalues are also discussed. Although the treatment differs from that for an impermeable crack reported in literature, the resulting governing equation still has a familiar structure. For the case of a point force, exact expressions for the full-space electro-elastic field are derived in terms of elementary functions with explicit stress and electric displacement intensity factors presented. The exact solution for a uniform loading is also given.     

A numerical Green's function for multiple cracks in anisotropic bodies

The numerical construction of a Green's function for multiple interacting planar cracks in an anisotropic elastic space is considered. The numerical Green's function can be used to obtain a special boundary-integral method for an important class of two-dimensional elastostatic problems involving planar cracks in an anisotropic body.     

Perturbation method of the piezoelectric fracture mechanics considering the permittivity of the medium in the crack gap

In this paper, the problem of the cracks with arbitrary forms in piezoelectric material is studied. The permittivity of the medium in the crack gap is considered. Except the collinear cracks, this boundary condition is too difficult to deal with; therefore, a perturbation method is recommended. By the way, the electric boundary conditions of electric fracture mechanics are discussed. For example, a small parameter solution of a crack is given and compared with the known `exact' (it will be discussed later) solution. This result shows that the impermeable or permeable conditions are only the boundary conditions for the first approximations of the perturbation solutions.     

Boundary element formulation for 3D transversely isotropic cracked bodies

The boundary traction integral representation is obtained in elasticity when the classical displacement representation is differentiated and combined according to Hooke's law. The use of both traction and displacement integral representations leads to a mixed (or dual) formulation of the BEM where the discretization effort for crack problems is much smaller than in the classical formulation. A boundary element analysis of three‐dimensional fracture mechanics problems of transversely isotropic solids based on the mixed formulation is presented in this paper. The hypersingular and strongly singular kernels appearing in the formulation are regularized by using two terms of the displacement series expansion and one term of the traction expansion, at the collocation point. All the remaining integrals are analytically evaluated or transformed by means of Stokes' theorem into regular or weakly singular integrals, which are numerically computed. The method is general and can be used for elements of any shape including quarter‐point crack front elements. No change of co‐ordinates is required for the integration. The formulation as presented in this paper is something as clear, general and easy to handle as the classical BE formulation. It is used in combination with three‐dimensional quadratic and quarter‐point elements to obtain accurate results for several different crack problems. Cracks in boundless and finite transversely isotropic domains are studied. The meshes are simple and include only discretization of the crack and the external boundary. The obtained results are in good agreement with those existing in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

横观各向同性土介质与结构物共同作用的有限元与边界元耦合分析

本文采用有限元与边界元耦合方法对上与结构物进行共同作用的分析．其中上部结构采用有限元子结构法，地基上考虑为横观各向同性特性，应用边界元法，通过基础界面上位移连续、面力平衡的条件进行耦合分析．编制了相应的计算程序，对某高层空间剪力培与地基的共同作用进行了计算，与有限元法和实测结果进行了对比分析，得到了满意的结果．说明：有限元和边界元的耦合方法能充分发挥两者的优越性，不失为共同作用分析的一种有效途径．     

Green's function expansion for exponentially graded elasticity

New computational forms are derived for Green's function of an exponentially graded elastic material in three dimensions. By suitably expanding a term in the defining inverse Fourier integral, the displacement tensor can be written as a relatively simple analytic term, plus a single double integral that must be evaluated numerically. The integration is over a fixed finite domain, the integrand involves only elementary functions, and only low‐order Gauss quadrature is required for an accurate answer. Moreover, it is expected that this approach will allow a far simpler procedure for obtaining the first and second‐order derivatives needed in a boundary integral analysis. The new Green's function expressions have been tested by comparing with results from an earlier algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Weight functions for interface cracks in dissimilar anisotropic piezoelectric materials

Weight functions proposed for interface cracks in dissimilar isotropic materials (Gao, 1991; Chen and Hasebe, 1994) are extended to treat those in piezoelectric materials. The difficulties in separating the eight distinct complex arguments are overcome. The pseudo-orthogonal properties of the eigenfunction expansion form found in isotropic dissimilar cases(Chen and Hasebe, 1994) are proved to be valid in the present cases although the mathematical manipulations performed here seem much more complicated than those in isotropic dissimilar materials. Several path-independent integrals are obtained and all the coefficients in the eigenfunction expansion form, including the K _I, K _II, K _III and K _e, could be calculated by the weight functions introduced in this paper. It is concluded that the weight functions presented here provide a powerful tool to calculate the dominant parameters at the interface crack tip without any special treatment to the singular stress field of the near-tip region.     

Transient analysis of three-dimensional dynamic interaction between multilayered soil and rigid foundation

The study of dynamic soil-structure interaction is significant to civil engineering applications, such as machine foundation vibration, traffic-induced vibration, and seismic dynamic response. The scaled boundary finite element method (SBFEM) is a semi-analytical algorithm, which is used to solve the dynamic response of a three-dimensional infinite soil. It can automatically satisfy the radiation boundary condition at infinity. Based on the dynamic stiffness matrix equation obtained by the modified SBFEM, a continued fraction algorithm is proposed to solve the dynamic stiffness matrix of layered soil in the frequency-domain. Then, the SBFEM was coupled with the finite element method (FEM) at the interface to solve the dynamic stiffness matrices of the rigid surface/buried foundation. Finally, the mixed-variable algorithm was used to solve the three-dimensional transient dynamic response of the foundation in the time domain. Numerical examples were performed to verify the accuracy of the proposed algorithm in solving the dynamic stiffness matrix of the infinite domain in the frequency domain and the dynamic transient displacement response of the foundation in the time domain. Compared with the previous numerical integration technique, the dynamic stiffness matrix in the frequency domain calculated by using the proposed algorithm has higher accuracy and higher efficiency.     

Unique real‐variable expressions of displacement and traction fundamental solutions covering all transversely isotropic elastic materials for 3D BEM

A general, efficient and robust boundary element method (BEM) formulation for the numerical solution of three‐dimensional linear elastic problems in transversely isotropic solids is developed in the present work. The BEM formulation is based on the closed‐form real‐variable expressions of the fundamental solution in displacements U_ik and in tractions T_ik, originated by a unit point force, valid for any combination of material properties and for any orientation of the radius vector between the source and field points. A compact expression of this kind for U_ik was introduced by Ting and Lee (Q. J. Mech. Appl. Math. 1997; 50 :407–426) in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector. Working from this expression of U_ik, and after a revision of their final formula, a new approach (based on the application of the rotational symmetry of the material) for deducing the derivative kernel U_{ik, j} and the corresponding stress kernel Σ_ijk and traction kernel T_ik has been developed in the present work. These expressions of U_ik, U_{ik, j}, Σ_ijk and T_ik do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex‐valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational‐symmetry axis. The expressions of U_ik, U_{ik, j}, Σ_ijk and T_ik have been presented in a form suitable for an efficient computational implementation. The correctness of these expressions and of their implementation in a three‐dimensional collocational BEM code has been tested numerically by solving problems with known analytical solutions for different classes of transversely isotropic materials. Copyright © 2007 John Wiley & Sons, Ltd.     

Approximate Green's functions for singular and higher order terms of an edge crack in a finite plate

An edge crack in a finite plate (FSECP) subjected to wedge forces is solved by the superposition of the analytical solution of a semi-infinite crack, and the numerical solution of a FSECP with free crack faces, which is solved by the Williams expansion. The unknown coefficients in the expansion are determined by a continuous least squares method after comparing it with the direct boundary collocation and the point or discrete least squares methods. The results are then used to validate the stress intensity factor (SIF) formula provided by Tada et al. that interpolates the numerical results of Kaya and Erdogan, and an approximate crack face opening displacement formula obtained in this paper by Castigliano's theorem and the SIF formula of Tada et al. These approximate formulae are accurate except for point forces very close to the outer edge, and can be used as Green's functions in the crack-closure based crack growth analysis, as well as in interpreting the size effect of quasi-brittle materials. Green's functions for coefficients relevant to the second to the fifth terms in the crack tip asymptotic field are also provided. Finally, a FSECP with a uniform pressure over a part of the crack faces is solved to illustrate the application of the obtained Green's functions and to further assess their accuracy by comparing with a finite element analysis.     

Dynamic stress intensity factors of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material under a time‐harmonic elastic P‐wave

The dynamic behavior of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material is investigated under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. The problem is formulated through double Fourier transform into three pairs of dual integral equations with the displacement jumps across the crack surfaces as the unknown variables. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the dynamic stress field and the dynamic electric displacement filed near the crack edges are obtained, and the effects of the shape of the rectangular crack, the characteristics of the harmonic wave, and the distance between two rectangular cracks on the stress and the electric intensity factors in a piezoelectric composite material are analyzed. Copyright © 2015 John Wiley & Sons, Ltd.