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.     

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.     

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.     

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。     

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.     

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.     

An elasticity solution for transversely isotropic,functionally graded circular plates

This article presents a new elasticity solution for transversely isotropic, functionally graded circular plates subject to axisymmetric loads. It is assumed that the material properties vary along the thickness of a circular plate according to an exponential form. By extending the displacement function presented by Plevako to the case of transversely isotropic material, we derived the governing equation of the problem studied. The displacement function was assumed as the sum of the Bessel function and polynomial function to obtain the analytical solution of a transversely isotropic, functionally graded circular plate under different boundary conditions. As a numerical example, the influence of the graded variations of the material properties on the displacements and stresses was studied. The results demonstrate that the graded variations have a significant effect on the mechanical behavior of a circular plate.     

A numerical integration Green's function model for ultrasonic field profiles in mildly anisotropic media

A numerical integration technique utilizing a point source Green's function is introduced to analyze the wave behavior in transversely isotropic-type anisotropic media allowing us to make fast and accurate computations of the acoustic field. The centrifugally cast stainless steel (CCSS) used in nuclear power plants is chosen as a sample medium because of its columnar grain character leading to material anisotropy. A representative number of field profiles are computed and plotted to illustrate the quasi-longitudinal, quasi-transverse, and horizontally-polarized shear wave propagation in a transversely-isotropic medium. Phenomena such as beam skewing, beam splitting, beam focusing, unsymmetrical beams, and other anisotropic effects, some of which are already known from earlier experimental observations, emerge as a computational result of the introduced technique.     

Evaluation of the three-dimensional elastic Green's function in anisotropic cubic media

By using the Fourier transforms method, the three-dimensional Green's function solution for a unit force applied in an infinite cubic material is evaluated in this paper. Although the elastic behavior of a cubic material can be characterized by only three elastic constants, the explicit solutions of Green's function for a cubic material are not available in the literatures. The central problem for explicitly solving the elastic Green's function of anisotropic materials depends upon the roots of a sextic algebraic equation, which results from the inverse Fourier transforms and is composed of the material constants and position vector parameters. The close form expression of Green's function is presented here in terms of roots of the sextic equation. The sextic equation for an anisotropic cubic material is discussed thoroughly and specific results are given for possible explicit solutions.     

Mixture rule for indentation derived Young's modulus in layered composites

The mixture rule for layered composites under the concentrated load condition has been formulated by the elastic energy method. Numerical boundary integral calculation reveals that the contour of strain is sensitive while that of stress is insensitive to the order and magnitude of elastic modulus, therefore stress could be an appropriate influence function for the present mixture rule. A simple mixture rule is obtained analytically by repeated integration of energy density, which is derived from Green's function of 2D and 3D half space. The prediction curve obtained by the present mixture rule shows good agreement with experimental indentation results.     

Lyapunov functions for quasi-Hamiltonian systems

A procedure for constructing the Lyapunov functions and studying their asymptotic Lyapunov stability with probability one for quasi-Hamiltonian systems is proposed. For quasi-non-integrable Hamiltonian systems, the Hamiltonian (the total energy) is taken as the Lyapunov function. For quasi-integrable and quasi-partially-integrable Hamiltonian systems, the optimal linear combination of the independent first integrals in involution is taken as the Lyapunov function. The derivative of the Lyapunov function with respect to time is obtained by using the stochastic averaging method for quasi-Hamiltonian systems. The sufficient condition for the asymptotic Lyapunov stability with probability one of quasi-Hamiltonian systems is determined based on a theorem due to Khasminskii and compared with the corresponding necessary and sufficient condition obtained by using the largest Lyapunov exponent. Three examples are worked out to illustrate the proposed procedure and its effectiveness.     

Application of hierarchical matrices to boundary element methods for elastodynamics based on Green's functions for a horizontally layered halfspace

This paper presents the application of hierarchical matrices to boundary element methods for elastodynamics based on Green's functions for a horizontally layered halfspace. These Green's functions are computed by means of the direct stiffness method; their application avoids meshing of the free surface and the layer interfaces. The effectiveness of the methodology is demonstrated through numerical examples, indicating that a significant reduction of memory and CPU time can be achieved with respect to the classical boundary element method. This allows increasing the problem size by one order of magnitude. The proposed methodology therefore offers perspectives to study large scale problems involving three-dimensional elastodynamic wave propagation in a layered halfspace, with possible applications in seismology and dynamic soil–structure interaction.     

Universal properties of kernel functions for probabilistic sensitivity analysis

Development of probabilistic sensitivities is frequently considered an essential component of a probabilistic analysis and often critical towards understanding the physical mechanisms underlying failure and modifying the design to mitigate and manage risk. One useful sensitivity is the partial derivative of the probability-of-failure and/or the system response with respect to the parameters of the independent input random variables. Calculation of these partial derivatives has been established in terms of an expected value operation (sometimes called the score function or likelihood ratio method). The partial derivatives can be computed with typically insignificant additional computational cost given the failure samples and kernel functions — which are the partial derivatives of the log of the probability density function (PDF) with respect to the parameters of the distribution. The formulation is general such that any sampling method can be used for the computation such as Monte Carlo, importance sampling, Latin hypercube, etc. In this paper, useful universal properties of the kernel functions that must be satisfied for all two parameter independent distributions are derived. These properties are then used to develop distribution-free analytical expressions of the partial derivatives of the response moments (mean and standard deviation) with respect to the PDF parameters for linear and quadratic response functions. These universal properties can be used to facilitate development and verification of the required kernel functions and to develop an improved understanding of the model for design considerations.     

Local Kronecker delta property of the MLS approximation and feasibility of directly imposing the essential boundary conditions for the EFG method

The element-free Galerkin (EFG) method is a promising method for solving many engineering problems. Because the shape functions of the EFG method obtained by the moving least-squares (MLS) approximation, generally, do not satisfy the Kronecker delta property, special techniques are required to impose the essential boundary conditions. In this paper, it is proved that the MLS shape functions satisfy the Kronecker delta property when the number of nodes in the support domain is equal to the number of the basis functions. According to this, a local Kronecker delta property, which is satisfying the Kronecker delta property only at boundary nodes, can be obtained in one- and two-dimension. This local Kronecker delta property is an inherent property of the one-dimensional MLS shape functions and can be obtained for the two-dimensional MLS shape functions by reducing the influence domain of each boundary node to weaken the influence between them. The local Kronecker delta property provides the feasibility of directly imposing the essential boundary conditions for the EFG method. Four numerical examples are computed to verify this feasibility. The coincidence of the numerical results obtained by the direct method and Lagrange multiplier method shows the feasibility of the direct method.     

Determination of weight functions for elastic T-stress from reference T-stress solutions

ABSTRACT This paper presents the application of the weight function method for the calculation of elastic T -stress. First, the background of the weight function method for the calculation of T -stress is summarized. Then an analysis of known weight functions for T -stress revealed that it is possible to approximate them with one universal mathematical form with three unknown parameters with high accuracy. The existence of this weight function form significantly simplified the determination of weight functions for T -stress. For any particular crack geometry, the unknown parameters can be determined from reference T -stress solutions. The general weight function expression, with suitable reference T -stress solutions, was used to derive the weight functions for single edge cracked plate, double edge cracked plate and center cracked plate specimens. These weight functions were then further used to calculate the T -stress solutions for cracked specimens under several nonlinear stress fields and were compared to available numerical data.     

近场数值波动分析中地震波输入的一种简化方法

在显式有限元方法结合黏弹性人工边界的时域波动方法的基础上,建立了地震波垂直输入时的一种简化输入方法。将近场有限元模型沿高度方向进行分层,将地震动的入射运动转化为作用于人工边界底面及分层后每层侧面上的均布力,以实现地震动的输入。与以等效节点力的方式实现的地震动输入相比,施加均布力的方式简化了地震波输入的前处理工作,且又能保证与等效节点力方式相同的精度。自由场数值算例表明：当本文方法中的分层高度与波动有限元网格离散要求的最大尺寸相等时,本文方法与等效节点力方法具有相同的精度;当局部区域按网格离散要求的最大尺寸进行分层而在其它区域放大分层高度时,局部区域上的近场波动响应仍可保证具有相当高的精度。另外,某隧道结构地震响应算例的计算结果同样说明了本文方法的有效性。