New Green’s function for stress field and a note of its application in quantum-wire structures

The elastic field caused by the lattice mismatch between the quantum wires and the host matrix can be modeled by a corresponding two-dimensional hydrostatic inclusion subjected to plane strain conditions. The stresses in such a hydrostatic inclusion can be effectively calculated by employing the Green’s functions developed by Downes and Faux, which tend to be more efficient than the conventional method based on the Green’s function for the displacement field. In this study, Downes and Faux’s paper is extended to plane inclusions subjected to arbitrarily distributed eigenstrains: an explicit Green’s function solution, which evaluates the stress field due to the excitation of a point eigenstrain source in an infinite plane directly, is obtained in a closed-form. Here it is demonstrated that both the interior and exterior stress fields to an inclusion of any shape and with arbitrarily distributed eigenstrains are represented in a unified area integral form by employing the derived Green’s functions. In the case of uniform eigenstrain, the formulae may be simplified to contour integrals by Green’s theorem. However, special care is required when Green’s theorem is applied for the interior field. The proposed Green’s function is particularly advantageous in dealing numerically or analytically with the exterior stress field and the non-uniform eigenstrain. Two examples concerning circular inclusions are investigated. A linearly distributed eigenstrain is attempted in the first example, resulting in a linear interior stress field. The second example solves a circular thermal inclusion, where both the interior and exterior stress fields are obtained simultaneously.     

The Dual-Reciprocity Boundary Element Analysis for Hydraulically Fractured Shale Gas Reservoirs Considering Diffusion and Sorption Kinetics

This work presents a new application of boundary element method (BEM) to model fluid transport in unconventional shale gas reservoirs with discrete hydraulic fractures considering diffusion, sorption kinetics and sorbed-phase surface diffusion. The fluid transport model consists of two governing partial differential equations (PDEs) written in terms of effective diffusivities for free and sorbed gases, respectively. Boundary integral formulations are analytically derived using the fundamental solution of the Laplace equation for the governing PDEs and Green’s second identity. The domain integrals arising due to the time-dependent function and nonlinear terms are transformed into boundary integrals employing the dual-reciprocity method. This transformation retains the domain-integral-free, boundary-integral-only character of standard BEM approaches. In the proposed solution, the free- and sorbed-gas flow in the shale matrix is solved simultaneously after coupling the fracture flow equation of free gas. Well production performance under the effect of relaxation phenomenon due to delayed responses of sorbed gas under nonequilibrium sorption condition is rigorously captured by imposing the zero-flux condition at fracture–matrix interface for the sorbed-gas transport equation. The validity of proposed solution is verified using several case studies through comparison against a commercial finite-element numerical simulator.

     

Green’s functions of an exponentially graded transversely isotropic half-space

By virtue of a complete set of displacement potential functions and Hankel transform, the analytical expressions of Green’s function of an exponentially graded elastic transversely isotropic half-space is presented. The given solution is analytically in exact agreement with the existing solution for a homogeneous transversely isotropic half-space. Employing a robust asymptotic decomposition technique, the Green’s function is decomposed to the closed-form Green’s function corresponding to the homogeneous transversely isotropic half-space and grading term with strong decaying integrands. This representation is very useful for numerical methods which are based on boundary-integral formulations such as boundary-element method since the numerically evaluated part is not responsible for the singularity. The high accuracy of the proposed numerical scheme is confirmed by some numerical examples.     

Inclusion of arbitrary polygon with graded eigenstrain in an anisotropic piezoelectric full plane

In this paper, an exact closed-form solution for the Eshelby problem of a polygonal inclusion with a graded eigenstrain in an anisotropic piezoelectric full plane is presented. For this electromechanical coupling problem, by virtue of Green’s function solutions, the induced elastic and piezoelectric fields are first expressed in terms of line integrals on the boundary of the inclusion. Using the line-source Green’s function, the line integral is then carried out analytically for the linear eigenstrain case, with the final expression involving only elementary functions. Finally, the solution is applied to the semiconductor quantum wire (QWR) of square, triangle, circle and ellipse shapes within the GaAs (0 0 1) substrate. It is demonstrated that there exists significant difference between the induced field by the uniform eigenstrain and that by the linear eigenstrain. Since the misfit eigenstrain in most QWR structures is actually non-uniform, the present solution should be particularly appealing to nanoscale QWR structure analysis where strain and electric fields are coupled and are affected by the non-uniform misfit strain.     

The non-homogeneous biharmonic plate equation: fundamental solutions

Real materials and structural components are often non-homogeneous, either by design or because of the physical composition and imperfections in the underlying material. Thus, analytical solutions for non-homogeneous materials under mechanical loads are of considerable interest to engineers and have widespread applications, given the prevalence of these materials in fields as diverse as aerospace, construction, electronics, etc. More precisely, those are essentially composites with carefully manufactured properties that yield desirable mechanical characteristics and properties, such as optimal arrangement of the material, minimum weight, etc. To this end, the displacement fundamental solution (or Green’s function) corresponding to a point force for the non-homogenous biharmonic equation in two dimensions are derived in this work by employing a conformal mapping technique in conjunction with the Radon transformation. These functions, besides being useful in their own right, can also be used within the context of integral equation formulations for the solution of boundary-value problems. Finally, a series of numerical examples that deal with the non-homogeneous plate on elastic foundation problem serve to illustrate the present method.     

Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing a plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite plane strain elastic body, which differs from that in earlier studies using the three-dimensional Green’s function. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is suppressed. The problem of a cylindrical inclusion embedded concentrically in a finite plane strain cylindrical elastic matrix of an enhanced continuum is analytically solved for the first time by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical elasticity-based Eshelby tensor for the cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are not considered. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

半无限弹性空间域内点加振格林函数的计算

本文给出了满足全部自由面边界条件的半无限弹性空间域内点加振的Ｇｒｅｅｎ函数，利用变形的Ｈａｎｋｅｌ函数，在复数域内进行无限积分的有限化，从而使Ｇｒｅｅｎ函数的计算变得比较简单和方便。     

Image singularities of Green’s functions for isotropic elastic bimaterials subjected to concentrated forces and dislocations

Green’s functions for isotropic materials in the two-dimensional problem for elastic bimaterials with perfectly bonded interface are reexamined in the present study. Although the Green’s function for an isotropic elastic bimaterial subjected to a line force or a line dislocation has been discussed by many authors, the physical meaning and the structure of the solution are not clear. In this investigation, the Green’s function for an elastic bimaterial is shown to consist of eight Green’s functions for a homogeneous infinite plane. One of the novel features is that Green’s functions for bimaterials can be expressed directly by knowing Green’s functions for the infinite plane. If the applied load is located in material 1, the solution for the half-plane of material 1 is constructed with the help of five Green’s functions corresponding to the infinite plane. However, the solution for the half-plane of material 2 only consists of three Green’s functions for the infinite plane. One of the five Green’s functions of material 1 and all the three Green’s functions of material 2 have their singularities located in the half-plane where the load is applied, and the other four image singularities of material 1 are located outside the half-plane at the same distance from the interface as that of the applied load. The nature and magnitude of the image singularities for both materials are presented explicitly from the principle of superposition, and classified according to different loads. It is known that for the problem of anisotropic bimaterials subjected to concentrated forces and dislocations, the image singularities are simply concentrated forces and dislocations with the stress singularity of order O(1/r). However, higher orders (O(1/r²) and O(1/r³)) of stress singularities are found to exist in this study for isotropic bimaterials. The highest order of the stress singularity is O(1/r³) for the image singularities of material 1, and is O(1/r²) for material 2. Using the present solution, Green’s functions associated with the problems of elastic half-plane with free and rigidly fixed boundaries, for homogeneous isotropic elastic solid, are obtained as special cases.     

Green’s functions for plane problem under various boundary conditions and applications

Green’s functions of a point dislocation as well as a concentrated force for the plane problem of an infinite plane containing an arbitrarily shaped hole under stress, displacement, and mixed boundary conditions are stated. The Green’s functions are obtained in closed forms by using the complex stress function method along with the rational mapping function technique, which makes it possible to deal with relatively arbitrary configurations. The stress functions for these problems consist of two parts: a principal part containing singular and multi-valued terms, and a complementary part containing only holomorphic terms. These Green’s functions can be derived without carrying out any integration. The applications of the Green’s functions are demonstrated in studying the interaction of debonding and cracking from an inclusion with a line crack in an infinite plane subjected to remote uniform tension. The Green’s functions should have many other potential applications such as in boundary element method analysis. The boundary integral equations can be simplified by using the Green’s functions as the kernels.     

Toughening by Phase Boundary Propagation

Transformation toughening has enhanced the fracture toughness of certain Zirconia-Toughened Ceramics (ZTC) by factors of 2–4. The primary explanation of toughening, by McMeeking and Evans [1] and Budiansky et al. [2], suggests that the main source of toughening is the energy stored by the transformed inclusions in the wake of a propagating crack. In the case of supercritical ZTC where the boundary of the transformed zone is a phase boundary (surface of strain discontinuity), this paper suggests an additional source of toughening – that due to propagation of the phase boundary. By extending the J-integral to cracked bodies containing surfaces of strain discontinuity, the transformation toughening for a steady Mode I crack is evaluated.     

SH波入射半空间双相介质界面附近圆形衬砌的动力分析

利用复变函数和Green函数法研究了无限半空间中双相介质界面附近圆形衬砌对SH波的散射与动应力集中问题。该问题的解答采用镜像法,首先构造出含有圆形衬砌的直角平面区域出平面问题的Green函数,然后利用“契合”技术,并根据界面处位移连续性条件将解答归结为具有弱奇异性的第一类Fredholm积分方程组的求解,结合散射波的衰减特性,直接离散该方程组,把积分方程组转化为线性代数方程组可得到该问题的数值结果。最后,通过算例分析了不同介质参数、几何参数和入射波时圆形衬砌界面的动应力集中情况。     

均布荷载作用下功能梯度简支梁弯曲的解析解

利用应力函数半逆解法,研究了均布载荷作用下、材料属性在厚度上任意变化的功能梯度简支梁弯曲的解析解,给出了各向应力应变与位移的解析显式表达式.首先根据平面应力状态的基本方程,得出了功能梯度梁的应力函数应满足的偏微分方程,并根据应力边界条件得出了各应力分布的表达式;进而根据功能梯度材料的本构方程和位移边界条件,得出了应变和位移的分布.最后,通过将本文的解退化到均质各向同性梁并与经典弹性解比较,证明了本文理论的正确性,并求解了材料组分呈幂律分布的功能梯度梁的应力和位移分布,分析了上下表层材料的弹性模量比λ与组分材料体积分数指数n对应力和位移分布的影响.     

Axisymmetric nuclei of strain and their applications for multilayered solids

In this paper, the effect of several axisymmetric elastic singularities (i.e., point forces, double forces, sum of two double forces and centers of dilatation) on the elastic response of a multilayered solid is investigated. The boundary conditions in an infinite solid at the plane passing through the singularity are derived first using Papkovich–Neuber harmonic functions. Then, a Green’s function solution for multilayered solids is obtained by solving a set of simultaneous linear algebraic equations using both the boundary conditions for the singularity and the layer interfaces. Finally, the elastic solutions in a single layer on an infinite substrate due to point defects and infinitesimal prismatic dislocation loops are presented to illustrate the application of these Green’s function solutions.     

Analysis of scattering waves in an elastic layered medium by means of the complete eigenfunction expansion form of the Green’s function

A scattering problem due to an object and a plane incident wave in an elastic layered half space is presented in this paper. The complete eigenfunction expansion form of the Green’s function developed by the author and the boundary integral equation method are introduced into the analysis. First, the complete eigenfunction expansion form of the Green’s function is investigated for its application to the scattering problem. A comprehensive explanation is also given for the fact that the complex Rayleigh wave modes exhibit standing waves. Next, a method for the analysis of scattering waves by means of the Green’s function is presented. The advantage of the present method is that the formulation itself is independent of the number of layers and the scattering waves can be decomposed into the modes for the spectra defined for the layered medium. Several numerical calculations are performed to examine the efficiency of the present method as well as the properties of the scattering waves. According to the numerical results, the complete eigenfunction expansion form of the Green’s function provides accurate values for application to a boundary element analysis. The spectral structure and radiation patterns of the scattering wave are presented and investigated. The differences in directionality can be found from the radiation patterns of the scattering waves decomposed into the modes for the spectra.     

Solution of the plane stress problems of strain-hardening materials described by power-law using the complex pseudo-stress function

In the present paper,the compatibility equation for the plane stress problems of power-law materials is transformed into a biharmonic equation by introducing the so-calledcomplex pseudo-stress function,which makes it possible to solve the elastic-plastic planestress problems of strain hardening materials described by power-law using the complexvariable function method like that in the linear elasticity theory.By using this generalmethod,the close-formed analytical solutions for the stress,strain and displacementcomponents of the plane stress problems’of power-law materials is deduced in the paper,which can also be used to solve the elasto-plastic plane stress problems of strain-hardeningmaterials other than that described by power-law.As an example,the problem of a power-law material infinite plate containing a circular hole under uniaxial tension is solved byusing this method,the results of which are compared with those of a known asymptoticanalytical solution obtained by the perturbation method.     

Time-harmonic Green’s functions for anisotropic magnetoelectroelasticity

Two-dimensional (2-D) and three-dimensional (3-D) time-harmonic Green’s functions for linear magnetoelectroelastic solids are derived in this paper by means of Radon-transform. Displacement field and electric and magnetic potentials in a fully anisotropic magnetoelectroelastic infinite solid due to a time-harmonic point force, point charge and magnetic monopole are obtained in form of line integrals over a unit circle in 2-D case and surface integrals over a unit sphere in 3-D case. This dynamic fundamental solution is then split into the sum of regular dynamic plus singular terms. The singular terms coincide with the Green’s functions for the static problem and may be further reduced to closed form expressions. The proposed Green’s functions can be used in the corresponding boundary element method (BEM) formulation.     

Singular boundary method for 2D dynamic poroelastic problems

This paper extends a strong-form meshless boundary collocation method, named the singular boundary method (SBM), for the solution of dynamic poroelastic problems in the frequency domain, which is governed by Biot equations in the form of mixed displacement–pressure formulation. The solutions to problems are represented by using the fundamental solutions of the governing equations in the SBM formulations. To isolate the singularities of the fundamental solutions, the SBM uses the concept of the origin intensity factors to allow the source points to be placed on the physical boundary coinciding with collocation points, which avoids the auxiliary boundary issue of the method of fundamental solutions (MFS). Combining with the origin intensity factors of Laplace and plane strain elastostatic problems, this study derives the SBM formulations for poroelastic problems. Five examples for 2D poroelastic problems are examined to demonstrate the efficiency and accuracy of the present method. In particular, we test the SBM to the multiply connected domain problem, the multilayer problem and the poroelastic problem with corner stress singularities, which are all under varied ranges of frequencies.     

On constructing a Green’s function for a semi-infinite beam with boundary damping

The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green’s functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green’s function method, and how damping properties can be obtained.     

双相介质半空间中圆孔对透射SH波的稳态分析

研究了平面SH波在半空间双相弹性介质中的传播。通过Green函数和积分方程方法,按照复变函数描述,对透射波被圆孔散射的情况进行稳态分析。将双相介质半空间沿界面剖分为1/4空间介质Ⅰ和含圆孔的1/4空间介质Ⅱ,分别构造了介质Ⅰ和介质Ⅱ中反平面点源荷载的Green函数,按双相介质中平面SH波的处理方法,给出介质Ⅰ和介质Ⅱ中的平面位移波,两种介质之间的相互作用力与对应Green函数的乘积沿界面的积分与平面位移波叠加得到介质Ⅰ和介质Ⅱ中的全部位移场。按照界面的位移连续条件,定解积分方程组,得到问题的稳态解,并给出圆孔位置和介质参数对散射的影响。     

Potential Method in the Linear Theory of Viscoelastic Materials with Voids

In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with voids is considered and some basic results of the classical theory of elasticity are generalized. Indeed, the basic properties of plane harmonic waves are established. The explicit expression of fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions. The Green’s formulas in the considered theory are obtained. The uniqueness theorems of the internal and external basic boundary value problems (BVPs) are proved. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The formulas of integral representations of Somigliana type of regular vector and regular (classical) solution are obtained. The Sommerfeld-Kupradze type radiation conditions are established. The basic properties of elastopotentials and singular integral operators are given. Finally, the existence theorems for classical solutions of the internal and external basic BVPs of steady vibrations are proved by using of the potential method (boundary integral method) and the theory of singular integral equations.