Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing an anti-plane strain inclusion of arbitrary cross-sectional shape 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 anti-plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite anti-plane strain elastic body. The solution reduces to that of the infinite-domain anti-plane strain inclusion problem when the boundary effect is not considered. The problem of a circular cylindrical inclusion embedded concentrically in a finite cylindrical elastic matrix undergoing anti-plane strain deformations is analytically solved 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 linear elasticity-based Eshelby tensor for the circular cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is small and that the boundary effect can dominate when the inclusion volume fraction is 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.     

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.     

Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory

A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an 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). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is expressed in terms of the Green's function for an infinite three-dimensional elastic body in the SSGET. It contains a volume integral term and a surface integral term. The former is the same as that for the infinite-domain inclusion problem based on the SSGET, while the latter represents the boundary effect. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is not considered. The problem of a spherical inclusion embedded concentrically in a finite spherical elastic body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. This Eshelby tensor depends on the position, inclusion size, matrix size, and material length scale parameter, and, as a result, can capture the inclusion size and boundary effects, unlike existing Eshelby tensors. It reduces to the classical Eshelby tensor for the spherical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. 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 as the inclusion becomes large enough, and the boundary effect is vanishing as the inclusion volume fraction gets sufficiently low.     

A boundary element method for calculating the SH waves scattering from arbitrarily shaped holes in an anisotropic medium

The scattering problem of elastic wave by arbitrarily shaped cavities in an infinite anisotropic medium is investigated by the boundary integral equation (BIE) method. The formulations of BIE are derived with the help of generalized Green's formula. The discretization of BIE is based upon constant elements. After confirmation of the accuracy of the present method, some numerical examples are given for various cavities in a full space, in which an isotropic body with a circular cylinder hole is used for comparison and good agreement is observed. It has been proved that the method developed in this paper is effective.     

各向异性体内含任意孔洞对反平面波散射的边界元方法

本文借助于广义格林公式导出了用位移表示的各向异性介质中SH波入射时的边界积分方程.根据本文作者在文献[8]给出的基本解,求解了各向异性介质中孔洞对SH波的散射问题.边界积分方程的离散基于常数元模式.文中给出了一个圆柱、一个椭圆柱和两个椭圆柱形式的孔洞周围的位移场和应力场的数值结果.最后,对入射波频率较高时的情形作了说明.     

On multiple circular inclusions in plane magnetoelasticity

A general series solution to the magnetoelastic problem of interacting circular inclusions in plane magnetoelasticity is provided in this paper. By the use of complex variable theory and Laurent series expansion method, the general expression of the magnetic and the magnetoelastic complex potentials for the circular inclusion problem is derived. Expanding the definition of the Airy’s stress function of pure elastic field into the magnetoelastic field and applying the superposition method, the general expression then can be reduced to a set of linear algebraic equations and solved in a series form. An approximate closed form solution for the case of two arbitrarily located inclusions is also provided. For illustrating the effect of the pertinent parameters, the numerical results of the interfacial magnetoelastic stresses are displayed in graphic form.     

Circular inclusions in anti-plane strain couple stress elasticity

The solution for a circular inclusion with a prescribed anti-plane eigenstrain is derived. It is shown that the components of the Eshelby tensor within the inclusion, corresponding to a uniform eigenstrain, can be either uniform or non-uniform, depending on the imposed interface conditions. The stress amplification factors due to circular void or rigid inclusion in an infinite medium under remote anti-plane shear stress are calculated. The failure of the couple stress elasticity to reproduce the classical elasticity solution in the limit of vanishingly small characteristic length is indicated for a particular type of boundary conditions. The solution for a circular inhomogeneity in an infinitely extended matrix subjected to remote shear stress is then derived. The effects of the imposed interface conditions, the shear stress and couple stress discontinuities, and the relationship between the inhomogeneity and its equivalent eigenstrain inclusion problem are discussed.     

圆内旋轮线形状异质夹杂热弹性问题的解析解

论文研究具有圆内旋轮线型形状夹杂域的平面热弹性体在夹杂域内非均匀温度场作用下对弹性场所产生的影响，其中考虑的夹杂与基体的材料不同但是具有相同的剪切模量。借助黎曼映射理论，将平面光滑闭合曲线外部区域映射到单位圆外部区域，进而利用解析函数性质，结合柯西型积分与Faber多项式，求解得到夹杂域内外场势函数的显式解。通过得到的势函数求出内外场应力，并对应力分布进行分析，结果表明：一般形状异质夹杂时，内场应力值与有限元计算值相吻合；退化到椭圆同质夹杂时，与相关文献中的结果相同，但是更具一般性与实际可操作性。     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

Diffraction of waves by semi-infinite breakwater using finite and infinite elements

A finite and infinite element model is derived to predict wave patterns around a semi-infinite breakwater in water of constant depth. Both circular and square meshes of elements are used. The wave theory used is that of Berkhoff. The appropriate boundary conditions for finite and infinite boundaries are described. The singularity in the velocity at the breakwater tip is modelled effectively using the technique of Henshell and Shaw originally developed in elasticity. The results agree well with the analytical solution. In addition the problem of waves incident upon a semi-infinite breakwater and parabolic shoal, where both diffraction and refraction are present, is solved. There is no analytical solution for this case. The combination of finite and infinite elements is found to be an effective and accurate technique for such problems.     

Scattering of harmonic waves by a disk-shaped rigid inclusion in a three-dimensional elastic matrix

We consider a three-dimensional problem on the interaction of harmonic waves with a thin rigid movable inclusion in an infinite elastic body. The problem is reduced to solving a system of two-dimensional boundary integral equations of Helmholtz potential type for the stress jump functions on the opposite surfaces of the inclusion. We propose a boundary element method for solving the integral equations on the basis of the regularization of their weakly singular kernels. Using the asymptotic relations between the amplitude-frequency characteristics of the wave farzone field and the obtained boundary stress jump functions, we determine the amplitudes of the shear plane wave scattering by a circular disk-shaped inclusion for various directions of the wave incident on the inclusion and for a broad range of wave numbers.     

三维裂纹分析的主值型面力边界积分方程法

给出了一组只包含Ｃａｕｃｈｙ主值积分、不含有强奇异积分的三维静动力边界积分方程及其应用于裂纹问题的具体列式，并给出了几何轴对称问题的相应半解析边界元求解方法，将三维问题降阶为一维数值问题．文中分析了无限、半无限介质中圆裂纹、平行圆裂纹系、球面裂纹等在静载及应力波作用下的静力或瞬态动力响应问题，求得了相应的应力强度因子．     

A Circular Inclusion with Inhomogeneously Imperfect Interface in Plane Elasticity

A general method is presented for the rigorous solution of a circular inclusion embedded within an infinite matrix in plane elastostatics. The bonding at the inclusion-matrix interface is considered to be imperfect with the assumption that the interface imperfections are circumferentially inhomogeneous. Using analytic continuation, the basic boundary value problem for four analytic functions is reduced to two coupled first order differential equations for two analytic functions. The resulting closed-form solutions include a finite number of unknown constants determined by analyticity and certain other auxiliary conditions. The method is illustrated using a particular class of inhomogeneous interface. The results from these calculations are compared to the corresponding results when the imperfections in the interface are circumferentially homogeneous. These comparisons illustrate, for the first time, how the circumferential variation of the parameter describing the imperfection has a pronounced effect on the average stresses induced within the inclusion. This revised version was published online in August 2006 with corrections to the Cover Date.     

Stress Distribution Around a Crack in Plane Micropolar Elasticity

In this paper we formulate the boundary value problem of plane micropolar elasticity for a domain containing a crack in Sobolev spaces and prove the existence and continuous dependence on the data of the corresponding weak solutions. We consider the cases of both finite and infinite domain and find the solutions in terms of modified single layer and modified double layer potentials with distributional densities.      

Three-dimensional elastostatic solutions for transversely isotropic functionally graded material plates containing elastic inclusion

Based on the generalized England-Spencer plate theory, the equilibrium of a transversely isotropic functionally graded plate containing an elastic inclusion is studied. The general solutions of the governing equations are expressed by four analytic functions α(ζ), β(ζ), φ(ζ), and ψ(ζ) when no transverse forces are acting on the surfaces of the plate. Axisymmetric problems of a functionally graded circular plate and an infinite func-tionally graded plate containing a circular hole subject to loads applied on the cylindrical boundaries of the plate are firstly investigated. On this basis, the three-dimensional (3D) elasticity solutions are then obtained for a functionally graded infinite plate containing an elastic circular inclusion. When the material is degenerated into the homogeneous one, the present elasticity solutions are exactly the same as the ones obtained based on the plane stress elasticity, thus validating the present analysis in a certain sense.     

Green's function for incremental nonlinear elasticity: shear bands and boundary integral formulation

An elastic, incompressible, infinite body is considered subject to plane and homogeneous deformation. At a certain value of the loading, when the material is still in the elliptic range, an incremental concentrated line load is considered acting at an arbitrary location in the body and extending orthogonally to the plane of deformation. This plane strain problem is solved, so that a Green's function for incremental, nonlinear elastic deformation is obtained. This is used in two different ways: to quantify the decay rate of self-equilibrated loads in a homogeneously stretched elastic solid; and to give a boundary element formulation for incremental deformations superimposed upon a given homogeneous strain. The former result provides a perturbative approach to shear bands, which are shown to develop in the elliptic range, induced by self-equilibrated perturbations. The latter result lays the foundations for a rigorous approach to boundary element techniques in finite strain elasticity.     

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.     

High-order local non-reflecting boundary conditions: a review

A common method for numerically solving wave problems in unbounded domains is based on truncating the infinite domain via an artificial boundary B, thus defining a finite computational domain, and using a special non-reflecting boundary condition (NRBC) on B. Low-order local NRBCs have been constructed and practiced since the 1970s. Exact non-local NRBCs were introduced in the 1980s. Only recently high-order local NRBCs have been devised. These NRBCs, despite being of an arbitrarily high-order, do not involve high derivatives owing to the use of specially defined auxiliary variables. This paper reviews the latter approach, explains its advantages compared to previous approaches, and discusses the different schemes which have been proposed in this context.     

Energy-consistent shear coefficients for beams with circular cross sections and radially inhomogeneous materials

An exact computational method for the shear stiffness of beams with circular cross sections and arbitrarily radially inhomogeneous Young’s modulus is presented. We derive the displacement and stress field of a cantilever beam according to 3D theory of elasticity, which requires to solve just a 1D linear boundary value problem. The shear stiffness is obtained by setting the shear strain energy from the exact solution equal to that from technical beam theory. Results and closed analytical formulae are given for several functionally graded and layered cross sections.     

Opening-Function Simulation of the Three-Dimensional Nonstationary Interaction of Cracks in an Elastic Body

Integral relations between three-dimensional dynamic displacements (stresses) in an infinite elastic body with arbitrarily located plane cracks and discontinuities in the displacements of the opposite crack faces are presented. The influence of opening cracks on each other is considered in the problem on crack faces loaded by pulse forces. This problem is reduced to a system of boundary integral equations of the wave-potential type in a time domain. The dynamic mode I stress intensity factors are determined for two coplanar elliptic cracks under forces in the form of the Heaviside function