Green's function solution and displacement potentials for Transformed Transversely Isotropic materials

A totally non-degenerate expression for the Green's function of infinite Transversely Isotropic (TI) materials is first deduced from the solutions given by Pan and Chou [Pan, Y.-C., Chou, T.-W., 1976. Point force solution for an infinite transversely isotropic solid. Trans. ASME, J. Appl. Mech. 43 (4), 608–612]. Then this solution and also the displacement potentials for TI materials are extended by a linear transformation to a larger family of anisotropic materials (Transformed Transversely Isotropic or TraTI materials). This family depends on 12 independent parameters and contains non-orthotropic materials and in this way a first explicit analytical solution for the Green's function for a non-orthotropic material is obtained. The TraTI materials which have orthotropic Symmetry (StraTI materials) constitute a sub-family depending on 6 independent parameters in the symmetry basis of the material. These materials present a 3D anisotropy (different stiffnesses in three orthogonal directions). General displacement potentials and the Green's function solution for STraTI materials can be deduced by a simple change and introducing one additional parameter in the well-known TI solutions.     

A point heat source on the apex of a transversely isotropic piezothermoelastic cone

We use the compact harmonic general solutions of transversely isotropic piezothermoelastic materials to construct the three-dimensional solutions of a steady point heat source on the apex of a transversely isotropic piezothermoelastic cone by four newly introduced harmonic functions. All components of coupled field are expressed in terms of elementary functions and are convenient to use. When the apex angle 2α equals to π, the solutions reduce to those of the semi-infinite body problem, which is called Green's function. Numerical results are given graphically by contours.     

Three-dimensional analytical solution for a rotating disc of functionally graded materials with transverse isotropy

Based on the basic equations for axisymmetric problems of transversely isotropic elastic materials, the displacement components are expressed in terms of polynomials of the radial coordinate with the five involved coefficients, named as displacement functions in this paper, being undetermined functions of the axial (thickness) coordinate. Five equations governing the displacement functions are then derived. It is shown that the displacement functions can be found through progressive integration by incorporating the boundary conditions. Thus a three-dimensional analytical solution is obtained for a transversely isotropic functionally graded disc rotating at a constant angular velocity.The solution can be degenerated into that for an isotropic functionally graded rotating disc. A prominent feature of this solution is that the material properties can be arbitrary functions of the axial coordinate. Thus, the solution for a homogeneous transversely isotropic rotating disc is just a special case that can be easily derived. An example is finally considered for a special functionally graded material, and numerical results shows that the material inhomogeneity has a remarkable effect on the elastic field.     

Magnetoelectric Green's functions and their application to the inclusion and inhomogeneity problems

Explicit expressions of magnetoelectric Green's functions are obtained for a transversely isotropic medium exhibiting coupling between the static electric and magnetic fields utilizing the contour integral representation. Four Green's functions exist which represent the coupled static electric and magnetic response to a unit point electric or magnetic charge. The Green's functions are applied to analyze the inclusion and inhomogeneity problems in an infinite magnetoelectric medium, and explicit, closed form expressions are obtained for the Eshelby type tensors. The magnetoelectric Eshelby's tensors can be readily used in the solution of numerous problems in the mechanics and physics of magnetoelectric solids.     

Three-dimensional Green's functions in anisotropic trimaterials

Three-dimensional elastostatic Green's functions in anisotropic trimaterials are derived, for the first time, by applying the generalized Stroh's formalism and Fourier transforms. The Green's functions are expressed as a series summation with the first term corresponding to the full-space solution and other terms to the image solutions due to the interfaces. The most remarkable feature of the present solution is that the image solutions can be expressed by a simple line integral over a finite interval [0,2π]. By partitioning the trimaterial Green's function into a full-space solution and a complementary part, the line integral involves only regular functions if the singularity is within one of the three materials, being treated analytically owning to the explicit expression of the full-space solution. When the singularity is on the interface, which occurs if the field and source points are both on the same interface, the involved singularity is handled with the interfacial Green's functions.A numerical example is presented for a trimaterial system made of two anisotropic half spaces bonded perfectly by an isotropic adhesive layer, showing clearly the effect of material layering on the Green's displacements and stresses. Furthermore, by comparing the present Green's solution to the direct (two-dimensional) 2D integral expression which is also derived in this paper, it is shown that, the computational time for the calculation of the Green's function can be substantially reduced using the present solution, instead of the direct 2D integral method.     

Revisiting displacement functions in three-dimensional elasticity of inhomogeneous media

The paper presents a three-dimensional solution to the equilibrium equations for linear elastic transversely isotropic inhomogeneous media. We assume that the material has constant Poisson’s ratios, and its Young’s and shear moduli have the same functional form of dependence on the co-ordinate normal to the plane of isotropy. We show, apparently for the first time, that stresses and displacements in such an inhomogeneous transversely isotropic elastic solid can be represented in terms of two displacement functions which satisfy the second- and fourth-order partial differential equations. We examine and discuss key aspects of the new representation; they include the relationship between the new displacement functions and Plevako’s solution for isotropic inhomogeneous material with constant Poisson’s ratio as well as the application of the new representation to some important classes of three-dimensional elasticity problems. As an example, the displacement function is derived that can be used to determine stresses and displacements in an inhomogeneous transversely isotropic half-space which is subjected to a concentrated force normal to a free surface and applied at the origin (Boussinesq’s problem).     

Anti-plane time-harmonic Green's functions for a circular inhomogeneity in piezoelectric medium with a spring- or membrane-type interface

The present work focuses on the two-dimensional anti-plane time-harmonic dynamic Green's functions for a circular inhomogeneity immersed in an infinite matrix with an imperfect interface, where both the inhomogeneity and matrix are assumed to be piezoelectric and transversely isotropic. Two types of imperfect interface, the spring-type interface with electromechanical coupling and the membrane-type interface, are considered. The former type is often used to model the electromechanical damage along the interface while the latter is largely employed to simulate surface/interface effect of nano-sized inhomogeneity. By using the Bessel function expansions, explicit solutions for the electromechanical fields induced by a time-harmonic anti-plane line force and line charge located in an unbounded matrix as well as the circular inhomogeneity are respectively derived. The present solutions can recover the anti-plane Green's functions for some special cases, such as the dynamic or quasi-static Green's functions of piezoelectricity with perfect interface as well as the dynamic or quasi-static Green's functions of pure elasticity with imperfect interface. For detailed discussions, selected analytical results are presented. Dependence of the electromechanical fields on circular frequency as well as interface properties is examined. The size effect related to interfacial imperfection is also discussed.     

Deformation of Euler-Bernoulli Beams by Means of Modified Green's Function: Application of Fredholm Alternative Theorem

This article deals with the determination of the static displacement function of an Euler-Bernoulli beam with two guided supports. To this end, the Green's function method is employed and exact solution is obtained. The Green's function of the problem is constructed, using pertinent boundary conditions of the problem. Nevertheless, the problem does not admit a Green's function due to a mathematical contradiction. In order to eliminate the trouble, the Fredholm Alternative Theorem is utilized and the Green's function is modified. In this case, application of this theorem adds a particular term to the Green's function which gives rise to an arbitrary constant in the Green's function. Moreover, it is shown that the problem may have no solution or an infinite number of solutions. Besides, the necessary condition for having any solution is investigated. This requirement, which states a significant rule in the mechanics of solids, is the static equilibrium of vertical forces acting on the beam. Some examples are presented and results are thoroughly discussed.     

Three-dimensional Green’s functions for transversely isotropic thermoelastic bimaterials

Green’s functions for transversely isotropic thermoelastic biomaterials are established in the paper. We first express the compact general solutions of transversely isotropic thermoelastic material in terms of harmonic functions and introduce six new harmonic functions. The three-dimensional Green’s function having a concentrated heat source in steady state is completely solved using these new harmonic functions. The analytical results show some new phenomena of temperature and stress distributions at the interface. The temperature contours are normal to the interface for the isotropic material but not for the orthotropic one. The normal stress contours are parallel to the interface at the boundary in the isotropic region only and shear failure is most likely at the heat source due to the highly degenerated direction of shear stress contours.     

Fundamental solution for transversely isotropic thermoelastic materials

We use the compact harmonic general solutions of transversely isotropic thermoelastic materials to construct the three-dimensional fundamental solutions for a steady point heat source in an infinite transversely isotropic thermoelastic material and a steady point heat source on the surface of a semi-infinite transversely isotropic thermoelastic material by three newly introduced harmonic functions, respectively. All components of coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results for hexagonal zinc are given graphically by contours.     

Inclined circular flat punch on a transversely isotropic piezoelectric half-space

Summary Utilizing the general solution of transversely isotropic piezoelectricity, the paper analyzes the problem of an inclined rigid circular flat punch indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to take into account the effect of the electric field in piezoelectric materials. Assuming that the punch is maintained at a constant electric potential, exact expressions for the elastoelectric field are derived in terms of elementary functions. It is noted that the solution corresponding to a flat circular punch centrally loaded by a concentrated force can be obtained as a special case. Received 15 December 1998; accepted for publication 9 March 1999     

各向同性和横观各向同性材料的统一点力解

本文根据横观各向同性弹性力学的通解获得了无限体的点力解，由它可以直接退化到各向同性情形的Ｋｅｌｖｉｎ解，利用这个点力解编制的边界元法程序，适用于横观各向同性材料也适用于各向同性材料，因此是真正的统一点力解。还用边界元法计算了两个数值例题。     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

横观各向同性饱和介质非轴对称动力响应分离变量解Separation of variables solution of non-axisymmetrical dynamic response of transversely isotropic saturated porous media

首先将横观各向同性饱和弹性多孔介质非轴对称问题的Bio t波动方程,变换为适宜于进行分离变量法求解的形式;然后在非轴对称简谐激励下,用分离变量法得到Bio t方程的一般解,即用分离变量法求得了多孔介质位移和应力分量的解析表达式;并给出了半空间横观各向同性饱和弹性多孔介质在表面竖向简谐荷载作用下表面竖向位移的数值分析结果,得出载荷对30倍受载半径以外的区域几乎无影响的结论。同时表明了本文的分析方法是切实可行的。     

Completeness of general solutions for three-dimensional transversely isotropic piezoelectricity

In this paper, a general solution for three-dimensional transversely isotropic piezoelectricity in terms of four quasi-quadri-harmonic functions is established first. Owing to complexity of the higher-order equation, it is difficult to obtain rigorous analytic solutions and in most cases this general solution is not applicable. By virtue of the generalized Almansi’s Theorem, the simplified generalized LHN solution and E–L solution expressed by lower order functions are achieved, respectively, by taking a decomposition and superposition technique. In the absence of piezoelectric coupling, these two simplified general solutions can be degenerated into those for transversely isotropic elasticity, i.e. LHN and E–L solutions. More importantly, the completeness of these two generalized solutions is proved if the domain is z-convex, no matter whether the characteristic roots are distinct or possibly equal to each other.     

A general solution and the application of space axisymmetric problem in piezoelectric material

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横观各向同性材料轴对称问题基本解

从横观各向同性材料的基本解出发,用积分的方法得到了轴对称问题的基本解,对于材料特征Ｓｉ互不相等的两种可能情形都给出了表达式,因此,可直接退化得到各向同性材料轴对称问题基本解。     

Axisymmetric micromechanics of elasticity-perfectly plastic fibrous composites under uniaxial tension loading

The uniaxial response of a continuous fiber elastic-perfectly plastic composite is modelled herein as a two-element composite cylinder. An axisymmetric analytical micromechanics solution is obtained for the rate-independent elastic-plastic response of the two-element composite cylinder subjected to tensile loading in the fiber direction for the case wherein the core fiber is assumed to be a transversely isotropic elastic-plastic material obeying Tsai-Hill's yield criterion, with yielding simulating fiber failure. The matrix is assumed to be an isotropic elastic-plastic material obeying Tresca's yield criterion. It is found that there are three different circumstances that depend on the fiber and matrix properties: (1) fiber yield, followed by matrix yielding; (2) complete matrix yield, followed by fiber yielding; and (3) partial matrix yield, followed by fiber yielding, followed by complete matrix yield. The order in which these phenomena occur is shown to have a pronounced effect on the predicted uniaxial effective composite response.     

POINT FORCE SOLUTION FOR A TRANSVERSELY ISOTROPIC ELASTIC LAYER

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功能梯度压电材料圆环位移和电势边值问题的精确解

圆环形的压电材料器件在智能结构中得到了广泛的应用。本文推导了横观各向同性功能梯度压电材料圆环在内、外边界上给定位移和电势情况下的一般解。极化方向在圆环的半径方向,材料常数的梯度方向也设定在半径方向,并可表示为半径r的幂,本构关系为线性。然后推导了压电圆环外壁固定、接地,内壁沿垂向有一微小位移、电势分别为余弦分布和均匀分布的问题的精确解,并计算了该问题在这两种电势情况下产生的无量纲形式的径向和环向位移、电势、应力及电位移沿径向分布的数值结果。计算中考虑了不同的材料梯度,以及内壁的位移与电势的不同比例。