A Generalized Piezoelectric Bernoulli–Navier Anisotropic Rod Model

We apply the asymptotic analysis procedure to the three-dimensional static equations of piezoelectricity, for a linear nonhomogeneous anisotropic thin rod. We prove the weak convergence of the rod mechanical displacement vectors and the rod electric potentials, when the diameter of the rod cross-section tends to zero. This weak limit is the solution of a new piezoelectric anisotropic nonhomogeneous rod model, which is a system of coupled equations, with generalized Bernoulli–Navier equilibrium equations and reduced Maxwell–Gauss equations.     

Generalization of the Stroh Formalism to 3-Dimensional Anisotropic Elasticity

A method of analysis for three-dimensional anisotropic elasticity is developed, based on Stroh's two-dimensional formalism. By use of the present method a three-dimensional solution can be constructed by the corresponding two-dimensional solution. The method is applied to solve several problems concerning a point force or a dislocation loop. Existing results for these problems are recovered in a unified manner and additional new results are obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

圆柱型各向异性弹性力学平面问题

本文对圆柱型各向异性弹性力学平面问题的基本方程进行了改写。在此基础上，导出了应力函数Ｇ和位移函数φ，它们满足相同的控制方程，比文〔１〕的应力函数Ｆ的控制方程要简单，便于求得特解，并有Ｆ＝ｒＧ的关系。还对若干经典问题进行了求解。     

Generalized stroh formalism for anisotropic elasticity for general boundary conditions

The Stroh formalism is most elegant when the boundary conditions are simple, namely, they are prescribed in terms of traction or displacement. For mixed boundary conditions such as there for a slippery boundary, the concise matrix expressions of the Stroh formalism are destroyed. We present a generalized Stroh formalism which is applicable to a class of general boundary conditions. The general boundary conditions include the simple and slippery boundary conditions as special cases. For Green's functions for the half space, the general solution is applicable to the case when the surface of the half-space is a fixed, a free, a slippery, or other more general boundary. For the Griffith crack in the infinite space, the crack can be a slit-like crack with free surfaces, a rigid line inclusion (which is sometimes called an anticrack), or a rigid line with slippery surface or with other general surface conditions. It is worth mention that the modifications required on the Stroh formalism are minor. The generalized formalism and the final solutions look very similar to those of unmodified version. Yet the results are applicable to a rather wide range of boundary conditions.     

Green's functions for a piezoelectric semi-infinite body with a fixed conductor surface

By using Stroh's formalism, simple explicit compact expressions of Green's functions for a piezoelectric semi-infinite body, with a fixed conductor surface electrode, subject to a singularity (i.e., a generalized line dislocation and a generalized line force at a point z°) are presented. Coulomb forces acted on the free line charge at z° due to the boundary polarization charges of the medium and the induction charges of the conductor together with the electromechanical coupling effects inside the region are analyzed in detail. The obtained results are valid not only for plane and antiplane problems but also for the coupled problems between inplane and outplane deformations.     

Optimal Fiber-Mesh Layout for a Composite Anisotropic Elastic Wedge

We consider a homogeneous orthotropic elastic wedge in plane stress loaded by a concentrated force at the apex. The orthotropic material is conceived of to be formed by a matrix reinforced by a triangular net of elastic fibers, so that the fiber size and the net shape determine the average elastic moduli of the composite material. It is found that maintaining fixed the amount of material used for the reinforcement, there exists a particular fiber disposal which maximizes the stiffness of the wedge. The optimal disposal is different-in-type for diverse opening-angles of the wedge. This revised version was published online in July 2006 with corrections to the Cover Date.     

Anisotropic Hyperelasticity Based Upon General Strain Measures

This paper presents stress-strain constitutive equations for anisotropic elastic materials. A special attention is given to the logarithmic strain. Assuming a constitutive equation for the specific internal energy the equation governing the Cauchy stress is derived. Mathematical relations presented take a relatively simple form and concern a very wide class of elastic materials. The dependence of third-order elastic constants on the choice of strain measure is shown. The third-order elastic constants measured experimentally in relation to the Green strain are recalculated here for the logarithmic strain. This revised version was published online in July 2006 with corrections to the Cover Date.     

各向异性弹性体光滑接触界面上的滑移脉冲波

利用Stroh公式,Fourier分析和奇异积分方程技术研究了两各向异性弹性半空间光滑接触可分离界面上滑移脉冲波的存在及其传播特性。结果表明,如果至少能在一种介质中存在Rayleigh波,且其波速小于两种介质中的最小极限速度,则滑移脉冲波就可以存在。这种脉冲波传播速度不确定,可在最小极限波速与较低的Rayleigh波速之间取值,而该取值范围又取决于无界面分离情况下的第一、第二滑移波的解。分离区大小取决于扰动的强度,界面法向力和质点速度在分离区两端有 1 /2奇异性。     

On the Interior Transmission Problem in Nondissipative, Inhomogeneous, Anisotropic Elasticity

In this paper, the interior transmission problem for the non absorbing, anisotropic and inhomogeneous elasticity is investigated. The direct scattering problem for the penetrable inhomogeneous, anisotropic and nondissipative scatterer is first studied and the existence and uniqueness of its solution are established. In the sequel, the interior transmission problem in its classical and weak form is presented and suitable variational formulations of it are settled. Finally, it is proved that the interior transmission eigenvalues constitute a discrete set. This revised version was published online in June 2006 with corrections to the Cover Date.     

GREEN''''S FUNCTIONS FOR 2D PROBLEMS OF ANISOTROPIC PIEZOELECTRIC MATERIAL WITH A STRIP REGION

By using Stroh's formalism and the conformal mapping technique, we derive the simple explicit elastic fields of a generalized line dislocation and a generalized line force in a general anisotropic piezoelectric strip with fixed surfaces, which are two fixed conductor electrodes. The solutions obtained are usually considered as Green's functions which play important roles in the boundary element methods. The Coulomb forces of the distributed charges along the region boundaries on the line chargeq atz ⁰ are analysed in detail. The results are valid not only for plane and antiplane problems but also for the coupled problems between inplane and outplane deformations.     

Hertzian Contact of Anisotropic Piezoelectric Bodies

The Fourier transform method is applied to the Hertzian contact problem for anisotropic piezoelectric bodies. Using the principle of linear superposition, the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations, one containing pressure and another containing electric displacement. By presupposing the forms of the pressure and electric displacement distribution over the contact area, the problem is solved successfully; then the generalized displacements, stresses and strains are expressed by contour integrals. Details are presented in the case of special orthotropic piezoelectricity whose material constants satisfy six relations, which can be easily degenerated to the case of transverse isotropic piezoelectricity. It can be shown that the result gained in this paper is of a universal and compact form for a general material.Supported by the National Natural Science Foundation of China (No.10372003).     

Directions of Coaxiality between Pure Strain and Stress in Linear Elasticity

For a linear anisotropic hyperelastic body subjected to a state of pure strain it is shown that there are at least three directions such that the stress and strain tensors are coaxial.     

浅析扭转理论的发展

按照扭转理论的发展过程分别分析了不同科学家在扭转理论上的贡献,从金属丝的 扭转到圆轴、圆管的扭转,再到任意截面的扭转问题,最后发现任意截面的棱柱扭 转问题是圣维南通过弹性理论的方法解决的.     

New Explicit Expression of Barnett-Lothe Tensors for Anisotropic Linear Elastic Materials

The three Barnett-Lothe tensors H, L, S appear often in the Stroth formalism of two-dimensional deformations of anisotropic elastic materials [1–3]. They also appear in certain three-dimensional problems [4, 5]. The algebraic representation of H, L, S requires computation of the eigenvalues p^v(v=1,2,3) and the normalized eigenvectors (a, b). The integral representation of H, L, S circumvents the need for computing p ^v(v=1,2,3) and (a, b), but it is not simple to integrate the integrals except for special materials. Ting and Lee [6] have recently obtained an explicit expression of H for general anisotropic materials. We present here the remaining tensors L, S using the algebraic representation. They key to our success is the obtaining of the normalization factor for (a, b) in a simple form. The derivation of L and S then makes use of (a, b) but the final result does not require computation of (a, b), which makes the result attractive to numerical computation. Even though the tensor H given in [6] is in terms of the elastic stiffnesses C^μ v while the tensors L, S presented here are in terms of the reduced elastic compliances s′ _μv , the structure of L, S is similar to that of H. Following the derivation of H, we also present alternate expressions of L, S that remain valid for the degenerate cases p ₁ p ₂ and p₁=p₂ = p ₃. One may want to compute H, L, S using either C _μv or s′ _μv v, but not both. We show how an expression in C^μ v can be converted to an expression in s′ _μv v, and vice versa. As an application of the conversion, we present explicit expressions of the extic equation for p in C^μ v and s′ _μv v. This revised version was published online in August 2006 with corrections to the Cover Date.     

用有限元广义混合法分析不可压缩或几乎不可压缩弹性体

不可压缩或几乎不可压缩问题在数学上表现为最小 势能原理中的某些项趋于无穷大,使得有限元方程产生病态。本文给出了不可压缩或几乎不可压缩弹性分析的广义混合变分原理,以此为基础建立了该类问题的有限元广义混合法。该变分原理的泛函中不含有上面这种奇异项,故其有限元方程不会产生病态。算例表明该有限元法可以同时进行可压缩、不可压缩或几乎不可压缩弹性分析,且精度良好;有限元常规位移法及Hermann法是该法的特例。     

各向异性弹性力学一般边值问题的广义Stroh公式

当边值问题是简单的,即是应力边值问题时,Stroh公式是很有效的。对于混合边值问题,倒如滑动边界条件,Stroh公式中的简洁的矩阵表达式就失效了。我们提出了一个广义的Stroh公式,它可应用于一大类一般的边界条件。简单的边界条件和滑动边界条件是这一类一般边界条件的特殊情形。值得指出的是,这个关于Stroh公式所作的修正并不大。广义的公式和最后的解答看起来很类似于未修正的原公式和原来的解。然而这个修正却可应用于相当广的边界条件。     

Simple Torsion of Isotropic, Hyperelastic, Incompressible Materials with Limiting Chain Extensibility

The purpose of this research is to investigate the simple torsion problem for a solid circular cylinder composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at large deformations are examined. These models involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The main mechanical quantities of interest in the torsion problem are obtained in closed form. In this way, it is shown that the torsional response of all three materials is similar. While the predictions of the models agree qualitatively with experimental data, the quantitative agreement is poor as is the case for the neo-Hookean material. In fact, by using a global universal relation, it is shown that the experimental data cannot be predicted quantitatively by any strain-energy density which depends solely on the first invariant. It is shown that a modification of the strain energies to include a term linear in the second invariant can be used to remedy this defect. Whether the modified strain-energies, which reflect material hardening, are a feasible alternative to the classic Mooney–Rivlin model remains an open question which can be resolved only by large strain experiments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Second-Order Fredholm Equations for the First Boundary-Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

A complete potential theory is constructed for the first boundary-value problem in the two-dimensional anisotropic theory of elasticity (the force vector is specified on the boundary) in a bounded domain on a plane with a Lyapunov boundary. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 85–94, March–April, 2006.     

Perturbation Formula for Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media

Herein we consider Rayleigh waves propagating along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.     

Spatial Estimates for the Constrained Anisotropic Elastic Cylinder

This paper establishes spatial estimates in a prismatic (semi-infinite) cylinder occupied by an anisotropic homogeneous linear elastic material, whose elasticity tensor is strongly elliptic. The cylinder is maintained in equilibrium under zero body force, zero displacement on the lateral boundary and pointwise specified displacement over the base. The other plane end is subject to zero displacement (when the cylinder is finite, say). The limiting case of a semi-infinite cylinder is also considered and zero displacement on the remote end (at large distance) is not assumed in this case. A first approach is developed by considering two mean-square cross-sectional measures of the displacement vector whose spatial evolution with respect to the axial variable is studied by means of a technique based on a second-order differential inequality. Conditions on the elastic constants are derived that show the cross-sectional measures exhibit alternative behaviour and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay. A second approach considers cross-sectional integrals involving the displacement and its gradient and furnishes information upon the spatial evolution, without restricting the range of strongly elliptic elastic constants. Such models are principally based upon a first-order differential inequality as well as on one of second order. The general results are explicitly presented for transversely isotropic materials and graphically illustrated for a cortical bone.