Strong Ellipticity of Transversely Isotropic Elasticity Tensors

The strong ellipticity of the elasticity tensor of a linearly hyperelastic, transversely isotropic material is investigated. The necessary and sufficient conditions for the elasticity tensor to be strongly elliptic are determined for the five constants characterizing it.     

The Representation Theorem for Linear, Isotropic Tensor Functions in Even Dimensions

In this note we give a proof of the representation theorem for linear, isotropic, tensor functions, which only assumes invariance under proper orthogonal tensors. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Representation Theorem for Linear, Isotropic Tensor Functions Revisited

Two recent notes [1–2] express ideas that can be combined in order to get an elementary proof of the representation theorem of the title, a proof stressing the geometrical action of 4-tensors in subspaces. Thus we examine direct consequences, for linear maps C of Lin into Lin, of the notion of invariance (we recall this concept and we recall other standard notations in the first section). Suppose we classify the invariant (under Orth) subspaces of Lin, that an invariant linear C maps invariant subspaces into invariant subspaces and that Lin = Skw Sph Dev expresses the unique decomposition of Lin into a direct sum of three-, one- and five-dimensional invariant subspaces. Then, if we assume that C is also invertible, the structure of C comes from the knowledge of the invariant linear maps in Skw and Dev. As they must be, in each case, multiples of the identity, CE=C(E _{_3}+E _{_1}+ E _{_5})=_{_3} E _{_3} +_{_1} E _{_1}+_{_5} E _{_5} for (nonzero) constants _{_3}, _{_1} and _{_5}. The noninvertible case results from considering C+I.     

On Small-Displacement Waves in Prestressed Bodies with Isotropic Incremental Elasticity Tensor

We consider a body at rest in a prestressed configurationwhich responds elastically to small incremental displacements fromthe incremental elasticity tensor is supposed isotropic. On the basis of the paper [1] we characterize the conditions for the propagation of longitudinal, transverse, and oblique small-displacement waves superimposed toFormulae for the propagation speeds of these waves are written in terms of the prestress components and Lamparameters. The amplitudes of longitudinal and transverse waves are eigenvectors for the prestress.     

横观各向同性圆柱体端部问题的三维弹性理论解

从位移的通解出发，用分离变量法得到横观各向同性圆柱体的位移和应力的特征函数展开式，并把位移势函数的解用付里叶积分的形式表示。利用留数运算，该积分解可以转换成类似于特征函数的展开式。通过混合端部边界问题，得到与特征函数解成双正交关系的另一组函数。利用这种双正交关系，可以处理不同的端部边界问题。     

A Concise Proof of the Representation Theorem for Fourth-Order Isotropic Tensors

We present a new proof of the representation theorem for fourth-order isotropic tensors that does not assume the tensor to have major or minor symmetries at the outset.     

On the Spectrum of the Interior Transmission Problem in Isotropic Elasticity

The present work concerns with the investigation of the interior transmission problem, which is naturally associated to the inverse elastic scattering problem of determining the support of an isotropic homogeneous penetrable body from a knowledge of the time harmonic incident plane waves and the far-field patterns of the corresponding scattered wave-fields. Our approach combines a boundary integral formulation of the problem and a compact perturbation argument to establish the discreteness of the set of transmission eigenvalues and the well-posedness of the interior transmission problem under the most general assumptions on the elastic parameters of the underlying media.      

On the Representation of Symmetric Isotropic 4-Tensors

This note provides short proof of the representation of a symmetric isotropic 4-tensor in an n-dimensional real Euclidean space. This revised version was published online in August 2006 with corrections to the Cover Date.     

正交异性扁壳的等价关系式和不变量

提出各向同性扁壳比拟法,分析满足条件D_3=D_(12)=(D_1D_2)~(1/2)的正交异性扁壳大挠度弯曲和超屈曲问题,导出了正交异性扁壳与各向同性扁壳之间,两种不同正交异性扁壳之间坐标变量、扁壳厚度和曲率半径、荷载、挠度、转角、弯矩、扭矩、中面应力的等价关系式,还证明了等价正交异性扁壳的几个等价不变量。     

Isotropic polynomial invariants of Hall tensor

The Hall tensor emerges from the study of the Hall effect, an important magnetic effect observed in electric conductors and semiconductors. The Hall tensor is third-order and three-dimensional, whose first two indices are skew-symmetric. This paper investigates the isotropic polynomial invariants of the Hall tensor by connecting it with a second-order tensor via the third-order Levi-Civita tensor. A minimal isotropic integrity basis with 10 invariants for the Hall tensor is proposed. Furthermore, it is proved that this minimal integrity basis is also an irreducible isotropic function basis of the Hall tensor.     

Laminations in Linearized Elasticity: The Isotropic Non-very Strongly Elliptic Case

The explicit computation of the effective elasticity tensor of the material produced by laminating two homogeneous elastic media is used to show that, in 2-dimensional and 3-dimensional linear elasticity, for any isotropic material a whose elasticity tensor is strongly elliptic, but not semipositive definite, we can select very strongly elliptic materials, so that through laminations between these with material a, we can create a nonstrongly elliptic media, whose existence contradicts properties concerning the propagation of elastic waves. This revised version was published online in August 2006 with corrections to the Cover Date.     

Multiple Natural States for an Elastic Isotropic Material with Polyconvex Stored Energy

We show a necessary and sufficient condition for the undistorted reference configuration y(x)=x, with ∇y=F=1, to be a minimizer of the total stored energy for an isotropic elastic body. Polyconvexity of the stored energy function is not sufficient, and we give an example which possesses two distinct natural (i.e., unstressed) states to illustrate this point. This revised version was published online in July 2006 with corrections to the Cover Date.     

Ellipticity Loss in Isotropic Elasticity

In this paper we prove that the equilibrium displacement of an isotropic elastic solid under imposed boundary displacement converges in the strong H1 topology when the shear modulus goes to zero. As a consequence, we show that the dependence on material constants may be dropped in the classical inequality expressing the continuous dependence of elastic equilibria on the boundary displacement. This revised version was published online in August 2006 with corrections to the Cover Date.     

多层横观各向同性圆柱壳的三维弹性理论解

本文从三维弹性理论出发,用特征函数法研究多层横观各向同性圆柱壳的轴对称问题.把位移和应力分量的齐次解表达成特征函数展开式,并把特解部分用Fourier级数表示.以多层圆柱壳的内、外柱面作为齐次边界,同时考虑层间的连续条件,推导出问题的特征方程并用Muller法求解.文中运用传递矩阵技术处理多层问题,并用边界型最小二乘配点法处理端部边界条件.作为实例,对双层圆柱壳作了数值计算.     

Invariance of the Equilibrium Equations of Finite Elasticity for Compressible Materials

For homogeneous, isotropic, compressible nonlinearly elastic materials, a wide class of strain-energy density functions are obtained that leave the equations of equilibrium invariant under simple scaling transformations of the material and spatial coordinates. These strain-energy densities are homogeneous functions of the principal stretches. Several illustrative examples of particular strain-energies are provided. For axisymmetric problems, the invariance discussed here ensures that the equations of equilibrium can be solved by quadratures and thus often leads to analytic solutions in parametric or closed-form. Mathematics Subject Classifications (2000) 74B20, 74G55.     

On the Boundary Value Problems of Linear Elasticity with Constraints

In this paper classic boundary value problems of linear elastostatics are studied. Displacement, mixed and traction type boundary conditions are considered for an internally constrained, non-homogeneous, anisotropic material. Existence of solutions and constraint stability results are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Torsin of Functionally Graded Isotropic Linearly Elastic Bars

The purpose of this research is to investigate the effects of material inhomogeneity on the torsional response of linearly elastic isotropic bars. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The classic approach to the torsion problem for a homogenous isotropic bar of arbitrary simply-connected cross-section in terms of the Prandtl stress function is generalized to the inhomogeneous case. The special case of a circular rod with shear modulus depending on the radial coordinate only is examined. It is shown that the maximum shear stress does not, in general, occur on the boundary of the rod, in contrast to the situation for the homogeneous problem. It is shown that the material inhomogeneity may increase or decrease the torsional rigidity compared to that for the homogeneous rod. Optimal upper and lower bounds for the torsional rigidity for nonhomogeneous bars of arbitrary cross-section are established. A new formulation of the basic boundary-value problem is given. The results are illustrated using specific material models used in the literature on functionally graded elastic materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Anisotropic Polynomial Relations for the Elasticity Tensor

In this paper, we explore new conditions for an elasticity tensor to belong to a given symmetry class. Our goal is to propose an alternative approach to the identification problem of the symmetry class, based on polynomial invariants and covariants of the elasticity tensor C, rather than on spectral properties of the Kelvin representation. We compute a set of algebraic relations which describe precisely the orthotropic ( $[\mathbb {D}_{2}]$ ), trigonal ( $[\mathbb {D}_{3}]$ ), tetragonal ( $[\mathbb {D}_{4}]$ ), transverse isotropic ([SO(2)]) and cubic ( $[\mathbb {O}]$ ) symmetry classes in $\mathbb {H}^{4}$ , the highest-order irreducible component in the decomposition of $\mathbb {E}\mathrm {la}$ . We provide a bifurcation diagram which describes how one “travels” in $\mathbb {H}^{4}$ from a given isotropy class to another. Finally, we study the link between these polynomial invariants and those obtained as the coefficients of the characteristic or the Betten polynomials. We show, in particular, that the Betten invariants do not separate the orbits of the elasticity tensors.     

Uniqueness of Equilibrium Solutions in Second-Order Gradient Nonlinear Elasticity

We consider a three-dimensional elastic body whose material response function depends not only on the gradient of the deformation, but also on its second gradient. Using the elastic energy-momentum tensor as derived by Eshelby [2] we generalize a well-known uniqueness result of Knops and Stuart [8] for a Dirichlet boundary value problem associated with this response function.