On the active part of the stress for elastic materials with internal constraints

Constitutive equations for constrained elastic materials prescribe a whole set of possible stresses for each deformation. We prove the existence of a normalization procedure for determining the active part of the stress through a single-valued response function which is both frame-indifferent and invariant under the same symmetry group of the given constitutive equation.     

On heat flux in simple media

For any simple body we prove that there is an infinite number of mathematical functions which can be added to the response functional for the heat flux vector without affecting the balance laws, the entropy inequality, and the boundary conditions on the normal heat flux. The presence of this indetermination cannot be detected by usual physical experiments, namely by experiments in which cuts of the body are not taken into account. The maximal class of this indetermination is fully characterized in the thermoelastic case.This work has been performed within the activity sphere of the Consiglio Nazionale delle Ricerche, Group no 3, in the academic year 1988/9.     

On a principle of virtual work for thermo-elastic bodies

A principle of virtual work is proposed for thermo-elastic bodies. From it are derived the equations of motion, the Cauchy stress principle and the Gibbs relations. The principle is also used to analyse the response of internally constrained bodies.     

On the separation of stress-induced and texture-induced birefringence in acoustoelasticity

In this paper we develop a simple micromechanical model of a prestressed polycrystalline aggregate, in which the texture-induced and stress-induced anisotropies of the aggregate are precisely defined; here the word texture always refers to the texture of the aggregate at the given prestressed configuration, not to that of a perhaps fictitious natural state of the aggregate. We use this model to derive, for a prestressed orthotropic aggregate of cubic crystallites, a birefringence formula which shows explicitly the effects of the orthotropic texture on the acoustoelastic coefficients. From this formula we observe that, generally speaking, we cannot separate the total birefringence into two distinct parts, one reflecting purely the influence of stress on the birefringence, and the other encompassing all the effects of texture. The same formula, on the other hand, provides for each material specific quantitative criteria under which the separation of stress-induced and texture-induced birefringence would become meaningful in an approximate sense.     

On the yield functions in Lagrangian stress space of elastic–plastic materials with internal constraints

In the present paper it is shown that the elastic range in the second Piola–Kirchhoff stress space can be chosen in a hyperplane which is through the origin of Lagrangian stress space and perpendicular to the normal of the constraint manifold at the plastic configuration, if the determinate stress response of the elastic–plastic material with simple internal constraints with some condition is correctly chosen, otherwise, it is in general in a hypersurface and the normal flow rule by Il yushin’s postulate will have an indeterminate part. The choice of determinate stress response is probable because of its indeterminacy. Therefore the yield function should be a function of the second Piola–Kirchhoff stress lying in the hyperplane so that it is more simple and the back stress as the geometric center of the elastic range in general is inside the elastic range. Finally some examples are concerned. The project supported by the National Natural Science Foundation of China (10272055).     

具有内部特征约束的四边形网格生成方法

提出一种具有内部特征约束的四边形网格自动生成方法,可以处理内部具有孔洞、约束线、约束点以及密度线、密度点等特征约束,满足数值分析中对网格生成的特殊要求。以区域分解法为基础,提出了确定最佳切割线的方法以及在切割线上生成过渡均匀网格节点的方法;将约束线、约束点以及密度线、密度点作为面积为零的孔洞,提出了对特征约束的处理方法;针对特征约束分隔区域的情况,提出了自动确定各子区域边界的方法。基于本文提出的方法已开发出具有特征约束的四边形网格自动生成程序,并成功应用于洪水分析系统和楼面设计分析系统。     

On the Galerkin vector and the Eshelby solution in linear elasticity

Displacement potentials in linear static elasticity consist of three functions which can be regarded as the three components of a vector, e.g., the Galerkin vector. This research note gives an explanation as to why the biharmonic equations govern these functions in isotropic elasticity as opposed to the sixth-order partial differential equations that govern them in anisotropic elasticity. This note also shows that the Eshelby solution in two-dimensional anisotropic elasticity can be derived from the method of displacement potentials.     

A directed rod subjected to internal constraints

In this paper, the field equations for a directed rod are given. The form of the reaction forces with respect to a general internal constraint imposed on the rod is also derived, based on the assumption that the reaction forces do no work in any deformation compatible with the constraint. Finally, the rod subjected to both special and fundamental internal constraints is treated.     

On Stroh orthogonality relations: An alternative proof applicable to Lekhnitskii and Eshelby theories of an anisotropic body

This paper establishes that the Stroh orthogonality relations for an anisotropic body are a direct consequence of the fact that the system of equations of equilibrium is self-adjoint and positive definite. It is demonstrated that, assuming a complex representation of displacements and boundary tractions, the Betti theorem of reciprocity implies the orthogonality, and positive definiteness of strain energy implies the full rank of the normalization matrix, in the Stroh orthogonality relations. The presented proof is applicable to both the Lekhnitskii and Eshelby theories of an anisotropic body.     

On the plane problem of orthotropic quasi-static thermoelasticity

The plane displacement boundary value problem of quasi-static linear orthotropic thermoelasticity is discussed. The thermoelastic system on a bounded simply-connected domain is decoupled. The decoupled temperature equation is investigated by using an accurate estimate and the contractive mapping principle. Representation of solution of the field equation is obtained, and some solvability results are proved. The results are of both theoretical and numerical interest.     

On the local measures of mean rotation in continuum mechanics

We implement Cauchy's concept of a rotation-angle function on an oriented plane, and characterize situations when a rotation-angle function exists, and hence when measuring mean rotations in the manner of Cauchy or Novozhilov makes sense. We also discuss in passing the role of the skew part of the deformation gradient in measuring the mean deformation.     

Saint-Venant's problem with Voigt's hypotheses for anisotropic solids

We seek for a solution of Saint-Venant's problem for inhomogeneous and anisotropic materials under the assumptions, introduced by Voigt, that the stress is either constant along the axis of the cylinder or depends linearly on the axial coordinate. We first prove the uniqueness of the solution in terms of resultants, then we exhibit an explicit formula for such a solution; we show finally how Clebsch's hypothesis, that the stress vector on axial planes is parallel to the axis, is compatible with Voigt's hypotheses provided that the symmetry group of the material comprising the cylinder contains the reflections on the cross-section.     

On the influence of cohesive stress-separation laws on elastic stress singularities

The nature of the stress field occurring at the vertex of an angular elastic plate under in-plane loading is reconsidered. An additional boundary condition is introduced. This boundary condition reflects the action of cohesive stress-separation laws. Companion asymptotic analysis proceeds routinely on employing coupled eigenfunction expansions. Results show that a number of configurations that had previously contained stress singularities become singularity free.     

On Self-Controlling Models in the Theory of Nonelastic Material Behavior of Metals

We discuss here solvability of systems modelling the nonelastic material behavior of metals under the assumption of monotonicity of the constitutive function and of the self-controlling property of the model under consideration. The main idea is to use the conception of coercive limits ([9, 10, 12]) and to prove a suitable convergence result. Examples of self-controlling models are presented at the end of the article. Received November 3, 1997