非局部微极线性弹性介质理论中的各种互易定理和变分原理

在本文的第一部份中.我们扩展了经典的卷积和I.Hlavácěk给出的“卷积数积”的概念,提出了“卷积向量”和“卷积向量点积”的概念.从而可使我们把具有算子系数的方程的初值问题和初值——边值问题推广到具有算子系数的方程组的相应问题中去.在本文的第二部份中,以卷积向量和卷积向量点积的概念为基础导出了非均匀的各向异性固体的非局部微极线性弹性动力学的两种基本型式的互易定理.在本文的第三部份中,利用一和二中卷积向量和卷积向量点积的概念和结论及由钱伟长提出的Lagrange乘子法给出了非局部微极线性弹性动力学的四种主要型式的广义变分原理.它们是与经典弹性理论中的胡海昌-鹫津久一郎型的、Hellinger-Reissner型的和Gurtin型的以及局部微极弹性理论和非局部弹性理论中的Hlavácěk型的和Iesan型的广义变分原理相应的各种变分原理.最后还指出了这里提出的后两种主要型式的广义变分原理是等价的.

Various Reciprocal Theorems and Variational Principles in the Theories of Nonlocal Micropolar Linear Elastic Mediums

In the first part of our paper, we have extended the concepts of the classical convolution and the "convolution scalar product" given by I. Hlavacek and presented the concepts of the "convolution vector" and the "convolution vector scalar product", which enable us to extend the initial value as well as the initial-boundary value problems for the equation with the operator coefficients to those for the system of equations with the operator coefficients.In the second part of this paper, based on the concepts of the convolution vector and the convolution vector scalar product, two fundamental types of reciprocal theorems of the non-local micro-polar linear elastodynamics for inhomogeneous and anisotropic solids are derived.In the third part of this paper, based on the concepts and results in the first and second parts as well as the Lagrange multiplies method which is presented by W. Z. Chien, four main types of variational principles are given for the nonlocal micropolar linear elastodynamics for inhomogeneous and anisotropic solids. These are the counterparts of the variational principles of Hu-Washizu type, Hellinger-Reissner type and Gurtin type in classical elasticity as well as Hlavacek type and lesan type in local micropolar and nonlocal elasticity. Finally, we have proved the equivalence of the last two main variational principles which are given in this paper.