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.