Determining equation of electric or magnetic deformation in isotropic materials

For isotropic materials, we construct and analyze the general expression and differential characteristics of the measure of “large” free [internal (inherent)] distortion depending on a single scalar parameter of state (e.g., thermal) and a single vector parameter (e.g., electric or magnetic). In particular, it is shown that if the initial value of the vector parameter is equal to zero, this distortion is equal to the deformation independent of the sign of the corresponding final value. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 6, pp. 37–41, November–December, 2005.     

Differential Relations between the Changes in the Strained State and the Level of Strains

We construct differential relationships between the variations of the parameters of the strained state (distortion tensor and the level of Cauchy strains) and the variations of the actual and normal configurations. The normal configuration is the objective reference configuration for the strained state and varies as a function all types of parameters of state (e.g., thermal, electric, and magnetic).     

Potentials and Equations of State for Two Special Reference Configurations

On the basis of the mathematical formulation of the principle of reference invariance of the functions of state of deformed bodies, we consider two special representations of the potential of state in which the reference configuration is variable and related to the current state of the body. The first of these representations (natural) follows from the general representation of the potential of state if a current state of the basis is regarded as the reference state and the corresponding stress-free configuration is taken as the reference configuration. The second (strain-free) representation is obtained if the role of the reference configuration is played by the current configuration. For both representations, we deduce the equations of state in the form of expressions for stresses and the parameters of state connected with the basis parameters. As a constructive proof of the unconditional existence of the elastic potential, we deduce the expression for the elastic potential via the potential of state by using the natural potential of state. It is proved that the general system of equations of state consists of four independent systems of equations, namely, of the equations of elasticity, piezoeffects, stress-free distortion, and stress-free state.     

Fundamental solution of balance equations for an anisotropic cylindrical shell with finite shear rigidity

We obtain a system of balance equations in displacements for an anisotropic nonflat cylindrical shell with finite shear rigidity, reduce this system to the tenth-order resolving equation, construct its fundamental solution, investigate the ranges of the roots of the characteristic polynomial of the resolving equation, and determine explicit expressions for the principal part of the fundamental solution.Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 2, pp. 56–61, March – April, 1995.     

Mathematical Description of Perturbations of the Stress-Strain State

We deduce expressions for the variations of initial stresses caused by small perturbations of a basic state of any type (thermal, electric, or magnetic). As a result, we get two kinetically different types of expressions. In the first case, the perturbation of stresses is connected with the perturbation of the stressed state. In the second case, it is connected with small additional strains. In both cases, the perturbation is also connected with perturbations of the other basis parameters of state. The physical meaning of the terms appearing in the proposed expressions is analyzed.