USING EIGENMODULI AND EIGENSTATES TO EVALUATE THE POSSIBILITY OF MARTENSITIC PHASE TRANSFORMATIONS

Journal of Applied Mechanics and Technical Physics - The possibility of phase transitions (martensitic transformations) in shape-memory alloys is evaluated using the concept of eigenmoduli...     

Functional relation between two symmetric second-rank tensors

The functional relationship between two symmetric second-rank tensors is considered. A new interpretation of the components of the tensors as projections onto an orthogonal tensor basis is given. It is shown that the constitutive relations can be written in the form of six functions each of which depends on one variable. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 134–137, September–October, 2007.     

Symmetry operators and general solutions of the equations of the linear theory of elasticity

Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 36, No. 5, pp. 98–104, September–October, 1995.     

General Solution for the Two-Dimensional System of Static Lame’s Equations with an Asymmetric Elasticity Matrix

We study a two-dimensional system of equations of linear elasticity theory in the case when the symmetric stress and strain tensors are related by an asymmetric matrix of elasticity moduli or elastic compliances. The linear relation between stresses and strains is written in an invariant form which contains three positive eigenmodules in the two-dimensional case. Using a special eigenbasis in the strain space, it is possible to write the constitutive equations with a symmetric matrix, i.e., in the same way as in the case of hyperelasticity. We obtain a representation of the general solution of two-dimensional equations in displacements as a linear combination of the first derivatives of two functions which satisfy two independent harmonic equations. The obtained representation directly implies a generalization of the Kolosov–Muskhelishvili representation of displacements and stresses in terms of two analytic functions of complex variable. We consider all admissible values of elastic parameters, including the case when the system of differential equations may become singular. We provide an example of solving the problem for a plane with a circular hole loaded by constant stresses.     

Affine transformations of the equations of the linear theory of elasticity

This paper deals with the general formulas of affine transformations that preserve invariance of the static equations of the linear theory of elasticity in the case of arbitrary anisotropic materials. The invariance of the equations with respect to affine transformations allows one to model a given anisotropic material by another material. All anisotropic materials are divided into classes of mutually congruent materials. The congruency conditions are obtained for orthotropic and isotropic materials and for orthotropic and transversely isotropic materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 124–134, July–August, 2006.     

On a model for anisotropic creep of materials

We apply the concepts of proper moduli and states, revealing the structure of the generalized Hooke’s law, to a model of anisotropic steady-state creep of materials. The steady-state creep equations for incompressible materials are expressed in invariant form. The matrix of anisotropy coefficients of these materials reduces to block form with the nine independent components. We consider the special case of an orthotropic incompressible material for which the matrix of anisotropy coefficients corresponds to a nonor.     

Extreme conditions of elastic constants and principal axes of anisotropy

This paper describes the derivation of extreme conditions of each elasticity coefficient (Young’s modulus, shear modulus, et al.,) for the general case of linear-elastic anisotropic materials. The stationarity conditions are obtained, and they determine the orthogonal coordinate systems being the principal axes of anisotropy, where the number of independent elasticity constants decreases from 21 to 18 and, in some cases of anisotropy, to 15 or lower. The example of a material with cubic symmetry is given.     

Diagonalization of a three-dimensional system of equations in terms of displacements of the linear theory of elasticity of transversely isotropic media

A dynamic three-dimensional system of linear equations in terms of displacements of the theory of elasticity of transversely isotropic media is given explicit expressions for phase velocities and polarization vectors of plane waves. All the longitudinal normals are found. For some values of the elasticity moduli, the system of equations is reduced to a diagonal shape. For static equations, all the conditions of the system ellipticity are determined. Two new representations of displacements through potential functions that satisfy three independent quasi-harmonic equations are given. Constraints on elasticity moludi, at which the corresponding coefficients in these representations are real, different, equal, or complex, are determined. It is shown that these representations are general and complete. Each representation corresponds to a recursion (symmetry) operator, i.e., a formula of production of new solutions.