Love solutions in the linear inhomogeneous transversely isotropic theory of elasticity

A general Love solution for the inhomogeneous transversely isotropic theory of elasticity with the elastic constants dependent on the coordinate z is proposed. This result may be considered as a generalization of the Love solutions we recently derived for the inhomogeneous isotropic theory of elasticity. The key steps of deriving the Love solution for the classical linear homogeneous transversely isotropic theory of elasticity are described for further use of the derivation procedure, which is then generalized to the inhomogeneous transversely isotropic case. Some particular cases of inhomogeneity traditionally used in the theory of elasticity are also examined. The significance of the derived solutions and their importance for the modeling of functionally graded materials are briefly discussed     

Solving the three-dimensional equations of the linear theory of elasticity

The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically     

Complete solutions of three-dimensional problems in transversely isotropic media

Continuum Mechanics and Thermodynamics - We present a systematic approach to the derivation of complete solutions for three-dimensional transversely isotropic elastic problems. The class of...     

One solution of an axisymmetric problem of the elasticity theory for a transversely isotropic material

A numerical-analytical method based on approximation of the sought solution by a system of basis functions is proposed to solve the boundary-value problem of axisymmetric deformation of articles made of a transversely isotropic material. An algorithm for constructing polynomial functions on the basis of invariant-group solutions is described.     

Theory and refined theory of elasticity for transversely isotropic plates and a new theory for tnick plates

A theory of elasticity for the bending of transversely isotropic plates has been developed from the basic equations of elasticity in terms of displacements for transversely isotropic bodies, which takes into account the loads distributed over the surfaces of the plates. Based on this theory, a refined theory of plates which can satisfy three boundary conditions along each edge of the plates and a new theory of thick plates are established. The solution of the refined theory for simply supported polygonal plates has been obtained; and its numerical result is very close to the exact solution of the three-dimensional theory of elasticity. A systematic comparison with the former theories of thick plates shows that the present theory of thick plates is closest to the result of the theory of elasticity.     

General love solution in the linear isotropic inhomogeneous theory of radius-dependent elasticity

A general Love solution for the inhomogeneous linear isotropic theory of elasticity with the elastic constants dependent on the coordinate r is proposed. The axisymmetric case is analyzed and cylindrical coordinates are used. This is the fourth publication in the series on general solutions in the inhomogeneous theory of elasticity. The new results are promising for the modern theory of functionally graded materials. The key steps of deriving the Love solutions are described for further use of the derivation procedure. The procedure of generalizing the Love solutions to the inhomogeneous theory of elasticity is detailed. The results obtained are discussed     

Elastic fields in rotating transversely isotropic media

Representation of elastic fields in terms of scalar functions, which permit reducing the problems of determining these fields to determining scalar potentials, are generalized to the case of transversely isotropic media rotating at a constant angular velocity. Relations for calculating the parameters of surface acoustic waves (SAW) propagating in a rotating transversely isotropic halfspace with various directions of the medium material symmetry axis with respect to the half-space surface are given.     

Solving the equilibrium equations for a transversely isotropic piezoceramic plate

The system of equilibrium equations for nonthin transversely isotropic piezoceramic plates subject to antisymmetric (about the mid-plane) deformation is set up by expanding the unknown functions into Fourier-Legendre series in the thickness coordinate. To find the general solution of the system of equations, a method is proposed, which is then used to solve the stress problem for a plate with a circular hole under mechanical loading __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 96–106, October 2006.     

Chirality in isotropic linear gradient elasticity

Chirality is, generally speaking, the property of an object that can be classified as left- or right-handed. Though it plays an important role in many branches of science, chirality is encountered less often in continuum mechanics, so most classical material models do not account for it. In the context of elasticity, for example, classical elasticity is not chiral, leading different authors to use Cosserat elasticity to allow modelling of chiral behaviour.Gradient elasticity can also model chiral behaviour, however this has received much less attention than its Cosserat counterpart. This paper shows how in the case of isotropic linear gradient elasticity a single additional parameter can be introduced that describes chiral behaviour. This additional parameter, directly linked to three-dimensional deformation, can be either negative or positive, with its sign indicating a discrimination between the two opposite directions of torsion. Two simple examples are presented to show the practical effects of the chiral behaviour.     

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.     

General solutions of weakened equations in terms of stresses in the theory of elasticity

The general solutions of some weakened systems of equations expressed in terms of stresses in the isotropic theory of elasticity are analyzed. These systems are not equivalent to the classical one and involve the equilibrium equations and only three of the six equations of compatibility (either diagonal or off-diagonal ones). In the framework of elasticity theory, an equivalence of the formulations of quasistatic boundary value problems based on such systems and expressed in terms of stresses is discussed.     

Particularizing the constitutive equations of elastoplasticity of a transversely isotropic material

A method for particularizing the constitutive equations describing the elastoplastic deformation of a transversely isotropic material is proposed. The method employs data of tension and internal-pressure base tests on tubular specimens. The calculated results and test data are compared