Version of the linear elasticity theory with a structural parameter

It is shown that, in the planar case, the system of constitutive equations of the linear elasticity theory should contain five independent equations. In the classical theory, only three equations are formulated, while the other two are contained in implicit form in the postulate of diffeomorphism, which is the assumption of smoothness of the displacement field. A closed elasticity model is constructed without the assumption of diffeomorphism, and it contains a structural parameter having a dimension of length. It is shown that, in a static version, macrodeformations depend on stresses and second derivatives of stresses with respect to the coordinates, while there is dispersion of longitudinal and transverse waves in the dynamics.     

Equations of the linear theory of elasticity of anisotropic materials,reduced to three independent wave equations

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 6, pp. 143–150, November–December, 1994.     

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     

Impulsive stress analysis of plates at the concentrated impact loading point by the three-dimensional dynamic theory of elasticity

In this paper, a method of stress analysis is proposed to analyze impulsive stresses at an impact loading point, since use of the classical plate theory cannot lead to a convergent result. In particular, under these impact conditions (regardless of mass of impactor, velocity of impactor, stiffness of plate, etc.), we used the three-dimensional dynamic theory of elasticity and potential theory of displacement to analyze the impulsive strains at an impact loading point. Also, by using Hertzian contact theory to account for local deformation, the impact loading could be analyzed, and this loading was applied to the impulsive stress analysis by allowing the impact loading to be approximated to an analyzable function. In the numerical analysis, we used a fast Fourier transform (FFT) algorithm and applied the numerical inverse Laplace transformation.     

On Saint-Venant's principle in the two-dimensional linear theory of elasticity

Archive for Rational Mechanics and Analysis -     

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.     

Construction of eigentensor columns in the linear micropolar theory of elasticity

The internal structure of the block tensor matrix of the elastic modulus tensors is studied for the case of micropolar theory. In particular, the problem of finding the eigenvalues and eigentensor columns of block tensor matrices is considered. The complete orthonormal system of eigentensor columns for a block tensor matrix is constructed. A number of definitions and theorems are formulated. Several newly introduced terms are used to propose various representations of the specific strain energy and the corresponding constitutive relations.     

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     

Coaxiality of strain and stress in anisotropic linear elasticity

For a linearly elastic anisotropic body there are at least two rotations of the principal axes of strain such that the stress and strain tensors become coaxial. These rotations correspond to critical points for the stored energy, viewed as a function of the relative orientation between the body and the strain tensor.Supported by Gruppo Nazionale per la Fisica Matematica of C.N.R. (Italy).     

Uniqueness of the concentrated-load problem in the linear theory of couple-stress elasticity

Summary It is pointed out that there exist at least two different solutions of the problem of concentrated loads in the two-dimensional, linear couple-stress theory when the formulation is based on the usual uniqueness theorem. An extension of this uniqueness theorem is proved. A set of conditions sufficient for uniqueness is found and is used in a formulation of the concentrated load problem which results in a unique solution. The significant new condition is that the order of the stress singularity is limited to O(r^–1), where r is the distance from the concentrated load.

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Sommario Si fa notare che esistono almeno due soluzioni diverse del problema dei carichi concentrati nella teoria lineare, a due dimensioni, delle coppie di volume quando la formulazione è basata sul teorema di unicità.Si dimostra una estensione di questo teorema di unicità. Si trova un gruppo di condizioni sufficienti per l'unicità; queste condizioni vengono usate nella formulazione del problema del carico concentrato che dà luogo ad un'unica soluzione.La nuova condizione significativa è che l'ordine della singolarità dello sforzo è limitato a O(r^–1), dove r è la distanza dal carico concentrato.

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This work is a result of research sponsored by the Office of Naval Research, U.S. Navy, under Contract Nonr-610(06).     

Couple stress based strain gradient theory for elasticity

The deformation behavior of materials in the micron scale has been experimentally shown to be size dependent. In the absence of stretch and dilatation gradients, the size dependence can be explained using classical couple stress theory in which the full curvature tensor is used as deformation measures in addition to the conventional strain measures. In the couple stress theory formulation, only conventional equilibrium relations of forces and moments of forces are used. The couple's association with position is arbitrary. In this paper, an additional equilibrium relation is developed to govern the behavior of the couples. The relation constrained the couple stress tensor to be symmetric, and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system. On the basis of this modification, a linear elastic model for isotropic materials is developed. The torsion of a cylindrical bar and the pure bending of a flat plate of infinite width are analyzed to illustrate the effect of the modification.     

Effective characteristics of fiber composites in the linear moment theory of elasticity

A special boundary value problem whose solution is used to determine the effective characteristics in the linear moment theory of elasticity is considered. A procedure of finding the effective characteristics is proposed by the example of a fiber composite whose matrix and inclusions are isotropic. The boundary effects of structure functions are also discussed.     

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