Integral equations of plane static boundary value problems of the elasticity theory for an inhomogeneous anisotropic medium

The static boundary value problems of plane elasticity for an inhomogeneous anisotropic medium in a simply connected domain are reduced to the Riemann–Hilbert problem for a quasianalytic vector. Singular integral equations over the domain are obtained, and their solvability is proved for a sufficiently wide anisotropy class. In the case of a homogeneous anisotropic body, the solutions of the first and second boundary value problems are obtained in closed form.     

Invariant recording of elasticity theory equations

An invariant (with respect to rotations) formalization of equations of linear and nonlinear elasticity theory is proposed. An equation of state (in the form of a convex generating potential) for various crystallographic systems is written. An algebraic approach is used, which does not require any geometric constructions related to the analysis of symmetry in crystals. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 127–142, September–October, 2008.     

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     

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     

Hypoelastic form of equations in nonlinear elasticity theory

It is shown that the complete system of equations of elasticity theory for an isotropic medium admits a unique representation in the hypoelastic form (the tensor of the rate of change of stresses is a linear function of the tensor of strain rates with coefficients depending on the invariants of the stress tensor). It is necessary to this end that the hypothesis be satisfied on the determination of strains by stresses which are unknown. Any arbitrariness in the choice of the coefficients of the hypoelastic relation may result in the thermodynamic identity being infringed.     

Cosserat spectrum of the first boundary-value problem of elasticity

The three-dimensional problem of calculating the Cosserat spectrum of the first boundaryvalue problem of elasticity for a body of revolution is considered. Using grids containing 900 and 3600 nodes, it is shown that the sequence of eigenvalues for a sphere found by E. Cosserat and F. Cosserat does not describe the entire range of eigenvalues.     

A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point

The theory of a Cosserat point has been used to formulate a new 3-D finite element for the numerical analysis of dynamic problems in nonlinear elasticity. The kinematics of this element are consistent with the standard tri-linear approximation in an eight node brick-element. Specifically, the Cosserat point is characterized by eight director vectors which are determined by balance laws and constitutive equations. For hyperelastic response, the constitutive equations for the director couples are determined by derivatives of a strain energy function. Restrictions are imposed on the strain energy function which ensure that the element satisfies a nonlinear version of the patch test. It is shown that the Cosserat balance laws are in one-to-one correspondence with those obtained using a Bubnov–Galerkin formulation. Nevertheless, there is an essential difference between the two approaches in the procedure for obtaining the strain energy function. Specifically, the Cosserat approach determines the constitutive coefficients for inhomogeneous deformations by comparison with exact solutions or experimental data. In contrast, the Bubnov–Galerkin approach determines these constitutive coefficients by integrating the 3-D strain energy function using the kinematic approximation. It is shown that the resulting Cosserat equations eliminate unphysical locking, and hourglassing in large compression without the need for using assumed enhanced strains or special weighting functions.     

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.     

Characteristic cones of the equations in the nonlinear theory of elasticity

The roots of the equation for the characteristic normals for two systems of differential equations in the nonlinear theory of elasticity are investigated. The first model is constructed using a thermodynamic identity. The second is a very simple hypoelastic model (the deviator of the stress-rate tensor is proportional to the deviator of the strain-rate tensor). It is shown that the roots of the equations for the normals to the characteristics for the second model are the same as the first-order terms in the expansion of the roots of the first model with respect to the strain-tensor deviator.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–132, May–June, 1974.The author is grateful to S. K. Godunov for discussions.     

Boundary integral equations for inverse problems in the elasticity theory

The paper is concerned with the application of the boundary integral equation method for the holomorphic vector to nonclassical problems in the static theory of elasticity. Problems are considered for the case when on part of a body conditions are overvalued, i.e. the vectors of displacements and loads are prescribed, and on the other part of the body conditions are unknown (so-called (u, p) problem). It is shown that the method proposed is efficient and can be applied to the problems of computer defectoscopy in stationary potential fields.