The peridynamic model in non-local elasticity theory

The peridynamic model is described and an overview of some recent results concerning the analysis of the peridynamic equation of motion is given. Moreover, its numerical solution is discussed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variant of the variable-modulus theory of elasticity

A variant of the variable-modulus theory — a generalization of the ideas of the classical theory of elasticity in which the observed difference in the moduli of elasticity in uniaxial tension and compression and homogeneous shear is taken into account — is considered. Quasilinear expressions are proposed for the stresses in terms of the strains and the strains in terms of the stresses.Tula Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 363–365, March–April, 1969.     

Boundary and variational problems for a combined mathematical model of the theory of elasticity

We construct a combined mathematical model of the theory of elasticity that describes the stress-strain state of an elastic body using the equations of the theory of elasticity in one part of the body and the equations of the theory of shells of Timoshenko type in the other part. We write the resolvent equations and conditions for elastic coupling. We study the variational formulation of the boundary-value problems of the combined model.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 92–95.     

Tomographic algorithms in the theory of elasticity

The problem of finding the distribution of the elastic moduli in a heterogeneous elastic from the results of a series of mechanical tests under given external effects and measurements of the reactions on the body surface is considered. Two solution algorithms are described and a justification is given.     

Symmetric and nonsymmetric equilibria of a rod-and-spring model

A simple rod-and-spring model is used to illustrate bifurcationin the presence of symmetry. Our aim is to demonstrate the relevanceof certain general mathematical ideas and computational techniquesin a comparatively simple physical example, a full appreciationof which can be intuitive. The model discussed here has twodistinct reflectional symmetries, i.e. Z₂Z₂ symmetry. The symmetry-grouprepresentations, invariant subspaces, projection matrices, projectedequa-tions, and extended systems used to compute loci of singularitiesare all given explicitly. A closely allied nonsymmetric modelis then developed and its behaviour is examined numerically.Connections between singularities in the symmetric and asymmetricmodels are explored.     

A problem in elasticity theory

The problem of determining the dimensions of the transverse cross-sections of a beam from the given frequencies of its natural vibrations is examined. Frequency spectra are indicated that determine the dimenions of the transverse cross-sections of the beam uniquely, an effective procedure is presented for solving the inverse problem, and a uniqueness theorem is proved. The method of standard models /1/ is used to solve the inverse problem.     

Neumann problem for the operator of elasticity theory

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1279–1281, September, 1989.     

On the frame invariance of linear elasticity theory

It is widely believed that classical linear elasticity theory does not conform to the Galilean frame invariance of general (non-relativistic) field theories. This view is traced here to an interpretation of the relationship between the deformation gradient and the displacement gradient which does not reflect the tensor character of the variables involved. Frame invariance is shown to follow if tensor character is imposed on this relationship at the outset.     

On the frame invariance of linear elasticity theory

It is widely believed that classical linear elasticity theory does not conform to the Galilean frame invariance of general (non-relativistic) field theories. This view is traced here to an interpretation of the relationship between the deformation gradient and the displacement gradient which does not reflect the tensor character of the variables involved. Frame invariance is shown to follow if tensor character is imposed on this relationship at the outset.     

Discrete models in the dynamic problem of linear elasticity and conservation laws

The dynamic problem of linear elasticity (a continuous model) is based on the momentum and moment of momentum conservation laws and hence has an additional conservation law for the total energy (the kinetic plus potential energy). Continuous models with this property are referred to as entropy models (S.K. Godunov). For difference schemes, this property is described with the use of the notion of complete conservativeness) (A.A. Samarskii). For the considered problem, we make a complete analysis of a two-parameter family of “conjugatecoordinated” two-layer difference schemes including a conservative scheme. For the latter, we construct efficient triangular-factorized implementations with the same parallelism degree as in ordinary explicit schemes. For difference schemes without the property of complete conservativeness, we discuss the role of the law of passage to the limit for the minimization of imbalance in the total energy.     

Symmetric hyperbolic equations in the nonlinear elasticity theory

The basic results are overviewed concerning the formulation of nonlinear elasticity equations in the form of symmetric hyperbolic systems in long-time studies performed under the direction of the first author. The underlying principles developed in those studies are stated, and some inaccuracies and errors are corrected. Procedures are described for computing the coefficient matrices of the equations from a given equation of state.     

On universal relations in elasticity theory

Zusammenfassung Es wird eine Methode zur Erhaltung allgemeingültiger Beziehungen für Verzerrungen kompressibler, isotroper, elastischer Körper dargestellt.     

An inverse contact problem in the theory of elasticity

In this paper, we discuss an inverse problem in elasticity for determining a contact domain and stress on this domain. We show that this problem is an ill‐posed problem, and we establish the uniqueness and L²‐conditional stability estimation for the stress. Copyright © 1999 John Wiley & Sons, Ltd.     

Statistical theory of elasticity of reinforced plastics

A method of solving statistical boundary value problems in the stresses is proposed for microinhomogeneous composite media simulating the structure and properties of reinforced plastics. The coefficients of variation of the physical properties of the medium are not assumed small. Numerical examples are presented.Mekhanika Polimerov, Vol. 4, No. 3, pp. 438–444, 1968Presented on 5 June 1967 at a scientific seminar held at the Institute of Polymer Mechanics of the Academy of Sciences of the Latvian SSR.