Concentrators of deformations and fracture of plastic bodies

An approach to the investigation of shape discontinuity regions as strain concentrators is proposed. The near-concentrator strain fields are determined on the basis of the theory of ideal rigid-plastic body; under the condition of plane deformation, their determination is reduced to integration of ordinary differential equations. The deformation as a function of the location of the plastic region and its shape evolution in the process of plastic flow is studied. The plastic flow is demonstrated to be not unique (within the framework of solution completeness). A deformation criterion for the choice of the preferred plastic flow is suggested. The fracture of a V-notched strip is considered. On the basis of the solutions obtained, an approach to the investigation of the fracture processes for more complicated models is formulated.     

On the theory of austenite-cementite phase equilibria in steels

A microscopic model is proposed, which describes the structure and thermodynamic properties of cementite (an ordered iron carbide with a composition of Fe₃C_{1 ? δ}) and the phase equilibria between cementite and austenite (a disordered solid solution of carbon in face-centered cubic iron). Based on this model, chemical and stress-induced (deformational) interactions of carbon atoms in cementite and austenite structures are quantitatively evaluated. The “lattice” contributions in the equations of phase equilibria, which are related to changes in the crystal structure as a result of the cementite-austenite phase transition, are estimated. The proposed model well describes available data on the thermodynamics of cementite and austenite and provides a basis for the further microscopic investigation of high-temperature phase transformations in steels.     

Classical and Generalized Solutions of a Mixed Problem for a Nonhomogeneous Wave Equation

Doklady Mathematics - A.N. Krylov’s ideas concerning convergence acceleration for Fourier series are used to obtain explicit expressions for the classical solution of a mixed problem for a...     

On the construction of approximations to continuous functions under integral boundary conditions

On the basis of the resolvent of a simple differential operator, a method for finding approximations to continuous functions is constructed. In this method, both the approximated function and its approximations satisfy the given integral boundary condition.     

The generating elements of certain Volterra operators connected with third- and fourth-order differential operators

Sufficient conditions are established forf (x) to be the generating function for the Volterra operator which is inverse to the Cauchy operator:l [y]=y⁽ⁿ⁾+p₂(x)y^(n–2) + ... +pn(x)y, y(0)=y(0)=...=y^(n–1)(0)=0 (n=3, 4), when the coefficients pi(x) are not analytic.Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 715–720, June, 1968.     

On the convergence of the Lavrent’ev method for an integral equation of the first kind with involution

The convergence in the mean-square metric of the Lavrent’ev regularizationmethod for an integral equation with involution is established. The proof of the convergence is based on studying the behavior of the resolvent of a certain integro-differential equation related to the original equation.     

Strain-energy failure criterion for rigid-plastic bodies

In the framework of the theory of strain-hardening rigid-plastic bodies, we suggest a strain-energy failure criterion. The strain states of a rigid-plastic body are depicted as points in the principal strain space and form a third-order hyperbolic surface for incompressible bodies. The strain processes are depicted by lines on this surface. The loading surface related to the strain surface lines is introduced, which permits calculating the energy dissipation as a parameter determining the failure.