Mixed problem of crack theory for antiplane deformation

The elastic equilibrium of an isotropic plane with one linear defect under conditions of longitudinal shear is considered. The strain field is constructed by the solution of a twodimensional boundary-value Riemann problem with variable coefficients. A special method that reduce the general two-dimensional problem to two one-dimensional problems is proposed. The strain field is described by three types of asymptotic relations: for the tups of the defect, for the tips of the reinforcing edge, and also at a distance from the closely spaced tips of the defect and the rib. The general form of asymptotic relations for strains with finite energy is deduced from analysis of the variational symmetries of the equations of longitudinal shear. A paradox of the primal mixed boundary-value problem for cracks is formulated and a method of solving the problem is proposed. Novosibirsk State Technical University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 163–172, May–June, 1998.     

SINGULAR PERTURBATION FOR A NONLINEAR BOUNDARY VALUE PROBLEM OF FIRST ORDER SYSTEM

ＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＦＯＲＡＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＦＩＲＳＴＯＲＤＥＲＳＹＳＴＥＭＣｈｅｎＳｏｎｇｌｉｎ（陈松林）（ＲｅｃｅｉｖｅｄＡｐｒｉｌ８,１９８４;ＲｅｖｉｓｅｄＡｐｒｉｌ１５,１９...     

On the first boundary value problem for the classical theory of elasticity in a three-dimensional domain with a singular boundary

An existence and uniqueness theorem is proved for the first boundary value problem of classical elasticity, relating to a broad class of three-dimensional domains whose boundaries may have edges and vertexes.The qualitative properties, up to the boundary, of the solution are investigated.

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Résumé On démontre un théorème d'existence et unicité pour le premier problème de valeurs au contour de l'élasticité classique, concernant une large classe de domaines de l'éspace à trois dimensions, avec des frontières présentant des arètes et des sommets.On étudie les propriétés qualitatives, jusque'au contour, de la solution.

     

Solution of a multidimensional impact deformation problem for an elastic half-space with curved boundary on the basis of a modified ray method

A generalization of the method for constructing approximate solutions of boundary value problems of impact deformation dynamics in the form of ray expansions for two-dimensional plane deformation problems is presented. For each shock wave, the solution near its front is determined on the basis of ray coordinates consistent with this wave. The nonlinear divergence of curvilinear rays is taken into account. A mechanism of transformation from one ray coordinate system to another, which is crucially important in the ray method, is described. The developed technique is illustrated by solving the impact deformation problem for a half-space with boundary of nonzero curvature.     

Approximate solutions of a nonlinear boundary value problem

Archive for Rational Mechanics and Analysis -     

Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary

Based on a modified Darcy's law, Stokes’ first problem was investigated for a second grade fluid in a porous half-space with a heated flat plate. Exact solutions of the velocity and temperature fields were obtained using Fourier sine transforms. In contrast to the classical Stokes’ first problem, there is a steady-state solution for the second grade fluid in the porous half-space, which is a damping exponential function with respect to the distance from the flat plate. The well-known solutions for Newtonian fluids in non-porous or porous half-space appear in limiting cases of our solutions.     

Plane contact problem for a half-space with boundary imperfections

In this paper the new micrmodelling approach to the contact problem for a half-space with boundary imperfections is proposed. The approach is based on a periodic distribution of micro-undulations along the space boundary and leads to the 2-D mathematical macro-model of the contact problem. The general idea of the modelling takes into account certain concepts used in the investigation of periodic composite materials (see e.g. Wozniak, 1993) . The resulting model constitutes a generalization of the known Winkler-type model (see e.g. Shtayerman, 1949) . The numerical solution to the special problem shows the boundary imperfections effect on the contact of bodies.     

Axisymmetric problem for a half-space in the micropolar theory of elasticity

The general solution of the equilibrium equations is obtained for a half-space subjected to arbitrary normal pressure. Four particular cases, including a concentrated force, are considered in detail. The singular parts of the displacements, rotations and stresses are obtained for the case of a concentrated force. The corresponding classical results have been derived. The graphs for various physical and geometrical quantities have been drawn to illustrate the micropolar effects.

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Zusammenfassung Die ellgemeine Lösung für Gleichgewicht eines Halbraums unter beliebigem Normaldruck wird gefunden. Vier Spezialfälle darunter eine konzentrieste Kraft werden genauer untersucht. Für diesen Fall werden die Verschiebungen, Drehungen und Spannungen gefunden. Die entsprechenden klassischen Lösungen wurden erhalten. Zur Illustration der Mikropolareffekte wurden Diagramme der verschiedenen physikalischen und geometrischen Grössen gezeichnet.

     

On the solution of nonlinear two-point boundary value problem

In this paper, a non-variational version of a max-min principle is proposed, andan existence and uniqueness result is obtained for the nonlinear two-point boundaryvalue problenl u"+g(t.u)=f(t),u(0)=u(2π)=0     

Singular perturbation of nonlinear vector boundary value problem

In this paper we study the perturbed boundary value problem of the form dx/dt=f(x,y,t;ε), εdy/dt=g(x,y,t;ε), a_1(ε)x(0,ε)+a_2(ε)y(0,ε)=a(ε) b_1(ε)x(1,ε)+εb_2(ε)y(1,ε)=β(ε)in whichx,f,β∈E~m, y,g,a∈E~n, 0<ε(?)1and a_1(ε), a_2(ε), b_2(ε)and b_2(ε) are matrices of the appropriate size. Under the condition that g_y(t) is nonsingular and other suitable restrictions, the existence of the solution is proved, the asymptotic expansion of solution of order n is constructed, and the remainder term is estimated.     

Solution of the Reissner-Sagoci problem for a half-space with a cylindrical cavity

Simferopol' University. Translated from Prikladnaya Mekhanika, Vol. 25, No. 7, pp. 24–30, July, 1989.     

A boundary model for the thermal creep problem

A new integral boundary condition model for determining the particle velocity distribution function in a Knudsen layer, which makes it possible to improve the accuracy of the moment approximations, is proposed. The results for various models are compared with the exact numerical solution of the problem.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 160–163, November–December, 1993.This research was made possible by financial support from the Exploratory Research Program, US EPA and by a Fulbright grant for Professor I. N. Ivchenko.     

A Riemann-Hilbert boundary value problem in neutron transport theory

Summary A Riemann-Hilbert boundary value problem, occurring in neutron transport theory, is presented and solved in this note for the case of an infinite straight line and zero index.

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Sommario Viene presentato e risolto in questa nota un problema al contorno di Riemann-Hilbert di indice zero e relativo ad un intervallo infinito. Esso ha origine da un problema di diffusione di neutroni, studiato nell'ambito della teoria del trasporto.