Mixed boundary value problem of heat conduction for infinite slab

Summary An axisymmetric steady state heat conduction boundary value problem having mixed boundary conditions on both faces of an infinite slab, is reduced to a pair of Fredholm integral equations of the second kind. For large values of h, the slab thickness, a solution correct to O(h ^–6) is obtained by expanding the kernels in power series.Presently at Imperial College, London.     

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.

     

A four-part boundary value problem in elasticity — indentation by a discontinuously concave punch

A solution is given for the frictionless indentation of an elastic half-space by a flat-ended cylindrical punch with a central circular recess, when the load is large enough to establish a circular region of contact in the recess. The problem is reduced to two simultaneous Fredholm equations using the method of complex potentials due to Green and Collins. Results are presented for the relationship between load, contact radius and penetration for various punch geometries.     

Mixed boundary value problems for a cylindrical shell

Two mixed boundary value problems in potential theory for a semi-infinite cylindrical shell are solved. The first is interpreted as a heat conduction problem for an insulated shell containing a circumferential obstruction. The second is the torsion of a cylindrical shell containing a circumferential crack. For both problems the normal derivative of the potential function is taken as zero on the inner and outer shell walls. The boundary conditions at the end of the shell are mixed with respect to the potential function and its normal derivative. The problems are formulated using integral transforms in a manner leading to a singular integral equation which can be solved by numerical means. Intensity factors along the circumference separating the mixed conditions are computed.     

Mixed boundary value problems for a penny-shaped cut

The axially symmetric problem for a penny-shaped cut is solved in the case the displacements are prescribed on its upper surface and stresses on its lower surface. The solution is achieved by using integral transforms along with certain representations that reduce the general problem to the solution of uncoupled Hilbert problems. For polynomial loadings the Hilbert problems can be solved exactly by methods given by Muskhelishvili. As in the related plane problem two singularities at the edge of the disc are noted in the solution.     

Phragmen–Lindelof alternative for the displacement boundary value problem in a theory of non-linear micropolar elasticity

In this note we investigate the spatial behavior of the solutions for the displacement boundary value problem in a theory of non-linear micropolar elasticity. Under suitable conditions on the non-linear terms we obtain estimates for the solutions. The main tool is the energy method.     

Multiplicity results for a fourth-order boundary value problem

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Dynamic problem of elasticity theory for a space with a moving semi-infinite crack

Taganrog Pedagogical Institute, Russia. Translated from Prikladnaya Mekhanika, Vol. 28, No. 12, pp. 56–63, December, 1992.     

Complex variable solution for boundary value problem with X-shaped cavity in plane elasticity and its application

A new type of displacement pile, the X-section cast-in-place concrete(XCC)pile, has recently been developed in China. Extensive field tests and laboratory experiments are undertaken to evaluate its performance and quantify the non-uniform deformation effect(NUDE) of the X-shaped cross section during installation. This paper develops a simplified theoretical model that attempts to capture the NUDE. Based on the theory of complex variable plane elasticity, closed-form solutions of the stress and displacement for the X-shaped cavity boundary value problem are given. Subsequently, the analytical solution is used to evaluate the NUDE, the concrete filling index(CFI), and the perimeter reduction coefficient of the XCC pile cross section. The computed results are compared with field test results, showing reasonable agreement. The present simplified theoretical model reveals the deformation mechanism of the X-shaped cavity and facilitates application of the newly developed XCC pile technique in geotechnical engineering.     

Mixed problem of the theory of elasticity for a truncated paraboloid of revolution

KVIRTU PVO, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 5, pp. 27–32, May, 1991.     

Axisymmetric displacement boundary value problem for a penny-shaped crack

Summary A solution is derived from equations of equilibrium in an infinite isotropic elastic solid containing a penny-shaped crack where displacements are given. Abel transforms of the second kind stress and displacement components at an arbitrary point of the solid are known in the literature in terms of jumps of stress and displacement components at a crack plane. Limiting values of these expressions at the crack plane together with the boundary conditions lead to Abel-type integral equations, which admit a closed form solution. Explicit expressions for stress and displacement components on the crack plane are obtained in terms of prescribed face displacements of crack surfaces. Some special cases of the crack surface shape functions have been given in the paper.     

A mixed problem of plane elasticity for a domain with partially unknown boundary

The paper addresses a problem of plane elasticity for a doubly connected body with outer and inner boundaries in the form of regular polygons with a common center and parallel sides. The neighborhoods of the vertices of the inner boundary are unknown equal full-strength smooth arcs symmetric about the rays coming from the vertices to the center. It is assumed that this elastic body is inserted into a hole of a rigid body, with the hole boundary coinciding with the outer boundary of the elastic body. Absolutely smooth rigid punches with rectilinear bases are pressed into all the rectilinear sections of the inner polygonal boundary of the elastic body. There is no friction between the elastic and rigid bodies. The unknown arcs are free from external stresses. Complex variable theory is used to determine the unknown arcs and the stress state of the elastic body __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 110–118, March 2006.     

Approximate solutions of a nonlinear boundary value problem

Archive for Rational Mechanics and Analysis -     

Initial value problem for the dynamic system of anisotropic elasticity

The main object of this paper is the Cauchy problem for the dynamic system of anisotropic elasticity. Existence and uniqueness theorems of weak and smooth solutions of this problem are established by the reduction of the original elasticity system into a symmetric hyperbolic system of the first order. The numerical method of the Cauchy problem solving for anisotropic elastic system with polynomial data is obtained and its correctness is established. The simulations of the numerical solutions are presented.     

Iterative solution of a non-linear boundary value problem for a rotating string

The problem of determining the motion of a rotating inextensible string of length l, free at one end and fixed at the other can, under certain assumptions, be treated by solving a non-linear boundary value problem. p]In this paper, two Picard-type iterative schemes are constructed and the sequences generated are proved to converge to a positive solution of that non-linear boundary value problem. Additionally, one sequence converges to the solution from above, the other from below. Moreover, the iterative scheme can be used to solve the inverse problem of determining the angular velocity of the rotating string, given two observations (at one point in time) of position of the string.     

An inverse boundary value problem for Maxwell's equations

We consider steady, two-dimensional motions of an incompressible, Newtonian fluid flowing under gravity down an inclined channel. If the bottom of the channel is flat, the flow is the classical Poiseuille-Nusselt flow and the free surface is then a plane parallel to the bottom. Motivated by the recent experimental and numerical studies of Pritchard, Scott & Tavener, we look at bottom configurations which possess some localized, non-uniform structure. We present an existence theory for steady, highly viscous flow over such configurations. An important consequence of our theory is that the steady flows whose existence is established decay exponentially rapidly to the unperturbed Poiseuille-Nusselt flow away from the local variation in the channel bottom profile.