Solution of a Riemann problem for elasticity

This paper describes a numerical algorithm for the Riemann solution for nonlinear elasticity. We assume that the material is hyperelastic, which means that the stress-strain relations are given by the specific internal energy. Our results become more explicit under further assumptions: that the material is isotropic and that the Riemann problem is uniaxial. We assume that any umbilical points lie outside the region of physical relevance. Our main conclusion is that the Riemann solution can be obtained by the iterative solution of functional equations (Godunov iterations) each defined in one- or two-dimensional spaces.Supported in part by AFOSR-88-0025.     

New formulations on the second fundamental problem in plane elasticity

Some general formulations of the second fundamental problem in plane elasticity are proposed here when the displacements given on the closed boundary contours of a multiconnected elastic region are relative to certain rigid motions which are different to each other for different boundary contours. In such case, it is proved that, for the unique existence of solution, there must be given in addition to the principal vector and the principal moment of the external forces on each boundary contour. A method of solution is given also together with some illustrative examples.Projects supported by the Science Fund of Academia Sinica.     

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.

     

Axisymmetric thermo elasticity problem for a cylinder with a slot

The axisymmetric thermoelastic state of an Isotropic, circular, infinite cylinder with an external annular groove is considered. It is assumed that uniformly distributed heat sources act on part of the slot surface, while the side surface of the cylinder is heat insulated. A formula is obtained to determine the normal stresses in the plane of the slot.Translated from Zhurnal Prikladnoi. Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 154–158, November–December, 1971.     

Approximate method for solving a two-dimensional problem of elasticity theory

A numerical-analytical method based on approximation by harmonic or biharmonic functions is proposed for solving a mixed two-dimensional problem of elasticity theory. This method allows one to decrease the geometric dimensionality of the boundary-value problem by reducing it to minimization of the boundary residual. The resultant approximate analytical solution satisfies all equations of elasticity theory. Kazan’ State Technical University, Kazan’ 420111. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 179–185, July–August, 1999.     

Mixed boundary value problem of elasticity for a quarter space

We consider the static elasticity problem for a quarter space with zero displacements on one of its surfaces and with given stresses on the other. The method for solving this problem is based on the use of newunknown functions in the formof a linear combination of the desired displacements, which reduces the system of three Lamé equations to two equations to be solved simultaneously and one equation to be solved separately. The exact solution of this problem was obtained earlier by the same method [1]. But it was shown in [2] that such a solution is exact only under certain restrictions on the given functions. In the present paper, the solution of this problem is constructed without restrictions on the given functions, which necessitates solving a one-dimensional integro-differential equation; this can be done approximately by the orthogonal polynomial method. We present numerical results obtained on the basis of our solution.     

Plane elasticity problem for a multi-wedge system with a thin wedge

The paper presents a method for studying a system of elastic wedges containing a thin wedge with the angle Θ₀, which may be arbitrary small. An analysis shows that the considered problem, involving 2-D vectors of tractions and displacements, cannot be solved by straight-forward extension of the method previously worked out by the authors for analogous scalar problems. The difficulty arises because of the disclosed feature of the dependences between the Mellin transformed displacements and tractions at the boundaries of a thin wedge: they are linearly dependent when their Taylor’s expansions in Θ₀ are represented by the first terms only. The difficulty is removed by using the consequences of the linear dependence and by an appropriate re-arrangement of variables. Then simple physical models, simulating the influence of a thin wedge on a multi-wedge system, become available. The models cover the cases of a very rigid and very compliant thin wedge and also intermediate cases. The ranges of the models applicability are studied analytically and illustrated by numerical results.     

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.