Mechanics

2005, No. 2

I.S. Akhmedjanov

APPLICATION OF THE QUADRATURE METHOD TO THE ANALYSIS OF MEMBRANE SHELLS OF REVOLUTION

The paper is devoted to the numerical quadrature method application to integration of differential equations of membrane shells of revolution. According to this method the initial system of differential equations first is transformed into a system of integral equations. Then the standard trapezoid formula is applied to computation of the involved integrals.

G.L. Gorynin, Y.V. Nemirovskiy

ASYMPTOTIC ANALYSIS OF THREE-DIMENSIONAL ELASTICITY PROBLEM FOR CIRCULAR AND RING-SHAPED PLATES

In the present paper the three-dimensional elasticity bending problem of layered plates is reduced to one-dimensional and two-dimensional boundary-value problems by the asymptotic splitting method. The obtained solutions permit to obtain the asymptotic approximations for all components of displacement vector and stress tensor. Under polyharmonic distributed external load the given high order approximations are the exact solutions of the corresponding three-dimensional elasticity problem with traditional integral conditions on a plate edge. The resolving equation for the first order approximation is the same due to S. Germen and for the second due to Timoshenko—Donnell.

Y.N. Radayev, V.A. Gudkov, Y.N. Bakhareva

ON THE SYMMETRY GROUPS AND LIE ALGEBRA OF THREE-DIMENSIONAL EQUATIONS OF THE MATHEMATICAL PLASTICITY

Group analysis of the system of partial differential equations of threedimensional plastic equilibrium is given. The Tresca yielding criterion is employed to formulate the involved system. Stress state is presumed correspond to an edge of the Tresca prism thus allowing formally consider the static equations independently on the flow rule. The system of static equilibrium equations is represented in the stress principal lines co-ordinate net (isostatic net). The symmetry group of this system is obtained. The Lie algebra and a first order optimal system of subalgebras of the symmetry group of partial differential equations of the three-dimensional mathematical theory of plasticity are studied.

L.V. Stepanova, M.E. Phedina, N.A. Kurnysheva

SELF-SIMILAR SOLUTION FOR A MODE I CRACK PROBLEM IN CREEPING DAMAGED MATERIALS IN THE COUPLED (CREEP–DAMAGE) FORMULATION

To evaluate mechanical behaviour near a Mode I crack tip the governing equations are formulated within the framework of Continuum Damage Mechanics. The asymptotic stress and integrity fields near the tip of a stationary crack are derived for non-linear creeping damaged materials deforming according to the creep power constitutive law. The conventional Kachanov–Rabotnov creep-damage theory is utilized and the scalar integrity parameter is incorporated into the constitutive equations. Stress analysis based on the similarity variable is carried out for Mode I crack under plane stress and plane strain conditions assuming the existence of a totally damaged zone near the crack tip. It is found that the Hutchinson—Rice—Rosengren solution can’t be used as the remote boundary condition and the actual far field stress is obtained. The shape of the totally damaged zone is analyzed.