Mechanics

2002, Special Issue

S.A. Lychev, Y.E. Senitskii

NONSYMMETRIC FINITE INTEGRAL TRANSFORMATIONS AND THEIR APPLICATION TO VISCO-ELASTICITY PROBLEMS

In this paper a new class of nonsymmetric finite integral transformations generated by nonselfconjugate differential pencils is proposed. By the introduced transformations the solutions of nonconjugate initial boundary-value problems in the space of square integrable vector-functions are obtained. Integral transformations technique are demonstrated by dynamic problem of rotating visco-elastic beem.

Y.N. Radayev, V.A. Gudkov

ON THE NULL LAGRANGIANS OF NONLINEAR ELASTIC FIELDS

In the present study the null Lagrangian theory for n-dimensional manifold (including 4-dimensional Minkowski space-time) is developed in an attempt to extend the canonical formalism of nonlinear field theory. All developments are presented in the frame of finite strains. By the aid of divergence formula for the null Lagrangians regular in n-dimensional star-shaped domains, a general representation of the null Lagrangian depending as maximum on the first order field gradients is obtained. A method of systematic derivation of the null Lagrangians in n-dimensional manifold is proposed. It is shown that in the case of nonlinear elastic field in 3-dimensional space the null Lagrangian is represented via 15 arbitrary independent field functions. The dual null Lagrangians, corresponding to the Piola inverse-motion description, are derived in the case of 3-dimensional space. Null Lagrangians of 4-dimensional material manifold are then analysed. Invariant under actual placement translations null Lagrangians are obtained. Finally, material forces acting on empty 4-dimensional manifold are studied.