Equations of bifurcation of equilibrium of an elastic isotropic body in terms of rates of change of lagrangean coordinates: PMM vol. 38, n≗4, 1974, pp. 693–702

Abstract

A new version of constructing the three-dimensional theory of elastic stability is advanced. Bifurcation is considered to be an interchange of material particles at a fixed point in space. As a kinematic variable we take the rate of change of Lagrangean particle coordinates. On the basis of obtained exact solutions, an approximate method is developed, valid in the case of small precritical deformations and rotations. Whenever the usual Lagrangean presentation is used for the motion of a continuous medium [1, 2], the equations which determine the changes in the stress tensor necessarily contain the rotations of material particles. As a result, the linearized deformation equations of equilibrium in the general case contain the sought critical stresses [1]. It is of interest to study that version of a boundary value stability problem for which the parameters enter essentially only into the boundary conditions. One of these versions was suggested by Leibenzon [3], and then independently by Ishlinskii [4], However, it cannot be obtained using the Lagrangean presentation while linearizing the original equations of the nonlinear theory of elasticity. In the present paper, nonlinear equations of the theory of elasticity in Eulerian representation are obtained using as a kinematic variable the rates of change in Lagrangean coordinates. On the basis of these equations, bifurcation of equilibrium of an isotropic elastic body is considered. The advantage of the suggested version is that the changes in the Cauchy stress tensor are related only to the deformation tensor of introduced velocities. Therefore, the differential equations of equilibrium contain only those parameters of the prebifurcation state which are related to the change in physical properties of the body during deformation. The components of the rotation tensor of these velocities enter only into the boundary conditions in connection with the change of shape of the body at the instant of bifurcation. The parameters which enter as factors of the components of the rotation tensor in the boundary conditions are the most essential part of parameters of the precritical state which enter into the structure of the obtained boundary value problem for neutral equilibrium. For an isotropic elastic body which is only slightly deformed in the precritical state, the indicated circumstance permits to suggest a simple approximate version of the equations of neutral equilibrium in which the sought parameter of the critical loading enters only into the boundary conditions. If the physical content of the sought functions which enter the differential equations and the boundary conditions is moved into the background, then it appears that the approximate version of the boundary value problem is close to the one used by Leiben zon.