 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| I. A. Brigadnov, "Discontinuous mappings and their approximation in the nonlinear theory of elasticity," Mech. Solids. 36 (2), 36-45 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
2 |
Pages |
36-45 |
| Title |
Discontinuous mappings and their approximation in the nonlinear theory of elasticity |
| Author(s) |
I. A. Brigadnov (St. Petersburg) |
| Abstract |
Two statements of boundary value problems for the mappings of the nonlinear
theory of elasticity are considered in the form of a variational problem of
global minimization and in the form of variational (stationary and
evolutionary) equations. The interrelations of these statements,
as well as their mathematical well-posedness and physical correctness, are
discussed. The statement of the elastostatic boundary value problem in
the form of an evolutionary variational equation allows one to study a
bifurcation process for an equilibrium configuration of a deformed elastomer.
The general existence theorem is formulated for the equilibrium
configuration allowing its branching.
For highly elastic materials with an ideal saturation, the variational
problems under consideration need an extension of the set of admissible
solutions to take into account possible discontinuities. For example,
hyperelastic materials are described by potentials growing linearly
with respect to the norm of the distortion tensor, and the existence of the
limiting load is characteristic of them. An example illustrating the
existence of mappings with slip-type discontinuities is given
for the classical Bartenev-Khazanovich potential.
To take into account discontinuous mappings one can use a partial
relaxation of evolutionary variational elastostatic problem based on a
special three-dimensional finite element approximation admitting fields
with discontinuities of the slip type. As a result, the original problem
for a continuous medium is reduced to an initial value problem for a stiff
nonlinear system
of ordinary differential equations. This system is solved numerically by
using an implicit Euler scheme.
The results of computational experiments are given. They show qualitative
advantage of the suggested approach over the standard methods based
on a continuous 3D finite element approximation. |
| References |
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|
| Received |
12 March 1998 |
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