Averaging of functionals of the calculus of variations and elasticity theory. (English. Russian original) Zbl 0599.49031

Math. USSR, Izv. 29, No. 1, 33-66 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 4, 675-710, 877 (1986).
The author begins his discussion with a comment that variational problems of mechanics appear to have increasingly more complex Lagrangians. Some change their properties in different regions, some are singular, some vary pointwise. In general, Lagrangians \(f(x, \xi)\) are measurable functions of \(x\) and convex in \(\xi\) for a.e. \(x\in\mathbf R^n\). \(\xi\) could be a vector or a symmetric matrix \(\{\xi_{ij}\}\), which is the case in classical elasticity. With each Lagrangian \(f(x,\xi)\) is associated a dual Lagrangian \(f^*(x, \xi)=\sup_\eta \{\eta\cdot\xi-f(x,\eta)\}\). This Young-type duality is used in a systematic manner to derive averaging results for nonlinear variational problems, particularly for the so-called regular Lagrangians obeying the inequalities \(c_1|\xi|^\alpha-c_0\leq f(x, \xi)\leq c_2|\xi|^\alpha+c_0\), \(c_i>0\), \(\alpha>1\).
These averaging methods are applied to random Lagrangians, to problems in plasticity with singular Lagrangians, and to Lagrangians representing the behavior of composite materials, or materials with inclusions. This is a development that follows the ingenious applications of the Young-Fenchel duality developed by Soviet authors such as V. L. Berdichevskiĭ, who has written a book “Variational principles of continuum mechanics” (Russian), “Nauka”, Moscow (1983)] and a series of articles.
This approach competes with regularization ideas in deciding when averaging is legal, particularly in problems dealing with critical loads and the corresponding designs of elastic systems.


49S05 Variational principles of physics
74S30 Other numerical methods in solid mechanics (MSC2010)
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