The general stress and strain relationship in non-linear materials

Modification of the Local Variation Method to Minimize the Functionals of the Elasticity Theory

“…where P δ , 2 P δ -first and second variations of functional (1). It means that ( ) i P u has a minimum value and it possible to use local variation method.…”

“…The expression (1) is the functional of total energy of elasticity and given surface force. It is possible to prove, that Euler-Langrange's equations for functional (1) gives the classic elasticity equations of Lame-Cauchy. It means that minimization of functional (1) by functions ( ) The next step -minimization of a functional by some method.…”

“…It means that minimization of functional (1) by functions ( ) The next step -minimization of a functional by some method. In the works [1,2,3] some nonlinear functionals of solid mechanics when Hook's law does not hold and relation between stress and strain components depends on three invariants of strain and stress tensors are proposed.…”