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Quasiconvexity at the boundary, positivity of the second variation and elastic stability. (English) Zbl 0552.73006

In this paper the relations connecting positivity of the second variation and elastic stability are studied in penetrating way. A crucial result shows that for nonlinear elasticity in n\(>1\) space dimensions positivity of the second variation does not imply a strong local minimum even under suitable convexity assumptions on the stored-energy function. A new necessary condition for a minimum, called by the author ”quasiconvexity at the boundary”, and valid for mixed problems of the calculus of variations is also given. This represents a contribution interesting in itself. In the last part of this paper it is shown that for nonlinear elasticity any proper local minimum of the energy in \(W^{1,1}\) lies in a potential well and thus is Lyapunov stable when the total energy is nonincreasing.
Reviewer: G.Cimatti

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74G99 Equilibrium (steady-state) problems in solid mechanics
74H99 Dynamical problems in solid mechanics
74B20 Nonlinear elasticity
49K27 Optimality conditions for problems in abstract spaces
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