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On the positivity of the second variation in finite elasticity. (English) Zbl 0657.73027
The problem of the minimization of the integral \(I(f)=\int_{\Omega}W(x,\nabla f(x))dx\) is investigated, where f is a vector-valued function on a bounded open domain \(\Omega \in {\mathbb{R}}^ n\) and satisfies \(f=d\) on a portion of the boundary \({\mathcal D}\), here d is a given function.
The authors deal with the following questions: (a) What are the necessary and sufficient conditions for the second variation of I to be uniformly positive? (b) What are the sufficient conditions for the second variation of I to be coercive on a certain domain?
In elasticity I(f) is the total stored energy of an elastic body when it is deformed by the deformation f. Applications of the proved results to the linear isotropic elastic body are given in detail. Some results are presented in connection with the operator \({\mathcal L}\) given by the formula \({\mathcal L}[u]=-div C_ f[\nabla u]+\lambda u\), where u is the displacement, \(C_ f\) is the elasticity tensor. The behaviour of the spectrum of \({\mathcal L}\) and the existence condition of the solution of the equation \({\mathcal L}[u]=b\) are also analyzed.
Reviewer: I.Ecsedi

MSC:
74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
49J27 Existence theories for problems in abstract spaces
49J45 Methods involving semicontinuity and convergence; relaxation
47A10 Spectrum, resolvent
47B38 Linear operators on function spaces (general)
49J20 Existence theories for optimal control problems involving partial differential equations
49K99 Optimality conditions
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