A New Variational Approach to Linearization of Traction Problems in Elasticity

Abstract

A new energy functional for pure traction problems in elasticity has been deduced in [23] as the variational limit of nonlinear elastic energy functional for a material body subject to an equilibrated force field: a sort of Gamma limit with respect to the weak convergence of strains when a suitable small parameter tends to zero. This functional exhibits a gap that makes it different from the classical linear elasticity functional. Nevertheless a suitable compatibility condition on the force field ensures coincidence of related minima and minimizers. Here we show some relevant properties of the new functional and prove stronger convergence of minimizing sequences for suitable choices of nonlinear elastic energies.

The Gap Between Linear Elasticity and the Variational Limit of Finite Elasticity in Pure Traction Problems

“…It is well known that such phenomenon takes place for pure traction problems in linear elasticity too, but in nonlinear elasticity this difficulty cannot be easily circumvented in general, since the fact that v j is a minimizing sequence does not entail that also v j − Pv j is minimizing sequence. In [25] we show that for some special integrand W, as in the case of Green-St-Venant energy density, if v j is a minimizing sequence then w j := v j − Pv j is a minimizing sequence too and there exist a (not relabeled) subsequence of functionals F h j such that the related minimizing subsequence w j converges weakly in…”

“…the classical linear elasticity formulation which achieves a finite minimum over H 1 (Ω, R N ) since the condition of equilibrated loads is fulfilled; nevertheless with exactly the same choices there is a sequence w h in [12], we cannot expect weak H 1 (Ω; R N ) compactness of minimizing sequences, not even in the simplest case of null external forces: although this fact is common to pure traction problems in linear elasticity, we emphasize that in nonlinear elasticity this difficulty cannot be easily circumvented in general by standard translations since F h (v h ) = F h (v h − Pv h ), with P projection on infinitesimal rigid displacements. We deal this issue in the paper [25], showing nonetheless that at least for some special…”

“…Concerning the structure of the new functional, we emphasize that actually F is different from the classical linear elasticity energy functional E, though there are many relations between their minimizers (see Theorem 4.1). Further and more detailed information about functional F (a suitable property of weak lower semicontinuity, lack of subadditivity, convexity in 2D, nonconvexity in 3D) are described and proved in the paper [25].…”

Variational linearization of pure traction problems in incompressible elasticity

“…In this case, it is crucial to impose suitable restrictions on the external forces. In particular, as done in [23,24], here we shall assume they have null resultant and null momentum with respect to the origin, namely…”

“…The first rigorous variational derivation of linearized elasticity from finite elasticity is given in [11], where Γ-convergence and convergence of minimizers of the associated Dirichlet boundary value problems are proven in the compressible case. We refer to [1][2][3]10,18,[23][24][25]37] for many other results of this kind, some of which including theories for incompressible materials [10,18,25]. The study of asymptotic properties of minimal energies, similar to (1.1)-(1.2), is also typical of dimension reduction problems, see for instance [4,21,[31][32][33][34][35].…”