Variational linearization of pure traction problems in incompressible elasticity
- PDF / 569,497 Bytes
- 26 Pages / 547.087 x 737.008 pts Page_size
- 99 Downloads / 171 Views
Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Variational linearization of pure traction problems in incompressible elasticity Edoardo Mainini
and Danilo Percivale
Abstract. We consider pure traction problems, and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads. Mathematics Subject Classification. 49J45, 74K30, 74K35, 74R10. Keywords. Calculus of variations, Linear elasticity, Finite elasticity, Gamma-convergence, Rubber-like materials.
1. Introduction Let us consider a hyperelastic body occupying a bounded open set Ω ⊂ R3 in its reference configuration. Equilibrium states under a body force field f : Ω → R3 and a surface force field g : ∂Ω → R3 are obtained by minimizing the total energy W I (x, ∇y(x)) dx − (y(x) − x) · f(x) dx − (y(x) − x) · g(x) dH2 (x). Ω
∂Ω
Ω
Here, y : Ω → R denotes the deformation field, H denotes the surface measure, and W I : Ω × R3×3 → [0, +∞] is the incompressible strain energy density. We require incompressibility by letting W I (x, F) = +∞ whenever det F = 1. Moreover, we assume that W I (x, ·) is a frame indifferent function that is minimized at F = I with value 0. If h > 0 is an adimensional small parameter, we rescale the displacement field and the external forces by letting f = hf, g = hg and y(x) − x = hv(x). We get EhI (v) := W I (x, I + h∇v) dx − h2 f · v dx − h2 g · v dH2 (x). 3
2
Ω
∂Ω
Ω
We aim at obtaining the behavior of rescaled energies as h → 0 and at showing that the linearized elasticity functional arises in the limit. More precisely, we aim at proving that inf EhI = h2 min E I + o(h2 ), and that if
EhI (vh )
− inf
EhI
(1.1)
2
= o(h ) (i.e., if vh is a sequence of almost minimizers for vh → v0 ∈ argmin E I
in a suitable sense, where E I (v) :=
QI (x, E(v)) dx −
Ω
0123456789().: V,-vol
then (1.2)
f · v dx −
Ω
EhI )
∂Ω
g · v dH2 (x).
146
Page 2 of 26
Here, E(v) := F ∈ R3×3 by
E. Mainini and D. Percivale
1 2 (∇v
ZAMP
+ ∇vT ) is the infinitesimal strain tensor field and QI (x, ·) is defined for every
QI (x, F) := lim h−2 W I (x, exp(hF)) = h→0
1
FT D2 W(x, I) F +∞ 2
if Tr F = 0, if Tr F = 0,
where W(x, F) := W I (x, (det F)−1/3 F) is the isochoric part of W I . Such a quadratic form is obtained by a formal Taylor expansion around the identity matrix (D2 denoting the Hessian in the second variable). Since QI (x, F) = +∞ if Tr F = 0, we see that E I (v) is finite only if div v = 0 a.e. in Ω. Therefore, E I is the linearized elastic energy with elasticity tensor D2 W(x, I) and div v = 0 is the linearized incompressibility constraint. Under Dirichlet boundary conditions, (1.1)–(1.2) have been obtained in [25], by means of a Γ-convergence analysis with respect to the weak topology of W 1,p (Ω, R3 ), where the exponent p is suitably related to the coercivity properties of W I (see Sect. 2). On the other hand, in this paper we shall consider natural Neumann boundary conditions, i.e., the pure traction p
Data Loading...
