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A variational approximation scheme for elastodynamic problems using a new class of admissible mappings

We consider a variational approximation scheme for the 3D elastodynamics problem. Our approach uses a new class of admissible mappings that are closed with respect to the space of mappings with finite distortion.

The motion of a deformable solid body can be written: where y : Ω×[t 0 , t 1 ] → R 3 is the displacement and S is the first Piola-Kirchgoff stress tensor. This equation can also be written as a system of conservation laws for the deformation gradient F = D x y and the velocity v = ∂ t y (1) i, α = 1, 2, 3.
In the case of hyperelastic materials, the tensor S can be expressed as the gradient of a scalar function W (F ). This function is called the stored energy function, W : M 3×3 → [0, ∞), where M 3×3 stands for (3 × 3)-matrices, i.e. S(F ) = ∂W ∂F ij (F ) . In order to prove the existence of solutions to the system of equations in 1, it would normally be necessary that W be convex. However, this is incompatible with the known physics of elastic materials such as the requirement of frame-indifference, i.e. the principle that certain properties of the system are invariant under arbitrary coordinate transformations [4]. This suggests the replacement of the condition of convexity with a weaker condition such as polyconvexity (for further details see, e.g. [1][2][3]). More precisely, we may assume that holds for some convex function G(F, Z, w), where cof F and det F are the cofactor matrix (i.e. transposed adjunctive matrix cof F = adj F T ) and determinant of the matrix F respectively.
At present, the question of the existence of a solution to the elastostatic problem has been thoroughly studied. A review of basic works and open problems can be found, for example, in [2]. Furthermore, the reader is referred to [5] for local existence of the classical solution of the elastodynamical system with rank-one convex and polyconvex stored energy functions. The existence of global weak solutions, excluding some particular cases [8], is still an open problem. Nevertheless, the existence of a global measure-valued solution was proven in [7] using a variational approximation scheme.
The variational approximation method or, in the nomenclature of E. De Giorgi [6], the minimizing movements method, is a method by which the limit of a minimizing sequence of iterations to the variational problem for an appropriate functional is found. The technique developed in [7] uses the variational approximation scheme to establish a link between elastostatics and elastodynamics. The method is based on the observation that the solution of the system (1) meets the additional conservation laws (the idea was suggested independently by P. G. Le Floch and T. Qin [12]) where ǫ ijk is the permutation symbol. Then, introducing new variables Z = cof F , w = det F and setting Ξ = (F, Z, w), we can use (1) together with (2) to derive the enlarged system and Φ(F ) = (F, cof F, det F ). Further, time discretization gives the variational problem , and the admissible set involves the "additional" constraints In this article, we consider the variational approximation scheme in a new class of admissible mappings, in function classes stemming from quasiconformal analysis, and derive the Euler-Lagrange equations in the cases of smaller regularity (p ≥ 3), finite distortion (|F | 3 ≤ Mw) and nonnegative Jacobian (w ≥ 0) requirements. Recall that a mapping f : Ω → R n is called the mapping with finite distortion, f ∈ F D(Ω), if f ∈ W 1 1,loc (Ω), J(x, f ) ≥ 0 almost everywhere (henceforth abbreviated as a.e.) in Ω and in Ω (see for example [9]). We also note that the problem of the approximation preserving the constraint det F > 0 is still open, except for the very special case of radial elastodynamics [10]. This condition on the deformation gradient is necessary to ensure that the mappings representing motion are orientation-preserving i.e. that the deformations are physical.
For the sake of simplicity we will work with periodic boundary conditions, i.e. the domain Ω is taken to be a three dimensional torus. Consider the stored energy function W : Ω × M 3 → R with the following properties: (H1) Polyconvexity: there exists a convex C 2 -function G : holds.
(H4) There is a constant C > 0 such that the inequality
Lemma 1. The admissible set A is nonempty.
In particular, if F 0 is a differential then so is F , and, thus, there exists the mapping y ∈ W 1 p (Ω) such that ∂ α y i = F iα . Consider the minimization problem for the functional Furthermore, if G is a strictly convex function then the minimizer (v, F, Z, w) ∈ A is unique.
The proof of Theorem 1 is based on the next theorem.
Theorem 2. Let {(v n , F n , Z n , w n )} n∈N ⊂ A and S = sup n∈N I(v n , F n , Z n , w n ) < ∞. Then there exist (v, F, Z, w) ∈ X and a subsequence (v µ , F µ , Z µ , w µ ) such that Moreover, (v, F, Z, w) ∈ A and This statement can be proven by applying the techniques and methods of papers [7,11,13].
We will now show that the minimizer of (6) over the admissible set A satisfies the weak form of the system of equations (4). To derive the Euler-Lagrange equations, we assume that the minimizer (v, F, Z, w) meets w(x) ≥ γ > 0 a.e. in Ω.
We fix "direction" θ = (θ 1 , θ 2 , For One can readily see that such variations fulfill the conditions (5) since columns of the matrix cof F 0 are divergence-free vector fields. Additional requirements on θ allow us to conclude that Ξ ε = (v ε , F ε , Z ε , w ε ) belongs to the admissible set A.
Furthermore, using the mean value theorem and Lebesgue's dominated convergence theorem we find that h∂ α θ i g iα (F, Z, w; F 0 ) dx for ε * ∈ [0, ε]. The last equality is the weak form of (4a). To apply the dominated convergence theorem we use the integrability properties of g iα derived from hypothesis (H4) and Young's inequality [7]. In the case p > 4, following to [7], we find the mapping y : (0, ∞) × Ω → R 3 , y ∈ W 1 ∞ ([0, ∞]; L 2 ) ∩ L ∞ ([0, ∞]; W 1 p ) such that the conditions ∂ t y = v, D x y = F , cof D x y = Z, det D x y = w are fulfilled. Moreover, they satisfy the weak form of the additional conservation laws (2) and y is the measure-valued solution of the system (3).