[Paper Review] Variational problems of nonlinear elasticity theory in certain classes of mappings with finite distortion

Anastasia Molchanova, S. K. Vodopyanov|arXiv (Cornell University)|Aug 27, 2015

Elasticity and Material Modeling26 references3 citations

TL;DR

This paper establishes the existence of minimizers for variational problems in nonlinear elasticity by introducing a new class of admissible deformations—mappings in $ W^1_n(\Omega) $ with finite distortion and integrable distortion characteristic $ M(x) \in L_s(\Omega) $, $ s > n-1 $. Under polyconvexity and coercivity of the stored-energy function, the authors prove weak lower semicontinuity and existence of a solution via compensated compactness and Mazur's lemma, extending Ball's framework to less restrictive summability and growth conditions.

ABSTRACT

We study the problem of minimizing the functional $$ I(φ)=\int\limits_Ω W(x,Dφ)\,dx $$ on a new class of mappings. We relax summability conditions for admissible deformations to $φ\in W^1_n(Ω)$ and growth conditions on the integrand $W(x,F)$. To compensate for that, we impose the finite distortion condition and the condition $\frac{|Dφ(x)|^n}{J(x,φ)} \leq M(x) \in L_{s}(Ω)$, $s>n-1$, on the characteristic of distortion. On assuming that the integrand $W(x,F)$ is polyconvex and coercive, we obtain an~existence theorem for the problem of minimizing the functional $I(φ)$ on a new family of admissible deformations. KEYWORDS: functional minimization problem, nonlinear elasticity, mapping with finite distortion, polyconvexity.

Motivation & Objective

• To extend the existence theory of minimizers in nonlinear elasticity beyond classical $ W^{1,p} $ spaces with $ p > n $.
• To relax the coercivity and summability conditions on the stored-energy function and deformation gradients.
• To incorporate the finite distortion condition and integrability of the distortion characteristic $ M(x) \in L_s(\Omega) $, $ s > n-1 $, as a replacement for stronger integrability assumptions.
• To prove existence of a minimizer in a new class of admissible deformations that includes mappings with potentially singular Jacobians but controlled distortion.
• To establish weak lower semicontinuity of the energy functional under these relaxed conditions using polyconvexity and compactness arguments.

Proposed method

• Define a new class of admissible deformations $ \mathcal{A} $ consisting of mappings $ \varphi \in W^1_n(\Omega) $ with finite distortion and $ \frac{|D\varphi(x)|^n}{J(x,\varphi)} \leq M(x) \in L_s(\Omega) $, $ s > n-1 $.
• Impose polyconvexity of the stored-energy function $ W(x,F) $, ensuring it can be represented as a convex function of $ F $, $ \operatorname{Adj}F $, and $ \det F $.
• Use coercivity in the form $ W(x,F) \geq \alpha(\|F\|^p + \|\operatorname{Adj}F\|^q + (\det F)^r) + g(x) $, with $ p > n-1 $, $ q \geq p/(p-1) $, $ r > 1 $, and $ g \in L_1(\Omega) $.
• Apply Mazur's lemma to extract a subsequence of minimizing mappings whose convex combinations converge weakly in $ W^1_n \times L_{n/(n-1)} \times L_r $.
• Leverage Fatou's lemma and the convexity of the representing function $ G $ to prove sequential weak lower semicontinuity of the energy functional.
• Establish that the limit mapping $ \varphi_0 $ satisfies $ J(x,\varphi_0) > 0 $ a.e. and is a homeomorphism under boundary conditions, using the distortion control and injectivity arguments.

Experimental results

Research questions

• RQ1 Can the existence of minimizers in nonlinear elasticity be established under weaker integrability and growth conditions on the stored-energy function?
• RQ2 Does the finite distortion condition combined with $ M(x) \in L_s(\Omega) $, $ s > n-1 $, suffice to control the Jacobian and ensure existence of a solution?
• RQ3 Can polyconvexity and coercivity be used to prove weak lower semicontinuity of the energy functional in this generalized setting?
• RQ4 Is the solution mapping a homeomorphism when the boundary data is a homeomorphism, even if the coercivity condition is relaxed?
• RQ5 Can the classical existence framework of Ball be extended to mappings in $ W^1_n(\Omega) $ without requiring $ p > n $?

Key findings

• The existence of a minimizer $ \varphi_0 \in \mathcal{A} $ is established for the functional $ I(\varphi) = \int_\Omega W(x,D\varphi)\,dx $ under polyconvexity and coercivity of $ W $, even when $ p = n $.
• The distortion characteristic $ \frac{|D\varphi|^n}{J(x,\varphi)} \leq M(x) \in L_s(\Omega) $ with $ s > n-1 $ replaces stronger integrability assumptions on $ D\varphi $, enabling the use of $ W^1_n $-mappings.
• Weak lower semicontinuity of the energy functional is proven via Fatou's lemma and convexity of the representing function $ G $, ensuring $ I(\varphi_0) \leq \varliminf_{k\to\infty} I(\varphi_k) $.
• The limit mapping $ \varphi_0 $ satisfies $ J(x,\varphi_0) > 0 $ a.e. in $ \Omega $, and under boundary conditions given by a homeomorphism, $ \varphi_0 $ is a homeomorphism.
• The result applies to Ogden-type materials with $ W_1(F) $ satisfying $ W_1(F) \geq \alpha\|F\|^p + \|\operatorname{Adj}F\|^q + c(\det F)^r + d(\det F)^{-s} $, even when $ p = n $.
• The framework extends Ball's existence theory to cases where $ W(F) \to \infty $ as $ \det F \to 0_+ $ is not required, showing existence is still possible under the new distortion control.