[Paper Review] Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity

Anastasia Molchanova, S. K. Vodopyanov|arXiv (Cornell University)|Apr 26, 2017

Analytic and geometric function theory34 references9 citations

TL;DR

This paper establishes conditions under which weakly differentiable mappings with finite distortion in nonlinear elasticity are injective almost everywhere, leveraging boundedness of pullback operators and polyconvex energy functions. The key result shows that under suitable coercivity and distortion bounds, minimizers of the energy functional are homeomorphisms, ensuring no material interpenetration.

ABSTRACT

We show that a sufficient condition for the weak limit of a sequence of $W^1_q$-homeomorphisms with finite distortion to be almost everywhere injective for $q \geq n-1$, can be stated by means of composition operators. Applying this result, we study nonlinear elasticity problems with respect to these new classes of mappings. Furthermore, we impose loose growth conditions on the stored-energy function for the class of $W^1_n$-homeomorphisms with finite distortion and integrable inner as well as outer distortion coefficients.

Motivation & Objective

• To establish injectivity almost everywhere for weakly differentiable mappings in nonlinear elasticity with finite distortion.
• To extend Ball’s existence theory for energy-minimizing deformations by ensuring global injectivity without assuming global invertibility a priori.
• To characterize conditions under which the inverse of a deformation belongs to a Sobolev space, ensuring regularity of the inverse mapping.
• To link the boundedness of the pullback operator $\varphi^*: L^1_p(\Omega') \to L^1_q(\Omega)$ to injectivity and finite distortion properties.
• To unify geometric conditions (quasi-isometric boundaries) with analytical conditions on the stored-energy function for injectivity.

Proposed method

• Utilizes polyconvexity of the stored-energy function $W(x,F)$ to ensure sequential weak lower semicontinuity of the energy functional.
• Applies coercivity inequalities involving $|F|^p$, $|\operatorname{Adj}F|^q$, $|\det F|^r$, and $|\det F|^{-m}$ to control blow-up near zero Jacobian.
• Imposes a finite distortion condition: $K_O(x,\varphi) = \frac{|D\varphi(x)|^3}{J(x,\varphi)} \leq K(x) < \infty$ a.e., ensuring no degenerate distortion.
• Establishes boundedness of the pullback operator $\varphi^*: L^1_p(\Omega') \to L^1_q(\Omega)$ as a key analytical tool for injectivity.
• Relies on the equivalence between quasi-isometric boundary and Lipschitz boundary in $\mathbb{R}^3$ to ensure geometric regularity.
• Uses Whitney-type extension theorems and intrinsic metric control to relate the geometry of domains to Sobolev regularity of inverses.

Experimental results

Research questions

• RQ1 Under what conditions on the stored-energy function $W$ and the distortion coefficient $K_O$ is a weakly differentiable deformation $\varphi$ injective almost everywhere?
• RQ2 How does the boundedness of the pullback operator $\varphi^*: L^1_p(\Omega') \to L^1_q(\Omega)$ imply injectivity of $\varphi$?
• RQ3 What geometric and analytic conditions on the domain $\Omega$ and the boundary $\partial\Omega$ ensure that the inverse of a deformation lies in $W^1_\sigma(\Omega')$?
• RQ4 Can the coercivity condition $W(x,F) \geq \alpha(|F|^p + |\operatorname{Adj}F|^q + |\det F|^r + |\det F|^{-m}) + g(x)$ guarantee injectivity when combined with finite distortion?
• RQ5 How are quasi-isometric boundaries related to the existence of bounded extension operators and the injectivity of deformations?

Key findings

• A deformation $\varphi$ minimizing the energy functional $I(\varphi) = \int_\Omega W(x,D\varphi)\,dx$ is a homeomorphism from $\Omega$ onto $\varphi(\Omega)$ if $W$ satisfies the coercivity condition (6) with $p>3$, $q>3$, $r>1$, $m>\frac{2q}{q-3}$, and $\sigma = \frac{q(1+m)}{q+m} > 3$.
• The inverse $\varphi^{-1}$ belongs to $W^1_\sigma(\overline{\Omega'})$ with $\sigma > 3$, ensuring sufficient regularity of the inverse mapping.
• Finite distortion is characterized by $K_O(x,\varphi) = \frac{|D\varphi(x)|^3}{J(x,\varphi)} < \infty$ a.e., which is essential for controlling the distortion of volume elements.
• The boundedness of the pullback operator $\varphi^*: L^1_p(\Omega') \to L^1_q(\Omega)$ is equivalent to the existence of a constant $C$ such that $\|\varphi^*u\|_{W^{1}_q(\Omega)} \leq C\|u\|_{W^1_p(\Omega')}$ for all $u \in W^1_p(\Omega')$.
• Quasi-isometric boundary mappings are locally bi-Lipschitz, and domains with quasi-isometric boundaries are equivalent to domains with Lipschitz boundaries in $\mathbb{R}^3$.
• The intrinsic metric $d_{\varphi(B)}(u,v)$ is comparable to the Euclidean metric $|u-v|$, which ensures the existence of bounded extension operators and supports the injectivity result.