[Paper Review] Relaxation and 3d-2d passage theorems in hyperelasticity

Omar Anza Hafsa, Jean-Philippe Mandallena|arXiv (Cornell University)|Jan 6, 2011

Elasticity and Material Modeling15 references5 citations

TL;DR

This paper establishes relaxation and 3D-2D dimension reduction theorems in hyperelasticity under determinant-type constraints using $γ(\pi)$-convergence and quasiconvexification. It extends Dacorogna's relaxation theorem and Le Dret-Raoul's 3D-2D passage theorem to hyperelastic settings with non-interpenetration and infinite energy for volume compression, proving that quasiconvexification preserves polynomial growth under 'ample' energy densities.

ABSTRACT

We give an overview of relaxation and 3d-2d passage theorems in hyperelasticity in the framework of the multidimensional calculus of variations. Some open questions are addressed. This paper, which is an expanded version of the outline-paper [AHM09b], comes as a companion to [AHM09a].

Motivation & Objective

• To extend Dacorogna's relaxation theorem to hyperelasticity by incorporating determinant-type constraints that enforce non-interpenetration and infinite energy for volume compression.
• To generalize the Le Dret-Raoul 3D-2D passage theorem to settings compatible with hyperelasticity, ensuring consistency with physical principles.
• To introduce and analyze 'ample energy densities'—those whose quasiconvexification has polynomial growth—enabling the extension of relaxation and dimension reduction theorems.
• To resolve open problems in relaxation and dimension reduction under determinant constraints, particularly in the case of strong determinant constraints.
• To provide a rigorous framework using $γ(\pi)$-convergence for the 3D-2D passage in thin structures under hyperelastic constraints.

Proposed method

• Utilizes $γ(\pi)$-convergence to analyze the $γ$-limit of 3D energy functionals as thickness $\varepsilon \to 0$, ensuring convergence to a 2D effective energy.
• Applies quasiconvexification via the quasiconvex envelope $\mathcal{Q}W$ to relax non-convex energy densities while preserving physical constraints.
• Introduces the concept of 'ample energy densities'—those for which the quasiconvexified energy has polynomial growth—enabling extension of classical theorems.
• Employs a parametrized family of test functions $H(\xi, t, a \otimes b)$ to characterize the relaxed energy, combining affine and nonlinear perturbations.
• Uses coercivity and continuity of the auxiliary function $h$ to prove lower semicontinuity and existence of minimizers in the relaxation process.
• Establishes representation formulas for the relaxed functional $\overline{I}(\phi) = \int_\Omega \mathcal{Q}W(\nabla\phi(x))\,dx$ under various determinant constraints.

Experimental results

Research questions

• RQ1 Can Dacorogda's relaxation theorem be extended to hyperelastic settings where the energy density blows up as $\det F \to 0^+$ and $\det F \leq 0$?
• RQ2 Does the Le Dret-Raoul 3D-2D passage theorem hold under strong determinant constraints consistent with non-interpenetration and infinite compressive energy?
• RQ3 Are there conditions under which the quasiconvex envelope $\mathcal{Q}W$ of a non-finite, determinant-constrained energy density retains polynomial growth?
• RQ4 Can the $\Gamma(\pi)$-convergence of 3D energy functionals to a 2D limit be established when the energy density satisfies hyperelastic constraints?
• RQ5 What is the role of 'ample energy densities' in ensuring consistency between relaxation and dimension reduction theorems in hyperelasticity?

Key findings

• The paper proves that the relaxed functional $\overline{I}(\phi)$ equals $\int_\Omega \mathcal{Q}W(\nabla\phi(x))\,dx$ for ample energy densities, even when $W$ is not finite or has singularities.
• For the 'weak-Determinant Constraint' ($\det F \leq 0 \Rightarrow W(F) = +\infty$, $W(F) \to \infty$ as $\det F \to 0^+$), the quasiconvex envelope $\mathcal{Q}W$ is shown to preserve polynomial growth.
• The $\Gamma(\pi)$-convergence of 3D energy functionals $I_\varepsilon$ to a 2D limit is established under the 'strong-Determinant Constraint', confirming consistency with hyperelasticity.
• The auxiliary function $\mathcal{H}(\xi)$, defined via a parametrized infimum over $t \in [0,1]$ and rank-one directions, is proven continuous and coercive, ensuring existence of minimizers.
• The quasiconvex envelope $\mathcal{Q}W$ is represented via a two-scale test function construction involving $\varphi \in W^{1,\infty}(Y)$, with $Y = (0,1)^N$.
• The proof establishes that $\mathcal{H}(\xi) = h(\xi)$ for all $\xi \in \mathbb{M}^{3\times 2}$, where $h$ is the original energy density, under the condition that $h$ is continuous and coercive.