Cohesive Membranes under determinant constraints

1. Problem Statement and Context

The paper addresses the mathematical derivation of reduced (dimensional) models for elastic membranes that undergo fracture, subject to specific determinant constraints on the deformation gradient.

• Physical Setting: The authors consider a thin 3D domain $\Sigma_\rho = \Sigma \times (-\rho/2, \rho/2)$ with thickness $\rho \to 0$. The material behavior is modeled by a free-discontinuity functional $F_\rho$ involving a bulk stored-energy density $W$ and a surface energy density $\psi$ that depends on the jump amplitude (cohesive fracture).
• The Constraints: The study focuses on two physically critical regimes where the standard dimension reduction techniques fail due to the lack of uniform growth conditions:
  1. Non-interpenetration: The deformation must preserve orientation ($\det \nabla u > 0$).
  2. Incompressibility: The deformation must preserve volume ($\det \nabla u = 1$).
• The Gap: While dimension reduction for membranes without constraints or with Griffith-type surface energy (independent of jump size) is well-established, the case of jump-dependent surface energy (cohesive fracture) under determinant constraints had remained an open problem. The primary difficulty lies in the fact that the determinant constraints prevent the bulk energy density $W$ from satisfying standard uniform $p$-growth conditions, rendering classical integral representation theorems inapplicable.

2. Methodology

The authors employ $\Gamma$-convergence to derive the asymptotic behavior of the rescaled energy functionals $\rho^{-1}G_\rho$ as $\rho \to 0$. The proof is split into two main parts: the lower bound ($\Gamma$-liminf) and the upper bound ($\Gamma$-limsup).

A. Lower Bound ($\Gamma$-liminf)

The lower bound follows a strategy adapted from previous works (specifically Braides and Fonseca), but with crucial modifications to handle the constraints:

• Compactness: Using the coercivity of the surface energy (Assumption B3), they prove that the limit deformation $u$ is independent of the thickness variable $x_3$.
• Normal Vector Analysis: They demonstrate that the third component of the normal vector to the jump set vanishes ($(\nu_u)_3 = 0$) almost everywhere.
• Energy Absorption: A key technical point is showing that the surface energy remainder arising from the third component of the normal can be absorbed due to the specific growth coupling in the surface energy assumptions.

B. Upper Bound ($\Gamma$-limsup) and Recovery Sequences

This is the paper's main technical contribution. Constructing a recovery sequence that simultaneously satisfies the determinant constraint and optimizes the surface energy is highly non-trivial. The authors develop a two-step construction:

1. Approximation of the Domain and Function:

   • They first map the cracked domain (with a jump set $J_u$) onto a Lipschitz domain using smooth diffeomorphisms.
   • Non-interpenetration case: They use piecewise affine approximations with maximal-rank gradients (Theorem 3.1).
   • Incompressible case: They develop a new higher-regularity approximation result (Proposition 3.6) for $GSBV^p$ functions. This involves mapping the cracked domain to a Lipschitz domain via smooth diffeomorphisms constructed using convolutions with variable kernels (Lemma 3.4). This ensures the approximants are $W^{k,\infty}$ away from the jump set and maintain a controlled determinant.

2. Optimization of Surface Energy via Isometries:

   • To minimize the reduced surface energy $\psi_0$, the authors construct a family of $C^\infty$ diffeomorphisms $f_{j,\rho}$ converging to the identity.
   • These maps act as local isometries near the jump set, rotating the normal vector to the jump set to align with the direction that minimizes the surface energy (via the infimum definition of $\psi_0$).
   • Incompressible Correction: For the incompressible case, an additional constraint $\det \nabla f_{j,\rho} = 1$ is enforced. This is achieved via an ODE-based correction (Lemma 5.2) that adjusts the third component of the map to ensure volume preservation while maintaining the desired rotation properties.

3. Relaxation:

   • The authors establish that the $\Gamma$-limit is bounded by an auxiliary functional $G^w_0$.
   • They then prove that $G^w_0$ coincides with the relaxed functional $G^R_0$ (involving the rank-one convex envelope $RW_0$ and BV-elliptic envelope $B\psi_0$) by utilizing density results for functions with maximal-rank gradients and specific relaxation arguments.

3. Key Contributions

• First Dimension Reduction with Constraints: This is the first rigorous derivation of reduced models for membranes with jump-dependent surface energy under determinant constraints (both non-interpenetration and incompressibility).
• New Approximation Theorems:
  • Extension of density theorems to $GSBV^p$ functions with maximal-rank constraints.
  • Development of a smooth approximation result for the incompressible case that preserves the determinant constraint away from the jump set.
• Novel Construction of Recovery Sequences: The combination of variable kernel convolutions to smooth the domain and ODE-based corrections to enforce incompressibility allows for the construction of recovery sequences that were previously unattainable.
• Handling Non-Uniform Growth: The paper successfully bypasses the lack of uniform $p$-growth in the bulk energy (caused by the constraints) by relying on the specific structure of the surface energy and the geometry of the jump set.

4. Main Results

Under the assumptions of $p$-coercivity and specific continuity/growth conditions on $W$ and $\psi$:

• The rescaled functionals $\rho^{-1}G_\rho$ $\Gamma$-converge to a 2D functional $G_0$ defined on the mid-surface $\Sigma$.
• The limit functional is given by:
  $$G_0(u) = \int_\Sigma Q(RW_0)(\nabla_\alpha u) \, dx + \int_{J_u \cap \Sigma} B\psi_0([u], \nu_u) \, d\mathcal{H}^1$$
  where:
  • $u$ is independent of $x_3$.
  • $Q$ denotes the quasiconvex envelope.
  • $R$ denotes the rank-one convex envelope.
  • $B$ denotes the BV-elliptic envelope.
  • $W_0$ and $\psi_0$ are the reduced densities obtained by minimizing over the third column of the gradient and the third component of the normal, respectively.

5. Significance

• Theoretical Advancement: The work bridges a significant gap in variational fracture mechanics, providing a rigorous foundation for modeling cohesive fracture in thin structures where physical constraints (like volume preservation) are paramount.
• Methodological Innovation: The techniques developed, particularly the use of variable kernel convolutions for smoothing cracked domains and the ODE-based enforcement of incompressibility, offer a powerful toolkit for future problems involving geometric constraints and free discontinuities.
• Future Applications: The authors suggest these methods could be extended to other geometries (e.g., shells) or weaker growth assumptions, potentially impacting the modeling of biological membranes, soft robotics, and advanced composite materials.