Derivation of variational membrane models in the context of anisotropic nonlocal hyperelasticity

1. Problem Statement

The paper addresses the mathematical challenge of deriving effective lower-dimensional models (specifically membrane models) for thin structures made of nonlocal hyperelastic materials.

• Context: Classical continuum mechanics often fails near singularities (e.g., cracks). Peridynamics and nonlocal hyperelasticity offer an alternative by replacing local gradients with nonlocal operators based on pairwise interactions within a finite "horizon."
• The Gap: While dimension reduction (thinning a 3D body to a 2D surface) is well-understood for classical local elasticity, rigorous variational derivations for finite-horizon nonlocal models are scarce. Existing literature often relies on heuristic reductions or focuses on infinite-horizon (fractional) models.
• The Specific Challenge: When a 3D domain with thickness $\varepsilon$ is rescaled to a fixed domain, a radially symmetric interaction horizon in the physical space transforms into an anisotropic (ellipsoidal) horizon in the rescaled space. Standard nonlocal theories assume isotropic (spherical) horizons. The authors must develop a theory for anisotropic nonlocal gradients where interaction radii in different directions can vanish independently (e.g., the out-of-plane radius vanishing as $\varepsilon \to 0$).

2. Methodology

The authors employ $\Gamma$-convergence, a variational framework for studying the asymptotic behavior of energy functionals, combined with a novel translation mechanism between nonlocal and local spaces.

A. Anisotropic Nonlocal Gradients

The core mathematical innovation is the definition of nonlocal gradients with ellipsoidal interaction neighborhoods.

• Definition: Instead of a standard ball $B_1(0)$, the interaction region is an ellipsoid defined by a diagonal matrix $T_\delta = \text{diag}(\bar{\delta}, \bar{\delta}, \delta_3)$, where $\bar{\delta}$ is the in-plane radius and $\delta_3$ is the out-of-plane radius.
• Operator: The nonlocal gradient $D_{\rho, \delta} u$ is defined via a convolution with a kernel $Q_{\rho, \delta}$, generalizing the isotropic case.
• Function Spaces: They introduce the space $H^p_{\rho, \delta}(\mathcal{O}; \mathbb{R}^3)$, consisting of functions with finite $L^p$ norm and finite nonlocal gradient, vanishing outside the expanded domain $\mathcal{O}_\delta$.

B. The Translation Mechanism

A critical tool developed in Section 2 is an isomorphism between the nonlocal space $H^p_{\rho, \delta}$ and the classical Sobolev space $W^{1,p}$.

• Operator $Q_{\rho, \delta}$: Maps a nonlocal function $u$ to a local function $v = Q_{\rho, \delta} u$ such that $\nabla v = D_{\rho, \delta} u$.
• Operator $P_{\rho, \delta}$: The inverse map, allowing the authors to pull back local estimates to the nonlocal setting.
• Uniformity: Crucially, the bounds for these operators are uniform with respect to the horizon parameters $\delta$, even as components of $\delta$ vanish (degenerate cases). This allows the reduction of nonlocal proofs to well-known local proofs.

C. Handling Degeneracy and Density

• Leibniz Rule: The authors establish a Leibniz rule for anisotropic gradients, carefully handling cases where interaction radii vanish (e.g., $\delta_3 = 0$).
• Density: They prove that smooth functions ($C^\infty_c$) are dense in $H^p_{\rho, \delta}$ even for degenerate horizons, which is essential for approximation arguments.
• Kernel Reduction: They show that if a function is independent of the out-of-plane variable, the 3D anisotropic gradient reduces to a 2D nonlocal gradient with a modified kernel $\bar{\rho}$.

3. Key Contributions

1. Unified Framework for Anisotropy: The paper provides the first rigorous theory for nonlocal gradients with anisotropic, finite horizons that can degenerate (radii vanishing). This bridges the gap between fully nonlocal, partially nonlocal, and purely local models.
2. Dimension Reduction for Finite Horizons: It derives the $\Gamma$-limit of thin-film nonlocal energies, showing that the limit is a 2D membrane energy.
3. Stabilization of Null-Spaces: The authors address a unique feature of nonlocal models: the existence of non-constant functions with zero nonlocal gradient (the space $N^p_{\rho, \delta}$). They introduce a stabilization term to the energy functional to penalize these "invisible" deformations, ensuring strong compactness and well-posedness of minimizers.

4. Main Results

Theorem 3.1: $\Gamma$-Convergence of Thin Films

Let the physical domain be $\Omega_\varepsilon = \omega \times (0, \varepsilon)$ with interaction radii $\delta(\varepsilon) = (\bar{\delta}, \delta_3)$. After rescaling, the energy functionals $I_\varepsilon$ $\Gamma$-converge to a limit functional $I$:
$$I(u) = \begin{cases} \int_\Omega W^{qc}(\bar{D}_{\rho, \delta_0} u) \, dx & \text{if } u \in H^p_{\rho, \delta_0}(\Omega), (D_{\rho, \delta_0} u)|_\Omega e_3 = 0 \\ \infty & \text{otherwise} \end{cases}$$

• Key Insight: The limit energy depends only on the in-plane nonlocal gradient $\bar{D}_{\rho, \delta_0}$. The out-of-plane component of the gradient vanishes in the limit.
• Recovery of Local Models: If the interaction radii vanish ($\delta_0 = 0$), the model recovers the classical local membrane energy involving the standard gradient $\nabla \bar{u}$.
• Compactness: Minimizing sequences converge (up to functions in the null-space $N^p_{\rho, \delta}$) to a limit that is independent of the thickness coordinate $x_3$.

Theorem 3.4: Stabilized $\Gamma$-Convergence

To overcome the issue of non-constant functions with zero energy (which makes minimizers non-unique and convergence weak), the authors add a stabilization term $S_\varepsilon$:
$$S_\varepsilon(u) = \int |u|^p + \iint \frac{|u(\bar{x}, x_3) - u(\bar{x}, x'_3)|^p}{|\varepsilon(x_3 - x'_3)|} \, dx \, dx'$$

• Result: The stabilized functionals $I_\varepsilon + \lambda S_\varepsilon$ $\Gamma$-converge to a limit where the minimizers converge strongly (without needing to subtract null-space functions).
• Limit Functional: The limit includes a penalty term $\lambda \int |u|^p$, ensuring the solution is unique and physically meaningful.

5. Significance and Impact

• Mathematical Rigor: This work moves beyond heuristic asymptotics to provide a rigorous variational derivation of nonlocal membrane models, filling a significant gap in the literature on peridynamics and nonlocal elasticity.
• Bridging Scales: The framework unifies local and nonlocal theories. By allowing interaction radii to vary and vanish, it demonstrates how classical local elasticity emerges as a specific limit of nonlocal models under dimensional reduction.
• Practical Modeling: The derivation of the anisotropic horizon is physically motivated by the geometry of thin films. The results provide a theoretical basis for simulating thin structures (membranes, plates) using nonlocal models without the computational cost of full 3D simulations, while correctly capturing size effects and fracture phenomena that local models miss.
• Handling Null-Spaces: The identification and stabilization of the "invisible" deformation modes (functions with zero nonlocal gradient) is a critical contribution, resolving a major theoretical obstacle in applying nonlocal models to boundary value problems.

In summary, the paper establishes a robust mathematical foundation for analyzing thin nonlocal structures, proving that they converge to effective 2D membrane models governed by anisotropic nonlocal gradients, and providing tools to ensure the well-posedness of the resulting variational problems.