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

Dominik Engl, Anastasia Molchanova, Hidde Schönberger

Abstract

Motivated by the analysis of thin structures, we study the variational dimension reduction of hyperelastic energies involving nonlocal gradients to an effective membrane model. When rescaling the thin domain, isotropic interaction ranges naturally become anisotropic, leading to the development of a theory for anisotropic nonlocal gradients with direction-dependent interaction ranges. Unlike existing nonlocal derivatives with finite horizon, which are defined via interaction kernels supported on balls of positive radius, our formulation is based on ellipsoidal interaction regions whose principal radii may vanish independently. This yields a unified framework that interpolates between fully nonlocal, partially nonlocal, and purely local models. Employing these tools, we present a $Γ$-convergence analysis for the nonlocal thin-film energies. The limit functional retains the structural form of the classical membrane energy, and the classical local model is recovered precisely when all interaction radii vanish.

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

Abstract

Motivated by the analysis of thin structures, we study the variational dimension reduction of hyperelastic energies involving nonlocal gradients to an effective membrane model. When rescaling the thin domain, isotropic interaction ranges naturally become anisotropic, leading to the development of a theory for anisotropic nonlocal gradients with direction-dependent interaction ranges. Unlike existing nonlocal derivatives with finite horizon, which are defined via interaction kernels supported on balls of positive radius, our formulation is based on ellipsoidal interaction regions whose principal radii may vanish independently. This yields a unified framework that interpolates between fully nonlocal, partially nonlocal, and purely local models. Employing these tools, we present a -convergence analysis for the nonlocal thin-film energies. The limit functional retains the structural form of the classical membrane energy, and the classical local model is recovered precisely when all interaction radii vanish.
Paper Structure (4 sections, 16 theorems, 147 equations, 1 figure)

This paper contains 4 sections, 16 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Anisotropic nonlocal gradients
  4. Thin films in nonlocal hyperelasticity

Key Result

Lemma 2.2

Let $\varphi\in C^\infty({\mathbb{R}}^3)$, $\delta\in[0,1]^2$, and $Q_{\rho}$ is given by Qrho, then the function ${\mathcal{Q}}_{\rho,\delta} \varphi$ given by satisfies ${\mathcal{Q}}_{\rho,\delta} \varphi\in C^\infty({\mathbb{R}}^3)$. Moreover, the corresponding (nonlocal) gradient and divergence satisfy the integration by parts formula

Figures (1)

  • Figure 1: An illustration of an ellipsoid with principal radii $\bar{\delta},\bar{\delta},\delta_3>0$.

Theorems & Definitions (43)

  • Example 2.1: Truncated fractional kernel
  • Lemma 2.2: Anisotropic nonlocal gradients
  • proof
  • Definition 2.3: Weak anisotropic nonlocal gradients
  • Remark 2.4: Anisotropic nonlocal gradients of affine functions
  • Remark 2.5: Anisotropic nonlocal gradients with positive horizons
  • Remark 2.6: On the set $N^{\rho,p}_{\delta}(O;{\mathbb{R}}^3)$
  • Lemma 2.7: Dimensionally reduced (non)local gradient
  • proof
  • Remark 2.8
  • ...and 33 more