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.
Peridynamics is a nonlocal extension of classical continuum mechanics introduced in [59], that naturally admits discontinuities such as fractures and cracks. Unlike classical elasticity, which relies on derivatives of deformations or displacements and hence becomes ill-posed near singularities, peridynamics formulates the governing equations of continuum mechanics in a nonlocal form. Thus, the energy is based on pairwise force interactions between material points within a defined interaction range known as the peridynamic horizon.
Peridynamics is typically classified into two categories. Bond-based peridynamics, introduced in [59], restricts force interactions to pairwise bonds between individual material points, where the force magnitude depends only on the relative displacement between two points. State-based peridynamics, proposed in [61], generalizes this framework by allowing interaction forces to depend on the deformation state of all bonds connected to the endpoints, not merely the bond stretch itself. The second approach allows for more realistic modeling of material behavior. However, the mathematical structure of state-based models is more complex, particularly in nonlinear and variational settings. For an overview of recent developments in theory, as well as their numerical implementations, the reader is referred to the reviews [28,29,35,40,50] and references therein.
Nonlocal hyperelasticity. This work addresses a specific class of state-based peridynamics called nonlocal hyperelasticity. In this framework, the material response is governed by an energy functional that mirrors the classical hyperelastic setting but replaces the local gradient ∇ by a nonlocal counterpart D ρ defined by
|y -x| ⊗ y -x |y -x| ρ(y -x) dy for x ∈ R n , with a suitable radial interaction kernel ρ : R n \ {0} → [0, ∞) and u : R n → R n . Precisely, we consider stored energies of the form
where u describes the deformation of the reference configuration Ω ⊂ R n , and W : R n×n → [0, ∞) denotes the material energy density. These models were already proposed in [61, Section 18], but have only recently been subjected to a rigorous variational analysis. The first work to investigate the Sobolev spaces associated with the nonlocal gradients was [56] (see also [21,57]), in which the Riesz fractional gradient was considered that arises by choosing the interaction kernel ρ = | • | -(n+s-1) for some fractional exponent s ∈ (0, 1). There, the authors established fractional Poincaré inequalities and compact embeddings, and addressed existence of minimizers for convex energy densities. These existence results were later generalized to polyconvex and quasiconvex energy densities [6,38], which are typically employed in applications of hyperelasticity. To better align with peridynamic models, the extension to finite-horizon fractional gradients was considered in [7,24]. More recently, a theory for general compactly supported interaction kernels ρ was developed [11], a setting that we also adopt in this paper. Beyond the results on the function spaces and the existence of minimizers, it has furthermore been shown in [25] (see also [9,45]) that nonlocal models Γ-converge to local ones as the horizon vanishes, in particular for families of kernels ρ δ := δ -n ρ(•/δ) for δ > 0.
Classical dimension reduction. We are particularly interested in the study of thin structures, which are known to show fundamentally different responses to external loading compared to bulk materials.
A natural way to analytically capture these effects within classical nonlinear hyperelasticity is to study the asymptotic behavior of three-dimensional energy functionals on domains whose thickness tends to zero and to identify the resulting lower-dimensional models. Depending on the underlying scaling, one obtains qualitatively different theories in the limit. Early rigorous results in this direction include limits that yield string [1] and membrane models [42]. The subsequent analysis of thin structures was strongly shaped by the seminal paper [33], which led, among other things, to a systematic derivation of plate theories in several scaling regimes [33,34] and the generalization to non-Euclidean elasticity [43], as well as the development of rod theories [46,47,52,53]. In the last two decades, variational dimension reduction has seen extensions in several different directions. A recurring theme is the incorporation of non-convex differential constraints, motivated for instance by incompressibility [22,23,30,31], non-selfinterpenetration of matter and (global) invertibility [12,41,49,54], or A-free fields [36,37] with applications in elasticity or micromagnetism. As the latter indicates, thin-structure limits have also been pursued well beyond the purely elastic setting. Further extensions include elastoplasticity [26,27,32], magnetoelasticity [16,17], liquid crystal elastomer sheets [63], or fracture mechanics [2,3,55], to name just a few. Much of the existing
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