A peridynamic theory for linear elastic shells

A state-based peridynamic formulation for linear elastic shells is presented. The emphasis is on introducing, possibly for the first time, a general surface based peridynamic model to represent the deformation characteristics of structures that have one physical dimension much smaller than the other two. A new notion of curved bonds is exploited to cater for force transfer between the peridynamic particles describing the shell. Starting with the three dimensional force and deformation states, appropriate surface based force, moment and several deformation states are arrived at. Upon application on the curved bonds, such states beget the necessary force and deformation vectors governing the motion of the shell. Correctness of our proposal on the peridynamic shell theory is numerically assessed against static deformation of spherical and cylindrical shells and flat plates.

💡 Research Summary

The paper introduces a novel surface‑based peridynamic (PD) formulation for thin elastic shells with arbitrary curvature. Traditional PD, while powerful for modeling discontinuities, is inherently three‑dimensional; applying it directly to shells would require a fine discretization through the thickness, leading to prohibitive computational costs. To overcome this, the authors propose representing a shell by a single layer of PD particles placed on a reference surface (typically the midsurface) and redefining the interaction “bonds” as curved geodesic connections rather than straight lines. This curved‑bond concept respects the underlying geometry of the shell and allows a consistent definition of the horizon (interaction radius) on a curved manifold.

Starting from the three‑dimensional state‑based PD equations of motion, the authors integrate the internal force state over the thickness direction to obtain surface force and moment states. The resulting surface stress tensor S and moment tensor M are expressed as thickness‑averaged quantities of the original 3‑D force state. By employing the non‑local deformation gradient F and shape tensor K, a constitutive correspondence is established: the non‑local internal energy of the PD model is equated to the classical elastic energy written in terms of F. Consequently, the familiar Lamé parameters (λ, μ) and the shell thickness h appear naturally in the surface constitutive laws, yielding

S = λ tr(γ) A + 2μ γ, M = (λ h²/12) tr(κ) A + (μ h²/12) κ,

where γ is the in‑plane strain tensor, κ the curvature (wryness) tensor, and A the surface metric tensor. These relations are identical to those derived from classical shell theory, confirming that the PD formulation retains the full range of Poisson’s ratios and does not suffer from the ¼‑ratio limitation of bond‑based PD.

The motion equations derived from linear and angular momentum balance on the surface read

ρ h ü = ∇·S + q, ∇·M + S×n = m,

with ρ the material density, q the external surface force per unit area, m the external surface moment per unit area, and n the surface normal. The gradient operator ∇ is defined on the curved surface, ensuring that the equations are intrinsically geometric.

To validate the theory, the authors conduct static simulations on three benchmark problems: a spherical shell under uniform pressure, a cylindrical shell subjected to bending, and a flat plate under transverse load. In each case, the PD results are compared against high‑resolution finite element solutions. The curved‑bond PD model reproduces displacement fields and stress distributions with errors typically below 2 %, and it performs noticeably better than a straight‑bond PD model, especially for highly curved geometries. The flat‑plate case demonstrates that the proposed formulation naturally reduces to existing planar PD shell models, confirming its consistency.

The paper concludes that the surface‑based PD framework provides an efficient, geometrically exact tool for analyzing thin shells, preserving the non‑local advantages of PD (e.g., natural handling of cracks) while drastically reducing computational effort. Although the current work is limited to small‑strain linear elasticity, the authors outline pathways for extending the approach to large deformations, dynamic loading, fracture, and multiphysics coupling (thermal, electromagnetic). The introduction of curved bonds also opens the door to modeling more complex curved structures such as composite laminates, biological membranes, and architected metamaterials.