Beyond Kirchhoff-Love: A Volumetric Approach to Thin-Shell Mechanics

We present a thin-shell energy formulation rigorously derived from first volumetric principles of 3D solid mechanics, overcoming limitations of prior methods that rely on geometric heuristics or restrictive assumptions such as those in Kirchhoff-Love shells. Our approach supports arbitrary nonlinear isotropic materials, reproduces volumetric behavior exactly under small deformations, and remains accurate for large deformations. Kirchhoff-Love thin shells constrain material lines normal to the mid-surface to remain straight, normal, and unstretched, suppressing through-thickness relaxation. As a result, they fail to reproduce the behavior of the corresponding 3D volumetric material such as Poisson contraction or bending-induced through-thickness nonlinearities, even under small deformations. We analytically derive the leading small-deformation through-thickness modes of an initially flat thin shell from volumetric elasticity: a linear normal mode that scales the local thickness, a quadratic normal mode that captures bending-induced thickness variation, and higher modes. We use the lowest modes as a compact kinematic ansatz for large-deformation thin-shell simulation, with a scalar thickness variable ρ and an optional, statically condensable bending amplitude ζ. Starting from an arbitrary isotropic hyperelastic energy density ψ , we derive a closed-form shell energy by expanding the deformation gradient through the thickness and analytically integrating the volumetric energy. The resulting energy separates stretching and bending contributions, avoids both volumetric meshing and through-thickness quadrature, and depends only on mid-surface quantities (the first fundamental form and the shape operator), together with ρ and ζ. We provide robust implementation-ready formulas for the energy, gradient, and Hessian, including systematic treatment of removable singularities that arise when principal stretches coincide. Experiments show close agreement with volumetric simulations across diverse materials and scenarios, greatly surpassing KL shells, while retaining the computational efficiency of thin shells.