Deformations of general parametric shells: Computation and robot experiment

A shell is a body enclosed between two closely spaced and curved surfaces. Classical theory of shells (Timoshenko and Woinowsky-Krieger, 1959; Saada, 1993; and Gould, 1999) assumes a parametrization along the lines of principal curvature on the middle surface of a shell. Such a parametrization, while always existing locally, is not known for many surfaces, and deriving one can be very difficult if not impossible. This paper generalizes the classical strain-displacement equations and strain energy formula to a shell with an arbitrary parametric middle surface. We show that extensional and shearing strains can all be represented in terms of geometric invariants including principal curvatures, principal vectors, and the related directional and covariant derivatives. Computation of strains and strain energy is also described for a general parametrization. The displacement field on a shell is represented as a B-spline surface. By minimization of potential energy, we have simulated deformations of algebraic surfaces under applied loads, and performed experiments on an aluminum soda can and a stretched cloth using a three-fingered Barrett hand. The measured deformations on each object match those in the simulation with good accuracy. The presented work is an initial step in our research on robot grasping of deformable objects.