Form-finding and analysis of hyperelastic tensegrity structures using unconstrained nonlinear programming

Abstract

This study presents a method for form-finding and analysis of hyperelastic tensegrity structures based on a special strut finite element and unconstrained nonlinear programming. The strut element can function as a hyperelastic truss element with an initial cut in its undeformed length or as a strut element that shows constant force irrespectively of its nodal displacements. For the hyperelastic strut element, the invariants of the Right Cauchy-Green deformation tensor are written in terms of the element’s nodal displacements and the cut in the element’s undeformed length. The structure’s total potential energy is expressed as function of its nodal displacements and the cuts in the elements’ undeformed lengths. The minimization of this function is a nonlinear programming problem where the displacements are the unknowns. The form-finding procedure is performed by a static analysis where the stiffness matrix maybe singular along the path to equilibrium without causing convergence problems. The mathematical model includes the element’s cross-sectional deformation while the element moves in space, fully modelling its three-dimensional character. The constraint for incompressibility is satisfied exactly, eliminating the need for a penalty or augmented Lagrangian method.