On a stress resultant geometrically exact shell model. Part IV: Variable thickness shells with through-the-thickness stretching

Abstract

This paper in concerned with the extension of the shell theory and numerical analysis presented in Part I, II and III to include finite thickness stretch and initial variable thickness. These effects play a significant role in problems involving finite membrane strains, contact, concentrated surface loads and delamination (in composite shells). We show that a direct numerical implementation of the standard single extensible director shell model circumvents the need for rotational updates, but exhibits numerical ill-conditioning in the thin shell limit. A modified formulation obtained via a multiplicative split of the director field into an extensible and inextensible part is presented, which involves only a trivial modification of the weak form of the equilibrium equations considered in Part III, and leads to a perfectly well-conditioned formulation in the thin-shell limit. In sharp contrast with previous attempts in the context of the degenerated solid approach, the thickness stretch is an independent field, not a dependent variable updated iteratively via the plane stress condition. With regard to numerical implementation, an exact update procedure which automatically ensures that the thickness stretch remains positive is presented. For the present theory, standard displacement models would exhibit ‘locking’ in the incompressible limit as a result of the essentially three-dimensional character of the constitutive equations. A mixed formulation is described which circumvents this difficulty. Numerical examples are presented that illustrate the effects of the thickness stretch, the performance of the proposed mixed interpolation, and the well-conditioned response exhibited by the present approach in the thin-shell (inextensible director) limit.