This chapter is an overview of a theory of a class of nonlinear elastic materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite plastic deformation of a variety of annealed metals. Research by Bell and his associates published since about 1979 is reviewed, and Bell's empirically deduced rules and laboratory data are compared with analytical results obtained within the context of nonlinear elasticity theory. First, Bell's empirical characterization of the constrained response of polycrystalline annealed metals in finite plastic strain is sketched. A few kinematical consequences of Bell's constraint, an outline of the constitutive theory developed to characterize the isoteopic, nonlinearly elastic response of Bell materials, and theoretical results that lead to Bell's empirical parabolic laws within the structure of isotropic, elastic and hyperelastic Bell constrained materials are presented. The study concludes with discussion of Bell's empirically based incremental theory of plasticity.

Introduction

It is common in technical writing to begin with a sketch of related research assembled to set the stage for the work ahead. But I'm not going to follow the usual path. There is more to this account than just its technical side - teachers and students, colleagues and associates, family and friends, places and events, life and death - the ingredients of the human side of the story. A reader who feels no interest in this sort of personal, anecdotal retrospection, however, will find immediate relief and surely suffer no loss in skipping ahead to Section 2.3 where Bell's important experiments and his internal material constraint are introduced. We'll return to this shortly.