Finite Deformation of Materials with an Ensemble of Defects

Abstract

The theory of large deformations developed here is closely related to continuum mechanics but it differs in several major respects, especially in considering the deformation associated with various types of physical behavior, making it possible to synthesize a general approach to formulating constitutive laws. One goal is to derive general concepts of strain, strain rate, stress, and stress rate that are somewhat more physics-based than in most standard works on continuum mechanics, and to demonstrate some new relations between these quantities. With these concepts it is possible to develop a generalized principle of superposition of strain rates (GSSR) that accounts for damage as well as plastic flow. The traditional superposition of strain rates allows for addition of elastic and plastic strain rates and is commonly thought to be valid only for small strains. The GSSR allows us to compute deformations involving plastic flow and, in addition, brittle failure, fragmentation, high-pressure effects and other types of behavior as necessary, and the theory is valid for arbitrarily large deformations. In fact, GSSR is derived from more basic ideas and has broader application than the standard superposition of strain rates. The physical basis for calculations of complex material response is developed in a separate report. The implementation into the SCRAM computer program is documented separately. The polar decomposition theorem is taken as a starting point for the theory of large deformation, an approach somewhat different from that usually taken in continuum mechanics. Two sets of orthogonal axes are distinguished, space axes that are fixed in ambient space, and polar axes that are related to material deformation. This clarifies several concepts; for example, it is shown that the Signorini and Green-St. Venant strains are actually measures of the same physical entity, one in space axes and the other in polar axes. It follows that they are not competing measures, as is often implied in traditional continuum mechanics. It also follows that Piola stress is a measure in polar axes, while Cauchy stress is a measure in space axes. Another consequence of polar decomposition is a proof that vorticity is not a measure of the rate of material rotation (as is often stated in the hydrodynamics literature) but that they are related. This allows us to develop an exact approach to computing rates of tensor quantities, called polar rates, that account for material rotation in an exact way. This leads to a simple relation between Signorini strain rate and stretching (the symmetric part of the velocity gradient). It also follows that the polar stress rate is the appropriate measure for the rate of change of Cauchy stress, and that the more traditional stress rate of Zaremba, Jaumann, and Noll is only an approximation, valid at small strains. Examples are described for materials undergoing simple shear, vortex motion, and torsion.