Theory of Deformation: Part 1, Kinematics

摘要

The purpose of this report is to develop the kinematic basis for a theory of deformation of materials with defects. One specific objective is to derive clear concepts of strain, strain-rate, stress and stress rate, and to demonstrate some new relations between measures of these quantities. A related objective is to clarify the concepts of strain, strain rate and stretching, so that the principle of superposition of strain rates can be developed in a systematic way in subsequent reports. This principle allows us to compute deformations involving both plastic flow and fragmentation. The theory of large deformation is addressed in a systematic manner with the polar decomposition theorem as a starting point. Two sets of orthogonal axes are distinguished, space axes that are fixed in ambient space, and polar axes that are related to material deformation. It is shown that in elasticity, Piola stress is a measure in polar axes. In addition, a special approach to computing rates of tensor quantities, called polar rates, is derived from several viewpoints. This allows us to compute rates in a manner that accounts for material rotation in an exact way. One consequence of this is a simple relation between Signorini strain rate and stretching, the symmetric part of the velocity gradient. A second result is to show that the polar stress rate is the appropriate measure of 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. 20 refs., 10 figs. (ERA citation 13:009063)