Hypo-Elastic Shear

Abstract

Part I reproduces the lecture given at the meeting of the Society of Rheology and furnishes a nonmathematical introduction to the theory of hypo-elasticity. Hypo-elasticity is a smooth, simple theory of elastic response based on time rates. For small strains it agrees with the classical linear theory of elasticity. To determine stress-strain relations for large deformation is a mathematical problem, the answer to which varies from one special case to another. Simple shear is taken as an example. Here it turns out that hypo-elastic materials may soften or stiffen in shear, depending on the value of a dimensionless constant which has no effect when the strain is small. For bodies which soften, a theoretical prediction of ``hypo-elastic yield'' is obtained. Part II concerns a new special type of hypo-elastic body in some ways more general, in other ways more special than that considered in Part I. According to this theory, yield of the von Mises type appears to follow if the stress intensity is sufficiently great. The equations of this theory are solved for the case of simple shear. It is shown that if von Mises yield occurs, hypo-elastic yield must occur at a lesser stress. For large values of a certain parameter, von Mises yield is imaginary and only hypo-elastic yield occurs. For moderate values of the parameter, hypo-elastic yield appears as primary yield, with von Mises yield as secondary yield at infinite strain. For small values of the parameter, hypo-elastic yield and von Mises yield are indistinguishable, and the stress-strain curve is similar to the idealized forms assumed at the outset in the conventional Prandtl-Reuss theory.