Convexity of internal energy of cubic crystal deformed to orthorhombic structure

Abstract

A generalized set of strain variablesq ^N_r , has been defined to develop the expression for a generalized set of second order and third-order elastic moduliC ^N_rs andC ^N_rst for a cubic crystal deformed to orthorhombic structure. The HessainC ^N_rs δq_rδq_s andC ^N_rst δq_rδq_sδq_t (r=1, 2……6; summation convention) are calculated in the new variables and compared withG-strength andS-strength, for both positive and negative loading environment.

The convexity of the internal energy relative to various choice of strain measure is examined considering up to third degree terms in the internal energy expression. The computational results forbcc iron is presented according to the new moduli. The stable ranges thus obtained for iron under hydrostatic compressive and tensile stresses is found to generate the classical stable range, green-stable range and stretch-stable range as the specific cases. However,bcc iron does not seem to follow any conventional stable ranges under hydrostatic compression, where the present generalized stable range is found satisfactory.