Failure Mechanics—Part II: The Central and Decisive Role of Graphene in Defining the Elastic and Failure Properties for all Isotropic Materials

Abstract

Continuing from Part I (Christensen, 2014, “Failure Mechanics—Part I: The Coordination Between Elasticity Theory and Failure Theory for all Isotropic Materials,â€‌ ASME J. Appl. Mech., 81(8), p. 081001), the relationship between elastic energy and failure specification is further developed. Part I established the coordination of failure theory with elasticity theory, but subject to one overriding assumption: that the values of the involved Poisson's ratios always be nonnegative. The present work derives the physical proof that, contrary to fairly common belief, Poisson's ratio must always be nonnegative. It can never be negative for homogeneous and isotropic materials. This is accomplished by first probing the reduced twodimensional (2D) elasticity problem appropriate to graphene, then generalizing to threedimensional (3D) conditions. The nanomechanics analysis of graphene provides the key to the entire development. Other aspects of failure theory are also examined and concluded positively. Failure theory as unified with elasticity theory is thus completed, finalized, and fundamentally validated.