Scaling Laws in Mechanics of Failure

Abstract

Scaling laws are the most fundamental aspect of every physical theory. Recently, the problem of scaling law and size effect in the theories of structural failure has received considerable attention, particularly with regard to distributed damage and nonlinear fracture behavior. The paper presents a rigorous mathematical analysis of scaling in various types of failure theories in structural mechanics. First it is shown that the scaling law is a power law if, and only if, a characteristic dimension is absent. For all the theories in which the failure condition is expressed in terms of stress or strain only, including elasticity with a strength limit, plasticity, and continuum damage mechanics, the nominal strength of the structure is shown to be independent of its size. For linear‐elastic fracture mechanics, in which the failure criterion is expressed in terms of energy per unit area, the scaling law for the nominal strength is shown to be (size)−1/2, provided that the cracks in structures of different size...