Scaling laws in mechanics of failure

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 sizes are geometrically similar. When the failure condition involves both the stress (or strain) and the energy per unit area, which is typical of quasibrittle materials, the scaling law represents a gradual transition from the strength theory, which is asymptotically approached for very small sizes, to LEFM, which is asymptotically approached for very large sizes. The size effect described by Weibull statistical theory of random material strength is also considered, and the reasons for its inapplicability to quasibrittle materials are explained. Finally, floating elastic plates with large bending fractures and a negligible process-zone size are shown to exhibit an anomalous scaling law, such that the nominal strength is proportional to (size)^-2/5, and another anomalous size effect of the type (size)^-3/8, pertinent to one recent theory of borehole breakout, is pointed out.