An Information Theoretic Argument on the Form of Damage Accumulation in Solids

An approach to modeling failure that is inspired by two experimentally observed facts is presented. These observations are: (1) cracks grow as the end result of a irreversible, dissipative process, and (2) fracture has an inherent lengthscale, timescale, and/or spatial hierarchy influenced by the (possibly dynamically changing) microstructural state. The second of these facts enables one to view the seemingly deterministic cracks observed at higher levels of hierarchy as resulting from uncertain events at lower-levels of hierarchy associated with microstructural variations. A key mathematical result developed in Information Theory together with the maximum entropy principle of Statistical Mechanics is utilized to derive a form of damage that is "maximally non-committal" about microstructural uncertainty in lower levels of fracture hierarchy. The irrecoverable energy that is expended in the creation of new surfaces or in plastic dissipation is associated with the microstructural damage using continuum thermodynamics and J2 plasticity theory. The formulated result is shown to provide an exponential form of damage accumulation under constant dissipation rate, and a form similar to the popular Smith-Ferrante traction-separation law of cohesive zone models under conditions of decreasing dissipation rate. Finally, the model is validated through comparisons with experimental observations of damage accumulation during cyclic fatigue testing of solder alloys.