Transport theorem and material derivatives in the description of crack growth process by thermomechanics

This publication derives mathematical expressions for two particular purposes in the theory of fracture mechanics. The first goal lies in the traditional fracture mechanics (i.e. Griffith & Rice). For this subject an expression that can be used in the derivation of a relationship between the derivative of the potential energy with respect to the crack length is derived. Second goal is the formulation of mathematical preliminaries for an extended theory that can be used in the simulation of the (curved) crack growth process in two-dimensional materials in general. Both investigations utilize the approach in which two coordinate systems are introduced: one is a fixed frame and the other is attached to the crack tip and therefore it translates and rotates with the growing crack. The obtained equations are formulated also for the finite deformation analyses