Phase field approach to mode-I fracture by introducing an eigen strain tensor: General theory

A thermodynamically consistent phase field theory for fracture is developed. An alternative approach to describe the reduction in the slope of the stress-strain curve and the reduction of the stresses to zero during damage is utilized by introducing transformation strain instead of the degradation function. The main requirements for the thermodynamic potential are presented and used to present the general energy terms. The analysis of the equilibrium solution is formulated which describes the desired stress-strain curves and for which the infinitesimal damage produces an infinitesimal change in the equilibrium elastic modulus. Analytical analysis of the equations is performed to calibrate phase field parameters for a general case. The explicit form of the Ginzburg-Landau equation is derived based on which the evolution of the order parameter is obtained. The importance of the analysis of the thermodynamic potential in terms of stress-strain curves is shown. The developed theory includes a broad spectrum of the shapes of stress-strain relationships.