Thermodynamically consistent and scale-dependent phase field approach for crack propagation allowing for surface stresses

A thermodynamically-consistent phase field approach for crack propagation which includes the following novel features is presented. (1) Scale dependency was included by relating the length scale to the number of cohesive interatomic planes at the crack tip. Because of this, the developed theory is applicable from the atomistic to the macroscopic scales. (2) The surface stresses (tension) are introduced by employing some geometrical nonlinearities even in small strain theory. They produce multiple contributions to the Ginzburg-Landau equation for crack propagation. (3) Crack propagation in the region with compressive closing stresses is eliminated by employing a stress-state-dependent kinetic coefficient in the Ginzburg-Landau equation. (4) The importance of 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. The finite element method is utilized to solve the complete system of crack phase field and mechanics equations. The effect of the above novel features is analyzed numerically for various model problems.