Revisiting Mindlin's theory with regard to a gradient extended phase‐field model for fracture

The application of generalized continuum mechanics is rapidly increasing in different fields of science and engineering. In the literature, there are several theories extending the classical first-order continuum mechanics formulation to include size-effects [1]. One approach is the strain gradient theory with the intrinsic features of regularizing singular stress fields occurring, e.g., near crack tips. It is crucial to realize that using this theory, the strain energy density is still localized around the crack tip, but does not exhibit any signs of a singularity. Therefore, these models seem to be appropriate choices for studying cracks in mechanical problems. Over the past several years, the phase-field method has gathered considerable popularity in the computational mechanics community, in particular in the field of fracture mechanics [2]. Recently, the authors have shown that integrating the strain gradient theory into the phase-field fracture framework is likely to improve the quality of the final results due to the inherent non-singular nature of this theory [3]. In the present work, we will focus on a general formulation of the first strain gradient theory. To this end, the homogenization approach introduced in Ref. [4] is employed. It is based on a series of systematic finite element simulations using different loading cases to determine the equivalent material coefficients on the macro-scale (i.e., for a strain gradient elastic material) by taking the underlying micro-structure into account