A phase-field fracture model based on strain gradient elasticity

•The conventional phase-field models for fracture exhibit stress singularities.•Two phase-field fracture formulations based on gradient elasticity are introduced.•The proposed gradient models successfully remove the singular response.•The fourth-order model significantly reduces the mesh sensitivity of the simulation. The conventional phase-field crack propagation models utilize the classical Cauchy continuum theory to approximate the elastic energy contribution to the total potential energy. It is widely known that the stress fields in the classical theory suffer from singularities at the crack tips. Therefore, the inherent purpose of this contribution is to demonstrate that the solution of fracture problems on the basis of a phase-field approach alone, is not sufficient to overcome the mentioned problem. On the contrary, it is demonstrated that the underlying continuum theory has to be adapted. To this end, the problems associated with the use of the classical continuum theory in the context of fracture analysis are investigated in a first step. Thereafter, two different phase-field fracture models based on the theory of strain gradient elasticity are introduced. It is shown that the proposed gradient models using the second- and the fourth-order phase-field formulations, respectively, improve the performance of the classical models by removing the singular response. Moreover, the numerical results indicate that the proposed fourth-order model is superior to the second-order one in that it provides more realistic solution characteristics. Another advantage of the fourth-order formulation is the significant reduction of the mesh sensitivity in numerical simulations. With this contribution, it has been demonstrated that future approaches directed towards phase-field fracture modelling need to take the effects of stress singularities into account to achieve reliable results.