led to a number of multi-physics applications, which include cracking in piezoelectric solids [@Miehe2010_jmps], fluid-driven fracture propagation [@Mikelic2015; @Miehe2015_jmps], thermal shock-induced cracks [@Bourdin2014] and fragmentation of battery electrode particles [@Miehe2015_ijnme]. This underscores the need for quantitative accuracy with regard to the fracture model, particularly in the case of crack nucleation which is often the critical failure mechanism for many such applications.

The remainder of this paper is structured as follows: We begin in Section 2 with a discussion of important fundamental concepts underlying the phase-field approach as well as our motivation for pursuing the current research direction. Specifics regarding the formulation used in the present work are given in Section 3 which also includes important details with regard to numerics; in particular for the case where the phase-field evolution equation is nonlinear, we outline a linearization scheme based on a truncated Taylor series approximation that avoids the implementation of nested loops in the solution scheme. The following two sections contain the main novelties of the current work: section 4 begins with a numerical example demonstrating the apparent contradiction that is often seen between simulation results and what is expected from the $\Gamma$-convergence property of the fracture phase-field model. This is 