  We address brittle fracture in anisotropic materials featuring two-fold and
four-fold symmetric fracture toughness. For these two classes, we develop two
variational phase-field models based on the family of regularizations proposed
by Focardi (Focardi, M. On the variational approximation of free-discontinuity
problems in the vectorial case. Math. Models Methods App. Sci., 11:663{684,
2001), for which Gamma-convergence results hold. Since both models are of
second order, as opposed to the previously available fourth-order models for
four-fold symmetric fracture toughness, they do not require basis functions of
C1-continuity nor mixed variational principles for finite element
discretization. For the four-fold symmetric formulation we show that the
standard quadratic degradation function is unsuitable and devise a procedure to
derive a suitable one. The performance of the new models is assessed via
several numerical examples that simulate anisotropic fracture under anti-plane
shear loading. For both formulations at fixed displacements (i.e. within an
alternate minimization procedure), we also provide some existence and
uniqueness results for the phase-field solution.
