  We propose a decomposition of constitutive relations into crack-driving and
persistent portions, specifically designed for materials with
anisotropic/orthotropic behavior in the phase field approach to fracture to
account for the tension-compression asymmetry. This decomposition follows a
variational framework, satisfying the orthogonality condition for anisotropic
materials. This implies that the present model can be applied to arbitrary
anisotropic elastic behavior in a three-dimensional setting. On this basis, we
generalize two existing models for tension-compression asymmetry in isotropic
materials, namely the volumetric-deviatoric model and the no-tension model,
towards materials with anisotropic nature. Two benchmark problems, single
notched tensile shear tests, are used to study the performance of the present
model. The results can retain the anisotropic constitutive behavior and the
tension-compression asymmetry in the crack response, and are qualitatively in
accordance with the expected behavior for orthotropic materials. Furthermore,
to study the direction of maximum energy dissipation, we modify the surface
integral based energy release computation, $G_\theta$, to account only for the
crack-driving energy. The computed energies with our proposed modifications
predict the fracture propagation direction correctly compared with the standard
G-theta method.
