  In this article, we describe the mathematical formulation and the numerical
implementation of an effective parametrization of the viscous anisotropy of
orthorhombic materials produced by crystallographic preferred orientations (CPO
or texture), which can be integrated into 3D geodynamic and materials science
codes. Here, the approach is applied to characterize the texture-induced
viscous anisotropy of olivine polycrystals, the main constituent of the Earth's
upper mantle. The parameterization is based on the Hill (1948) orthotropic
yield criterion. The coefficients of the Hill yield surface are calibrated
based on numerical tests performed using the second order Viscoplastic
Self-consistent (SO-VPSC) model. The parametrization was implemented in a 3D
thermo-mechanical finite-element code developed to model large-scale
geodynamical flows, in the form of a Maxwell rheology combining isotropic
elastic and anisotropic non-linear viscous behaviors. The implementation was
validated by comparison with results of the analytical solution and of the
SO-VPSC model for simple shear and axial compression of a homogeneous
anisotropic material. An application designed to examine the effect of
texture-induced viscous anisotropy on the reactivation of mantle shear zones in
continental plates highlights unexpected couplings between localized
deformation controlled by variations in the orientation and intensity of the
olivine texture in the mantle and the mechanical behavior of the
elasto-viscoplastic overlying crust. Importantly, the computational time only
increases by a factor 2-3 with respect to the classic isotropic Maxwell
viscoelastic rheology.
