We extend the isotropic non-linear damage rheology model with a scalar damage parameter to a more complex formulation that accounts for anisotropic damage growth under true triaxial loading. The model takes account of both the anisotropy of elastic properties (associated with textural rock structure) and the stress- and damage-induced anisotropy (associated with loading). The scalar, isotropic model is modified by assuming orthotropic symmetry and introducing a second-order damage tensor, the principal values of which describe damage in three orthogonal directions associated with the orientations of the principal loading axes. Different damage components, accumulated under true triaxial loading conditions, allows us to reproduce both stress-strain curves and damage- and stress-induced seismic wave velocity anisotropy. The suggested model generalization includes a non-classical energy term similar to the isotropic non-linear scalar damage model, which allows accounting for the abrupt change in the effective elastic moduli upon stress reversal.

For calibration and verification of the model parameters, we use experimental stress-strain curves from deformation of dry sandstone under both conventional and true triaxial stress conditions. Cubic samples were deformed in three orthogonal directions with independently controlled stress paths. To characterize crack damage, changes in ultrasonic P-wave velocities in the three principal directions were measured, together with the bulk acoustic emission output. The parameters of the developed model were constrained using the conventional triaxial test data, and provided good fits to the stress-strain curves and P-wave velocity variations in the three orthogonal directions. Numerical simulation of the true triaxial test data demonstrates that the anisotropic damage rheology model adequately describes both non-linear stress-strain behavior and P-wave velocity variations in the tested Darley Dale sandstone.

我们将具有标量损伤参数的各向同性非线性损伤流变模型扩展到一个更复杂的公式，该公式考虑了真实三轴载荷下各向异性损伤的增长。该模型考虑了弹性特性的各向异性（与纹理岩石结构有关）以及应力和损伤引起的各向异性（与载荷有关）。通过假设正交各向异性并引入二阶损伤张量来修改标量各向同性模型，该损伤张量的主值描述了与主载荷轴方向相关的三个正交方向上的损伤。在真实的三轴载荷条件下累积的不同损伤分量使我们能够重现应力-应变曲线以及损伤和应力引起的地震波速度各向异性。

为了校准和验证模型参数，我们使用了常规和真实三轴应力条件下干燥砂岩变形产生的实验应力-应变曲线。立方样品在三个正交方向上变形，并具有独立控制的应力路径。为了表征裂纹损伤，测量了超声波在三个主要方向上的P波速度变化以及整体声发射输出。使用常规的三轴测试数据来约束所开发模型的参数，并且可以很好地拟合三个正交方向上的应力-应变曲线和P波速度变化。