Factors that control the angle of shear bands in geodynamic numerical models of brittle deformation

Numerical models of brittle deformation on geological timescales typically use a pressure-dependent (Mohr–Coulomb or Drucker–Prager) plastic flow law to simulate plastic failure. Despite its widespread usage in geodynamic models of lithospheric deformation, however, certain aspects of such plasticity models remain poorly understood. One of the most prominent questions in this respect is: what are the factors that control the angle of the resulting shear bands? Recent theoretical work suggest that both Roscoe (45°), Coulomb angles (45 +/− ϕ/2, where ϕ is the angle of internal friction) and Arthur angles (45° +/− (ϕ + Ψ/4) where Ψ is the dilation angle), as well as all intermediate angles are possible. Published numerical models, however, show a large range of shear band angles with some codes favoring Arthur angles, whereas others yield Coulomb angles.In order to understand what causes the differences between the various numerical models, here I perform systematic numerical simulations of shear localization around an inclusion of given length scale. Both numerical (element type), geometrical and rheological (viscoplastic versus viscoelastoplastic) effects are studied. Results indicate that the main factor, controlling shear band angle, is the non-dimensional ratio between the length scale of the heterogeneity d and the size of the numerical mesh Δx. Coulomb angles are observed only in cases where the inclusion is resolved well (d/Δx > 5–10), and in which it is located sufficiently far from the boundary of the box. In most other cases, either Arthur or Roscoe orientations are observed. If heterogeneities are one element in size, Coulomb angles are thus unlikely to develop irrespective of the employed numerical resolution.Whereas differences in element types and rheology do have consequences for the maximum obtainable strain rates inside the shear bands, they only have a minor effect on shear band angles. Shear bands, initiated from random noise or from interactions of shear bands with model boundaries or other shear bands, result in stress heterogeneities with dimensionless length scales d/Δx ~ 1–2. Such shear bands are thus expected to form Roscoe or Arthur orientations, consistent with the findings in previous numerical models.