Orientation of shear bands for a rigid plastic frictional material in simple shear

Abstract

The orientation of shear bands is investigated analytically and numerically for a rigid plastic frictional material in simple shear. The model is based on co-axial flow rule, incompressible deformations and a friction factor which depends on the strain history. Since we are focussing on geological timescales, the influence of elasticity is neglected. Firstly, a linear stability analysis is performed confirming Rice's 1976 assertion [The localization of plastic deformation, in Proceedings of the 14th International Congress on Theoretical and Applied Mechanics, W.T. Koiter, ed., North Holland, Amsterdam, 1976, p.207] that, in the hardening regime, bifurcation is possible at every stage. Orientation of shear bands against the less compressive principal axis lies anywhere between the Roscoe and Coulomb angles, namely between π/4+ψ/2 and π/4 + ϕ/2, where ϕ and ψ are the mobilised angles of friction and dilatancy, respectively (in our study, we assume ψ = 0). The linear stability analysis leaves open the question of which orientation will actually emerge in a boundary value problem that consider all nonlinearities. This question is addressed in a finite element study of simple shear with periodic boundary conditions in the shear direction. Our simulations show temporary inclined shear bands in the hardening regime followed by a persistent horizontal shear band. A sensitivity study with respect to geometric and constitutive parameters indicates that the height of the sample controls the orientation of the inclined shear bands. Finally, we extend our analytical and numerical studies to Cosserat plasticity. It turns out that inclined shear bands are suppressed for large values of the internal length R (narrow bands). The case of a standard continuum is gradually recovered for small R (wide bands).

Keywords:

• finite element method
• simple shear
• plasticity
• shear band
• orientation
• linear stability analysis
• Cosserat theory

Acknowledgements

We would like to acknowledge support from the ARC Discovery Grant DP0985662 and the ongoing support through Auscope/NCRIS. We are also grateful to Cihan Altinay, Joel Fenwick, Lin Gao and Vince Boros of ESSCC at the University of Queensland for their help during the preparation of the manuscript.

Notes

Note

1. Elastic and plastic strain rates do not share the same principal axes whereas viscous and plastic strain rates do.