  Simulation of frictional contact and shear failure of fractures in fractured
media is of paramount important in computational mechanics. In this work, a
preconditioned mixed-finite element (FE) scheme with Lagrange multipliers is
proposed in the framework of constrained variational principle, which has the
capability to handle frictional contact and slip of the multiple crossing
fractures. The slippage, opening and contact traction on fractures are
calculated by the resulted saddle-point algebraic system. A novel treatment is
devised to guarantee physical solutions at the intersected position of crossing
fractures. A preconditioning technique is introduced to re-scale the resulting
saddle-point algebraic system, to preserve the robustness of the system. An
iteration strategy, namely monolithic-updated contact algorithm, is then
designed to update the two primary unknowns (displacement and Lagrange
multiplier) in one algebraic block. Then, a series of numerical tests is
conducted to study the frictional contact and shear failure of single- and
multi-crossing fractures. Benchmark study is performed to verify the presented
mixed-FE scheme. Two tests with crossing fractures are studied, in which the
slippage and opening can be calculated. The effects of crossing fractures on
the deformation field are observed in the simulation, in which the variation of
slippage, opening and stress intensity factor are analyzed under different
loading conditions.
