Rake dependent rate and state friction

Abstract

The sliding direction within the fault plane, conventionally defined as the rake, changes with time during earthquakes. To aid studies of this process, the formalism for rate and state friction is extended by using tensors to include anisotropy of the fault surface from sliding with variable rake. The evolution of the state variable ψ and porosity within a fault zone are considered to be decoupled from anisotropic damage, which is represented by a tensor Lijkl, which has the usual symmetries of elastic constants. Letting the strain rate from sliding within the fault zone be ε̇ij, a permitted evolution equation for L is ∂Lijkl/∂t = ε̇ij/ε̇kl/(|ε̇|ε 2) - |ε̇|/ε1 where t is time, and ε2 and ε1 are strains to create new anisotropic damage and to undo preexisting anisotropic damage, and the second invariant of the strain rate tensor is |ε̇|2 ≡ 1/2ε̇ijε̇ij. The instantaneous coefficient of friction for unidirectional sliding is μ0 + a ln (ε̇/ε̇0) + b ln (ψ/ψ0) where μ0 is the steady state coefficient of friction at shear strain rate ε̇0, a and b are small constants, ε̇ is the shear strain rate, ψ is a state variable that represents damage, and ψ0 a normalizing factor. The tensor flow law is then ε̇ij = ε̇X[D-1ijklτkl/|τ|] ψ0/ψ]b/a exp[(τres - μ0ΔP)/aΔP] where ε̇X is a normalization strain rate, Dijkl ≡ δikδjl + δilδjk - Lijkl, where δij is the Kronecker delta, ΔP is the effective normal traction, |τ|2 ≡ 1/2τijτij is the second invariant of the shear traction, and τres is resolved shear traction in the sliding direction. Transient strengthening of a fault zone followed by gradual weakening occurs when the rake is suddenly changed.