RAKE DEPENDENT RATE AND STATE FRICTION

The sliding direction within the fault plane, conventionally defined a s the rake, changes with time during earthquakes. To aid studies of th is process, the formalism for rate and state friction is extended by u sing tensors to include anisotropy of the fault surface from sliding w ith variable rake. The evolution of the state variable psi and porosit y within a fault zone are considered to be decoupled from anisotropic damage, which is represented by a tensor L-ijkl, which has the usual s ymmetries of elastic constants. Letting the strain rate from sliding w ithin the fault zone be (epsilon) over dot(ij), a permitted evolution equation for L is partial derivative L-ijkl/partial derivative t = (ep silon) over dot(ij)(epsilon) over dot(kl)/(\(epsilon) over dot\epsilon (2)) - \(epsilon) over dot\/epsilon(1) where t is time, and epsilon(2) and epsilon(1) are strains to create new anisotropic damage and to un do preexisting anisotropic damage, and the second invariant of the str ain rate tensor is \(epsilon) over dot\(2) = 1/2(epsilon) over dot(ij) (epsilon)(ij). The instantaneous coefficient of friction for unidirect ional sliding is mu(0) + a ln ((epsilon) over dot/(epsilon) over dot(0 )) + b ln(psi/psi(0)) where mu(0) is the steady state coefficient of f riction at shear strain rate (epsilon)(0), a and b are small constants , (epsilon) over dot is the shear strain rate, psi is a state variable that represents damage, and psi(0) a normalizing factor. The tensor f low law is then (epsilon) over dot(ij) = (epsilon) over dot(X)[D(ijkl) (-1)tau(kl)/\tau\][psi(0)/psi](b/a) exp [(tau(res) - mu(0) Delta P)/a Delta P] where (epsilon) over dot(X) is a normalization strain rate, D -ijkl = delta(ik)delta(il) + delta(il)delta(jk) - L-ijkl, where delta( ij) is the Kronecker delta, Delta P is the effective normal traction, \tau\(2) = 1/2 tau(ij)tau(ij) is the second invariant of the shear tra ction, and tau(res) is resolved shear traction in the sliding directio n. Transient strengthening of a fault zone followed by gradual weakeni ng occurs when the rake is suddenly changed.