The effects of friction between crack faces on nonlinear wave propagation in microcracked media

Abstract

The effective behaviour of a composite consisting of an elastic matrix containing a dilute array of aligned, randomly positioned, ‘closing’, penny-shaped cracks is considered. In Smyshlyaev and Willis (1996) and Woolfries (1998), the faces of the cracks were taken to be frictionless. A similar formulation is studied here when the crack faces permit no sliding when closed, i.e. they have infinite friction. These conditions are written down mathematically and their implications explored under “low-frequency” assumptions. The resulting effective constitutive relations, depending on a field of internal variables, contain conditional inequalities. The tangential components contain a convolution term which describes the effect of the cracks' history on the present tangential stress. The Fourier transform of this term was given in Smyshlyaev and Willis (1994). Here the inverse transform is carried out. The mean wave is taken to be at an angle to the common crack normal. The behaviour of tangential components now interacts with the rest of the system, which was not the case in Smyshlyaev and Willis (1996) and Woolfries (1998) when the mean wave was parallel to the crack normal. The resulting equations on effective variables are reduced by evaluating the Green's function for the system. This allows a simple numerical method to solve initial- or boundary-value problems. The case of no friction may easily be compared to the case of infinite friction by relaxing the condition of no slip, and using the variational inequality formulation of Smyshlyaev and Willis (1996). Numerical routines are carried out for this case also.