Wave Propagation in a Heteromodular Elastic Medium

Rocks, soils, and oil and tar sands are complex materials containing pores, cracks, and other defects. If this is the case, their constitutive behaviour can be nonlinear and stress-dependent, which implies that loading changes the properties of the material. If materials react differently to compression and tension, this can have a strong influence on the propagation of seismic waves. A possible approach to describe such materials is heteromodular elastic theory: a piece-wise linear theory with different elastic moduli depending on the stress state. The formulation of such a theory for the 2D and 3D cases is a difficult task. Even for the 1D case, there are only a few dynamical problems solved. One needs a simple model problem to imagine how the signal behaves when passing through such a medium. We consider a 1D problem with a small heteromodularity, obtain its analytical (asymptotical) solution for the case of a suddenly applied harmonic load, and compare it with numerical results. This gives us a general idea of the character of the wave propagation we may expect in such media, and the technique to apply in more complex cases.