Plane shear waves under a periodic boundary disturbance in a saturated granular medium (Q1808401)

The paper describes the plane shear wave propagation in a saturated granular medium. The problem is governed by two essentially nonlinear equations for velocity \(v\) and stress \(\sigma\), \(\partial_t v= \partial_x \sigma\), \(\partial_t\sigma= a\partial_x v+b|\partial_x v|\), where \(a\) and \(b\) are constants. The solution is located in the domain \(\{x,t\geq 0\}\) and obeys initial and boundary values \(v(x,0)=0\), \(\sigma (x,0)= \sigma_0 < 0\), \(v(0,t)= f(t)\), where \(f(t)\) is a periodic signal whose restriction over the period gives a square wave with two values. Such problems occur in the theory of liquefaction of soils. The authors derive an asymptotic solutions in the case when \(\partial_xv\) is of definite sign. It is shown that this solution saturates away from the boundary. The authors also prove existence and uniqueness theorems, and present some numerical computations which confirm the results.