Explicit secular equations for surface waves in monoclinic materials with the symmetry plane at x1=0, x2=0 or x3=0

The traditional way of deriving the secular equation for surface waves propagating in the direction of the x(1)-axis in an anisotropic elastic half-space x(2) greater than or equal to 0 is to find a general steady-state solution for the displacement that vanishes at x(2) = infinity. This involves the computation of three Stroh eigenvalues p and three associated eigenvectors a that depend on the surface wave speed v. Three more vectors b that depend on p and a provide the stress. The secular equation for v is then obtained by the vanishing of the surface traction at x(2) = 0. One could find the vectors b without finding the vectors a, but in either case this approach provides an explicit expression of the secular equation only for isotropic and orthotropic materials. A direct derivation of the explicit secular equation without finding p, a, b was given for orthotropic materials by Mozhaev using the first integrals and by Ting using the amplitudes of the surface displacement. Destrade has recently employed the first integrals to obtain an explicit secular equation for monoclinic materials with the symmetry plane at x(3) = 0. However, the first-integral approach does not seem to lead to an explicit secular equation for monoclinic materials with the symmetry plane at x(1) = 0 or x(2) = 0. We present here a new approach that not only recovers Destrade's secular equation easily, but also provides an explicit expression of p and the vectors a, b without solving the quartic equation for p. They are needed in the surface wave solution. Employing this new approach, explicit secular equations for monoclinic materials with the symmetry plane at x(1) = 0 or x(2) = 0 are presented. The secular equations are given in terms of the elastic stiffness as well as the elastic compliance.