An asymptotic solution of an initial value problem for a non-linear viscoelastic rod

Stress wave propagation in a one-dimensional materai, described by the non-linear constitutive law

$\text{?σ}\text{?t}\text{? E}\text{?ε}\text{?t}\text{+ kσ ? E'g3 + βε}{}^2\text{] = 0}$

, is analyzed by a two-time variable perturbation method. Secular terms are “cast out” by employing special orthogonality conditions. The resulting expansion reveals a “mode coupling”, due to the non-linearity, and displays the significance of the three parameters present in the constitutive law. Numerical values for the parameters are assumed and the resulting displacements computed.