The paper considers the problem of torsion of an infinite viscoelastic layer of finite thickness by a round rigid washer in the formulation of the theory of creep of inhomogeneously ageing media. Solving the problem reduces to an integral equation with a kernel generated by Weber-Sonin integral with Bessel functions of the first kind of index 1. A quadrature rule was derived for the evaluation of integrals with the mentioned kernel. By the method of mechanical quadrature, numerical analysis of the stated problem was carried out, and the dependence of the angle of twist upon time, stipulated by the property of creep of the layer material, was determined.We discuss dispersion relations for an infinite elastic rod considering some non-classical models of the continuum. Among the latter, we analyse classical and non-classical theories of bars, stress and strain gradient elasticity, nonlocal (integraltype) elasticity and peridynamics. A comparison of the dispersion relations for these models is given.