  In this paper we provide a formal derivation of both the Camassa-Holm
equation and the fractional Camassa-Holm equation for the propagation of
small-but-finite amplitude long waves in a nonlocally and nonlinearly elastic
medium. We first show that the equation of motion for the nonlocally and
nonlinearly elastic medium reduces to the improved Boussinesq equation for a
particular choice of the kernel function appearing in the integral-type
constitutive relation. We then derive the Camassa-Holm equation from the
improved Boussinesq equation using an asymptotic expansion valid as
nonlinearity and dispersion parameters tend to zero independently. Our approach
follows mainly the standard techniques used widely in the literature to derive
the Camassa-Holm equation for shallow water waves. The case where the Fourier
transform of the kernel function has fractional powers is also considered and
the fractional Camassa-Holm equation is derived using the asymptotic expansion
technique.
