×

Transverse cylindrical simple waves and shock waves in elastic non- conductors. (English) Zbl 0734.73014

Summary: A modulated simple wave theory is developed for transverse cylindrical motions of an unstrained incompressible isotropic elastic non-conductor with the aid of a modified version of J. K. Hunter and J. B. Keller’s “weakly nonlinear geometrical optics” method [Commun. Pure Appl. Math. 36, 547-569 (1983; Zbl 0547.35070)]. This theory is then used to construct shock wave solutions using the shock-fitting method. The evolution law thus derived shows that the effect of nonlinearity on the evolution of transverse cylindrical shock waves is cumulative, but that by the time it becomes most pronounced, geometrical spreading has already attenuated the shock amplitude until it is exponentially small. It follows that the linear theory gives satisfactory results for the propagation of transverse cylindrical shock waves. This is in sharp contrast to the situation for plane transverse shock waves whose amplitudes decay in the presence of material nonlinearities whilst the linear theory predicts constant amplitudes. Where it is present, geometrical spreading would appear to be a more potent decay mechanism than material nonlinearity.

MSC:

74M20 Impact in solid mechanics
74J99 Waves in solid mechanics
35L67 Shocks and singularities for hyperbolic equations
78A05 Geometric optics
74B20 Nonlinear elasticity
35Q60 PDEs in connection with optics and electromagnetic theory
PDFBibTeX XMLCite
Full Text: DOI