Fu, Y. B.; Scott, N. H. Transverse cylindrical simple waves and shock waves in elastic non- conductors. (English) Zbl 0734.73014 Int. J. Solids Struct. 27, No. 5, 547-563 (1991). 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. Cited in 2 Documents 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 Keywords:approximations; perturbations; evolution equation for amplitude of second-order discontinuity; third-order discontinuity; transverse cylindrical motions; unstrained incompressible isotropic elastic non- conductor; weakly nonlinear geometrical optics; shock wave solutions; shock-fitting method; evolution law Citations:Zbl 0691.73017; Zbl 0547.35070 PDFBibTeX XMLCite \textit{Y. B. Fu} and \textit{N. H. Scott}, Int. J. Solids Struct. 27, No. 5, 547--563 (1991; Zbl 0734.73014) Full Text: DOI