Features of the reflection of waves with strong discontinuities from thin compressible obstacles

Dynamical problem for an incompressible laminated cylinder with helical anisotropy. Part 2. Numerical analysis

“…Since the stresses are functions not only of u but also of ∂ ∂ u r, it is clear that the uniform convergence in u does not always guarantee that the convergence in ∂ ∂ u r and, hence, in stresses is also uniform. This is clearly true for the wave solutions with strong discontinuities always observed under impact loads of type (34) with discontinuities on the leading edge [6,7]. Nevertheless, as follows from Figs.…”

“…For incompressible composite materials, the values of the coefficients of transverse deformation υ ij are specified, according to (7), as follows: υ ϕ x = 0 125 . , υ ϕr = 0 5 .…”

“…This is why, for the analyzed period of time, the input signal (34) undergoes a great number of reflections across the thickness of the two-layer cylinder, both from free surfaces and from the interface of the layers. For a load of type (34) with discontinuity on the leading edge, this leads to the formation of a saw-tooth distribution of stresses both in time and across the thickness of the shell [6,7]. In compressible materials, the sound velocity is much lower and, hence, the indicated effect is much weaker (curves 3 in Figs.…”

Solutions of dynamic problems for incompressible and slightly compressible helically orthotropic non-uniform thick-walled cylinders

“…When condition (12) is satisfied, it follows from relations (2), (11) and the second equation of (9) that (13) Substituting expressions (11) and (13) into the equation of motion (1), we have…”

“…Now the stress is found from relation (13), while x and x are found from the last two equations of (9). The strains are found from relations (2) and (11)…”