Influence of elastic material compressibility on parameters of the expanding spherical stress wave. I. Analytical solution to the problem

We investigated the influence of elastic material compressibility on parameters of the expanding spherical stress wave. The material compressibility is represented by Poisson's ratio $ \nu$. The stress wave is generated by pressure created inside the spherical cavity. The isotropic elastic material surrounds this cavity. Analytical closed form formulae determining the dynamical state of mechanical parameters (displacement, particle velocity, strains, stresses, and material density) in the material have been derived. These formulae were obtained for surge pressure $ p(t)=p_0=\text{const }$ inside the cavity. From analysis of these formulae it results that Poisson's ratio $ \nu$ substantially influences the course of material parameters in space and time. All parameters intensively decrease in space together with increase of the Langrangian coordinate $ r$. On the contrary, these parameters oscillate versus time around their static values. These oscil lations decay with a lapse of time. We can mark out two ranges of the parameter $ \nu$ values in which vibrations of the parameters are damped with a different degree. Thus, a decrease in Poisson's ratio in the range $ \nu\leq0.4$ causes an intense decay of oscillation of parameters. On the other hand, in the range $ 0.4<\nu<0.5$, i.e. in quasi-compressible materials the damping of parameters vibrations is very low. In the limiting case when $\nu=0.5$, i.e. in the incompressible material damping vanishes, and the parameters harmonically oscillate around their static values. The abnormal behaviour of the material occurs in the range $ 0.4<\nu\leq0.5$. In this case an insignificant increment of Poisson's ratio causes considerable an increase of the parameters vibration amplitude. The specific influence of Poisson's ratio on the parameters of the expanding spherical stress wave in elastic media is the main result of this paper. As we see it, this fact may be the contributio n supplementing the description of properties of the expanding spherical stress wave in elastic media.