Converging shocks in elastic-plastic solids

We present an approximate description of the behavior of an elastic-plastic material processed by a cylindrically or spherically symmetric converging shock, following Whitham's shock dynamics theory. Originally applied with success to various gas dynamics problems, this theory is presently derived for solid media, in both elastic and plastic regimes. The exact solutions of the shock dynamics equations obtained reproduce well the results obtained by high-resolution numerical simulations. The examined constitutive laws share a compressible neo-Hookean structure for the internal energy $e=e_s\left(I_1\right)+e_h(\rho ,\varsigma )$, where $e_s$ accounts for shear through the first invariant of the Cauchy–Green tensor, and $e_h$ represents the hydrostatic contribution as a function of the density $\rho$ and entropy $\varsigma$. In the strong-shock limit, reached as the shock approaches the axis or origin $r=0$, we show that compression effects are dominant over shear deformations. For an isothermal constitutive law, i.e., $e_h=e_h\left(\rho \right)$, with a power-law dependence $e_h\propto {\rho }^{\alpha }$, shock dynamics predicts that for a converging shock located at $r=R\left(t\right)$ at time $t$, the Mach number increases as $M\propto {\left[\log (1/R)\right]}^{\alpha }$, independently of the space index $s$, where $s=2$ in cylindrical geometry and 3 in spherical geometry. An alternative isothermal constitutive law with $p\left(\rho \right)$ of the $\text{arctanh}$ type, which enforces a finite density in the strong-shock limit, leads to $M\propto R^{-(s-1)}$ for strong shocks. A nonisothermal constitutive law, whose hydrostatic part $e_h$ is that of an ideal gas, is also tested, recovering the strong-shock limit $M\propto R^{-(s-1)/n\left(\gamma \right)}$ originally derived by Whitham for perfect gases, where $\gamma$ is inherently related to the maximum compression ratio that the material can reach, $(\gamma +1)/(\gamma -1)$. From these strong-shock limits, we also estimate analytically the density, radial velocity, pressure, and sound speed immediately behind the shock. While the hydrostatic part of the energy essentially commands the strong-shock behavior, the shear modulus and yield stress modify the compression ratio and velocity of the shock far from the axis or origin. A characterization of the elastic-plastic transition in converging shocks, which involves an elastic precursor and a plastic compression region, is finally exposed.

Figure 1

Spherically symmetric ($s=3$) converging shock initially started at $R=R_i$ with $J_i=0.9$ (i.e., $M_i\approx 1.14$) and propagating from left to right into a purely elastic solid medium described by the isothermal constitutive law (8) with polynomial pressure form (13): (a) density radial profiles obtained from the numerical simulation at equally spaced times (dashed lines) and density ratio immediately behind the shock ($r=R{\left(t\right)}^{+}$) given by WSD (solid line); (b) $u+a$-characteristics obtained from numerical simulation (dashed lines) and shock trajectory $r=R\left(t\right)$ vs $t$ obtained from WSD (solid line); (c) shock Mach number $M$ as a function of the shock position $R\left(t\right)$ plotted in a log-log scale, from the simulation (dashed line) and WSD (solid line).

Figure 2

Spherically symmetric ($s=3$) converging shock initially started at $R=R_i$ with $J_i=0.9$ (i.e., $M_i\approx 1.02$) and propagating from left to right into a purely elastic solid medium described by the isothermal constitutive law (8), using the arctanh pressure form (16) with the choice $J_{\infty }=1/6$, $p_0=10\text{GPa,}$ and $\beta =5$. See Fig. 1 for keys.

Figure 3

Spherically symmetric ($s=3$) converging shock initially started at $R=R_i$ with $J_i=0.9$ (i.e., $M_i\approx 1.07$) and propagating from left to right into a purely elastic solid medium described by the nonisothermal constitutive law (19) with $\gamma =1.4$ (i.e., $J_{\infty }=1/6$). See Fig. 1 for keys.

Figure 4

Spherically symmetric ($s=3$) converging shock initially started at $R=R_i$ with $J_i=0.9$ (i.e., $M_i\approx 1.01$) and propagating from left to right into an elastic-plastic solid medium following the the isothermal constitutive law (26) with the polynomial pressure form (13), and given the von Mises constraint (28) with ${\sigma }_Y=0.29$ GPa (aluminum). See Fig. 1 for keys.

Figure 5

Density radial profiles obtained from the numerical simulation for (a) spherically symmetric converging and (b) planar motion. Elastic-plastic deformations follow the isothermal constitutive law (26), using the polynomial pressure form (13) with $c_1=c_2=c_3=1\text{GPa}$, and given the von Mises constraint (28) with ${\sigma }_Y=7\text{GPa}$. Note that for the planar case an initial shock Mach number cannot be defined since the shock is started beyond the elastic-plastic transition. The elastic precursor Mach number is $M^e\approx 1.02$ for both simulations

Figure 6

$|{\sigma }_{rr}-{\sigma }_{\theta \theta }|/{\sigma }_Y$ vs $J(={\rho }_0/\rho )$ immediately behind the shock for an elastic-plastic material. The Hugoniot curve (i.e., the locus of the possible postshock states of the material for a given initial condition) is completed by some Rayleigh lines (i.e., the thermodynamic path connecting the initial state with a postshock state). Isothermal polynomial pressure form is considered, but the shape of the Hugoniot curve and the different regions can be reproduced for other constitutive laws.

Figure 7

(a) Shock trajectory and (b) $J$ immediately behind the shock vs $R$ for a spherically symmetric ($s=3$) converging shock initially stated at $r=R_i$ with $J_i=0.9$. Comparisons between the purely elastic, elastic-plastic, and zero-shear solid simulations using the isothermal constitutive law (8) or (26), with the polynomial pressure form (13). ${\sigma }_Y=0.29$ GPa was used for the elastic-plastic case.