Shock-Wave Attenuation and Energy-Dissipation Potential of Granular Materials

Abstract

The propagation of uniaxial-stress planar shocks in granular materials is analyzed using a conventional shock-physics approach. Within this approach, both compression shocks and decompression waves are treated as (stress, specific volume, particle velocity, mass-based internal energy density, temperature, and mass-based entropy density) propagating discontinuities. In addition, the granular material is considered as being a continuum (i.e., no mesoscale features like grains, voids, and their agglomerates are considered). However, while the granular material is treated as a (smeared-out) continuum, it is recognized that it contains a solid constituent (parent matter), and that the structurodynamic properties (i.e., Equations of State (EOS) and Hugoniot relations) of the granular material are related to its parent matter. Three characteristic shock loading regimes of granular material are considered and, in each case, an analysis is carried out to elucidate shock attenuation and energy dissipation processes. In addition, an attempt is made to identify a metric (a combination of the material parameters) which quantifies the intrinsic ability of a granular material to attenuate a shock and dissipate the energy carried by the shock. Toward that end, the response of a typical granular material to a flat-topped compressive stress pulse is analyzed in each of the three shock loading regimes.

Appendix: Strong-Shock Temperature Hugoniot

The procedure described in this appendix enables calculation of the temperature Hugoniot from the known stress/pressure Hugoniot for the granular material in the strong shock regime, as well as for its parent material. First, the Rankine Hugoniot equation is defined as:

$$ \upvarepsilon^{{ ( {\text{H)}}}} \left( v \right) = \upvarepsilon^{ - } + \frac{1}{2}\left[ {p^{{ ( {\text{H)}}}} \left( v \right) + p^{ - } } \right](v^{ - } - v ) $$

(A.1)

and differentiated to get:

$$ {\frac{{{\text{d}}\upvarepsilon^{(H)} \left( v \right)}}{{{\text{d}}v }}} = \frac{1}{2}\left[ {\left( {v^{ - } - v } \right){\frac{{{\text{d}}p^{{\left( {\text{H}} \right)}} \left( v \right)}}{{{\text{d}}v }}} - p^{{\left( {\text{H}} \right)}} \left( v \right) - p^{ - } } \right] $$

(A.2)

Then, the following expression for the combined First and Second Law of Thermodynamics along the Hugoniot:

$$ {\frac{{{\text{d}}\upvarepsilon^{(H)} \left( v \right)}}{{{\text{d}}v }}} = \uptheta^{{\left( {\text{H}} \right)}} \left( v \right)\;{\frac{{{\text{d}}\upeta^{{\left( {\text{H}} \right)}} \left( v \right)}}{{{\text{d}}v }}} - p^{{\left( {\text{H}} \right)}} \left( v \right) $$

(A.3)

is combined with Eq A.2 to eliminate ${\frac{{{\text{d}}\upvarepsilon^{(H)} \left( v \right)}}{{{\text{d}}v }}}$ and get:

$$ {\frac{{{\text{d}}\upeta^{{\left( {\text{H}} \right)}} \left( v \right)}}{{{\text{d}}v }}} = {\frac{\kappa \left( v \right)}{{2\uptheta^{{ ( {\text{H)}}}} \left( v \right)}}} $$

(A.4)

where $\kappa \left( v \right) = p^{{\left( {\text{H}} \right)}} \left( v \right) - p^{ - } + \left( {v^{ - } - v } \right)\;{\frac{{{\text{d}}p^{{\left( {\text{H}} \right)}} \left( v \right)}}{{{\text{d}}v }}}.$

To obtain the temperature Hugoniot, $\upeta^{{\left( {\text{H}} \right)}} \left( v \right)$ must be eliminated from Eq A.4. Toward that end, the total derivative of the temperature is defined using the known thermodynamic derivatives (Ref 1) as:

$$ {\text{d}}\uptheta = \left. {{\frac{\partial \uptheta }{\partial v }}} \right|_{\upeta } {\text{d}}v + \left. {{\frac{\partial \uptheta }{\partial \upeta }}} \right|_{v } {\text{d}}\upeta = - \uptheta \;{\frac{\upgamma \left( v \right)}{v }}{\text{d}}v + {\frac{\uptheta }{{c^{v } \left( \upeta \right)}}}{\text{d}}\upeta $$

(A.5)

or

$$ {\frac{{{\text{d}}\uptheta^{\left( H \right)} \left( v \right)}}{{{\text{d}}v }}} = - {\frac{\upgamma \left( v \right)}{v }}\;\uptheta^{{\left( {\text{H}} \right)}} \left( v \right) + {\frac{1}{{C^{v } \left( {\upeta^{{\left( {\text{H}} \right)}} \left( v \right)} \right)}}} \;\uptheta^{{\left( {\text{H}} \right)}} \left( v \right)\;{\frac{{{\text{d}}\upeta^{\left( H \right)} \left( v \right)}}{{{\text{d}}v }}} $$

(A.6)

using Eq A.4, (Eq A.6) becomes

$$ {\frac{{{\text{d}}\uptheta^{({\text{H}})} \left( v \right)}}{{{\text{d}}v }}} + {\frac{\upgamma \left( v \right)}{v }}\;\uptheta^{{\left( {\text{H}} \right)}} \left( v \right) = {\frac{\kappa \left( v \right)}{{2C_{\text{R}}^{v } }}} $$

(A.7)

Equation A.7 is a first-order ordinary differential equation (ODE) in the quadrature format (since its second term is a product of a function of the independent variable v and the dependent variable, $\uptheta^{{\left( {\text{H}} \right)}} \left( v \right)$. These types of ODE’s are integral in a disguised format and can be readily integrated. The integrated form of Eq A.7 is given in Eq 15.