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Converging shocks in elastic-plastic solids

López Ortega, A. and Lombardini, M. and Hill, D. J. (2011) Converging shocks in elastic-plastic solids. Physical Review E, 84 (5). 056307. ISSN 1539-3755.

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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)(I_1)+e_(h)(ρ,ς), 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 ρ and entropy ς. 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)(ρ), with a power-law dependence e_(h) ∝ ρ_(α), shock dynamics predicts that for a converging shock located at r=R(t) at time t, the Mach number increases as M ∝ [log(1/R)]^α, independently of the space index s, where s=2 in cylindrical geometry and 3 in spherical geometry. An alternative isothermal constitutive law with p(ρ) of the arctanh type, which enforces a finite density in the strong-shock limit, leads to M ∝ R^(−(s−1)) for strong shocks. A nonisothermal constitutive law, whose hydrostatic part eh is that of an ideal gas, is also tested, recovering the strong-shock limit M∝R^(−(s−1)/n(γ)) originally derived by Whitham for perfect gases, where γ is inherently related to the maximum compression ratio that the material can reach, (γ+1)/(γ−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.

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Additional Information:© 2011 American Physical Society. Received 13 July 2011; published 14 November 2011. We thank D. I. Pullin and D. I. Meiron for their useful comments on this manuscript. This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28613.
Funding AgencyGrant Number
Department of Energy (DOE) National Nuclear Security AdministrationDE-FC52- 08NA28613
Issue or Number:5
Classification Code:PACS: 47.40.Nm, 62.50.Ef, 52.50.Lp
Record Number:CaltechAUTHORS:20120105-121751503
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28673
Deposited By: Jason Perez
Deposited On:06 Jan 2012 00:18
Last Modified:03 Oct 2019 03:34

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