# Shock compression modeling of metallic single crystals: comparison of finite difference, steady wave, and analytical solutions

- Jeffrey T Lloyd
^{1, 3}Email author, - John D Clayton
^{1}, - Ryan A Austin
^{2}and - David L McDowell
^{3, 4}

**2**:14

**DOI: **10.1186/s40323-015-0036-6

© Lloyd et al. 2015

**Received: **20 January 2015

**Accepted: **12 June 2015

**Published: **10 July 2015

## Abstract

### Background

The shock response of metallic single crystals can be captured using a micro-mechanical description of the thermoelastic–viscoplastic material response; however, using a such a description within the context of traditional numerical methods may introduce a physical artifacts. Advantages and disadvantages of complex material descriptions, in particular the viscoplastic response, must be framed within approximations introduced by numerical methods.

### Methods

Three methods of modeling the shock response of metallic single crystals are summarized: finite difference simulations, steady wave simulations, and algebraic solutions of the Rankine–Hugoniot jump conditions. For the former two numerical techniques, a dislocation density based framework describes the rate- and temperature-dependent shear strength on each slip system. For the latter analytical technique, a simple (two-parameter) rate- and temperature-independent linear hardening description is necessarily invoked to enable simultaneous solution of the governing equations. For all models, the same nonlinear thermoelastic energy potential incorporating elastic constants of up to order 3 is applied.

### Results

Solutions are compared for plate impact of highly symmetric orientations (all three methods) and low symmetry orientations (numerical methods only) of aluminum single crystals shocked to 5 GPa (weak shock regime) and 25 GPa (overdriven regime).

### Conclusions

For weak shocks, results of the two numerical methods are very similar, regardless of crystallographic orientation. For strong shocks, artificial viscosity affects the finite difference solution, and effects of transverse waves for the lower symmetry orientations not captured by the steady wave method become important. The analytical solution, which can only be applied to highly symmetric orientations, provides reasonable accuracy with regards to prediction of most variables in the final shocked state but, by construction, does not provide insight into the shock structure afforded by the numerical methods.

### Keywords

Crystal plasticity Finite difference method Dislocations Shock waves## Introduction

An understanding of the thermomechanical response of metallic crystals at high strain rates and high pressures is important for research and development of technologies involving impact, as occurring in crashworthiness applications and ballistic collisions, for example. Detailed constitutive models for single crystal thermoelastic–viscoplastic response enable prediction of effects of microstructure—e.g., lattice orientation, dislocation content, grain structure—on the performance of metals in such dynamic loading regimes. For modeling shocks of significant magnitude in single crystals, nonlinear elasticity, thermoelastic coupling, and material anisotropy become important. Models for the shock response of solids have witnessed continuous development and refinement since the mid-twentieth century [1–3], with theories involving various levels of detail, complexity, and efficiency available.

The finite difference (FD) approach to modeling shock wave propagation involves discretization of the solution domain in both space and time. Applications of FD methods towards descriptions of wave propagation in metals include [3–6]. Advantages of the method developed in Refs. [5, 6] include the following: crystals of any symmetry and orientation can be studied (i.e., transverse waves are captured), material properties may be heterogeneous in the (longitudinal) direction of wave propagation, and sophisticated rate- and temperature-dependent crystal plasticity models are enabled. Relative disadvantages are the time required for calculation of solutions and the need for artificial viscosity to regularize the shock width in the strong shock regime.

The steady wave (SW) approach to modeling shock waves presented in this work, which is strictly valid only for uniaxial strain conditions, involves transformation of governing partial differential equations to ordinary differential equations relative to a coordinate frame that moves along with a steady shock wave. Applications of the steady wave method towards descriptions of plastic shocks in metallic crystals include [7–11]. Advantages of the method developed in Ref. [10], which is the first known implementation of the SW approach for anisotropic elastic–plastic crystals, include the following: a detailed description of the steady shock structure (and associated material state) is obtained, solutions are obtained at relatively low computational cost, no artificial viscosity is used, and sophisticated rate- and temperature-dependent crystal plasticity models are enabled. Disadvantages are that effects of transverse waves for non-symmetric crystal orientations are ignored, unsteady waves cannot be addressed, and material properties must be spatially homogeneous.

The present analytical approach to modeling shocked metals involves simultaneous solution of the Rankine–Hugoniot jump conditions for conservation of mass, momentum, and energy, along with rate-independent constitutive equations for thermoelastic–plastic response. Previous work includes [12–15]. The present method, which can be applied only for symmetric crystal orientations (e.g., shocks propagating along [100] and [111] directions in FCC crystals), essentially reduces the problem to simultaneous solution of the yield condition and energy balance for the cumulative plastic slip and entropy, with the remaining conservation and constitutive laws sufficient for determination of the downstream material state. In this paper, “downstream” refers to material behind the plastic shock wave, “upstream” to material ahead of the shock. Advantages of this method are its simplicity (few material parameters are needed, and solutions are obtained nearly instantly) and ability to incorporate various nonlinear anisotropic thermoelastic potentials [16]. Disadvantages are the following: only highly symmetric orientations can be modeled as noted above, time dependence (e.g., explicit strain rate effects on strength) is ignored, and the shock is treated as a perfect jump discontinuity such that no further information regarding its structure (e.g., transitional values of state variables between upstream and downstream states) is obtained.

The remainder of this paper is outlined as follows. The FD model, the SW model, and the analytical model are described in “Finite difference model”, “Steady wave model”, and “Analytical model”, including governing equations, constitutive theory and parameters, and numerical methods. Because these models have been described at length in prior publications [6, 10, 15], only essential features are provided herein. Quantitative comparison and evaluation of the numerical approaches (FD and SW) are given in “Numerical methods comparison”. Comparison of these results with the limited scope of results available from analytical solutions is given in “Comparison of numerical and analytical solutions”. Concluding discussion follows in “Conclusion”. The material of study is pure aluminum [Al, face centered cubic (FCC) structure], which is advantageous because of the extensive data available for its thermoelastic and shock response [17–19], and because it typically does not undergo twinning which would require more elaborate constitutive theory [20] than that employed herein.

Although all three models have been presented individually and validated versus experimental data in prior work [6, 10, 15], previous papers have not included any comparisons of results among the three methods or any evaluations of computational efficiency. Explicit method comparisons identifying material orientations and loading regimes for which each method may be most appropriate are the primary new contributions of this paper. The only shocks considered herein are stable planar shocks as encountered in traditional plate impact experiments with null obliquity. Numerical methods developed to capture the behavior of converging and diverging shocks and their associated applications may be found elsewhere [21].

## Finite difference model

The FD model evaluated in this paper incorporates constitutive theories for nonlinear anisotropic thermoelasticity and crystal plasticity described in detail in Ref. [6, 10]. Many, if not most, features are also used in the SW and analytical models described later in “Steady wave model” and “Analytical model”.

*U*is internal energy per unit reference volume.

Thermoelastic properties of Al (\(\theta _0 = 300\)K)

Property | Value | Units |
---|---|---|

\(C_{11}, \, C_{12}, \, C_{44}\) | 106.7, 60.4, 28.3 | GPa |

\(C_{111}, \, C_{112}, \, C_{123}\) | −1,076, −315, 36 | GPa |

\(C_{144}, \, C_{155}, \, C_{456}\) | −23, −340, 30 | GPa |

\(\Gamma \) | 2.30 | – |

\( c_0\) | 2.35 | MPa/K |

\( \rho _0 \) | 2.71 | g/cm |

*k*with initial slip direction \(\varvec{s}^k\) and plane normal \(\varvec{m}^k\),

*b*, the mobile dislocation density is \(N^k_m\) with glide velocity \(\bar{v}^k\), and the rate of homogeneous nucleation is \(\dot{N}^k_{hom}\) with a mean glide displacement \(\bar{x}\). For FCC Al, slip occurs on up to \(k=1,2,\ldots ,12\) \(\{111\}\langle 110 \rangle \) systems, and the corresponding ambient shear modulus is \(\mu _0 = C_{44}+\frac{1}{3}(C_{11}-C_{12}-2C_{44})\), leading to an initial shear wave velocity of \(c_{s} = \sqrt{\mu _0/\rho _0}\); in the model, \(c_s\) and \(\mu \) are also updated with temperature and elastic strain [10]. The total dislocation density is \(N^k = N^k_m + N^k_i\), where \(N^k_i\) is the immobile density. Constitutive relations for the crystal-level mobile and immobile dislocation density evolution, as well as their associated mean velocity, build upon on previously developed isotropic constitutive models [3, 8, 9]. Letting \(\tau ^k = \pmb {\sigma }:(\varvec{F}^E \varvec{s}^k \otimes \varvec{m}^k \varvec{F}^{E \, -1})\) denote the resolved Cauchy stress on system

*k*, evolution equations are [6]

Plastic properties of Al (\(\theta _0 = 300\)K)

Property | Definition | Value | Units |
---|---|---|---|

| Burgers vector | 0.286 | nm |

\(N^k_0\) | Initial dislocation density | 0.56 | \(1/\upmu\mathrm{m}^2 \) |

\(f_0\) | Initial mobile disloc. fraction | 0.3 | – |

\(\dot{N}_0\) | Homogeneous gen. factor | \(7.2 \times 10^7\) | \(1/(\upmu\mathrm{m}^2\) \(\upmu \)s) |

\(g_{0hom}\) | Homogeneous gen. parameter | 0.04125 | – |

\(\tau _{0hom}/\mu _0\) | Homogeneous gen. stress | 0.05 | – |

\(\chi \) | Mobile hom. disloc. fraction | 0.08 | – |

\(\bar{x}/b\) | Generation displacement | 13.3 | – |

\(\alpha _{het}\) | Heterogeneous gen. factor | 320 | \(1/\upmu\mathrm{m}^2 \) |

| Heterogeneous gen. exponent | 0.8 | – |

\(\tau _{min} / \mu _0\) | Heterogeneous gen. bound | 0.004 | – |

\(\tau _{max} / \mu _0\) | Heterogeneous gen. bound | 0.04 | – |

\(p_{mul}\) | Multiplication probability | 0.088 | – |

\(\alpha _{ann}\) | Annihilation factor | 0.25 | – |

\(\alpha _{tra}\) | Trapping factor | 0.051 | – |

\(\alpha _{pas}\) | Passing strength factor | 0.1 | – |

\(\alpha _{cut}\) | Cutting strength factor | 0.9 | – |

\(\nu _G \) | Obstacle attempt frequency | \(1 \times 10^5\) | 1/\(\upmu \)s |

\(B_0\) | Drag coefficient | 18.0 | Pa \(\upmu \)s |

\(p, \, q\) | Strength exponents | 0.5, 2 | – |

*q*; correspondingly, from (2) and (3), for deformation of the form (21),

## Steady wave model

*U*via the partial Legendre transformation

*D*in the (\(X_1\)-) direction of shock propagation, partial differential equations in (32) can be transformed to the ordinary differential equations

## Analytical model

*D*. As in “Steady wave model”, let \((\cdot )^+\) and \((\cdot )^-\) label quantities in the material ahead (i.e., upstream) and behind (i.e., downstream) from the shock. Let \([\![(\cdot ) ]\!]= (\cdot )^- - (\cdot )^+\) and \(\langle (\cdot ) \rangle = \frac{1}{2}[(\cdot )^- + (\cdot )^+] \) denote the jump and average of a quantity across the shock. Let \(\varvec{n}\) be a unit normal vector to the planar shock, i.e., \(\varvec{n}=\partial \varvec{x} / \partial x_1\). The only nonvanishing component of particle velocity is \(\upsilon = \pmb {\upsilon } \cdot \varvec{n}\). The Cauchy stress component normal to the shock front is \(\sigma = \pmb {\sigma }:(\varvec{n} \otimes \varvec{n}) = \sigma _{11}\). The relative velocity of the material with respect to the shock is \(\mathsf {v} = \upsilon - D\). Internal energy per unit mass is \(u = U/ \rho _0\). Using (37), (38) can be rewritten as [12]

*D*if the HEL is exceeded. For overdriven shocks, there is no precursor. Total deformation is

*n*systems experience the same resolved shear stress \(\tau = \tau ^k\). In lieu of the viscoplastic model implemented in “Finite difference model” and “Steady wave model”, for the analytical treatment a two-parameter yield criterion in the plastically deforming regime is prescribed:

*H*to the total slip on all

*n*active systems. The factor of \(J = J^E\) in (43) accounts for work conjugacy of Kirchhoff stress and plastic slip in the intermediate configuration implied by the multiplicative decomposition of \(\varvec{F}\) in (1) [23].

*D*and downstream particle velocity \(\upsilon ^-\) can be obtained from the Hugoniot equations for mass and momentum conservation in (37), leading to [7]

*H*will be explained later in “Comparison of numerical and analytical solutions”. Much of the foregoing discussion applies for weak shocks; for strong shocks, conditions \(P_H \rightarrow P^+ = 0\), \(\lambda ^+ = 1\), \(\upsilon ^+=0\), and \(U^+=0\) are enforced.

## Numerical methods comparison

Numerical simulations and approximations

Simulation | SW approximations | FD approximations |
---|---|---|

5 GPa [100] | None | None |

5 GPa low symmetry | Uniaxial \(\varvec{F}\) | None |

25 GPa [100] | None | Artificial viscosity |

25 GPa low symmetry | Uniaxial \(\varvec{F}\) | Artificial viscosity |

### Velocity profiles

For the [100] orientation, Figure 2a indicates that the SW and FD simulations give nearly identical results. This agreement is expected as neither method introduces intrinsic approximations for this orientation and shock strength (Table 3). For the low symmetry orientation, Figure 2b shows that although the SW method approximates deformation as uniaxial, it predicts a nearly identical longitudinal component of the velocity profile as the FD simulation. Although the wave profiles are nearly identical, the SW method under-predicts the peak resolved shear stress that occurs on slip systems by approximately 10% because it does not include the shear components of the quasi-longitudinal wave [22]. However, this appears to negligibly influence the longitudinal component of the velocity profile at the low impact stress.

For the [100] orientation, wave profiles for the two methods are shown in Figure 3a. Because the SW method begins to track the solution at an adiabatic elastic compression for which the longitudinal elastic wave speed equals the SW speed (in this case, at a particle velocity of approximately 0.55 km/s), it gives no additional information concerning the wave profile up to this velocity; however, any wave structure calculated by the FD method up to this velocity is due to the artificial viscosity, so no physical insight is gained in FD simulations up to this velocity either. Above this velocity, the SW method predicts a slightly sharper rise than the FD method. This is because even though an extremely fine mesh resolution is employed there is still a smearing effect from the artificial viscosity, which decreases the peak strain rate experienced in the material. This decrease in peak strain rate decreases the rate of homogeneous dislocation by two orders of magnitude, which in turn alters the wave profile at elevated velocities, as fewer mobile dislocations are available to relax the deviatoric response through glide. Even though the coupling between viscoplasticity and viscous regularization is undesirable, the FD and SW methods predict nearly identical strength and accumulated plastic strain, although their wave profiles differ slightly.

Computed wave profiles for the low symmetry orientation in Figure 3b differ in several respects. The FD method captures transverse components of formation of the quasi-longitudinal wave which cannot be considered using the SW method. Additionally, the FD method predicts a single wave structure, whereas the SW method indicates deformation preceding the main rise. As discussed previously, local adiabatic treatment of overdriven shocks provides no information until the point where the elastic wave speed equals the SW speed. However, for the low symmetry shock, since SW simulation (longitudinal wave) has a different elastic stiffness than the FD simulation (quasi-longitudinal wave), the point where viscoplastic deformation occurs differs. This causes the two methods to predict differing shock structures, where the SW method predicts a dual shock structure and the FD method predicts an overdriven shock. Additionally, the FD method experiences elevated shear stresses due to the transverse deformation components, which gives a slightly different viscoplastic response as well. Based on these observations, the SW method appears unsuitable for simulations in which transverse wave components have a relatively large magnitude, i.e., simulation of strong shocks in single crystals with low symmetry orientations.

### Computational efficiency

Total computation times on a single 2.67 GHz Intel Xeon X5650 processor

Simulation | Machine time (s) | Speedup factor | |
---|---|---|---|

SW | FD | ||

5 GPa [100] | 32.38 | \(6.69 \times 10^4\) | 2006 |

5 GPa low symmetry | 58.24 | \(1.69 \times 10^5\) | 2902 |

25 GPa [100] | 15.99 | \(1.27 \times 10^5\) | 7942 |

25 GPa low symmetry | 70.43 | \(1.28 \times 10^5\) | 1817 |

For the FD simulations presented in this work, one reason that the total computation times are several orders of magnitude longer than SW simulations is that a sufficiently fine mesh is employed so that dissipation is primarily due to the viscoplastic constitutive equation and not from artificial viscous regularization. To illustrate the effect of mesh resolution and viscous regularization on the wave profile, FD simulations were performed on a [100] single crystal shocked at \(P^-=5~\mathrm {GPa}\). The resultant wave profiles are given in Figure 4 whereas computation times and computed dislocation densities in the shocked state are given in Table 5. Figure 4 illustrates that when an artificial viscosity is employed in conjunction with a mesh resolution that is too coarse to resolve the shock width predicted from the viscoplastic constitutive relations alone, the shock is smoothed. Consequently, Table 5 indicates that although computation times approach those associated with the SW method, the viscoplastic behavior, indicated by the total dislocation density in the shocked state, is altered due to the decrease in peak strain rate associated with the shock. When this damping occurs the viscoplastic behavior becomes highly mesh-dependent. Consequently, physical meaning associated with internal state variables that govern viscoplastic deformation and the ability to predict detailed wave profile evolution are lost in the FD approach with nonzero artificial viscosity.

Computational time and total dislocation density for [100] shock at \(P^-=5~\mathrm {GPa}\) computed using the FD method with varying mesh resolutions

Mesh resolution \(\Delta X_1\) (\(\upmu \mathrm {m}\)) | Machine time (s) | Total dislocation density (\(\upmu {\mathrm {m}}^{-2}\)) |
---|---|---|

0.83 | \(6.69 \times 10^4\) | 270 |

2.5 | \(8.06 \times 10^3\) | 212 |

5 | \(2.08 \times 10^3\) | 192 |

10 | 562 | 176 |

20 | 159 | 177 |

40 | 51.5 | 172 |

## Comparison of numerical and analytical solutions

*D*for the overdriven shock at \(P^- = 25\) GPa. Also shown for comparison are hydrodynamic predictions obtained using the Birch–Murnaghan pressure–volume equation of state (EOS) [33], with compressibility properties of Al from the literature [34]. Temperature rise in the EOS was calculated assuming compression is isentropic and internal energy is first order in entropy. The hydrodynamic approximation, which by construction omits shear/deviatoric stress components, is often used as a simple model for shocks in materials whose strength is low relative to shock pressure.

Shocked state of Al [100] (\(P^- = 5\) GPa)

Variable (units) | FD | SW | Analytical | Birch–Murnaghan EOS |
---|---|---|---|---|

\(\lambda ^-\) | 0.944 | 0.944 | 0.945 | 0.944 |

\(\tau ^-\) (MPa) | 56.7 | 55.8 | 120.4 | 0 |

\((\bar{\epsilon }^P_{eff})^-\) | 0.037 | 0.037 | 0.034 | 0 |

\((\Delta \theta )^-\) (K) | 40.5 | 39.2 | 42.3 | 38.36 |

\(\upsilon ^-\) (km/s) | 0.323 | 0.323 | 0.319 | 0.321 |

| 5.739 | 5.717 | 5.798 | 5.771 |

Shocked state of Al [100] (\(P^- = 25\) GPa)

Variable (units) | FD | SW | Analytical | Birch–Murnaghan EOS |
---|---|---|---|---|

\(\lambda ^-\) | 0.810 | 0.805 | 0.805 | 0.816 |

\(\tau ^-\) (GPa) | 1.023 | 1.047 | 0.417 | 0 |

\((\bar{\epsilon }^P_{eff})^-\) | 0.113 | 0.110 | 0.131 | 0 |

\((\Delta \theta )^-\) (K) | 256.1 | 269.1 | 250.8 | 126.56 |

\(\upsilon ^-\) (km/s) | 1.324 | 1.341 | 1.343 | 1.303 |

| 7.000 | 6.882 | 6.879 | 7.105 |

*H*) is required, rather than an extensive list as in Table 2,

*H*must still be prescribed via comparison with shear strength data from experiments or other more physically descriptive model output. Here, following the latter approach,

*H*has been calibrated such that cumulative plastic deformation

Because the Birch–Murnaghan EOS assumes a spherical stress state, shear stress and plastic deformation are unresolved in Tables 6 and 7, and temperature rise is under-predicted since there is no contribution to dissipation from plastic slip. The EOS does, however, predict reasonably accurate values of relative volume, particle velocity, and shock velocity.

## Conclusion

Analytical, FD, and SW numerical solutions have been compared for shock loading of single crystals using identical thermoelastic frameworks, but with rate- and temperature-independent shear strength constitutive relations in the analytical approach, and rate- and temperature-dependent shear strength constitutive relations in the latter two methods. Scenarios exist in which each of these methods is most appropriate. These method comparisons have not been published in previous papers which have focused on each model and its results in isolation.

Given a material for which there are limited strength data and incomplete understanding of physical mechanisms governing dissipation during shock loading, the analytical approach provides rapid shock response characterization based on thermoelastic properties, which are often available in literature, and a simple empirical hardening relation. In this case wave profile information predicted by the SW and FD methods would be largely speculative unless physical or experimental insights could be used to suggest more realistic, and presumably more complex representations of dissipation mechanisms and their relation to material strength.

On the other hand if there is data that quantifies the spatio-temporal of the velocity profile, or if rate-dependent micromechanical mechanisms that govern the viscoplastic material response are well characterized, the SW or FD methods may be more appropriate. In particular, the SW and FD methods can be used to predict the steady shock structure as well as give information regarding evolution of internal state variables and the thermodynamic state prior to, within, and after the shock. Due to its computational efficiency, the SW method is especially useful for developing constitutive equations prior to their implementation in FD frameworks. Additionally, in “Computational efficiency” it was shown that predicted rate-dependent behavior may be unphysically altered in FD simulations due to viscous damping effects unless a sufficiently fine mesh resolution is employed, whereas viscous damping is not required in the SW method. Only the FD method is capable of quantifying transient aspects of evolving shock waves, which is necessary to model spatio-temporal shock wave evolution data such as elastic precursor decay.

All three methods can be used to model highly symmetric single crystal orientations subjected to shock loading, but only the FD method can be used to capture quasi-longitudinal and quasi-transverse waves that arise in low symmetry crystal orientations. For weak shock loading in Al, approximating deformation as uniaxial was shown to be reasonable. Therefore, the SW method should be preferred due to its computational efficiency and lack of artificial viscosity. For strong shocks, however, the response of low symmetry crystal orientations was poorly captured using the SW method. Therefore, when modeling strong shocks for low symmetry crystal orientations relative to the loading direction, the finite-difference method should be employed.

These conclusions can be extended to other cubic metals with similar elastic anisotropy. However, additional investigations are required before generalizing these conclusions to materials that exhibit significantly higher elastic anisotropy or materials with lower crystal symmetry.

## Declarations

### Authors’ contributions

JTL performed finite difference and steady wave simulations. JDC derived and calculated the analytical shock response. RAA and DLM helped create, derive, and evaluate the viscoplastic formulation in conjunction with JTL. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank Dr. Richard Becker for stimulating discussions concerning implementation of the FD method. This work was performed, in part (RAA), under the auspicies of the US Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 (LLNL-JRNL-663743). DLM is grateful for the support of the Carter N. Paden, Jr. Distinguished Chair in Metals Processing.

### Compliance with ethical guidelines

**Competing interests** The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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