Material Strength under Shock and Shock-Free Loading Conditions

D.L. Preston, R.T. Olson (P-22), A. Kaul, R.J. Faehl (X-1)
Excerpted from LA-14202-PR

Introduction

Numerical simulations of explosively driven deformation and high-velocity impacts require rate-dependent models of material strength. The main challenge in constructing such models is the wide range of thermodynamic and mechanical conditions that occur in particular solid-flow processes: plastic strains to several hundred percent, plastic strain-rates up to 10¹¹ s^-1, pressure to several megabars, and temperatures up to melt. Ideally, a material-strength model would be based on internal-state variables that provide a complete representation of the microstructural state and its evolution, but our limited knowledge of dislocation, grain-boundary dynamics, and phase-transformation kinetics currently precludes the construction of such a model. Nevertheless, steps towards developing such a material-strength model have been taken by Follansbee and Kocks who used the mechanical threshold stress (MTS, plastic-flow stress at 0 K) as a structure parameter.¹ Their MTS model only accounts for thermally activated dislocation motion, thus it cannot be reliably applied at strain rates much above 10⁴ s^-1. This limitation was overcome by Preston, Tonks, and Wallace (PTW) who developed a model of material strength² applicable at strain rates from 10^-3 s^-1 to 10¹² s^-1. The PTW model has been implemented in several hydrodynamic codes at LANL and has been successfully used to simulate Taylor cylinder impacts, explosively driven systems, and high-velocity impact cratering.³ It should be mentioned that the 25-year-old rate-independent Steinberg-Guinan (SG) material-strength model is also still used in certain numerical simulations at LANL.⁴

Adiabatic Shear-Band Formation and Time-Dependent Material Strength

The PTW, MTS, and SG models are all founded on the assumption that the plastic flow is spatially homogeneous, that is, the plastic deformation is not localized. However, recent experiments on copper by V.A. Raevsky et al. at the All-Russian Research Institute of Experimental Physics (VNIIEF) show that this assumption can break down under both shock and shock-free loading conditions.⁵ (These experiments were funded by LANL under the Gordon/Ryabev agreement.) The shock experiments are at strain rates above 10⁹s?^-1, while the shock-free experiments are used to reduce strain rates to 10^5–7s?^-1. The Raevsky experiments, which are described in more detail below, involve loading copper plates with surface perturbations by explosive detonation and then measuring the perturbation amplitude as a function of time. In recent experiments at VNIIEF, some of the plates were recovered for subsequent metallographic analysis. Remarkably, shear bands, localized regions of large plastic deformation, were present in the recovered samples loaded to pressures above 27 GPa. The shear-band density increases in the direction of the compression wave's propagation into the sample, probably as a result of the increase in the strain rate as the compression wave steepens into a shock wave. Raevsky et al. have also carried out experiments in collaboration with Lawrence Livermore National Laboratory (LLNL) that show shear-band formation in 6061-T6 aluminum.⁶

We can deduce the implications of these findings for material strength by considering the formation and structure of a shear band. Approximately 90% of the work done during plastic deformation is converted into heat. If the strain rate in a region is sufficiently high, as is the case in a high-pressure shock front, then very little heat diffuses out of the region during its deformation, thus the heating is nearly adiabatic. If the decrease in strength due to the increase in temperature exceeds the increase in strength from work hardening, then the plastic deformation becomes unstable, resulting in the formation of an adiabatic shear band. The total plastic strain in the shear band can be quite large: 500%–800%. The corresponding temperature rise is of order 10³ K. The time scale for heat flow away from the hot shear band is τ_Q = w²/κ, where w is the width of the shear band and κ is the thermal diffusivity. As an example, for w = 10 μm and κ = 0.1 cm² s^-1 (characteristic of steels) we find τ_Q = 10 μs. The macroscopic (spatially averaged) strength (flow stress) will change during the time, τ_Q, required for temperature equilibration around the shear bands. Therefore, the heterogeneity causes the macroscopic material strength to be time dependent.

In view of these considerations, it is no surprise that none of our strength models agree with the Raevsky data on copper (Figure 1). The observed rate of perturbation growth is much greater than the predicted growth rate, which implies that the actual strength is less than that predicted by the models. This is consistent with the presence of hot shear bands in the copper. Unknowns in the Raevsky experiments such as the absolute pressure determination, experiment-to-experiment variability, and the unrealistically small error bars as determined from x-radiographs, raise questions about the overall accuracy of the Raevsky data. Even so, refinements in the Raevsky technique are not expected to bring the data into agreement with calculations based on current strength models.

Future Experiments Based on the Raevsky Technique

Efforts are currently underway in Physics (P) and Applied Physics Divisions to incorporate the effects of shear bands in the PTW model. Experimental data to validate the generalized model are absolutely essential. P Division is currently planning experiments that leverage upon the technique and data of Raevsky in order to create a validation database. These experiments will provide validation data precisely where the PTW model and other rate-dependent models (e.g., MTS and the LLNL Steinberg-Lund model) are least reliable, that is, at strain rates between 10⁵ s^-1 and 10⁸ s^-1 and strains up to several hundred percent, which are the conditions commonly achieved under explosive loading or high-velocity impact of metals. Furthermore, the development of this experimental capability will allow us to investigate materials not studied by Raevsky. The first experiments will be done in collaboration with Raevsky at VNIIEF and then a large suite of additional experiments will be conducted at LANL.

The Raevsky technique utilizes a flat metal plate with perturbations of known wavelength and amplitude machined into the surface. High explosive (HE) is used to generate either shock or shock-free planar loading of the plate. The amplitude of the Rayleigh-Taylor unstable perturbations is measured from x-radiographs acquired as a function of time (Figure 2). This technique can be used to generate pressures in the metal sample in excess of 80 GPa with strain rates ranging from 10⁴ to 10¹⁰ s^-1.

The preliminary experiments to be performed at VNIIEF will use diagnostics and well-characterized copper samples from LANL. These experiments will duplicate exactly those previously performed by Raevsky and, as a result, will allow for a direct comparison to the reported Raevsky data. This comparison is essential, for if LANL is to utilize the Raevsky data for model validation purposes, we must corroborate the results and analyze their uncertainties. We currently lack information about repeatability of sample preparation, drive conditions, and absolute peak pressure. It is for this reason that LANL will supply the samples and field a high-precision velocity-measurement diagnostic. Further, the reported accuracy of perturbation amplitudes as determined from radiographic analysis seems to be unrealistically high based upon the resolution limitations present in the VNIIEF x-ray system. The LANL x-ray system to be fielded will provide multiple radiographs per experiment with a small improvement in amplitude resolution. Successful completion of these joint experiments will allow us to define a level of confidence in the reported Raevsky data. The shock-free copper experiments will be continued at LANL to obtain a complete validation data set throughout the range of conditions where no other reliable data are available. The HE arrangement and quantity will be altered to control the strain rate and peak pressure through the range 10⁵–10⁸ s?^-1 and 20–70 GPa, respectively. Sample strain (up to ~ 250%) will be controlled through the wavelength and amplitude of the perturbations. Repeatability of the drive conditions and sample preparation will be ensured with sample velocimetry and material characterization. Amplitude growth will be measured via multiple repetition, low-energy flash x-radiography and will provide the measurable quantity to compare with hydrodynamic computations.
The results of hydrodynamic code simulations using the generalized PTW model will be compared to the extensive validation data set obtained at LANL and VNIIEF. Progressively more realistic representations of shear bands will be incorporated in the generalized model until good agreement with the data is achieved. We should emphasize that the Raevsky technique is currently the only means of obtaining accurate validation strength data at plastic strain-rates exceeding 10⁵ s^-1, rates that are typical of explosively driven systems and high-velocity impacts.

References

1. P.S. Follansbee and U.F. Kocks, "A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable," Acta Metallurgica 36, 82–93 (1988).
2. D.L. Preston, D.L. Tonks, and D.C. Wallace, "Model of plastic deformation for extreme loading conditions," Journal of Applied Physics 39, 211–220 (2003).
3. R.F. Davidson and M.L. Walsh, "Constitutive modeling for hypervelocity cratering," AIP Conference Proceedings 370, 1159–1162 (1996).
4. D.J. Steinberg, S.G. Cochran, and M.W. Guinan, "Constitutive model for metals applicable at high-strain rate," Journal of Applied Physics 51, 1498–1504 (1980).
5. V.A. Raevsky, All-Russian Institute of Experimental Physics (VNIIEF), Report to LANL on Task 3.1 under Agreement 37713-000-02-35, (2003).
6. V.A. Raevsky, All-Russian Institute of Experimental Physics (VNIIEF), Final Report to LLNL under Agreement B512964, (2002).

Acknowledgment

We would like to thank Dr. Victor A. Raevsky and his collaborators at the VNIIEF for several very informative discussions of their perturbation growth experiments.

For further information, contact Dean Preston, 505-667-8968, dean@lanl.gov.