Three groups of single-crystal disks approximately 5-mm thick with surface normals along [100], [110], and [111] crystallographic directions were prepared from 99.99+ at.% pure copper. These specimens were shock loaded to about 50 kbar in a state of uniaxial strain by nitroguanidine explosive plane-wave generators, and the propagated wave profiles were measured with quartz gauges. Elastic wavefronts for the single crystals exhibited sharp risetimes (of the order of 10 nsec) to dynamic yield points, and subsequent stress relaxations preceding the plastic wavefronts. For the propagation distances of about 5 mm, the measured yield point normal stresses were about 2.0, 1.3, and 1.3 kbar, respectively, for wave propagation in the [100], [110], and [111] directions. Although the principal stress states at the yield points differed, analysis reveals that the shear stresses on {111} 〈110〉 slip systems were about the same for all orientations. Single-crystal disks prestrained by about 3½% exhibited essentially zero yield stresses and ramp-like elastic waves. Similar behavior observed for polycrystalline specimens indicates the importance of initial dislocation density on dynamic yielding. In all cases the plastic wave velocities were the same. Constitutive relations derived on the basis of dislocation dynamics are given for the three single crystal orientations. From these relations the decay of the respective dynamic yield points with increasing propagation distance can be predicted as a function of the dislocation mobility and the initial mobile dislocation density. Within the framework of this theory it is shown that simple dislocation damping models for the mobility are not consistent with the experimental results.
Rate-dependent constitutive relations for single crystals are derived in terms of dislocation dynamics. Contributions from slip on the individual glide planes are assumed to superpose linearly to give the total plastic strain. As an application of the theory, equations describing elastic precursor decay are developed for longitudinal plane-wave propagation in fcc, bcc, and rocksalt structures with wave propagation in the [100], [110], and [111] directions. In addition, expressions for precursor decay in zinc (hcp structure) are derived for wave propagation both parallel and perpendicular to the c-axis. Calculated theoretical results are compared with experimental data on precursor amplitudes for single-crystal copper (fcc), tungsten (bcc), NaCl (rocksalt), and LiF (rocksalt). Dislocation mobilities determined from direct observation of dislocations are used in these calculations. In general the theory predicts the proper relative order of the precursor amplitudes for different propagation directions. The comparisons show that in order for theoretically determined amplitudes to agree with experimental data, initial mobile dislocation densities must be two or three orders-of-magnitude greater than initial total densities which are measured in the material prior to shock loading. Possible explanations for this discrepancy are discussed.
A common objective in designing a postmortem type of plate-impact experiment is to be able to attribute the observable residual effects (such as residual strain, hardness, or dislocation density) primarily to the conditions which existed while the material was in a state of uniaxial strain. In the past it has generally been assumed that effects due to radial stress release phenomena, which are always present in such an experiment, are of secondary importance. In order to test the validity of this assumption, a two-dimensional Lagrangian finite-difference computer program is used to model physical experiments representative of common practice. Target plate dimensions, the target and flyer plate material, and the impact velocity are systematically varied for circular target plates with, and without, guard rings. The results show that in many cases the effects of radial release phenomena are too large to ignore. Conclusions are presented which serve as guidelines for designing experiments to minimize radial release effects.
Transient longitudinal wave propagation in a semi-infinite, circular, elastic bar loaded by a radially distributed pressure-step end stress is investigated on the basis of the exact equations of motion. The stress applied to the end of the bar has a radial dependence which can be continuously varied, by means of a loading parameter, from a uniform distribution to a load concentrated at the bar axis. Both analytical and numerical techniques are employed to obtain a complete description of the pulse head strain (ezz + eθθ), as a function of the nonuniformity of the loading, the radial coordinate, the distance from the bar end, and time. The analytic solution, which is valid asymptotically at large distances from the bar end, describes the first mode and shows only very small effects from even a high degree of radial nonuniformity in the applied stress. Near the bar end, solutions for (ezz + eθθ) and the axial stress τzz are obtained by direct numerical integration of the equations of motion. Good agreement between the numerical and analytic results at a propagation distance of 20 dia demonstrates the accuracy of the numerical technique. At distances less than 20 dia from the bar end, the effect of increasing the nonuniformity of the end loading is to greatly enhance the contributions of the higher modes, especially at the bar axis. With regard to a dynamic Saint Venant’s principle, differences in average dynamic stresses and strains resulting from statically equivalent but different radial end stress distributions are negligible at distances greater than 5 bar dia from the end. Differences in peak values are insignificant only at distances greater than 20 bar dia from the end.
Release adiabats and Hugoniot curves centered at shock states can be readily determined by impacting a projectile disk onto a stationary reverberation disk made of a linear elastic material of known shock properties. The reverberation disk may have a free back surface or may be backed by a buffer disk made of the specimen or some other material. The reverberation disk is very thin compared to the thicknesses of the other disks so that many wave reverberations occur in it during the experiment. Depending on the impedance of the reverberation disk relative to the other disks, each reverberation successively unloads, or loads, the projectile disk, thus establishing points on a release adiabat or on recentered Hugoniot curves of the specimen material. The technique is particularly valuable for measurements on compressible nonlinear materials, and it generates a large amount of information in a single experiment. Experiments have been performed with X-cut quartz, Lucalox, and 60° orientation sapphire reverberation disks which illustrate the technique, and results are presented for an epoxy resin and a porous tuff.
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