I. Rigid body penetration into brittle materials. II. Phase change effect on shock wave propagation

Part I.

We have developed a technique for measuring the depth time history of rigid body penetration into brittle materials (hard rocks and concretes) under a deceleration of ~ 10⁵ g. The technique includes bar-coded projectile, sabot-projectile separation, detection and recording systems. Because the technique can give very dense data on penetration depth time history, penetration velocity can be deduced. Error analysis shows that the technique has a small intrinsic error of ~ 3-4 % in time during penetration, and 0.3 to 0.7 mm in penetration depth. A series of 4140 steel projectile penetration into G-mixture mortar targets have been conducted using the Caltech 40 mm gas/ powder gun in the velocity range of 100 to 500 m/s.

We report, for the first time, the whole depth-time history of rigid body penetration into brittle materials (the G-mixture mortar) under 10⁵ g deceleration. Based on the experimental results, including penetration depth time history, damage of recovered target and projectile materials and theoretical analysis, we find:

1. Target materials are damaged via compacting in the region in front of a projectile and via brittle radial and lateral crack propagation in the region surrounding the penetration path. The results suggest that expected cracks in front of penetrators may be stopped by a comminuted region that is induced by wave propagation. Aggregate erosion on the projectile lateral surface is < 20% of the final penetration depth. This result suggests that the effect of lateral friction on the penetration process can be ignored.

2. Final penetration depth, P_max, is linearly scaled with initial projectile energy per unit cross-section area, e_s , when targets are intact after impact. Based on the experimental data on the mortar targets, the relation is P_max(mm) 1.15e_s (J/mm² ) + 16.39.

3. Estimation of the energy needed to create an unit penetration volume suggests that the average pressure acting on the target material during penetration is ~ 10 to 20 times higher than the unconfined strength of target materials under quasi-static loading, and 3 to 4 times higher than the possible highest pressure due to friction and material strength and its rate dependence. In addition, the experimental data show that the interaction between cracks and the target free surface significantly affects the penetration process.

4. Based on the fact that the penetration duration, t_max, increases slowly with e_s and does not depend on projectile radius approximately, the dependence of t_max on projectile length is suggested to be described by t_max(μs) = 2.08e_s (J/mm² + 349.0 x m/(πR²), in which m is the projectile mass in grams and R is the projectile radius in mm. The prediction from this relation is in reasonable agreement with the experimental data for different projectile lengths.

5. Deduced penetration velocity time histories suggest that whole penetration history is divided into three stages: (1) An initial stage in which the projectile velocity change is small due to very small contact area between the projectile and target materials; (2) A steady penetration stage in which projectile velocity continues to decrease smoothly; (3) A penetration stop stage in which projectile deceleration jumps up when velocities are close to a critical value of ~ 35 m/s.

6. Deduced averaged deceleration, a, in the steady penetration stage for projectiles with same dimensions is found to be a(g) = 192.4v + 1.89 x 10⁴, where v is initial projectile velocity in m/s. The average pressure acting on target materials during penetration is estimated to be very comparable to shock wave pressure.

7. A similarity of penetration process is found to be described by a relation between normalized penetration depth, P/P_max, and normalized penetration time, t/t_max, as P/P_max = f(t/t_max, where f is a function of t/t_max. After f(t/t_max is determined using experimental data for projectiles with 150 mm length, the penetration depth time history for projectiles with 100 mm length predicted by this relation is in good agreement with experimental data. This similarity also predicts that average deceleration increases with decreasing projectile length, that is verified by the experimental data.

8. Based on the penetration process analysis and the present data, a first principle model for rigid body penetration is suggested. The model incorporates the models for contact area between projectile and target materials, friction coefficient, penetration stop criterion, and normal stress on the projectile surface. The most important assumptions used in the model are: (1) The penetration process can be treated as a series of impact events, therefore, pressure normal to projectile surface is estimated using the Hugoniot relation of target material; (2) The necessary condition for penetration is that the pressure acting on target materials is not lower than the Hugoniot elastic limit; (3) The friction force on projectile lateral surface can be ignored due to cavitation during penetration. All the parameters involved in the model are determined based on independent experimental data. The penetration depth time histories predicted from the model are in good agreement with the experimental data.

9. Based on planar impact and previous quasi-static experimental data, the strain rate dependence of the mortar compressive strength is described by σ_f/σ⁰_f = exp(0.0905(log(έ/έ_0) ^1.14, in the strain rate range of 10^-7/s to 10³/s (σ⁰_f and έ are reference compressive strength and strain rate, respectively). The non-dispersive Hugoniot elastic wave in the G-mixture has an amplitude of ~ 0.14 GPa and a velocity of ~ 4.3 km/s.

Part II.

Stress wave profiles in vitreous GeO₂ were measured using piezoresistance gauges in the pressure range of 5 to 18 GPa under planar plate and spherical projectile impact. Experimental data show that the response of vitreous GeO₂ to planar shock loading can be divided into three stages: (1) A ramp elastic precursor has peak amplitude of 4 GPa and peak particle velocity of 333 m/s. Wave velocity decreases from initial longitudinal elastic wave velocity of 3.5 km/s to 2.9 km/s at 4 GPa; (2) A ramp wave with amplitude of 2.11 GPa follows the precursor when peak loading pressure is 8.4 GPa. Wave velocity drops to the value below bulk wave velocity in this stage; (3) A shock wave achieving final shock state forms when peak pressure is > 6 GPa. The Hugoniot relation is D = 0.917 + 1.711u (km/s) using present data and the data of Jackson and Ahrens [1979] when shock wave pressure is between 6 and 40 GPa for ρ₀ = 3.655 gj cm³ . Based on the present data, the phase change from 4-fold to 6-fold coordination of Ge⁺⁴ with O^-2 in vitreous GeO₂ occurs in the pressure range of 4 to 15 ± 1 GPa under planar shock loading. Comparison of the shock loading data for fused SiO₂ to that on vitreous GeO₂ demonstrates that transformation to the rutile structure in both media are similar. The Hugoniots of vitreous GeO₂ and fused SiO₂ are found to coincide approximately if pressure in fused SiO₂ is scaled by the ratio of fused SiO₂to vitreous GeO₂ density. This result, as well as the same structure, provides the basis for considering vitreous Ge0₂ as an analogous material to fused SiO₂ under shock loading. Experimental results from the spherical projectile impact demonstrate: (1) The supported elastic shock in fused SiO₂ decays less rapidly than a linear elastic wave when elastic wave stress amplitude is higher than 4 GPa. The supported elastic shock in vitreous GeO₂ decays faster than a linear elastic wave; (2) In vitreous GeO₂ , unsupported shock waves decays with peak pressure in the phase transition range (4-15 GPa) with propagation distance, x, as α 1/x^-3.35 , close to the prediction of Chen et al. [1998]. Based on a simple analysis on spherical wave propagation, we find that the different decay rates of a spherical elastic wave in fused SiO₂ and vitreous GeO₂ is predictable on the base of the compressibility variation with stress under one-dimensional strain condition in the two materials.

Large-eddy simulations of fully developed turbulent channel and pipe flows with smooth and rough walls

Studies in turbulence often focus on two flow conditions, both of which occur frequently in real-world flows and are sought-after for their value in advancing turbulence theory. These are the high Reynolds number regime and the effect of wall surface roughness. In this dissertation, a Large-Eddy Simulation (LES) recreates both conditions over a wide range of Reynolds numbers Re_τ = O(10²)-O(10⁸) and accounts for roughness by locally modeling the statistical effects of near-wall anisotropic fine scales in a thin layer immediately above the rough surface. A subgrid, roughness-corrected wall model is introduced to dynamically transmit this modeled information from the wall to the outer LES, which uses a stretched-vortex subgrid-scale model operating in the bulk of the flow. Of primary interest is the Reynolds number and roughness dependence of these flows in terms of first and second order statistics. The LES is first applied to a fully turbulent uniformly-smooth/rough channel flow to capture the flow dynamics over smooth, transitionally rough and fully rough regimes. Results include a Moody-like diagram for the wall averaged friction factor, believed to be the first of its kind obtained from LES. Confirmation is found for experimentally observed logarithmic behavior in the normalized stream-wise turbulent intensities. Tight logarithmic collapse, scaled on the wall friction velocity, is found for smooth-wall flows when Re_τ ≥ O(10⁶) and in fully rough cases. Since the wall model operates locally and dynamically, the framework is used to investigate non-uniform roughness distribution cases in a channel, where the flow adjustments to sudden surface changes are investigated. Recovery of mean quantities and turbulent statistics after transitions are discussed qualitatively and quantitatively at various roughness and Reynolds number levels. The internal boundary layer, which is defined as the border between the flow affected by the new surface condition and the unaffected part, is computed, and a collapse of the profiles on a length scale containing the logarithm of friction Reynolds number is presented. Finally, we turn to the possibility of expanding the present framework to accommodate more general geometries. As a first step, the whole LES framework is modified for use in the curvilinear geometry of a fully-developed turbulent pipe flow, with implementation carried out in a spectral element solver capable of handling complex wall profiles. The friction factors have shown favorable agreement with the superpipe data, and the LES estimates of the Karman constant and additive constant of the log-law closely match values obtained from experiment.

Entrainment of fine sediments by turbulent flows

A study was made of the means by which turbulent flows entrain sediment grains from alluvial stream beds. Entrainment was considered to include both the initiation of sediment motion and the suspension of grains by the flow. Observations of grain motion induced by turbulent flows led to the formulation of an entrainment hypothesis. It was based on the concept of turbulent eddies disrupting the viscous sublayer and impinging directly onto the grain surface. It is suggested that entrainment results from the interaction between fluid elements within an eddy and the sediment grains.

A pulsating jet was used to simulate the flow conditions in a turbulent boundary layer. Evidence is presented to establish the validity of this representation. Experiments were made to determine the dependence of jet strength, defined below, upon sediment and fluid properties. For a given sediment and fluid, and fixed jet geometry there were two critical values of jet strength: one at which grains started to roll across the bed, and one at which grains were projected up from the bed. The jet strength K, is a function of the pulse frequency, ω, and the pulse amplitude, A, defined by

K = Aω^-s

Where s is the slope of a plot of log A against log ω. Pulse amplitude is equal to the volume of fluid ejected at each pulse divided by the cross sectional area of the jet tube.

Dimensional analysis was used to determine the parameters by which the data from the experiments could be correlated. Based on this, a method was devised for computing the pulse amplitude and frequency necessary either to move or project grains from the bed for any specified fluid and sediment combination.

Experiments made in a laboratory flume with a turbulent flow over a sediment bed are described. Dye injection was used to show the presence, in a turbulent boundary layer, of two important aspects of the pulsating jet model and the impinging eddy hypothesis. These were the intermittent nature of the sublayer and the presence of velocities with vertical components adjacent to the sediment bed.

A discussion of flow conditions, and the resultant grain motion, that occurred over sediment beds of different form is given. The observed effects of the sediment and fluid interaction are explained, in each case, in terms of the entrainment hypothesis.

The study does not suggest that the proposed entrainment mechanism is the only one by which grains can be entrained. However, in the writer’s opinion, the evidence presented strongly suggests that the impingement of turbulent eddies onto a sediment bed plays a dominant role in the process.

The structure of liquid argon as determined by x-ray diffraction

X-ray diffraction measurements and subsequent data analyses have been carried out on liquid argon at five states in the density range of 0.91 to 1.135 gm/cc and temperature range of 127 to 143°K. Duplicate measurements were made on all states. These data yielded radial distribution and direct correlation functions which were then used to compute the pair potential using the Percus-Yevick equation. The potential minima are in the range of -105 to -120°K and appear to substantiate current theoretical estimates of the effective pair potential in the presence of a weak three-body force.

The data analysis procedure used was new and does not distinguish between the coherent and incoherent absorption factors for the cell scattering which were essentially equal. With this simplification, the argon scattering estimate was compared to the gas scattering estimate on the laboratory frame of reference and the two estimates coincided, indicating the data normalized. The argon scattering on the laboratory frame of reference was examined for the existence of the peaks in the structure factor and the existence of an observable third peak was considered doubtful.

Numerical studies of the effect of truncation, normalization, the subsidiary peak phenomenon in the radial distribution function, uncertainties in the low angle data relative to errors in the direct correlation function and the distortion phenomenon are presented.

The distortion phenomenon for this experiment explains why the Mikolaj-Pings argon data yielded pair potential well depths from the Percus-Yevick equation that were too shallow and an apparent slope with respect to density that was too steep compared to theoretical estimates.

The data presented for each measurement are: empty cell and cell plus argon intensity, absorption factors, argon intensity, smoothed argon intensity, smoothed argon intensity corrected for distortion, structure factor, radial distribution function, direct correlation function and the pair potential from the Percus-Yevick equation.