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.

Investigation of hypervelocity impact phenomena using real-time concurrent diagnostics

Hypervelocity impact of meteoroids and orbital debris poses a serious and growing threat to spacecraft. To study hypervelocity impact phenomena, a comprehensive ensemble of real-time concurrently operated diagnostics has been developed and implemented in the Small Particle Hypervelocity Impact Range (SPHIR) facility. This suite of simultaneously operated instrumentation provides multiple complementary measurements that facilitate the characterization of many impact phenomena in a single experiment. The investigation of hypervelocity impact phenomena described in this work focuses on normal impacts of 1.8 mm nylon 6/6 cylinder projectiles and variable thickness aluminum targets. The SPHIR facility two-stage light-gas gun is capable of routinely launching 5.5 mg nylon impactors to speeds of 5 to 7 km/s. Refinement of legacy SPHIR operation procedures and the investigation of first-stage pressure have improved the velocity performance of the facility, resulting in an increase in average impact velocity of at least 0.57 km/s. Results for the perforation area indicate the considered range of target thicknesses represent multiple regimes describing the non-monotonic scaling of target perforation with decreasing target thickness. The laser side-lighting (LSL) system has been developed to provide ultra-high-speed shadowgraph images of the impact event. This novel optical technique is demonstrated to characterize the propagation velocity and two-dimensional optical density of impact-generated debris clouds. Additionally, a debris capture system is located behind the target during every experiment to provide complementary information regarding the trajectory distribution and penetration depth of individual debris particles. The utilization of a coherent, collimated illumination source in the LSL system facilitates the simultaneous measurement of impact phenomena with near-IR and UV-vis spectrograph systems. Comparison of LSL images to concurrent IR results indicates two distinctly different phenomena. A high-speed, pressure-dependent IR-emitting cloud is observed in experiments to expand at velocities much higher than the debris and ejecta phenomena observed using the LSL system. In double-plate target configurations, this phenomena is observed to interact with the rear-wall several micro-seconds before the subsequent arrival of the debris cloud. Additionally, dimensional analysis presented by Whitham for blast waves is shown to describe the pressure-dependent radial expansion of the observed IR-emitting phenomena. Although this work focuses on a single hypervelocity impact configuration, the diagnostic capabilities and techniques described can be used with a wide variety of impactors, materials, and geometries to investigate any number of engineering and scientific problems.

Probing the thermodynamic properties of mantle rocks in solid and liquid states

Our understanding of the structure and evolution of the deep Earth is strongly linked to knowledge of the thermodynamic properties of rocky materials at extreme temperatures and pressures. In this thesis, I present work that helps constrain the equation of state properties of iron-bearing Mg-silicate perovskite as well as oxide-silicate melts. I use a mixture of experimental, statistical, and theoretical techniques to obtain knowledge about these phases. These include laser-heated diamond anvil cell experiments, Bayesian statistical analysis of powder diffraction data, and the development of a new simplified model for understanding oxide and silicate melts at mantle conditions. By shedding light on the thermodynamic properties of such ubiquitous Earth-forming materials, I hope to aid our community’s progress toward understanding the large-scale processes operating in the Earth’s mantle, both in the modern day and early in Earth’s history.

Advanced density functional theory methods for materials science

In this work we chiefly deal with two broad classes of problems in computational materials science, determining the doping mechanism in a semiconductor and developing an extreme condition equation of state. While solving certain aspects of these questions is well-trodden ground, both require extending the reach of existing methods to fully answer them. Here we choose to build upon the framework of density functional theory (DFT) which provides an efficient means to investigate a system from a quantum mechanics description.

Zinc Phosphide (Zn₃P₂) could be the basis for cheap and highly efficient solar cells. Its use in this regard is limited by the difficulty in n-type doping the material. In an effort to understand the mechanism behind this, the energetics and electronic structure of intrinsic point defects in zinc phosphide are studied using generalized Kohn-Sham theory and utilizing the Heyd, Scuseria, and Ernzerhof (HSE) hybrid functional for exchange and correlation. Novel 'perturbation extrapolation' is utilized to extend the use of the computationally expensive HSE functional to this large-scale defect system. According to calculations, the formation energy of charged phosphorus interstitial defects are very low in n-type Zn₃P₂ and act as 'electron sinks', nullifying the desired doping and lowering the fermi-level back towards the p-type regime. Going forward, this insight provides clues to fabricating useful zinc phosphide based devices. In addition, the methodology developed for this work can be applied to further doping studies in other systems.

Accurate determination of high pressure and temperature equations of state is fundamental in a variety of fields. However, it is often very difficult to cover a wide range of temperatures and pressures in an laboratory setting. Here we develop methods to determine a multi-phase equation of state for Ta through computation. The typical means of investigating thermodynamic properties is via ’classical’ molecular dynamics where the atomic motion is calculated from Newtonian mechanics with the electronic effects abstracted away into an interatomic potential function. For our purposes, a ’first principles’ approach such as DFT is useful as a classical potential is typically valid for only a portion of the phase diagram (i.e. whatever part it has been fit to). Furthermore, for extremes of temperature and pressure quantum effects become critical to accurately capture an equation of state and are very hard to capture in even complex model potentials. This requires extending the inherently zero temperature DFT to predict the finite temperature response of the system. Statistical modelling and thermodynamic integration is used to extend our results over all phases, as well as phase-coexistence regions which are at the limits of typical DFT validity. We deliver the most comprehensive and accurate equation of state that has been done for Ta. This work also lends insights that can be applied to further equation of state work in many other materials.

Methods for melting temperature calculation

Melting temperature calculation has important applications in the theoretical study of phase diagrams and computational materials screenings. In this thesis, we present two new methods, i.e., the improved Widom's particle insertion method and the small-cell coexistence method, which we developed in order to capture melting temperatures both accurately and quickly.

We propose a scheme that drastically improves the efficiency of Widom's particle insertion method by efficiently sampling cavities while calculating the integrals providing the chemical potentials of a physical system. This idea enables us to calculate chemical potentials of liquids directly from first-principles without the help of any reference system, which is necessary in the commonly used thermodynamic integration method. As an example, we apply our scheme, combined with the density functional formalism, to the calculation of the chemical potential of liquid copper. The calculated chemical potential is further used to locate the melting temperature. The calculated results closely agree with experiments.

We propose the small-cell coexistence method based on the statistical analysis of small-size coexistence MD simulations. It eliminates the risk of a metastable superheated solid in the fast-heating method, while also significantly reducing the computer cost relative to the traditional large-scale coexistence method. Using empirical potentials, we validate the method and systematically study the finite-size effect on the calculated melting points. The method converges to the exact result in the limit of a large system size. An accuracy within 100 K in melting temperature is usually achieved when the simulation contains more than 100 atoms. DFT examples of Tantalum, high-pressure Sodium, and ionic material NaCl are shown to demonstrate the accuracy and flexibility of the method in its practical applications. The method serves as a promising approach for large-scale automated material screening in which the melting temperature is a design criterion.

We present in detail two examples of refractory materials. First, we demonstrate how key material properties that provide guidance in the design of refractory materials can be accurately determined via ab initio thermodynamic calculations in conjunction with experimental techniques based on synchrotron X-ray diffraction and thermal analysis under laser-heated aerodynamic levitation. The properties considered include melting point, heat of fusion, heat capacity, thermal expansion coefficients, thermal stability, and sublattice disordering, as illustrated in a motivating example of lanthanum zirconate (La₂Zr₂O₇). The close agreement with experiment in the known but structurally complex compound La₂Zr₂O₇ provides good indication that the computation methods described can be used within a computational screening framework to identify novel refractory materials. Second, we report an extensive investigation into the melting temperatures of the Hf-C and Hf-Ta-C systems using ab initio calculations. With melting points above 4000 K, hafnium carbide (HfC) and tantalum carbide (TaC) are among the most refractory binary compounds known to date. Their mixture, with a general formula Ta_xHf_1-xC_y, is known to have a melting point of 4215 K at the composition Ta₄HfC₅, which has long been considered as the highest melting temperature for any solid. Very few measurements of melting point in tantalum and hafnium carbides have been documented, because of the obvious experimental difficulties at extreme temperatures. The investigation lets us identify three major chemical factors that contribute to the high melting temperatures. Based on these three factors, we propose and explore a new class of materials, which, according to our ab initio calculations, may possess even higher melting temperatures than Ta-Hf-C. This example also demonstrates the feasibility of materials screening and discovery via ab initio calculations for the optimization of "higher-level" properties whose determination requires extensive sampling of atomic configuration space.

High pressure states in condensed matter: I. High pressure behavior of the iron-sulphur system with applications to the earth's core. II. Empirical equation of state for organic compounds at high pressures

Part I:

The earth's core is generally accepted to be composed primarily of iron, with an admixture of other elements. Because the outer core is observed not to transmit shear waves at seismic frequencies, it is known to be liquid or primarily liquid. A new equation of state is presented for liquid iron, in the form of parameters for the 4th order Birch-Murnaghan and Mie-Grüneisen equations of state. The parameters were constrained by a set of values for numerous properties compiled from the literature. A detailed theoretical model is used to constrain the P-T behavior of the heat capacity, based on recent advances in the understanding of the interatomic potentials for transition metals. At the reference pressure of 10⁵ Pa and temperature of 1811 K (the normal melting point of Fe), the parameters are: ρ = 7037 kg/m³, K_S0 = 110 GPa, K_S' = 4.53, K_S" = -.0337 GPa-1, and γ = 2.8, with γ α ρ^-1.17. Comparison of the properties predicted by this model with the earth model PREM indicates that the outer core is 8 to 10 % less dense than pure liquid Fe at the same conditions. The inner core is also found to be 3 to 5% less dense than pure liquid Fe, supporting the idea of a partially molten inner core. The density deficit of the outer core implies that the elements dissolved in the liquid Fe are predominantly of lower atomic weight than Fe. Of the candidate light elements favored by researchers, only sulfur readily dissolves into Fe at low pressure, which means that this element was almost certainly concentrated in the core at early times. New melting data are presented for FeS and FeS₂ which indicate that the FeS₂ is the S-hearing liquidus solid phase at inner core pressures. Consideration of the requirement that the inner core boundary be observable by seismological means and the freezing behavior of solutions leads to the possibility that the outer core may contain a significant fraction of solid material. It is found that convection in the outer core is not hindered if the solid particles are entrained in the fluid flow. This model for a core of Fe and S admits temperatures in the range 3450K to 4200K at the top of the core. An all liquid Fe-S outer core would require a temperature of about 4900 K at the top of the core.

Part II.

The abundance of uses for organic compounds in the modern world results in many applications in which these materials are subjected to high pressures. This leads to the desire to be able to describe the behavior of these materials under such conditions. Unfortunately, the number of compounds is much greater than the number of experimental data available for many of the important properties. In the past, one approach that has worked well is the calculation of appropriate properties by summing the contributions from the organic functional groups making up molecules of the compounds in question. A new set of group contributions for the molar volume, volume thermal expansivity, heat capacity, and the Rao function is presented for functional groups containing C, H, and O. This set is, in most cases, limited in application to low molecular liquids. A new technique for the calculation of the pressure derivative of the bulk modulus is also presented. Comparison with data indicates that the presented technique works very well for most low molecular hydrocarbon liquids and somewhat less well for oxygen-bearing compounds. A similar comparison of previous results for polymers indicates that the existing tabulations of group contributions for this class of materials is in need of revision. There is also evidence that the Rao function contributions for polymers and low molecular compounds are somewhat different.

Effect of Microstructural Interfaces on the Mechanical Response of Crystalline Metallic Materials

Advances in nano-scale mechanical testing have brought about progress in the understanding of physical phenomena in materials and a measure of control in the fabrication of novel materials. In contrast to bulk materials that display size-invariant mechanical properties, sub-micron metallic samples show a critical dependence on sample size. The strength of nano-scale single crystalline metals is well-described by a power-law function, σαD^-n, where D is a critical sample size and n is a experimentally-fit positive exponent. This relationship is attributed to source-driven plasticity and demonstrates a strengthening as the decreasing sample size begins to limit the size and number of dislocation sources. A full understanding of this size-dependence is complicated by the presence of microstructural features such as interfaces that can compete with the dominant dislocation-based deformation mechanisms. In this thesis, the effects of microstructural features such as grain boundaries and anisotropic crystallinity on nano-scale metals are investigated through uniaxial compression testing. We find that nano-sized Cu covered by a hard coating displays a Bauschinger effect and the emergence of this behavior can be explained through a simple dislocation-based analytic model. Al nano-pillars containing a single vertically-oriented coincident site lattice grain boundary are found to show similar deformation to single-crystalline nano-pillars with slip traces passing through the grain boundary. With increasing tilt angle of the grain boundary from the pillar axis, we observe a transition from dislocation-dominated deformation to grain boundary sliding. Crystallites are observed to shear along the grain boundary and molecular dynamics simulations reveal a mechanism of atomic migration that accommodates boundary sliding. We conclude with an analysis of the effects of inherent crystal anisotropy and alloying on the mechanical behavior of the Mg alloy, AZ31. Through comparison to pure Mg, we show that the size effect dominates the strength of samples below 10 μm, that differences in the size effect between hexagonal slip systems is due to the inherent crystal anisotropy, suggesting that the fundamental mechanism of the size effect in these slip systems is the same.

Fabrication, Characterization, And Deformation of 3D Structural Meta-Materials

Current technological advances in fabrication methods have provided pathways to creating architected structural meta-materials similar to those found in natural organisms that are structurally robust and lightweight, such as diatoms. Structural meta-materials are materials with mechanical properties that are determined by material properties at various length scales, which range from the material microstructure (nm) to the macro-scale architecture (μm – mm). It is now possible to exploit material size effect, which emerge at the nanometer length scale, as well as structural effects to tune the material properties and failure mechanisms of small-scale cellular solids, such as nanolattices. This work demonstrates the fabrication and mechanical properties of 3-dimensional hollow nanolattices in both tension and compression. Hollow gold nanolattices loaded in uniaxial compression demonstrate that strength and stiffness vary as a function of geometry and tube wall thickness. Structural effects were explored by increasing the unit cell angle from 30° to 60° while keeping all other parameters constant; material size effects were probed by varying the tube wall thickness, t, from 200nm to 635nm, at a constant relative density and grain size. In-situ uniaxial compression experiments reveal an order-of-magnitude increase in yield stress and modulus in nanolattices with greater lattice angles, and a 150% increase in the yield strength without a concomitant change in modulus in thicker-walled nanolattices for fixed lattice angles. These results imply that independent control of structural and material size effects enables tunability of mechanical properties of 3-dimensional architected meta-materials and highlight the importance of material, geometric, and microstructural effects in small-scale mechanics. This work also explores the flaw tolerance of 3D hollow-tube alumina kagome nanolattices with and without pre-fabricated notches, both in experiment and simulation. Experiments demonstrate that the hollow kagome nanolattices in uniaxial tension always fail at the same load when the ratio of notch length (a) to sample width (w) is no greater than 1/3, with no correlation between failure occurring at or away from the notch. For notches with (a/w) > 1/3, the samples fail at lower peak loads and this is attributed to the increased compliance as fewer unit cells span the un-notched region. Finite element simulations of the kagome tension samples show that the failure is governed by tensile loading for (a/w) < 1/3 but as (a/w) increases, bending begins to play a significant role in the failure. This work explores the flaw sensitivity of hollow alumina kagome nanolattices in tension, using experiments and simulations, and demonstrates that the discrete-continuum duality of architected structural meta-materials gives rise to their flaw insensitivity even when made entirely of intrinsically brittle materials.

Dynamics of Granular Crystals with Elastic-Plastic Contacts

We study the behavior of granular crystals subjected to impact loading that creates plastic deformation at the contacts between constituent particles. Granular crystals are highly periodic arrangements of spherical particles, arranged into densely packed structures resembling crystals. This special class of granular materials has been shown to have unique dynamics with suggested applications in impact protection. However, previous work has focused on very low amplitude impacts where every contact point can be described using the Hertzian contact law, valid only for purely elastic deformation. In this thesis, we extend previous investigation of the dynamics of granular crystals to significantly higher impact energies more suitable for the majority of applications. Additionally, we demonstrate new properties specific to elastic-plastic granular crystals and discuss their potential applications as well. We first develop a new contact law to describe the interaction between particles for large amplitude compression of elastic-plastic spherical particles including a formulation for strain-rate dependent plasticity. We numerically and experimentally demonstrate the applicability of this contact law to a variety of materials typically used in granular crystals. We then extend our investigation to one-dimensional chains of elastic-plastic particles, including chains of alternating dissimilar materials. We show that, using the new elastic-plastic contact law, we can predict the speed at which impact waves with plastic dissipation propagate based on the material properties of the constituent particles. Finally, we experimentally and numerically investigate the dynamics of two-dimensional and three-dimensional granular crystals with elastic-plastic contacts. We first show that the predicted wave speeds for 1D granular crystals can be extended to 2D and 3D materials. We then investigate the behavior of waves propagating across oblique interfaces of dissimilar particles. We show that the character of the refracted wave can be predicted using an analog to Snell's law for elastic-plastic granular crystals and ultimately show how it can be used to design impact guiding "lenses" for mitigation applications.