Predictions for Multi-Scale Shock Heating Of a Granular Energetic Material

1. Predictions for Multi-Scale Shock Heating Of a Granular Energetic Material By Venugopal Jogi (M.S Candidate) Advisor: Dr. Keith A. Gonthier Support Air Force Research Laboratory, MNMW, Eglin AFB, FL Mechanical Engineering Department, LSU

2. Outline • Introduction to Granular Material Compaction • Problem Description • Solution Methodology & Numerical Procedure • Representative Results & Analysis • Summary

3. Introduction Compaction wave Propagation Localized stresses & grain heating at the contact surfaces Length scales • Bulk / Macro scale : 5 -10 cm • Localization / Grain scale : 0.1 – 200 µm Grain scale phenomena important for ignition and DDT transition : high frequency stress & temperature fluctuations

4. Relevant applications • Safe handling & storage of damaged explosives • Mechanical loading response of propellants, pyrotechnics and solid rocket fuels • Synthesis of high strength materials (Powder metallurgy) • Importance of Heterogeneity Modeling • Heterogeneities due to density discontinuities and porosity - More sensitive • Disparate time & length scales • Multi-phase flows • Difficult to perform meso scale experiments at dynamic compaction speeds • Existing models and their limitations • Predict only bulk response; fail to capture grain scale phenomena • Bulk models - based on quasi-static experiments • Either don’t track hot-spots or are inconsistent in coupling multi-scale phenomena

5. Steady wave structure – Bulk model predictions D Solid volume fraction Bulk temperature • Novel aspects of present study • An energetically consistent 1-D model to couple bulk & grain scale phenomena • Track the formation & growth of hot-spots for subsonic & supersonic compaction waves • Solid compressibility, effect of phase change on thermal energetics • Comparison of predictions with detailed meso scale simulations

6. Mathematical Model (i) Bulk model (ii) Localization model Bulk model Conservation equations of mass, momentum, total energy and grain density of granular solid Evolution equations of and where is Recoverable compaction energy

7. Constitutive relations (Hayes EOS) Evolution of Internal Energy for Granular Material where, and

8. Comparison of bulk model behavior with dynamic experiments Hugoniot Curves Predict the after-shock conditions of the compaction process Hugoniot curves for HMX in (a) P-v plane and (b) D-Vp plane • For same initial conditions, closely match with experiments • Bulk model parameter β0 is modified to better replicate the similar experiments (Sheffield et.al.) • Plots also include data for PBXs

9. Localization model Bulk compaction energy localization strategy • Adopted Gonthier’s Localization strategy - Bulk compaction energy deposited at the intergranular contact surfaces, within the localization sphere • Localization strategy compatible with grain contact mechanics & bulk energetics • Uniform grain sizes • Equate the integrated volumetric plastic flow work to bulk dissipated compaction energy where,

10. Meso-scale response Track grain temperature evolution or liquid phase formation within the localization sphere where is thermal conduction Coupling terms between bulk & grain scales is bulk dissipated compaction energy is volumetric bulk compression work Temperature evolution during pure solid or liquid phase During phase change region • Isothermal phase change at melting temperature Tm0 = 520 K (Ref. Menikoff & Kober) • Constant latent heat of fusion qm0 = 0.22 MJ/kg.

11. 3. Solution Procedure & Numerical Methodology • Approximations & assumptions • Steady compaction waves • Inert material • Only solid phase with porosity (phase change energetics) • Only plastic deformation mechanism Coordinate transformation to moving wave frame Transformed equations of bulk & grain scale models are: I.Cs and B.Cs • Non-dimensionalized; Nr = 100 • Solved numerically by method of lines (Runge-Kutta 4th order implicit solver ODE15s in MATLAB)

12. 4. Representative results & analysis • Predictions include bulk properties like and , solid pressure Ps, and ; • localization scale parameters such as rc , r0 , R and grain temperature • Case1: Subsonic compaction wave (Up = 106 m/s) • To illustrate some of the key features of low speed compaction • With and without phase change • Case2: Supersonic compaction wave (Up = 1153 m/s with phase change) Subsonic compaction wave (a) (b) Viscoplastic region D Viscoelastic region Bulk model predictions of (a) & and (b) solid pressure Ps, for Up = 106 m/s

13. (a) (b) (c) (d) Predictions of localization parameters (a) localization radii (b) and (c) & (d) grain temperature with and without phase change for Up = 106 m/s

14. Supersonic compaction wave (a) (b) Shock (c) (d) Model predictions of (a) & (b) Ps (c) localization radii and (d) grain temperature for Up = 1153 m/s

15. Observations • Compaction zone is thinner at higher compaction speeds • The remnants of two-wave structure is seen at lower piston speed (Up = 106m/s) • Role of compressive heating is insignificant compared to the bulk compaction energy which is dominant over the range of piston speeds studied here, i.e., • Phase change reduces the grain temperature by a significant 100 K at Up = 106m/s • Thermal conduction is not important for high rate compaction (adiabatic deformation)

16. Comparison of model predictions with detailed numerical simulations (Menikoff & Kober) Detailed numerical simulations for temperature predictions for Up = 1000 m/s (Ref. Menikoff & Kober) (b) (a) Comparison of (a) plastic strain and (b) solid pressure predictions for Up = 200, 500 and 1000 m/s

17. Comparison Contd, • At Up = 200 m/s, the detailed simulations may be under resolved hence not able to estimate correct amount of plastic strain • At Up = 500 m/s, both the results match well • At Up = 1000 m/s, we over-predict plastic strain, which is limited by phase change Summary • Thermal energetics of granular compaction occurring at grain scale are captured in a manner consistent with thermodynamics and grain contact mechanics • Coupling between bulk model and localization model is obtained in an energetically consistent manner • Subsonic and supersonic steady compaction waves are studied to predict the bulk as well as grain scale parameters such as solid volume fraction, pressure, grain temperature • Effect of phase change is studied at different compaction speeds • Compared with detailed meso scale simulations to develop more sophisticated models to capture these grain-scale phenomena