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Predictions for Multi-Scale Shock Heating Of a Granular Energetic

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					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
Outline




1. Introduction to Granular Material Compaction
2. Problem Description
3. Solution Methodology & Numerical Procedure
4. Representative Results & Analysis
5. Summary
1. 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
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
                         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
2. Mathematical Model
 (i) Bulk model (ii) Localization model

   Bulk model

                          u          
                                        
          u            u 2  P       
                                                                             Conservation equations of
            u  
               2
                              u 2           0,                               mass, momentum, total
 t   (e  )  x  u (e       P / )                                    energy and grain density
              2               2          
                                                                             of granular solid
          n                nu          
       (1   )
    u              ( Ps   ),
 t    x    c
                                                                               Evolution equations of 
                                                                                   ~
                                                                               and 
  ~     ~ 1        ~                          ~
       ~ ( f   ) if                  f 
    u                                             ,
 t    x 
                0                        otherwise

 where

                                                             ~
                                                              
    s ,     P  Ps ,       e  es  B,        B             d    is Recoverable compaction energy
                                                          0
                                                                 
Constitutive relations (Hayes EOS)

                                                   ~                      ~
P  Ps (  s , T ), e  es (  s , T ),    ( ,  ), f  f ( ),  c , 


Evolution of Internal Energy for Granular Material


 de des dB                          where,
            ,
 dt   dt   dt
                       ~
des ( Ps   ) d    d   Ps d s   de de
                        2            
        s dt  s dt
                        s 
 dt                            dt     dt   dt
              
               
                 Compaction                Compression

 and

dB     d         ~
            (   )
dt    s dt
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
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 0  r  r0
• Localization strategy compatible with grain contact mechanics & bulk energetics
• Uniform grain sizes
• Equate the integrated volumetric plastic flow work to bulk dissipated compaction energy

                      drc    s de                              R * Pc   where,
                                       ;           rc ( x,0) 
                      dt 4nc rc2 PY dt                            2E   *


  R*  R / 2;   E *  E / 2(1  2 );   Plastic flow stress , PY  3.0Y ;   Contact stress , Pc  1.6Y
Meso-scale response
                                  ˆ
Track grain temperature evolution T ( x, r , t ) or liquid phase formation  ( x, r , t ) within
                                                                           ˆ
the localization sphere

                         ˆ
                        de       1  2            ˆ
                                                 Sφ    ˆ
                                                      Sρ
                                       (r q)         ,              where
                        dt     s 0 r r
                                     2
                                                 ρs
                                                    
                                                      ρs
                                                 0 0
           ˆ
    q  kT / r      is thermal conduction                                    Coupling terms between
                                                                                  bulk & grain scales
         r  3 de
                           
    ˆ              s               for 0  r  rc ( x, t )
              0
    S  r ( x , t )         ( x, t )
           c            dt
        0                              for rc ( x, t )  r  r0
                                                                   is bulk dissipated compaction energy
        

   ˆ       de       P d s
   S  s       s s2                 0  r  r0                 is volumetric bulk compression work
            dt       s dt
       dT  ˆ
                          ˆ
                     for T  Tm
                              0              Temperature evolution during pure solid or liquid phase
   de  cv dt
    ˆ 
     
   dt  0 d ˆ           ˆ
       qm            for T  Tm
                              0
                                         0   1
                                             ˆ                             During phase change region
      
          dt
  • Isothermal phase change at melting temperature Tm0 = 520 K (Ref. Menikoff & Kober)
  • Constant latent heat of fusion qm0 = 0.22 MJ/kg.
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           x  Dt , v  u  D
   Transformed equations of bulk & grain scale models are:               I.Cs and B.Cs

                                     ~                                          ~
   d  (1   )                   d      1       ~                     (0)   (0)  0
                  ( Ps   );           ~ ( f   );                   ˆ
   d     v c                     d v                                T (0, r )  T0 (300 K )
     ˆ           ˆ         ˆ        ˆ          ˆ                         (0, r )  0
                                                                         ˆ
   T    2T 2 T               S         S
                                                                                   R * Pc
         2                                   ,                    rc (0) 
       v  r
                     r r   s 0 cv v  s 0 cv v
                                                                                   2E *
                    ˆ         ˆ        ˆ         ˆ                         ˆ
                                                                        T r ( ,0)  0
    cv   2T 2 T 
     ˆ                                 S        S
              2                                   ,                  ˆ
                                                                        T r ( , r0 )  0
    q m v  r
                        r r   s 0 q m v  s 0 q m v
                                
   drc        s      de
       
   d    4nc rc2 PY d
   • Non-dimensionalized; Nr = 100
   • Solved numerically by method of lines (Runge-Kutta 4th order implicit solver ODE15s in MATLAB)
4. Representative results & analysis
                                                    ~
                                              
          Predictions include bulk properties like      and              ˆ      ˆ
                                                              , solid pressure Ps,
                                                                        S     S
          and     ;                                                        ˆ
                                                                          T ( x, r , t )
          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)

                       (a)             Subsonic compaction wave                        (b)
                      Viscoplastic region




                                                                                             D

                                  Viscoelastic region




                                              ~
            Bulk model predictions of (a)  &  and (b) solid pressure Ps, for Up = 106 m/s
          (a)                                                           (b)




       (c)                                                              (d)




                                                                  ˆ      ˆ
Predictions of localization parameters (a) localization radii (b) S and S  (c) & (d)
       grain temperature with and without phase change for Up = 106 m/s
                (a)
                             Supersonic compaction wave                      (b)




                                                                             Shock




              (c)                                                            (d)




                             ~
Model predictions of (a)  &  (b) Ps (c) localization radii and (d) grain temperature for Up = 1153 m/s
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
                                                                    ˆ
    is dominant over the range of piston speeds studied here, i.e., S  Sˆ
                                                                         

    • 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)
         Comparison of model predictions with detailed numerical
                    simulations (Menikoff & Kober)




Detailed numerical simulations for temperature predictions for Up = 1000 m/s (Ref. Menikoff & Kober)
                      (a)                                                     (b)




Comparison of (a) plastic strain and (b) solid pressure predictions for Up = 200, 500 and 1000 m/s
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

				
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posted:11/26/2011
language:English
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