Thermal Explosion Theory for Reactive Shear Localizing Solids - PowerPoint PPT Presentation

Thermal Explosion Theory for Reactive Shear Localizing Solids University of Notre Dame Joseph M. Powers University of Notre Dame Department of Aerospace and Mechanical Engineering Notre Dame, Indiana 46556-5637 USA powers@nd.edu 27 th International Symposium on Combustion Boulder, Colorado 6 August 1998

Acknowledgments University of Notre Dame AFOSR, Summer Research Extension Program, RDL-96-0870 Los Alamos National Laboratory James J. Mason, Notre Dame Richard J. Caspar, Notre Dame Deanne J. Idar, Los Alamos Jonathan L. Mace, Los Alamos

Partial Review University of Notre Dame • Powers, J. M., 1998, Thermal explosion theory for shear localizing ener- getic solids, Combustion Theory and Modeling , submitted. • Caspar, R. J., Powers, J. M., and Mason, J. J., 1998, Investigation of reactive shear localization in energetic solids, Combustion Science and Technology , to appear. • Frey, R. B., 1981, The initiation of explosive charges by rapid shear, Seventh Symposium (International) on Detonation , NSWC: Annapolis, pp. 36-42. • Mohan, V. K., Bhasu, V. C. J., and Field, J. E., 1989, Role of adiabatic shear bands in initiation of explosives by drop-weight impact, Ninth Sym- posium (International) on Detonation , ONR: Arlington, VA, pp. 1276-83. • Chen, H. C., Nesterenko, V. F., Lasalvia, J. C., and Meyers, M. A., 1997, Shear-induced exothermic chemical reactions, Journal de Physique IV , 7: 27-32.

Motivation University of Notre Dame Development of insensitive solid explosives Development of transient detonation models for solid explosives • steady detonation relatively well-characterized, • late-time hydrodynamic transients relatively well-characterized, • early time ignition events poorly understood • thermal stimuli • mechanical stimuli, e.g. shear localization, also known as shear band- ing

Shear Localization University of Notre Dame (a) (b) (c) a) Initial homogeneous unstrained state, b) Applied shear force induces uniform strain, c) Shear localization induced by local inhomogeneity • Shear localization occurs when thermal softening dominates over strain and strain rate hardening • Hypothesized hot spot location for reaction initiation

Approach and Novelty University of Notre Dame • Approach – Obtain data for high strain rate constitutive theory from Notre Dame torsional split-Hopkinson bar – Use simple model to predict ∗ spatially homogeneous time-dependent solutions ∗ spatially inhomogeneous time-independent solutions • Novelty – Experimental stress-strain-strain rate characterization of inert simu- lant (Mock 900-20) of heterogeneous explosive LX-14 (95.5 % HMX, 4.5 % Estane 5703-P; C 1 . 52 H 2 . 92 N 2 . 59 O 2 . 66 ) – Extension of Frey’s (1981) analysis to include strain rate effects – new application of thermal explosion theory – sensitivity analysis performed

Experimental Method University of Notre Dame Torsional Split Hopkinson Bar-Notre Dame Solid Mechanics Laboratory Incident Bar Torsional Pulley Transmission Bar Specimen A B Clamp Strain Gages Incident Pulse Incident Pulse Shear Strain (m/m) Transmitted Pulse Reflected Pulse Transmitted Pulse Reflected Pulse Time ( µ s) 6 m

Experimental Results University of Notre Dame Torsional Split Hopkinson Bar Test Results for LX-14 simulant (Mock 900-20) and theoretical model predictions . -1 γ = 2800 s measurement prediction Average Shear Stress, τ (MPa) failure at stress overshoot at γ = 0.20, t = 75 µ s γ = 0.015 where τ ∝ γ η � � µ ∂γ . ∂t γ = 300 s -1 prediction measurement peak stress at γ = 0.065 failure at γ = 0.09, t = 350 µ s Average Shear Strain, γ

Model Assumptions University of Notre Dame v = v v = 0 θ θ L thin-walled cylindrical geometry θ r initially unreacted, unstressed, and cold z v r = v z = u r = u z = 0 ∂θ = ∂ ∂ ∂r = 0 v = v L θ one step Arrhenius chemistry z = L z = 0 incompressible constant properties

Model Equations University of Notre Dame ρ∂v θ ∂τ = momentum conservation ∂z , ∂t ρ∂e τ ∂v θ ∂z − ∂q z = energy conservation ∂z , ∂t � � ∂λ − E = a (1 − λ ) exp , reaction kinetics ∂t ℜ T ∂u θ = displacement definition v θ , ∂t � T � ν � ∂u θ � η � ∂v θ � µ L τ = α , constitutive equation for stress T 0 ∂z ∂z v L − k∂T = Fourier’s Law q z ∂z , = c T − λ ˜ caloric state equation e q . Differential-Algebraic system can be shown to be parabolic.

Boundary and Initial Conditions University of Notre Dame Boundary Conditions specified velocity and displacement at both ends, thermally insulated v θ ( t, 0) = 0 , = u θ ( t, 0) = 0 , u θ ( t, L ) = v L t, v θ ( t, L ) v L , ∂T ∂T ∂z ( t, 0) = 0 , ∂z ( t, L ) = 0 . Initial Conditions spatially homogeneous strain rate, unstrained, unreacted, temperature per- turbation near center z v θ (0 , z ) = u θ (0 , z ) = 0 , λ (0 , z ) = 0 , v L L, z �∈ � L ǫ L ) � � T 0 , ǫ L ) , L 2 (1 − ˆ 2 (1 + ˆ , � L � T (0 , z ) = ǫ L ) , L T 0 (1 + ˆ ǫ T ) , z ∈ 2 (1 − ˆ 2 (1 + ˆ ǫ L ) .

Scaling University of Notre Dame Scaled independent variables z ∗ = z t ∗ = v L L, L t, Scaled dependent variables v ∗ = v θ T ∗ = T u ∗ = u θ λ ∗ = λ, , , L . v L T 0 Dimensionless Parameters Ec = v 2 ˜ α Pe = ρc q a = L E � � L ˆ α = ˆ q = ˆ ˆ Θ = , , k v L L, , a, . ρv 2 cT 0 cT 0 v L ℜ T 0 L

Reduced Dimensionless Equations, Initial and Boundary Conditions University of Notre Dame � � ∂u ∗ � η � ∂v ∗ � µ � ∂v ∗ ∂ T ν = ˆ α , ∗ ∂t ∗ ∂z ∗ ∂z ∗ ∂z ∗ � ∂u ∗ � η � ∂v ∗ � µ +1 � � ∂ 2 T ∗ ˆ + 1 Θ ∂T ∗ α � Ec T ν = ˆ + ˆ a ˆ q (1 − λ ∗ ) exp − , ∗ � ∂z 2 ∂t ∗ ∂z ∗ ∂z ∗ T ∗ Pe ∗ � � ˆ Θ ∂λ ∗ = a (1 − λ ∗ ) exp ˆ − , ∂t ∗ T ∗ ∂u ∗ = v ∗ . ∂t ∗ v ∗ ( t ∗ , 0) = 0 , v ∗ ( t ∗ , 1) = 1 , u ∗ ( t ∗ , 0) = 0 , u ∗ ( t ∗ , 1) = t ∗ , ∂T ∗ ∂T ∗ ( t ∗ , 0) = 0 , ( t ∗ , 1) = 0 , v ∗ (0 , z ∗ ) = z ∗ , u ∗ (0 , z ∗ ) = 0 , λ ∗ (0 , z ∗ ) = 0 , ∂z ∗ ∂z ∗ � 1 � � 1 , ǫ L ) , 1 z ∗ �∈ 2 (1 − ˆ 2 (1 + ˆ ǫ L ) , � 1 � T (0 , z ∗ ) = ǫ L ) , 1 1 + ˆ ǫ T , z ∗ ∈ 2 (1 − ˆ 2 (1 + ˆ ǫ L ) . . .

Thermal Explosion Theory University of Notre Dame • Unsteady solutions indicate early time behavior is largely spatially homo- geneous • Formally examine such behavior by assuming • negligibly small temperature perturbation ǫ T • T ∗ = T ∗ ( t ∗ ) (requires � Pe >> 1) • λ ∗ = λ ∗ ( t ∗ ) • v ∗ = z ∗ • u ∗ = z ∗ t ∗ Result is two non-autonomous ordinary differential equations in T ∗ and λ ∗ : � � ˆ dT ∗ Θ α � Ec T ν ∗ t η = ˆ ∗ + ˆ a ˆ q (1 − λ ∗ ) exp T ∗ (0) = 1 , − , dt ∗ T ∗ � � ˆ Θ dλ ∗ = a (1 − λ ∗ ) exp ˆ λ ∗ (0) = 0 . − , dt ∗ T ∗

Approximate Solution University of Notre Dame Neglect reaction in favor of plastic work at early time dT ∗ α � Ec T ν ∗ t η = ˆ T ∗ (0) = 1 . ∗ , t ∗ < t ∗ i , dt ∗ Exact solution available: � 1 − ν � 1 1 − ν α � Ec t η +1 T ∗ ( t ∗ ) = 1 + η ˆ + 1 , t ∗ < t ∗ i . ∗ Determine time when reaction balances plastic work: � � � 1 − ν � � 1 − ν � − 1 ν 1 − ν 1 − ν Ec t η +1 t η Ec t η +1 α � α � α � − ˆ ˆ 1 + η ˆ + 1 ∗ i = ˆ a ˆ q exp Θ 1 + η ˆ + 1 Ec . ∗ i ∗ i

Asymptotic Solution University of Notre Dame Consider high activation energy limit T ∗ = 1 + 1 λ ∗ = 1 Θ T ∗ 1 + · · · , Θ λ ∗ 1 + · · · , ˆ ˆ Energy equation reduces to � β 2 e T ∗ 1 � ˆ a ˆ dT ∗ 1 q α � = ˆ t η ∗ + ˆ β 1 = ˆ ˆ ˆ β 1 , T ∗ 1 (0) = 0 , Θ ˆ Ec, β 2 = Θ . α � Ec e ˆ dt ∗ ˆ For η = 0, induction time is � � Ec e ˆ α � Θ 1 ˆ t ∗ i 0 = ln ( η = 0) . , α � ˆ ˆ a ˆ q Θ ˆ Ec

Numerical Solution Method University of Notre Dame • ODE’s from thermal explosion theory solved with NDSolve in Mathemat- ica 3.0 to 16 digits of accuracy; solution time less than one minute on Sun UltraSparc1 workstation. • PDE’s from full equations solved with method of lines marching technique embodied in Fortran 77 code on Sun UltraSparc1 workstation – forty-nine spatial nodes – second order centered spatial finite difference technique – implicit time integration of ODE’s which result from discretization using DLSODE package – convergence of error norms consistent with order of numerical method – solution time ten minutes – extremely stiff near shear localization events

Thermal Explosion Theory Predictions University of Notre Dame Results from numerical solution of spatially homogeneous ordinary differential equations away from asymptotic limits (a) T* λ * (b) 20 T = 19.74 * 1 (T = 5883 K) 0.001 15 1. · 10 -6 T = 2.18 10 * i (T = 649 K) 1. · 10 -9 i t = 15.13 * i (t = 5.40 ms) 5 i 1. · 10 -12 1. · 10 -15 t * t * 0 5 10 15 20 5 10 15 20 τ * (c) (d) T - T *num *app 500 100 0.1 Peak error = 17.5 400 0.0001 τ = 536 300 * ( τ = 48.5 MPa) τ = 24.6 1. · 10 -7 * 200 ( τ = 2.23 MPa) 1. · 10 -10 100 1. · 10 -13 t * 5 10 15 20 t * 0 5 10 15 20