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Thermal Explosion Theory for Reactive Shear Localizing Solids - - PowerPoint PPT Presentation
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
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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.
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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
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Shear Localization
University of Notre Dame
(b) (c) (a)
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
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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; C1.52H2.92N2.59O2.66) – Extension of Frey’s (1981) analysis to include strain rate effects – new application of thermal explosion theory – sensitivity analysis performed
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Experimental Method
University of Notre Dame
Torsional Split Hopkinson Bar-Notre Dame Solid Mechanics Laboratory
Torsional Pulley Clamp Specimen Incident Bar Strain Gages A B Transmission Bar Incident Pulse Transmitted Pulse Reflected Pulse
6 m
Time (µs) Shear Strain (m/m) Incident Pulse Transmitted Pulse Reflected Pulse
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Experimental Results
University of Notre Dame
Torsional Split Hopkinson Bar Test Results for LX-14 simulant (Mock 900-20) and theoretical model predictions
γ = 300 s
- 1
Average Shear Stress, τ (MPa)
.
Average Shear Strain, γ prediction measurement
.
γ = 2800 s
- 1
measurement prediction peak stress at γ = 0.065 failure at γ = 0.09, t = 350 µs stress overshoot at γ = 0.015 failure at γ = 0.20, t = 75 µs
where τ ∝ γ η
∂γ ∂t
µ
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Model Assumptions
University of Notre Dame
z = 0 v = 0 θ
z
z = L r θ θ vL v = θ v = v
L
thin-walled cylindrical geometry initially unreacted, unstressed, and cold vr = vz = ur = uz = 0
∂ ∂θ = ∂ ∂r = 0
- ne step Arrhenius chemistry
incompressible constant properties
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Model Equations
University of Notre Dame
ρ∂vθ ∂t = ∂τ ∂z , momentum conservation ρ∂e ∂t = τ ∂vθ ∂z − ∂qz ∂z , energy conservation ∂λ ∂t = a (1 − λ) exp
- − E
ℜT
- ,
reaction kinetics ∂uθ ∂t = vθ , displacement definition τ = α
T
T0
ν ∂uθ
∂z
η ∂vθ
∂z L vL
µ
, constitutive equation for stress qz = −k∂T ∂z , Fourier’s Law e = c T − λ ˜ q . caloric state equation Differential-Algebraic system can be shown to be parabolic.
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Boundary and Initial Conditions
University of Notre Dame
Boundary Conditions specified velocity and displacement at both ends, thermally insulated vθ (t, 0) = 0, vθ (t, L) = vL, uθ (t, 0) = 0, uθ (t, L) = vLt, ∂T ∂z (t, 0) = 0, ∂T ∂z (t, L) = 0. Initial Conditions spatially homogeneous strain rate, unstrained, unreacted, temperature per- turbation near center vθ (0, z) = vL z L, uθ (0, z) = 0, λ (0, z) = 0, T (0, z) =
T0,
z ∈ L
2(1 − ˆ
ǫL), L
2(1 + ˆ
ǫL) , T0(1 + ˆ ǫT), z ∈
L
2(1 − ˆ
ǫL), L
2(1 + ˆ
ǫL)
- .
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Scaling
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Scaled independent variables z∗ = z L, t∗ = vL L t, Scaled dependent variables v∗ = vθ vL , T∗ = T T0 , λ∗ = λ, u∗ = uθ L . Dimensionless Parameters ˆ α = α ρv2
L
,
- Ec = v2
L
cT0 ,
- Pe = ρc
k vLL, ˆ q = ˜ q cT0 , ˆ a = L vL a, ˆ Θ = E ℜT0 .
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Reduced Dimensionless Equations, Initial and Boundary Conditions
University of Notre Dame
∂v∗ ∂t∗ = ˆ α ∂ ∂z∗
- T ν
∗
∂u∗
∂z∗
η ∂v∗
∂z∗
µ
, ∂T∗ ∂t∗ = ˆ α Ec T ν
∗
∂u∗
∂z∗
η ∂v∗
∂z∗
µ+1
+ 1
- Pe
∂2T∗ ∂z2
∗
+ ˆ a ˆ q (1 − λ∗) exp
- −
ˆ Θ T∗
- ,
∂λ∗ ∂t∗ = ˆ a (1 − λ∗) exp
- −
ˆ Θ T∗
- ,
∂u∗ ∂t∗ = v∗. v∗(t∗, 0) = 0, v∗(t∗, 1) = 1, u∗(t∗, 0) = 0, u∗(t∗, 1) = t∗, ∂T∗ ∂z∗ (t∗, 0) = 0, ∂T∗ ∂z∗ (t∗, 1) = 0, v∗ (0, z∗) = z∗, u∗ (0, z∗) = 0, λ∗ (0, z∗) = 0, T (0, z∗) =
1,
z∗ ∈
1
2(1 − ˆ
ǫL), 1
2(1 + ˆ
ǫL)
- ,
1 + ˆ ǫT, z∗ ∈
1
2(1 − ˆ
ǫL), 1
2(1 + ˆ
ǫL)
- .
. .
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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 ǫT
- 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∗: dT∗ dt∗ = ˆ α Ec T ν
∗ tη ∗ + ˆ
a ˆ q (1 − λ∗) exp
- −
ˆ Θ T∗
- ,
T∗(0) = 1, dλ∗ dt∗ = ˆ a (1 − λ∗) exp
- −
ˆ Θ T∗
- ,
λ∗(0) = 0.
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Approximate Solution
University of Notre Dame
Neglect reaction in favor of plastic work at early time dT∗ dt∗ = ˆ α Ec T ν
∗ tη ∗,
t∗ < t∗i, T∗(0) = 1. Exact solution available: T∗(t∗) =
1 − ν
1 + η ˆ α Ec tη+1
∗
+ 1
- 1
1−ν
, t∗ < t∗i. Determine time when reaction balances plastic work: ˆ α Ec
1 − ν
1 + η ˆ α Ec tη+1
∗i
+ 1
- ν
1−ν
tη
∗i = ˆ
a ˆ q exp
- − ˆ
Θ
1 − ν
1 + η ˆ α Ec tη+1
∗i
+ 1
−
1 1−ν
- .
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Asymptotic Solution
University of Notre Dame
Consider high activation energy limit T∗ = 1 + 1 ˆ ΘT∗1 + · · · , λ∗ = 1 ˆ Θλ∗1 + · · · , Energy equation reduces to dT∗1 dt∗ = ˆ β1
- tη
∗ + ˆ
β2eT∗1 , T∗1(0) = 0, ˆ β1 = ˆ Θ ˆ α Ec, ˆ β2 = ˆ a ˆ q ˆ α Ec e ˆ
Θ.
For η = 0, induction time is t∗i0 = 1 ˆ Θ ˆ α Ec ln
- ˆ
α Ec e ˆ
Θ
ˆ a ˆ q
- ,
(η = 0).
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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
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Thermal Explosion Theory Predictions
University of Notre Dame
Results from numerical solution of spatially homogeneous ordinary differential equations away from asymptotic limits
5 10 15 20 5 10 15 20
t = 15.13 (t = 5.40 ms)
* i i * i
T = 2.18 (T = 649 K)
i
T = 19.74 (T = 5883 K) * T* t * (a)
5 10 15 20
- 1. · 10-13
- 1. · 10-10
- 1. · 10-7
0.0001 0.1 100
T - T
*num
t * Peak error = 17.5 (d)
5 10 15 20 100 200 300 400 500
τ* τ = 536 (τ = 48.5 MPa) * t * τ = 24.6 (τ = 2.23 MPa) * (c)
5 10 15 20
- 1. · 10-15
- 1. · 10-12
- 1. · 10-9
- 1. · 10-6
0.001 1
λ * t * (b)
*app
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Comparison with Spatially Inhomogeneous Solutions
University of Notre Dame
induction/localization time predicted well by spatially homogeneous thermal explosion theory
0.5 1 5 10
−6 −4 −2
z* t*
*
10 10 10 10
(c)
0.5 1 5 10 −0.1 0.1 z* t* ∆ v* 0.5 1 5 10 −0.05 0.05 z* t* ∆ u* 0.5 1 5 10
−20 −10
z* t* λ*
10 10
10
(a) (b) (d) ∆T
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Conclusions
University of Notre Dame
- Spatially homogeneous thermal explosion theory predicts ignition time ac-
curately, even in presence of shear-localization-inducing inhomogeneities.
- While strong experimental evidence exists detailing the importance of
localized hot spots in accelerating ignition in high explosives, the present theory, confined to the shear initiation mechanism, indicates the shear localization is a consequence, and not a cause, of an already imminent reaction.
- Sensitivity analysis indicates ignition time is generally more sensitive to