This document simulates the generation of high pressure and temperature in metals under shock loading. It presents equations to model the total pressure and energy behind a shock front in metals. The pressure is modeled as a function of density ratio and temperature, taking into account cold, thermal, and excitation pressures. Results show that as the shock wave converges towards the center of a spherical metal sample, excitation pressure dominates over elastic and thermal pressures. Graphs show the variation of pressure and temperature in different metals like aluminum, iron, stainless steel, molybdenum, and iridium as the shock wave moves inward. Thermal pressure initially increases then levels off, while excitation pressure rises very fast before decreasing as the shock nears the center

1. International Journal of Research in Computer Science
eISSN 2249-8265 Volume 2 Issue 1 (2011) pp. 7-10
© White Globe Publications
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SIMULATION OF GENERATION OF HIGH PRESSURE AND
TEMPERATURE IN METALS UNDER SHOCK LOADING
Dr. V.P.Singh
Ex Scientist F (Retd), CASSA, MOD, Bangalore
Director, Sat Kabir Institute of Technology and Management,
Ladrawan, Bahadurgarh, Haryana (INDIA)
Abstract: Pressure and Temperature in different shown that discontinuities in pressure can be removed
metals including radioactive materials behind at ultra high pressure. In this process, a very strong
converging shock waves, is simulated using compression of a condensed medium generates a
generalized form of equation of state. Tait’s equation colossal internal pressure, even in the absence of
of state of metals, is valid for pressures of the range of heating due only to the repulsive forces between the
few mega bars and takes into account only elastic atoms. The material is also very strongly heated by
pressures. At such high pressures, metal undergo shock waves and, this result in the appearance of a
phase change and normal equation of state no more is pressure associated with the thermal motion of atoms.
valid. At such pressures, temperature in metals It is known that thermal energy is not linear function
becomes very high and thermal and excitation of temperature and CV also varies with temperature. In
pressures dominate over elastic pressure. It is the present paper we have taken Einstein Equation for
observed that as shock approaches the center of thermal energy and have studied the pressure and
sphere, excitation pressure dominates elastic as well temperature variations behind the converging shock
as thermal pressure. waves. Results of variation of pressure by
conventional method and by present method are
Keywords: Phase change, shocks in metals and radio-
compared in table-3.
active materials.
II. FORMULATION OF THE PROBLEM
I. INTRODUCTION
It is assumed that a spherical shock wave is moving
Simulation of high pressure and temperature
from the surface of a metal (or radioactive material)
generated by imploding shock waves through
sphere towards its center. Generalized thermal l energy
condensed media, such as metals and radioactive
of materials given by Einstein Equation of state,
materials, is of great theoretical and practical
importance. Behavior of metals under high pressures hν
has been analyzed by many authors [1-4], by taking ET = 3 N (1)
into account, the general Hugoniot relationship [exp(hν / kT ) − 1]
between shock velocity and particle velocity. This It is used along with jump conditions [2] to get
linear equation fails when we analyze the material expressions for pressure and temperature behind
undergoing crystallographic phase change under high
imploding shock waves. In equation (1) N, h, ν, k are
pressure. Thus, knowledge of the thermodynamic
Avagadro’s Number, Plank’s constant, natural
properties of the materials is necessary to study their
frequency of solid material, and Boltzman’s constant
behavior under high pressures and temperatures.
respectively.
Although, no appreciable [2] difficulties are
encountered in calculating the thermodynamic Excitation energy of metals is given by[2,5]
properties of gases, a theoretical description of the 2/3
1 ρ   
1/ 2
k 2m
thermodynamic properties of solids and liquids at high E = β  0 T 2
and β = 21.179 e N 1/ 3  1 
e 2 o ρ  o e ρ 
pressure generated by very strong shocks, present a   h2  1
very complex problem. (2a)
Variation of material parameters under shock where me is the electron mass, and Ne is the number of
loading has been studied by Singh and Renuka [5], by free electrons per unit volume of the metal.
taking thermodynamic properties of the materials into Cold energy is given by,
account. In this paper, total energy of system is taken
as sum of cold, excitation and thermal energy. By Ec = − ∫ Pc dV (2b)
taking thermal energy as CV T ,with CV constant, it was
Equations for cold, thermal and excitation pressure are,
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2. 8 Dr.V.P.Singh
 ρ n   3 NΓ0 ρ 0 k x 2 (δ − 1) ρ β T (δ − 5) δ 6 NkΘ 2 ρ 0δ 
g (T , δ ) =  + 0 0 + 2 x  (8)
Pc = A(S)   − 1 (3a) (e − 1)
x 2
T (e − 1) 2 
 ρ 0 

 2 
  
 hν
x= , Γ = Γ0 / δ (ref [6])
E kT
T
P = Γ(V) (3b)
T V We get T in terms of δ, from this equation, and then
using equation (4) for P, other parameters can be
1 Ee computed from,
Pe = (3c)
2 V
Pδ
U= (9)
where V is specific volume and [5], ρ1(δ − 1)
2 2
2 V (d Pc /dV )
Γ (V) = − − (3d) P(δ − 1)
3 2 (dPc /dV) u2 = (10)
ρ1 δ
Using equations(1)-(3), total pressure P and energy E
behind the shock front is obtained, In the above equations, natural frequency of the
metal is slightly complex parameter. This is computed
3 NΓρ 0 δ k Θ 1 by replacing it by Debye’s temperature given by,
P = A(δ n − 1) + Θ/T
+ ρ 0 β 0δ 1 / 2T 2 (4a)
(e − 1) 4 Θ = hν / k
(11)
A  (δ n + n − 1)  3 Nk Θ 1 Debye’s temperature values for different type of
E=   + Θ/T + β 0δ −1 / 2T 2 (4b) metals are given in table -2, and basic parameters are
ρ 0  δ (n − 1)  (e − 1) 4
given in table-1.
Here Θ = hν / k , is Debye’s constant and III. SOLUTION OF THE PROBLEM
ρ The equations (1) to (4) are solved to get the
δ = = Shock strength . We observe from
ρ0 pressure, temperature, shock velocity and particle
velocity of the material behind the shock front. These
relations (4) that pressure P is function of density ratio parameters are expressed in terms of δ and T. It is not
δ and temperature T. We take δ as the independent possible to solve these equations explicitly in terms of
parameter and express all other paraameters in terms one parameter, say δ. Therefore temperature is
of δ. Elliminating U and u from equations (1) of [2], expressed in differential form and evaluated by
integrating equation (7) using Runge –Kutta method of
one gets,
fourth order.
(δ − 1)( P + 2 P0 ) = 2 ρ 0δE (5) In order to express pressure and temperature in
terms of the distance ‘r’ of the shock front from the
where P=P2-P0 and E=E2-E0.
center of the sphere, we use Energy Hypothesis [6].
Transmitted pressure due to the initial compression of
Differentiating this equation with respect to δ, we get,
the metal by the shock transmitted from explosive to
material at its outer boundary is calculated using
dP dE
( P + 2 P0 ) + (δ − 1) = 2 ρ 0 E + 2 ρ 0δ (6) mismatch conditions [6].
dδ dδ
Variation of δ as the shock moves from the surface
Using equations (4) in (5) and after some algebraic of metallic sphere to its apex is calculated by using
calculations, we get, energy hypothesis [6] as,
dT f (T , δ ) δ E (δ )
3
= (7)  ro 
dδ g (T , δ )   = (12)
r  δ i E (δ i )
where
where index ‘i’ denotes, values at the explosive-
 A
{
 ρ δ nδ
n +1
− (n + 2)δ n + 2 + } 
 material boundary. Energy E ( δ) is given by
f (T , δ ) = − ρ1  2 
1
T β 0 
 {δ + 3} − 2 ρ1 E + P2 (T , δ ) / ρ 0 + P0 / ρ 0  1 2
8 δ  E (δ) = E(δ) + u2
2 (13)
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3. Simulation of Generation of High Pressure and Temperature in Metals under Shock Loading 9
E (δ ) VI. ACKNOWLEDGEMENT
where is given by equation (4b).
Present work was funded by Defence Research and
Knowing δ as a function of r/r0, other parameters are Development,, Ministry of Defence.
evaluated from equations (2)-(4).
IV. RESULTS AND DISCUSSION VII. FIGURES USED
In this paper, we have considered five materials, of
which two are radioactive in nature (Table 1). 1.00E+04
PxE+10(Kg/ms2)
Ir
Variation of pressure for Al, Fe, Stainless Steel 1.00E+03
Mo
(Type 404), Molybdenum (Mo), and Iridium is shown
1.00E+02 SS
in Figure 1, whereas variation of temperature for these
Fe
metals is given in figures 2. Transmitted pressure in 1.00E+01
Al
Aluminum is low at the metal-explosive interface but
rises fast and becomes infinite at radius equal to 0.3. 1.00E+00
Similarly pressure in Iron grows faster than Stainless
35
45
55
65
75
85
95
0.
0.
0.
0.
0.
0.
0.
steel. In Molybdenum (Mo), pressure rise is slow as
Radius
compared to lighter metals, and becomes infinite at
radius ratio 0.1. In figure -3, we have plotted variation
of cold, thermal and elastic pressures in Iron. It is seen Figure-1: Variation of pressure VS dimensionless radius for
different materials
that in the beginning thermal and elastic pressures are
less than cold pressure but ultimately, thermal pressure
and excitation pressure increase and become more than 10000000
elastic pressure. Excitation pressure increases very fast
Temperature(oK)
Ir
and get maximum at radius 0.3 and then starts 1000000
Mo
reducing. 100000 SS
It has been observed that as the density of material Fe
10000
increases, its maximum compression as well as Al
transmitted pressure from explosive decrease (Table- 1000
2).
00
3
4
5
6
7
8
9
0.
0.
0.
0.
0.
0.
0.
1.
V. CONCLUSION Radius
In studying the pressure variation across the shock
Figure-2: Variation of temperature VS dimensionless radius
using Hugoniot equation of state, it has been observed
for different materials
that at high pressures, certain materials show sudden
increase in pressure values or even a discontinuity in
the pressure curve as the shock wave converges from Variations of different Pressures
the surface of the sphere to its center [5]. This is
PressurexE+07(Kg/ms2)
because this equation does not take into account the 1.00E+16
thermal and excitation energy of metals. Figure-3 1.00E+14
1.00E+12
gives three pressure components for a typical metal, 1.00E+10 Pc
say Iron. It shows that, PT and Pe are of same order 1.00E+08 PT
initially, but ultimately Pe becomes much more that PT. 1.00E+06 Pe
Near the centre, effect of excitation pressure is to 1.00E+04
expand where as it is resisted by elastic pressure, 1.00E+02
1.00E+00
which creates a uncertainty at such points, resulting in
35
45
55
65
75
85
95
3
the abnormal behavior of Pe curve. Thermal pressure
0.
0.
0.
0.
0.
0.
0.
0.
first increases and then becomes almost constant and Radius
when shock approaches near the centre, again rises.
Figure-1 gives pressure variations, where as figure- 2 Figure 3: Comparison of three components of pressure
gives variation of temperature as shock converges.
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4. 10 Dr.V.P.Singh
A Γ at β ρ E0
Material ΓO n
Bars boundary erg/g.deg2 g/cm3 108 erg/g
1.8200e
Al 2.0 1.35 4.352 500 2.785 16.1
+010
3.3233e
SS 2.17 1.49 4.96 540.6 7.896 12.9
+010
1.5010e
Fe 1.69 1.56 6.68 541 7.85 12.85
+010
6.8150e
Mo 1.520 1.39 3.932 495 10.206 7.2
+010
Table 1 Data used in this paper
Sr.
Metal Debey’s Temperature Θ δ δi δmax
No o
K
1. Aluminum 428 2785 1.25213 1.31154
2. Iron 470 7850 1.2146 1.2437
3. Stainless Steel 470 7896 1.1689 1.1899
4. Molybdenum 450 10206 1.09422 1.1035
5. Iridium 420 10484 1.08848 1.1035
Table 2. Comparison table showing pressure values at few points
VIII. REFERENCES
[1] H.S.Yadav and V.P.Singh, Pramana, 18, No.4, 331-
338, April, (1982).
[2] Ya.B.Zeldovich et al.,“Physics of shock waves and high
temperature hydrodynamic phenomena”, Academic
Press, New York, (1967).
[3] Ray Kinslow, “High velocity impact phenomena”,
Academic Press, New York, (1970), R.G.Mcqueen,
S.P.Marsh, et al.,“The Equation of state of solids from
shock wave studies”, page no. 294-415.
[4] V.N.Zharkov et. al., “Equations of state for solids at
high pressures and temperatures”, Consultants Bureau,
New York, (1971).
[5] V.P.Singh and Renuka DV, Study of phase change in
materials under high pressure, J. Appl. Physics, 86, no.
9, 1st Nov 99, pp. 4881-4.
[6] V.P.Singh et al., Proc.Indian Acad.Sci.(Engg.Sci.), 3,
Pt.2, (1980).
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