International Journal of Research in Computer ScienceeISSN 2249-8265 Volume 2 Issue 1 (2011) pp. 7-10© White Globe Publica...
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Simulation of Generation of High Pressure and Temperature in Metals under Shock Loading                                   ...
10                                                                                                              Dr.V.P.Sin...
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SIMULATION OF GENERATION OF HIGH PRESSURE AND TEMPERATURE IN METALS UNDER SHOCK LOADING

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Pressure and Temperature in different metals including radioactive materials behind converging shock waves, is simulated using generalized form of equation of state. Tait’s equation of state of metals, is valid for pressures of the range of few mega bars and takes into account only elastic pressures. At such high pressures, metal undergo phase change and normal equation of state no more is valid. At such pressures, temperature in metals becomes very high and thermal and excitation pressures dominate over elastic pressure. It is observed that as shock approaches the center of sphere, excitation pressure dominates elastic as well as thermal pressure.

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SIMULATION OF GENERATION OF HIGH PRESSURE AND TEMPERATURE IN METALS UNDER SHOCK LOADING

  1. 1. International Journal of Research in Computer ScienceeISSN 2249-8265 Volume 2 Issue 1 (2011) pp. 7-10© White Globe Publicationswww.ijorcs.org 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 removedmetals including radioactive materials behind at ultra high pressure. In this process, a very strongconverging shock waves, is simulated using compression of a condensed medium generates ageneralized form of equation of state. Tait’s equation colossal internal pressure, even in the absence ofof state of metals, is valid for pressures of the range of heating due only to the repulsive forces between thefew mega bars and takes into account only elastic atoms. The material is also very strongly heated bypressures. At such high pressures, metal undergo shock waves and, this result in the appearance of aphase 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 functionbecomes very high and thermal and excitation of temperature and CV also varies with temperature. Inpressures dominate over elastic pressure. It is the present paper we have taken Einstein Equation forobserved that as shock approaches the center of thermal energy and have studied the pressure andsphere, excitation pressure dominates elastic as well temperature variations behind the converging shockas thermal pressure. waves. Results of variation of pressure by conventional method and by present method areKeywords: 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 energycondensed media, such as metals and radioactive of materials given by Einstein Equation of state,materials, is of great theoretical and practicalimportance. Behavior of metals under high pressures hνhas 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 getlinear equation fails when we analyze the material expressions for pressure and temperature behindundergoing crystallographic phase change under high imploding shock waves. In equation (1) N, h, ν, k arepressure. Thus, knowledge of the thermodynamic Avagadro’s Number, Plank’s constant, naturalproperties of the materials is necessary to study their frequency of solid material, and Boltzman’s constantbehavior under high pressures and temperatures. respectively.Although, no appreciable [2] difficulties areencountered 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 2mthermodynamic 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 ofloading 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 takenas sum of cold, excitation and thermal energy. By Ec = − ∫ Pc dV (2b)taking thermal energy as CV T ,with CV constant, it wasEquations for cold, thermal and excitation pressure are, www.ijorcs.org
  2. 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 TP = 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 δUsing equations(1)-(3), total pressure P and energy Ebehind 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 ofE=   + 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 theδ = = Shock strength . We observe from ρ0 pressure, temperature, shock velocity and particle velocity of the material behind the shock front. Theserelations (4) that pressure P is function of density ratio parameters are expressed in terms of δ and T. It is notδ and temperature T. We take δ as the independent possible to solve these equations explicitly in terms ofparameter and express all other paraameters in terms one parameter, say δ. Therefore temperature isof δ. Elliminating U and u from equations (1) of [2], expressed in differential form and evaluated by integrating equation (7) using Runge –Kutta method ofone 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 thewhere P=P2-P0 and E=E2-E0. center of the sphere, we use Energy Hypothesis [6]. Transmitted pressure due to the initial compression ofDifferentiating 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 usingcalculations, 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 byf (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) www.ijorcs.org
  3. 3. Simulation of Generation of High Pressure and Temperature in Metals under Shock Loading 9 E (δ ) VI. ACKNOWLEDGEMENTwhere is given by equation (4b). Present work was funded by Defence Research andKnowing 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, ofwhich 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 SSin Figure 1, whereas variation of temperature for these Femetals is given in figures 2. Transmitted pressure in 1.00E+01 AlAluminum is low at the metal-explosive interface butrises fast and becomes infinite at radius equal to 0.3. 1.00E+00Similarly 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 Radiuscompared to lighter metals, and becomes infinite atradius ratio 0.1. In figure -3, we have plotted variationof cold, thermal and elastic pressures in Iron. It is seen Figure-1: Variation of pressure VS dimensionless radius for different materialsthat in the beginning thermal and elastic pressures areless than cold pressure but ultimately, thermal pressureand excitation pressure increase and become more than 10000000elastic pressure. Excitation pressure increases very fast Temperature(oK) Irand get maximum at radius 0.3 and then starts 1000000 Moreducing. 100000 SS It has been observed that as the density of material Fe 10000increases, its maximum compression as well as Altransmitted pressure from explosive decrease (Table- 10002). 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 radiususing Hugoniot equation of state, it has been observed for different materialsthat at high pressures, certain materials show suddenincrease in pressure values or even a discontinuity inthe pressure curve as the shock wave converges from Variations of different Pressuresthe 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+16thermal and excitation energy of metals. Figure-3 1.00E+14 1.00E+12gives three pressure components for a typical metal, 1.00E+10 Pcsay Iron. It shows that, PT and Pe are of same order 1.00E+08 PTinitially, but ultimately Pe becomes much more that PT. 1.00E+06 PeNear the centre, effect of excitation pressure is to 1.00E+04expand where as it is resisted by elastic pressure, 1.00E+02 1.00E+00which creates a uncertainty at such points, resulting in 35 45 55 65 75 85 95 3the abnormal behavior of Pe curve. Thermal pressure 0. 0. 0. 0. 0. 0. 0. 0.first increases and then becomes almost constant and Radiuswhen shock approaches near the centre, again rises.Figure-1 gives pressure variations, where as figure- 2 Figure 3: Comparison of three components of pressuregives variation of temperature as shock converges. www.ijorcs.org
  4. 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). www.ijorcs.org

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