Thermochemical modeling of temperature controlled shock-induced chemical reactions in multifunctional energetic structural materials under shock compression X. F. Zhang, A. S. Shi, J. Zhang, L. Qiao, Y. He et al. Citation: J. Appl. Phys. 111, 123501 (2012); doi: 10.1063/1.4729048 View online: http://dx.doi.org/10.1063/1.4729048 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i12 Published by the American Institute of Physics.
Related Articles Lithiation of silica through partial reduction Appl. Phys. Lett. 100, 243905 (2012) Detection of filament formation in forming-free resistive switching SrTiO3 devices with Ti top electrodes Appl. Phys. Lett. 100, 223503 (2012) Strength of hydrogen bonds of water depends on local environment J. Chem. Phys. 136, 144305 (2012) Modeling of an ionic liquid electrospray using molecular dynamics with constraints J. Chem. Phys. 136, 124507 (2012) Epitaxial oxide bilayer on Pt (001) nanofacets J. Chem. Phys. 136, 044704 (2012)
Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors
JOURNAL OF APPLIED PHYSICS 111, 123501 (2012)
Thermochemical modeling of temperature controlled shock-induced chemical reactions in multifunctional energetic structural materials under shock compression X. F. Zhang,1,a) A. S. Shi,1 J. Zhang,1 L. Qiao,1 Y. He,1 and Z. W. Guan2 1
School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, People Republic of China 2 School of Engineering, University of Liverpool, Brownlow Street, Liverpool L69 3GQ, United Kingdom
(Received 31 August 2011; accepted 10 May 2012; published online 18 June 2012) Multifunctional energetic structural materials (MESMs) are a new class of energetic materials which release energy due to exothermic chemical reactions initiated under shock loading conditions. In order to analyze shock-induced chemical reactions (SICR) for MESMs, theoretical models have been developed to calculate the Hugoniot data which include the heat released by shock temperature controlled reactions. The temperature rise of porous materials due to shock compression is first calculated using a constant volume and pressure adjustment. Then the Arrhenius reaction rate and Avrami-Erofeev kinetic models are used to calculate the extent of reaction of MESMs under shock compression. Thermochemical models for shock-induced reactions, in which the reaction efficiency is considered, are given by combining the shock temperature rise with the chemical reaction kinetics. The Hugoniot relations and temperatures are calculated by using the proposed method. The models developed have been validated against the experimental SICR data involving Fe2O3=Al, Al=Ni, and Ti=Ni mixtures. It has been shown that the theoretical calculations correlate reasonably well with the corresponding experimental and simulation results. The models presented can be used to predict the C 2012 American Institute of reaction results of MESMs over a wide range of pressure satisfactorily. V Physics. [http://dx.doi.org/10.1063/1.4729048]
I. INTRODUCTION
Multifunctional energetic structural materials (MESMs) are evolving as a class of materials that integrate desirable characteristics of high energy density and rapid energy release properties along with at least one other designed functionality, for example, mechanical strength. Most of these materials under consideration include thermites, intermetallics, metal-polymer mixtures, metastable intermolecular composites (MICs), matrix materials as well as hydrides.1,2 Such mixtures are inert under ambient conditions, but will be triggered into reactions in the case of energy supplied by the passage of sufficiently strong shock waves.3 Due to the reaction characteristics, most MESMs are assembled without sintering and melting, which are related to extremely high temperatures. Molding process is the most common method used to manufacture this type of materials and therefore, most of the MESMs are typically multi-component mixtures with porosity. Under a shock compression, MESMs experience rapid mixing of constituent materials behind the shock discontinuity due to plastic flow and void collapse. Such process can result in temperature rise in the mixtures, which causes continuous chemical reactions. The challenge in designing MESMs also lies in improving their safety, reliability, and load bearing capability. In order to analyze the dynamic behavior of a particular porous mixture of MESMs by shock compression, a thermochemical model is required to describe the shock-induced chemical reaction (SICR) process.4 a)
Electronic mail:
[email protected].
0021-8979/2012/111(12)/123501/9/$30.00
In the case of homogeneous models for the shockinduced chemical reactions of MESMs, a few Hugoniot based models have been proposed to analyze the global effects of chemical reactions by shock wave propagation. The well known CONMAH model developed by Graham5 describes the reaction initiation mechanisms as: (1) large degrees of fluid-like plastic flow, (2) turbulent mixing, (3) defect generations, (4) cleansing of existing reacting surfaces, (5) formation of new reactant surfaces, and (6) elevated surface temperatures. Batsanov et al.6 proposed a Hugoniot model, in which only the change in internal energy of reaction from reactants and products was accounted. In the models proposed by Song and Thadhani,7 Yu and Meyers,8 and Boslough,9 the Mie-Gru¨neisen equation of state (EOS) was used to describe the change in specific internal energy and the change in specific volume due to chemical reactions and EOS of both reactants to products. A Ballotechnic model was developed by Bennett,10,11 in which integration paths over constant volume and pressure was applied for the adjustment of the reference curve during the SICR process of MESMs. As discussed before, most MESMs are multicomponent mixtures with porosity properties and the SICR processes are controlled by multi-scale properties of discrete systems of MESMs. Some models on heterogeneous chemical reactions are described here. Do and Benson12,13 presented a Mesoscale simulation, in which a heterogeneous chemical reaction with homogenization approach was employed to prevent numerical “quenching”. A heterogeneous chemical reaction model was proposed by Reding,14–16 which was used to describe shock-
111, 123501-1
C 2012 American Institute of Physics V
123501-2
Zhang et al.
induced chemical reactions that took place in reactive granular mixtures during shock compression. The results show that the pressure dependence on reaction is qualitatively different between the homogeneous and heterogeneous reactions. The multiscale chemical reaction (MSR) model is incorporated with the discrete particle simulation (DPS). This provides the details of phase morphology and arrangement, which is an integral part of designing this class of materials.15 During shock load compression or intense dynamic loading, a temperature rise is resulted from rapid mixing of constituents behind the shock front due to plastic deformation, jetting, fracture, and pore collapse. It is obviously that the shock temperature is controlled by the input stress which is directly influenced by impact velocity, whereas the reaction rate and efficiency are controlled by the shock temperature. This phenomenon was observed from the experimental results obtained by Ames,17,18 which show that the reaction efficiency is a strong function of impact velocity (shock pressure) for most MESMs. This behavior indicates that the energy supplied by the fracture process (which increases with increasing of impact velocity) is the primary determinant of the reactivity of a material. The data also indicate that once a critical threshold is reached, increases in efficiency are relatively smaller. Eakins19,20 also showed some partial reactions in their investigation on shock-induced reactions in NiþAl powder mixtures under shock compression. In order to analyze the shock-induced chemical reaction for MESMs, this paper presents a theoretical model to be used to calculate the Hugoniot data of shock induced chemical reactions of MESMs. The temperature rise due to shock compression is first calculated using an adjustment on constant volume and constant pressure. Then the Arrhenius reaction rate and Avrami-Erofeev kinetic models that are controlled by shock temperature are used to calculate the extent of reaction of MESMs. A thermochemical model for shock-induced reactions, which includes the reaction efficiency, is given by combining shock temperature rise with chemical reaction kinetics. The Hugoniot relation and temperature are calculated based on the above methods. Theoretical calculations are compared with the experimental results on shock compression of Fe2O3=Al, Al=Ni, and Ti=Ni granular mixtures.
II. SHOCK THERMODYNAMICS OF MESMs
During the shock-induced chemical reaction of reactive powder mixtures, some basic effects that must be accounted for are7: (i) the change in specific internal energy, (ii) the change in specific volume, and (iii) the change of EOS from reactants to products. Some basic Hugoniot calculations that include some of the above effects are available in the related literatures. Batsanov6 recommended a Hugoniot calculation with consideration of the change of specific internal energy due to heat of reaction. In their model neither the change of specific volume nor the EOS of the products are considered. Song7 and Yu8 assumed that a porous material was fully crushed-up into solid at a low shock pressure and the heat of reaction was simply added to the change in energy to yield
J. Appl. Phys. 111, 123501 (2012)
the specific internal energy along the Hugoniot of the reactive granular mixtures. Boslough9 found that reactions occurred at extremely fast rates as evidenced by temperature increases that could not be explained by the plastic work alone. For a reactive material, a chemical reaction occurs during shock process and the energy of the Hugoniot state is different by an amount equal to the heat of reaction (DH). The computed temperatures were compared with experimentally measured shock temperature, demonstrating that a shock induced reaction took place. Other models for shock-induced chemical reactions were developed by Bennett,10,11 which gave derivations of formulas for both constant volume (Mie-Gru¨neisen) and pressure. Those models included changes in specific internal energy, specific volume and EOS of reactants and products. The formulas show the emphasis of the importance of choosing the Hugoniot or isentropic compression curve of a solid product for the reference curve. Unfortunately, some basic errors appear when apply the above derivations of the energy jump due to shock induced reactions. The difficulty is evident that the complete reactions occur during the SICR process and that the errors could cause the heat of a complete reaction to be accounted even at a very low pressure. A. Equation of state (EOS) for MESMs
As discussed before, most MESMs are multi-component mixtures and have some porosities due to their reaction properties and manufacture processes. In order to analyze the dynamic behavior and temperature rise of a granular mixture with porosities in any given situation, a mixture theory of EOS that can be used to describe MESMs is required. In the present paper, a cold energy mixture theory is applied for the equation of state of MESMs subjected to shock compression. It is a combination of the cold energy mixture theory and the extension of Wu and Jing’s work by Geng.2–23 The full details of the relevant mixture theories summarized here can be referred to the work undertaken by McQueen and Marsh,24 Wu and Jing,25,26 Geng,21–23 and Zhang et al.27 1. Equation of state for solid multi-component mixtures
One popular model that is used extensively is the MieGru¨neisen EOS,28 which is given by V Pc ðVÞ Ec ðVÞ cðVÞ ; PðVÞ ¼ V 1 ðV0 VÞ cðVÞ 2
(1)
where V0 is the initial specific volume of solid, V is the postshock specific volume, cðVÞ is the Gru¨neisen coefficient which is assumed to be a function of the specific volume V, Pc ðVÞ, and Ec ðVÞ are the cold (elastic) pressure and cold (elastic) internal energy, respectively. From the Hugoniot theory of multi-component mixtures proposed by Krueger and Vreeland,29 two preliminary
123501-3
Zhang et al.
J. Appl. Phys. 111, 123501 (2012)
assumptions are made, i.e., (1) components are at an equal pressure and (2) particle velocity and an average method are to be used to determine the properties of mixtures. VðPÞ ¼
N X
mi Vi ðPÞ;
(2)
mi ECi ðVÞ;
(3)
i¼1
EC ðVÞ ¼
N X i¼1
N X
mi ¼ 1;
(4)
i¼1
where mi is the mass fraction of constituent materials. The cold (elastic) internal energy can be calculated by using the zero-mixture theory in Eq. (3). From the Born-Mayer theory for cold (elastic) pressure and internal energy,30 the Pc ðVÞ and Ec ðVÞ are expressed in the form of 3Q 1 1 1=3 1=3 EC ðdÞ ¼ exp½qð1 d Þ d þ 1 ; (5) q0K q q PC ðdÞ ¼ Qd2=3 fexp½qð1 d1=3 Þ d2=3 g; d¼
q V0K ; ¼ V q0K
(6) (7)
where Q and q are two parameters for cold energy, q 0K(V0K) is the initial density of a material or mixture at 0K. d(V0K= V) is the compressibility. Following the descriptions given by Jinbiao,31 the parameters Q and q can be written as: k¼
1 q2 þ 6q 18 ; 12 q2
(8)
Qðq 2Þ ; 3q0K
(9)
C20 ¼
c2 C0 ¼ C00 1 þ 2k0 0 1 av T0 ; 4 0 k 1 c20 0 1 av T0 ; k¼k 1þ 2 8 k0
(10)
(11)
h i 13 2 13 q d exp q 1 d 6d 1 h i cðdÞ ¼ : 1 6 q exp q 1 d3 2d
(13)
2. Equation of state for porous multi-component mixtures
The EOS of porous materials has been studied extensively in terms of theoretical models and experiments,21,22 where the shock temperature is several thousands of degrees Kelvin. The influence of thermo-electrons becomes important in such a high temperature range. According to Geng’s results,23 there is Rs b0 T 2 Vs ð1=2Þ ðV00 V0 Þ þ : (14) V ¼ Vs þ 2 Rs 4P V0K Here Vs is the specific volume of the corresponding solid material under the same pressure, which can be determined from the solid Hugoniot relation. b0 is the initial electronic specific heat. T is temperature. Rs is the Wu–Jing parameter of the lattice part. Using an analysis similar to that used in the Mie– Gru¨neisen EOS development of the Gru¨neisen coefficient, WuJing parameter R can be written as13–15 d ln HD d ln V cP cP @V R¼ ¼ ¼ ; d ln V T d ln P T VK T V @P T (15) where HD is the Debye temperature and KT is the isothermal bulk modulus. Here, by using the zero temperature isothermal bulk modulus as the isothermal bulk modulus and substituting the Born-Mayer potential in Eq. (6) into Eq. (15), the item ð@V=@PÞT can then be expressed as @V d1 V : ¼ 2 q @P T Q exp½qð1d1=3 Þ d1=3 þ d2=3 43 d1=3 3 3 (16) B. Temperature rise of MESMs under shock compression 1. Shock temperature along constant volume
where C0 is the zero-pressure bulk sound speed, k is related to the binding energy and repulsive potential of a solid material, av is the coefficient of cubic expansion, C00 and k0 are the material constants at 0K. It has been experimentally demonstrated for most metallic materials that the Gru¨neisen coefficient c can be described by the Dugdale-MacDonald relationship24 as
dE ¼ TdS PdV:
2=3
VC d2 ðPV C Þ=dV 2C 1 : cðdÞ ¼ 2 dðPV 2=3 Þ=dV C 3
As described by Meyers,32 steady-state shock-wave propagation at a constant velocity in a thermo-viscous solid results in compression of the material and a subsequent temperature rise. The thermodynamics process at the shock front is assumed to be adiabatic. The first law of thermodynamics is then expressed by
(12)
C
Substituting the Born-Mayer potential in Eq. (6) into Eq. (12) c can be expressed as
(17)
An appropriate thermodynamic expression of TdS is @P dV; TdS ¼ Cv @T þ T @T
(18)
123501-4
Zhang et al.
J. Appl. Phys. 111, 123501 (2012)
where Cv is the heat capacity at constant volume and c is the Gru¨neisen coefficient. A differential form of the above equation can be written as dT cTCv 1 dP P ¼ (19) þ ðV0 VÞ þ : Cv V dV H 2 dV H 2 By solving the above differential equation, the standard solutions can be obtained for temperature as a function of the specific volume on any point along the Hugoniot curve of a material.
TABLE I. The material parameters related to the calculations. Material Irona Coppera Aluminum-2024a Nickela Tungstena Hematiteb Titaniuma Material
2. Shock temperature along constant pressure
According to Geng’s theory22,33 the differential of the specific enthalpy can be written as dH VdP ¼ Cp dT Cp T
R dP; P
(20)
where Cp is the heat capacity at constant pressure. From the Hugoniot relation and H ¼ E þ PV, an alterative expression for dH VdP can be written as 1 dV V0 V þ P dP: (21) dH VdP ¼ 2 dP
q0(g=cm3)
C00 (km=s)
k0
c0
av =(105=K)
7.856 8.924 2.784 8.875 19.200 5.274 4.510
3.780 3.973 5.370 4.590 4.040 4.450 4.695
1.652 1.498 1.290 1.440 1.230 1.323 1.146
1.81 1.97 2.18 2.00 1.78 1.99 1.33
3.51 5.01 6.93 3.81 1.38 1.50 3.51
Q=(GPa)
Ironc 39.964 11.280 Copperc 59.717 9.888 Aluminum-2024 37.890 8.417 Nickelc 81.740 8.991 Tungsten 129.588 8.583 Hematiteb 51.475 8.150 Titaniuma 50.990 7.585
b0 CV CP (erg g1 K2) (J g1 K1) (J g1 K1) 193.95 174.47 415.30 180.32 83.85 150.3 271.43
0.414 0.372 0.850 0.398 0.130 0.662 0.496
0.448 0.386 0.903 0.444 0.133 0.684 0.523
a
Xu and Zhang40 Reding and Hanagud3 c Huayun23
b
ln By combining the above two relations, an ordinary differential equation for the shock temperature along constant temperature is derived as dT RT 1 dV ¼ : (22) V00 V þ P dP ð3R þ 1ÞP 2CP dP
q
b Ru A Ea 1 ¼ ln ; 2 R Tr Tr Ea
(25)
where b is the heating rate, Tr is the peak temperature of the differential scanning calorimetry (DSC) curve. From the above description, the SICR process is assumed to be a temperature controlled reaction process. In the present research, the Avrami-Erofeev kinetic model,35 which is based on the controlled rate thermal analysis, is
C. The extent of reaction of MESMs controlled by temperature
TABLE II. The calculated and experimental shock temperature results of solid and porous iron.
Solid-state reaction kinetics is based on the activated state theory, with the reaction rate being written as
Material
dy ¼ k f ðyÞ; dt
(23)
Pressure (GPa)
T1Cal (K)
T2Cal (K)
Texp(K) (Refs. 38 and 41)
Solid iron (q0 ¼ 7856 kg=m3)
40 60 80 100 120 140 160 180 200 220 240 260
628 946 1359 1850 2412 3040 3711 4430 5223 6007 6850 7702
592 898 1303 1788 2347 2968 3640 4347 5104 5857 6660 7486
657 1007 1444 1953 2523 3145 3812 4520 5264 6040 6846 7680
Porous iron (q00 ¼ 6904 kg=m3)
80 85 100 116 134 145 171
1159 1254 1559 1909 2354 2655 3354
2711 2911 3513 4202 4998 5505 6732
2700 2890 3400 4063 4457 5016 5440
where k is a rate constant and has the unit of s1, y is extent of reaction and t is the reaction time. For the kinetics of thermally stimulated solid-state reactions, the reaction rate constant (k) is exponentially dependent on the inverse of absolute temperature T. The following formula has been widely accepted34: Ea ; (24) k ¼ A exp Ru T where Ru is the universal gas constant, T is the absolute temperature, A and Ea are apparent pre-exponential factor and apparent activation energy, respectively. For most of the solid reaction samples, the preexponential factor can be computed using experimental data as
123501-5
Zhang et al.
J. Appl. Phys. 111, 123501 (2012)
TABLE III. Reaction parameters for three typical MESMs. MESMs a
AlþFe2O3 (25.3=74.7) AlþNi(23.88=76.12)b NiþTi(55.1=44.9)c
q0 (g=cm3)
Ea (KJ mol1)
DHR (KJ g1)
n
4.249 5.823 5.724
145.00 348.40 201.00
3.97 1.38 0.63
0.25 0.10 0.20
a
Fan et al.42 White et al.43 c Rodriguez et al.44 b
developed by McQueen et al.,38 is used to calculate the EOS of mixtures of reactants and products, i.e., VðPÞ ¼ ð1 yÞVr ðPÞ þ yV p ðPÞ; V V0 V0 ¼ ð1 yÞ þy ; c0 r c0 p c
(28) (29)
where Vr ðPÞ and Vp ðPÞ are the specific volume for reactants and products, respectively.
used to describe the solid state reaction under a high rate of temperature rising. According to Umbrajkar’s results36 of exothermic reactions of AlþCuO nano-composites, for temperature rising rates up to 1000K=s, a n-dimensional nuclear=growth controlled reaction is suitable to model the solid state reaction, i.e. f ðyÞ ¼ nð1 yÞ½lnð1 yÞðn1Þ=n ;
(26)
where n is the coefficient related to boundary conditions and reaction mechanisms. The emphasis of the present research is to determine the relationship between the shock temperature and SICR results. Hence the reaction rate is taken as a linear function of time, i.e. Ct ¼ dy=dt. According to Ortega’s results,37 the Avrami-Erofeev equation can be written as the first derivative of absolute temperature T with respect to extent of reaction “y” as follows. dT Ru T 2 1 n lnð1 yÞ þ n 1 ¼ : (27) Ea 2y nð1 yÞ½lnð1 yÞ dy
D. Hugoniots for partial reaction of MESMs 1. Mixture theory for the equation of state of reactants and products in a partial chemical reaction
In order to calculate Hugoniot curves for the partial chemically reactive MESMs, a multi-component mixture,
FIG. 1. Extent of reaction against temperature for different MESMs.
FIG. 2. Calculated temperature rises for no reaction, partial reaction and complete reaction: (a) Al=Fe2O3(25.3=74.7); (b) Al=Ni(23.9=76.1); and (c) Ni=Ti(55.1=44.9).
123501-6
Zhang et al.
J. Appl. Phys. 111, 123501 (2012)
TABLE IV. Computational and experimental shock temperatures (Ref. 9) with chemical reaction. Material AlþFe2O3 (25.3=74.7)
q00 (g=cm3)
P* (GPa)
TS (K)
TR (K)
TR ðKÞ
y
2.22 2.20
4.4–4.7 3.9–4.1
886–924 826–854
3300–3750 2612–2944
3400 3000
0.43 0.32
2. Pressure and temperature rise for a partial reaction of MESMs
Following the assumption of the SICR model for MESMs, the energy of the Hugoniot state is different by an amount equal to the heat of reaction for a reactive material undergoing a partial reaction during the shock compression. Here we use the constant volume (Mie-Gru¨neisen) adjustment method presented by Bennett10 to calculate Hugoniot curves for MESMs. Thus, the general Hugoniot pressure of a reactive material is written as: ðV0 VÞ V yDH þ PH 2:0 c P¼ : (30) 1 V ðV00 VÞ 2 c In Eq. (30), the Gru¨neisen ratio, V=c, and the specific volume V refer to the mixture of reactants and products. Based on Eq. (19), the temperature that includes the heat of reaction can be written as dT cT 1 dP P þ ¼ ðV0 VÞ þ yDH=Cv : (31) dV V 2Cv dV H 2Cv
III. APPLICATIONS OF SHOCK-INDUCED CHENMICAL REACTIONS OF MESMs
In the present research, the Hugoniot relation for a solid mixture of MESMs is firstly determined by using the mixture theory associated with the cold internal energy (EC ). The material constants Q and q are obtained by using the non-linear iterative method known as the Levenberg-Marqyardt algorithm in order to obtain the best fit of EC -V curve. Then the cold pressure PC is calculated using Eq. (6). The Hugoniot curves for MESMs with porosity are further determined by applying the method extended by Geng,21,23,39 based on the Wu-Jing’s theory. The material parameters related to the calculations for seven elements are tabulated in Table I, which are taken from Xu and Zhang,40 Reding3 and Geng23 The Gru¨neisen coefficient c and Wu-Jing parameter R for a single element and multi-components are calculated using Eqs. (13) and (15).
are used to determine the shock temperatures along constant volume and pressure, respectively. Table II shows the calculated results of solid and porous Iron under compression loadings. T1Cal and T2Cal are the results using the constant volume and pressure methods, respectively. It can be seen from Table II that the calculated temperature rises for solid Iron obtained from both methods are fairly similar, which agree closely with the results from McQueen et al.38 However, the constant volume method gives lower prediction values than those given by Zhang41 for porous Iron and the predicted values from the constant pressure method agree well with the corresponding simulation results (Table II). The results also indicate that a higher temperature is reached for a given pressure due to the thermoelectronic contribution of the porous Iron. B. Shock-induced chemical reactions of MESMs
In order to evaluate the present model, the Hugoniots of the reactive granular mixtures were calculated for three typical MESMs, AlþFe2O3, AlþNi, and NiþTi. The material parameters related to the calculations are already introduced in Table I, and some reaction parameters are listed in Table III. Fig. 1 shows extent of reaction against shock temperature for the three MESMs. Here, the first derivative of the AvramiErofeev equation gives a direct approach to calculate extent of reaction, which is depended on active energy, temperature, and parameters related to reaction mechanisms. Fig. 2 shows that, for Hugoniot pressure up to 10 GPa, the shock temperature rises corresponding to no reaction, partial reaction and complete reaction of the three typical MESMs. The temperature vs. pressure curve related to the no reaction is parallel to that corresponding to the complete
A. Calculations of temperature rise using different methods
As this paper aims to present thermochemical models for the SICR process of MESMs, a suitable method is required to determine the temperature rise of solid and porous materials under shock compression. Constant volume method and constant pressure method in Eqs. (19) and (22)
FIG. 3. Comparisons between the calculated and the experiment results (Refs. 19 and 20) for Al=Ni (23.9=76.1).
123501-7
Zhang et al.
J. Appl. Phys. 111, 123501 (2012)
TABLE V. Computational and experimental shock velocity (Refs. 19 and 20) with chemical reaction. Material AlþNi (23.9=76.1)
TMD (%)
Input stress P(GPa)
Pr (GPa)
US (km=s)
UR (km=s)
UR ðkm=sÞ
60.7 55.3 58.8 60.0 59.3 60.2 60.4
0.510 1.550 2.850 3.590 4.090 5.310 5.480
0.512 1.566 2.970 3.853 4.500 6.350 6.640
0.599 1.024 1.401 1.572 1.682 1.905 1.944
0.599 1.033 1.430 1.632 1.764 2.080 2.137
0.611 1.027 1.355 1.462 1.749 1.865 1.790
reaction, as the heat of the complete reaction is added in the Hugoniot equations. The curve of the partial reaction shows a nonlinear relation to the Hugoniot pressure. The curves of the no reaction and the partial reaction are indistinguishable at low pressures (
