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AD-772 919
SENSITIVITY OF MATERIAL RESPONSE CALCU- LATIONS TO THE EQUATION OF STATE MODEL
William 0 . Wray
Systems, Science and Software
Prepared for:
Army Ballistic Research Laboratories
December 1973
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um National Technical Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road, Springfield Va. 22151
UNCLASSIFIED tif ciirity CI«nific«tion ihll2Sl3.
DOCUMENT CONTROL DATA R&D (SrcutHy claitllhiilon ol llllr, body ol mbitntl und Indtxlng tnnolmllen mu»l 6» mfrrd mMn Ihm urmrmll npoil I» rlmfllltd)
I OHICINATINIS ACTIVITv (Caiparat« aufftorj
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1 REPORT TITLE
Sensitivity of Material Response Calculations to the Equation of State Model
4 DESCRIPTIVE NOTES CTVp« o/»pert «nil/neliM<r* da>«t;
Final Report - May 73 - October 73 s *u THORISI (Flfl nrnan, mlddlt Inlllml, Imtl nmmm)
William 0. Wray
a REPORT DATE
DECEMBER 1973
7«. TOTAL NO. OF PACES
59 76. NO. OF REFS
JX •a. CONTRACT OR GRANT NO.
DAAD05-73-C-0489 b. PROJECT NO.
RDT^E Project 1W062105A659
M. ORICINATOR'S REPORT NUMMCRI*)
SSS-R-73-1910
tb. OTHER REPORT NOIt) (Any olhmr number« dtml mmy b» m»ml0tmd Oil* nporl)
BRL CONTRACT REPORT NO. 130 10. DISTRIBUTION STATEMENT
Approved for public release; distribution unlimited.
II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
US Army Ballistic Research Laboratories Aberdeen Proving Ground, MD 21005
This report examines the sensitivity of material response calculations to the choice of equation of state model. Three equation of state models, all of which are available as subroutines in the RIP code, are considered: 1) modified PUFF; 2) RIP mixed phase; and 3) GRAY. Each of these models is used to calculate the one-dimensional response of aluminum, beryllium and titanium; this series of calculations is performed for two distinct x-ray sources. The sensitivity of the calculated material response to the choice of equation of state model is characterized in terms of the generated impulse and the peak propagating stress at the time the radiation source is cut off. For the calculations presented in this report, the three equation of state models are in fairly good agreement.
NATIONAL TECHNICAI INfORMATlON SERVICE U c nppartrm>nt of Cnmmerr.p
SpnnRfieW VA 22151
1%^ fMM 4 M TO "«PLAC«» DO POHM (471. I JAN I %j\j I M*V •■14 10 OUOLBT« POR ANMV «*•■■ |
•4, «MICH I«
I. UMOASSI cunty ClaMlflcatioa
UNCLASSIFIED Security CUttl.lcdlon
KEY WORD*
Equation of State GRAY Equation of State RIP Mixed Phase Equation of State Modified PUFF Equation of State Aluminum Beryllium Titanium
l(k.
UNCLASSIFIED
Security CUasification
BALLISTIC RESEARCH LABORATORIES
BRL CONTRACT REPORT NO. 130
SSS-R-73-1910
DECEMBER 1973
SENSITIVITY OF MATERIAL RESPONSE CALCULATIONS TO THE EQUATION OF STATE MODEL
FINAL REPORT
William 0. Wray Systems, Science § Software
LaJolla, California
Contract No. DAAD05-73-C-0489
and
RDTfiE Project No. 1W062105A6S9
Approved for public release; distribution unlimited.
ABERDEEN PROVING GROUND, MARYLAND
ABSTRACT
This report examines the sensitivity of material response calculations to the choice of equation of state model. Three equation of state models, all of which are available as sub- routines in the RIP code, are considered: 1) modified PUFF; 2) RIP mixed phase; and 3) GRAY. Each of these models is used to calculate the one-dimensional response of aluminum, beryllium and titanium; this series of calculations is per- formed for two distinct x-ray sources. The sensitivity of the calculated material response to the choice of equation of state model is characterized in terms of the generated impulse and the peak propagating stress at the time the radiation source is cut off. For the calculations presented in this report, the three equation of state models are in fairly good agreement.
in
ACKNOWLEDGEMENT
This report and the associated research was supported by Contract No. DAAD05-73-C-0489 under the sponsorship of the Ballistic Research Laboratory in Aberdeen, Maryland. The pro- ject was funded and monitored through the efforts of Joseph F. Lacetera of the Effects Analysis Branch.
Mr. Robert A. Cecil and Mr. C. Michael Archuleta of the Systems, Science and Software (S3) scientific staff made signi- ficant technical contributions to the performance of the research and calculations reported herein. The EOSGEN code, which is currently undergoing refinement and documentation, is primarily the result of their efforts. Additionally, John R. Triplett, also a member of the S3 scientific staff, provided invaluable theoretical support by interpreting and evaluating the GRAY equation of state.
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CONTENTS
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Page
1. INTRODUCTION
1.1 OBJECTIVE AND PROCEDURE 1
1.2 EQUATION OF STATE MODELS 1
1.3 SOURCE DESCRIPTION 3
2. EQUATION OF STATE DATA 6
2.1 MODIFIED PUFF 6
2.2 RIP MIXED PHASE 6
2.3 GRAY 7
3. RESULTS AND DISCUSSION 26
4. CONCLUSIONS AND RECOMMENDATIONS 31
REFERENCES 36
APPENDIX A - THE EOSGEN CODE 37
A.l THE VIRIAL EXPANSION' 37
A. 2 DEFINITION OF THE MIXED PHASE BOUNDARY 39
A.3 THE JOIN FUNCTION 42
APPENDIX B - CORRECTIONS TO EOS2 47
APPENDIX C - CORRECTIONS TO EOS3 50
VI i
LIST OF FIGURES
Page
Figure 1. Incident energy spectrum for SPEC 1 radia- tion source (normalized to unity).
Figure 2. Normalized energy flux vs time for SPEC 1 radiation source.
Figure 3. Intersection of isotherms with mixed phase boundary. 32
Figure 4. Schematic pressure-volume diagram illustrating new join procedure for GRAY equation of state. 34
Figure A.l Intersection of isotherms with mixed phase boundary. 40
Figure A.2 Schematic representation of the RIP mixed phase equation of state. 43
Figure B.l Schematic representation of the RIP mixed phase equation of state. 48
Figure C.l Schematic representation of GRAY EOS. 51
Preceding page blank
ix
LIST OF TABLES
TABLE 1 Modified PUFF Parameters for Aluminum
TAbI.E 2 Modified PUFF Parameters for Berylliua
TABLE 3 Modified PUFF Parameters for Titanium
TABLE 4 EOSGEN Input Data for Aluminum
TABLE 5 EOSGEN Input Data for Beryllium
TABLE 6 EOSGEN Input Data for Titanium
TABLE 7 EOSGEN Output for Storage in /EQST/COMMON
TABLE 8 /EOSTAB/COMMON
TABLE 9 EOSTAB Input: Aluminum
TABLE 10 EOSTAB Input: Beryllium
TABLE 11 EOSTAB Input: Titanium
TABLE 12 Gray Equation of State Parameters for Aluminum
TABLE 13 Gray Equation of State Parameters for Beryllium
TABLE 14 Gray Equation of State Parameters for Titanium
TABLE 15 Summary of RIP Calculations
TABLE 16 Results of RIP Calculations on Aluminum
TABLE 17 Results of RIP Calculations on Beryllium
TABLE 18 Results of RIP Calculations on Titanium
Page
8
9
10
11
12
13
14
15
17
19
21
23
24
25
27
28
29
30
Preceding page blank
1. INTRODUCTION
1.1 OBJECTIVE AND PROCEDURE
The purpose of this report is to determine the sensi- tivity of material response calculations to the choice of equation of state model. Three equation of state models, all of which are available as subroutines in the RIP1 J code, are considered: 1) modified PUFF; 2) RIP mixed phase; 3) GRAY. Each of these models is used to calculate the one-dimensional response of aluminum, beryllium and titanium; this series of calculations is performed for two distinct x-ray sources. The sensitivity of the calculated material response to the choice of equation of state model is characterized in terms of the generated impulse and the peak propagating stress at the time the radiation source is cut off. Whenever possible, handbook values are used for the required equation of state data; no attempt was made to adjust the data in order to reduce the differences between the three models.
1.2 EQUATION OF STATE MODELS
The first equation of state considered in this report is referred to as the modified PUFF equation of state. I1! The modified PUFF equation of state is the simplest of the three models; it consists of a Mie-Gruneisen solid equation of state for the compressed region which is joined at the ambient density to a function which limits to an ideal gas equation of state as the density approaches zero. Pressure continuity is maintained at the ambient density for all ener- gies, but the sound speed is continuous only at the zero reference energy; the discontinuity in sound speed becomes more severe as the energy is increased.
1h& RIP mixed phase equation of state is based on a virial expansion which was applied by Tuerpe, Keeler and McCarthy 1*1 to the study of aluminum at high temperature and pressure. The equation of state was later generalized to other metals by D. Steinberg and reported in an internal memo dated August 22, 1967 at Lawrence Livermore Laboratory. Incorporation of the Steinberg equation of state into the RIP code involved the following tasks:
1. The caloric equation of state was inverted and the resulting expression for the tem- perature was substituted into the thermal
equation of state, thereby yielding a formula for the pressure as a function of energy and density. Seven independent coefficients appear in this formula and they are evaluated by the E0S6EN* code and subsequently input to EOSTAB common in the RIP code.
2. A Maxwell construction was required in order to determine corresponding values of pressure, energy, sound speed and density on the mixed phase boundary. This task is also performed by the EOSGEN code, and points on the mixed phase boundary are subsequently input to EOSTAB common in the RIP code.
3. A join function was developed which provides for pressure and sound speed continuity between the virial expansion for expanded states and the Mie-Gruneisen solid equation of state for the compressed region. The coefficients in the join function are evaluated by the EOSGEN code and subsequently input to EQST common in the RIP code.
A brief discussion of the EOSGEN code and associated corrections** to the E0S2 subroutine in the RIP code is pre- sented in Appendix A. All of the work presented in Appendix A was performed in support of Contract DAAD05-73-C-0489 and was required in order to perform hydrodynamic calculations using the RIP mixed phase equation of state in the RIP code.
The third equation of state considered in this research is the GRAY three-phase equation of state for metals developed by E.B. Royce 13] at Lawrence Livermore Laboratory. Material near normal density, in the solid-liquid region, is described by a scaling-law equation of state for metals developed by Grover.t*) The scaling-law equation of state includes a Gruneicen description of the solidt^J normalized to the experi- mental Hugoniot. Material in the liquid-vapor region is treated according to an equation of state developed by Young and Alder;[6] their van der Waals model uses an analytic repre- sentation of the classical hard-sphere equation of state with a van der Waals attractive term added. The complete equation
* Documentation of the EOSGEN code is in progress.
** It was necessary to develop a new join function in order to allow pressure and sound speed continuity between the com- pressed and expanded equations of state.
of state, as described in Reference 3, Is developed by analytically joining the Grover scaling law and the Young-Alder models at a volume in the range 1.3 to 1.5 times normal volume. This is accom- plished by adding correction terms to the Young-Alder equation of sate employed on the low density side of the join volume.
It was necessary to make several corrections to the E0S3 subroutine in the RIP code before performing calcu.atlons with the GRAY equation of state. These corrections are presented in Appendix B.
1.3 SOURCE DESCRIPTIONS
The first radiation source (SPEC 1) was prescribed by the Ballistic Research Laboratory.[7] The energy spectrum and the time dependence of the energy flux are presented in Figure 1 and Figure 2 respectively. The second peak In the energy spectrum is due to a typographical error in Reference 7. The calculations were performed before this error was recognized, but since the same spectrum was used throughout, the calculations still represent a valid sensitivity study. A fluence level of 5952 cal/cm2 was used for all calculations performed with this radiation source.
The second radiation source (SPEC 2) considered 1n this study Is a 2-keV blackbody with a 10-nsec square pulse shine time. Appropriate fluence levels were selected in order to guarantee that several zones of material are vaporized at the front surface of each metal considered. The fluence levels selected are:
Aluminum: 100 cal/cnr
Beryllium: 2000 cal/cm2
Titanium: 100 cal/cm2
9.0
8.0 -
% 7.0
r*
§
o •n i o
^
6.0 -
5.0 *
6 8
«
3.0
2.0
50 100 150
Energy (keV)
300
Figure 1. Incident energy spectrum for SPEC 1 radiation source (normalized to unity) .
1.0
0.8 1.0 Time (10"6 sec)
Figure 2. Normalized energy flux vs time for SPEC 1 radia- tion source.
2.1 MODIFIED PUFF
The modified PUFF equation of state parameters for alurinum, beryllium, and titanium are presented in Tables 1, 2, and 3 respectively. All of the parameters appearing in the tedales were obtained from Reference 8 with the exceptions of ELV» BLM' and ESM which were calculated as described in the following paragraph.
The incipient melt energy, Eg«, is obtained by multi- plying the difference between the melt temperature (Refer- ence 9, page D-103) and the reference temperature, 20° C, by the heat capacity at constant pressure (Reference 9, D-97) . The complete melt energy. ELM» is obtained by adding the heat of fusion (Reference 10) to the incipient melt energy. Finally, an approximate value for the incipient vaporization energy, ELV» is obtained by multiplying the difference be- tween the melt temperature and the boiling temperature (Refer- ence 9, page D-103) by the heat capacity and adding the result to the complete melt energy. These three energy parameters are used for the momentum edits (the momentum of each thermo- dynamic state is displayed) and for determining the energy density, EgM» at which the material should have zero strength. All of the data for modified PUFF is stored in the ESTCON array in EQST common.
2.2 RIP MIXED PHASE
The first thirteen parameters listed in Tables 1, 2, and 3 are required as input to EQST common in order to apply the EOS2 equation of state subroutine in hydrodynamic calcula- tions on aluminum, beryllium and titanium, respectively. The remaining equation of state data are generated by the EOSGEN code which is described in Appendix A.
Fundamental material properties of aluminum, beryllium, and titanium which are required as input data to the EOSGEN code are presented in Tables 4, 5, and 6 respectively. The ambient densities are the same values used for the modified PUFF equation of state and the reference temperature for all three materials is assumed to be 300° K. The bulk modulus
of aluminum was calculated according to the relation
ft - Y
where 6» is the bulk modulus, Y is Young's modulus and a is Poisson's ratio. Numerical values for Y and a were obtained from Table 3.1 on page 44 of Reference 1. The bulk moduli of beryllium and titanium were taken equal to the linear Hugoniot coefficients, Al, used in the modified PUFF equation of state. Constant pressure heat capacities obtained from page D-97 of Reference 9 were used for Cvor the constant volume heat capacity at the reference state; heat capacities at constant pressure and volume are almost identical for solids at low temperatures. The critical pressure, tempera- ture, and volume of aluminum were obtained from Table 3.1 on page 44 of Reference 1. The critical poiuts of beryllium and titanium were calculated using a procedure presented in Refer- ence 6. The required relations are
v = (2.417X1024) a3
Tc =
.7232
RVc
a
Pc - .2596 a
where Vc Tc , and Pc are the critical volume, temperature, and pressure, respectively, a is the hard sphere diameter, ay is the coefficient of the attractive potential for the vapor, and R is the universal gas constant. The hard sphere diameter is related to the excluded volume, Vb, by the equation
.3 6Vb
where N is Avogadro's number. Numerical values of ay and V^ are presented in Table 3.2 on page 52 of Reference 1.
2.3 GRAY
The GRAY equation of state parameters for aluminum/ beryllium, and titanium are presented in Tables 11, 12 and 13 respectively; the parameters also used in the modified PUFF equation of state are not repeated in these tables. The entropies of melting were obtained from Reference 9; all other GRAY equation of state parameters were obtained from the original Lawrence Livermore Laboratories report by E.B. Royce.[5]
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e u e u 0 0 ü i 0) 0) •H •H •^
•H •H M p ■P ■P +• ,0 ,0 H (d •H •H -H ^ fi 3 <u U u ^
^ a s CJ u u
14
w
9
i u
a
a
I o E-i Ui
05 O
D
8
w o w
^ ^ m (N E iH r-i H •H 3 r- o Q o O •H oo H H rH rH C ^ X x X X id • CO yo vo a\ vo r«- *i ^ a o\ o o\ OJ 00 rH •H o en f-i CN •* r> m H t • • • • • •
CM r-H m (N 1
a\ <N rH
1
00
• •
0 CM | »n 0) ■M N O O o o 3 H •^ H iH H H
■H H 00 X X X x (0 >i • in VO 00 rH a\ T •*• > M H a\ r~ ro •* vo in en
0) *o in in in 00 r^ 00 ■-( 0) • • • • • • • (0 ot r-i 00 rH n m n u | | I •H M 0)
CM CM CN rH 2 E rH H H rH 3 m O O O O
c H H rH rH H •H H X X X X E O CM VO m rH m rH o 3 • ^ "«», <^ rH TT 00 o iH <M vo r^ LO H vo vo r* •< • • • • • • •
CM n iH <N 1
in CN CN
en n n CM fN CN g 3 B
e ro E ro E n o 0 CO e ü E ü 1 \ \ "N.
H B N Ö N 0 \ 1 3 3 0 0 in Ui \ to \ Ui S. tr 1 a>
c e 0) 1 0) 6 0)
c 3 -^*
s > > > > < In ^U U) TJ -0 TJ -o U u U
o ^-^ ^-^ *—' 0 iH n fO T m vo r^ a Ü u O u u 0 o
•H 0)
(0 ■P c
Ui c
>1 e a) 0
(Hi 3 •0
•H Ü IH 0)
•H C -P
0) 0< m IM
0 5 •H Ü 0 s o «M •H •r-t 3 u 0 S •M
04
u
0 u
15
Purpose;
TABLE 8
/EOSTAB/COMMON
Stores the constants and the tabular values which describe the mixed phase region in the RIP mixed- phase equation of state model.
Used In: CARDS, DREAD, EDIT, EOS2, RIP, SPRINT
Location Variable Definition
THETA(I) 1=1,100
Yl
PU(I) 1=1,100
Y101
RHOU(I) 1=1,100
Y201
EU (I) 1-1,100
Y301
Temperature (0K) of the points for
the mixed phase region table. List-
ed in ascending order from T = 0 to
T = T . . These values are deter- cnc mined by the EOSGEN code and are not
used in subroutine EOS2. 2
Pressure (dynes/cm ) of the points in
the mixed phase region table. These
values correspond to THETA values and
are in ascending order from P = 0 to
P = P ... cnt
Density (g/cm ) of the points for the
range p > p > p .. of the mixed co cnt phase region: the variable p co is
the value of the density at the cut-
off point for the mixed phase region
which is entered into ESTCON(16,N).
The variable p .. corresponds to the
density at P These values corre- cnt spend to the values appearing in the PU(I) array.
Specific internal energy (erg/g) corresponding to the points in the RHOU(I) array.
16
TABLE 8, /EOSTAB/COMMON (Continued)
Variable Location Definition
CU(I) 1=1,100
Y401 Sound speed (cm/sec) corresponding to
the points in the RHOU(I) array.
RHOL(U) I=:l,100
Y501 Density (g/cm ) of the points for the
range p . > p > 0 of the mixed
phase region. These values correspond
to the values appearing in the PU(I)
array.
EL(I) 1=1,100
¥601 Specific internal energy (erg/g)
corresponding to the points in the
RHOL(I) array.
CL(I) 1=1,100
Y701 Sound speed (cm/sec) corresponding
to the points in the RHOL(I) array.
Z(I) 1=1,15
Y801 Z(l) through 7(8) are defined in the
EOSGEN code. These are coefficients
used in the calculation of pressures
and sound speeds for state points
above and below the mixed phase
region. Z(9) through Z(15) are coeffi-
cients calculated within subroutine
E0S2 and are used for state points
above and below the mixed phase.
17
TABLE 9
EOSTAB INPUT; ALUMINUM
n
noir
rzji-
rsoir
f«3I =
3.0000*02t 1.8540*03» 2.0 760*03» 2.2980*03* 2.5200*03* 2.7«20«C3t 2.9643*33» 3.1360*03» 3.4333*33* 3.5333*33. 3.e52C«03t 4.C74C*r3» 4.??60*03» 4.5180*03* 4.74CP*"3. «.962Q*C3* 5.1843*33» 5.4363*03» 5.6233*33* 5.3503*33. 6,0170*03» 6.294 0*03» 6.5160*03» 6.7380*03* 6.9600*03. 7.1820»C3t 7.4043*33» 7.6260*03» 7.8433*33* 3.0703*03. 8.292C«03f 8.5140*03» 8.7360*03» 8.9580*03» 9.1800*03. 9.«t020*03f 9.6243*33» 9.8160*03» 1.0358*34* 1.3293*34. 1.0512*0«i. 1.0734*04» 1.0956*04» 1.1178*04* 1.1430«0<i
3.3333 > 1.5397*33» 1.1142*04» 5.4831*34* 2.3249*35. 5.S992*0e. 1.'005*06* 3.2346*06» 6.4628*06* 1.1663*07. 1.9C3e*07f 3.1378*37* 4.6363*07» 6.7821*37* 9.4657*07. 1.2820*08* 1.6902*08* 2.1787*08» 2.7518*08* 3.414B*r,8. «.1710C8* 5.3253*03* 5.97 96*08» 7.3355*33* 3.1963*33. 9.%61«»03t 1.0833*09* 1.2310*09» 1.3895*09* 1.556e*',9. 1.7392*09» 1.9283*39* 2.1299*09» 2.3398*39* 2.5630*39, 2.7920*09» 3.0330*09* 3.2838*0'. • 3.5444*09. 3.ei4n*r!j. 4.3935*09» 4.3819*39* 4.6792*09. 4.9853*39* 5.3000*0 9
?.7DCC*0C. 2.1086*00* 2.0425*00» 1.9799*00* 1.9202*00. 1.963M*33» 1.8391*33* 1,7573*00* 1.7371*30* 1.6592*00. 1.6111*00» 1.5687*00* 1.5257*00* 1.4843*00* 1.4440*00. 1.(1351*00» 1.3672*30* 1.3IC4*00* 1.2945*30* 1.2595*32. 1.22*4*00» 1.I92OC0* 1.1593*00* 1.1272*00* 1.0957*00* 1.0647*00» 1.3341*33* 1.3343*00» 9.7413-11* 3.4481-01* 9.1525-01» 8.6604-01* 8.5689-01« 8.2771-01. 7.9841-01. 7.»385-01» 7.3888-31» 7.0127-01» 5.7635-31* 6.4423-31. 6.0975-01. 5.7260-01» 5.3090-01* 4.8003-01* 3.7C37-ri
3.0033 » 2.2583*13» 2.6249*13* 2.9931*10* 3.3737*10, 3.7480*10» 4.1273*10* 4.5079*10. 4.6895*10. 5.2715*10. S.:5IE*13-» 6.0357*13. 6.4173*10* 6.799?*10. 7.1AC«>*10, 7.5613*10» 7.9419*10* 8.3216*10* B.Tßü.S*lC* 9.0800*10, 9.4587*10» 9.8371*10* 1.3n5*ll» 1.0594*11. 1.3972*11* 1.1350*11» 1.1729*11* 1.2109*11» 1.2490*11. 1.2371*11, 1.3255*11» 1.3543*11* 1.4328*11» 1.4418*11* I .4913*11* 1.5711*11» 1.5616*11* 1.6C28*11» 1.6446*11» 1.6651*11* 1.7333*11* 1.7803*11* 1.3315*11* 1.8933*11. 1.9969*11
5.0000*05» 4.3552*05» 4.2724*05* 4.1913*05. 4.1116*CE. 4.3331*05» 3.9355*35» 3.3787*05* 3.8324*35* 3.7253*35, 3.6516*05» 3.5771*05. 3.5324*05* 3.42o4*C5* Z.3cjZj*~t, 3.2799*05» 3.2054*35» 3.1313*05* 3.C3i9*35* 2,3121*35.
18
TABLE 9, EOSTAB INPUT: ALUMINUM (Continued)
f53l;
Y6D1 =
r73X:
YBOlr
2.93r0«05 2.52M9*05 2.123J*SS l.GB7e*05 1.1909*05 7. B535«0<l
P.COCO 5.813Q-35 1.36M6-03 7.m?t-03 2.0069-02 if.1557-02 7.3595-02 1.19S5-C1 1.9213-01 3.7037-01
1.319<<«11 1.5T91*H 1.5378«11 1.7323*li 1.32:3*11 i.90o«i*n 1.9712*11 2.0?36*li 2.?6ol*n 1.9969*11
3.3612*04 1.2396*03 1.^227*05 1.5720*05 t.65'r*Qe
1.5679*05 I.Sl«0*0f i.<ism*c5 1.2335*05 7.8535*0(1
3.7658*:6 9.0000*06
•5.'»376*10
2.8316*33 2.<tMC<t*C5 2.2395*35 1.59H:*05 1.0339*35
2.1915-07 1.3139-31 2.C531-C3 9.C930-3J 2.36E1-C2 1.7153-32 8.1179-02 1,3129-31 2.1336-C1
1.1592*11 1.55«»C*11 1.6571*11 1.7505*11 1.8373*11 1.9153*11 1.9311*11 2.C3«>5*11 2.0673*11
9.9676*01 1.2573*35 1.4578*05 1.5937*35 1.6617*05 1.6623*35 1.5»11«C5 1.1133*35 1.1605*C5
•2.025ii*3S 1.2921*11 2.1933*13
2.7358*05 2.3672*05 1 .9512*05 1.19e:*C5 3.6723*01
1.124 8-06 2.7^39-34 2.9516-03 1.1J51-32 2.7595-02 S.3361-32 8.9953-02 1 .H35-01 2.3957-C1
1.4792*11 1.5789*11 1.6763*11 1.7664*11 1.8333*11 1.9299*11 1.9351*11 2.0482*11 2.3354*11
1.0531*05 1.3320*05 1.4304*05 1.6127*05 i,6 673*05 1.6543*05 1.5715*05 1.43^2*05 1.0740*05
9.2^22*10 4.5000*01 8.5331*10
2.679Ü05 2.2871*05 1.9573*:5 1.3993*05 3.333i|*34
6,3m-Cf S.C53S-:i M,oebO-C3 1.3973-32 3.1932-02 5.9122-32 9.9C31-C2 1.S3.9-31 2.7539-C1
1.4932*11 1.5986*11 1.6952*11 1.7860*11 1.33J3*I1 1.9441*11 2.0332*11 2.0557*11 2.0551*11
1.1078*05 1.31*i*35 1.52C3*C5 1.6233*35 1.67C2*0E 1.6111*35 1,?453*C5 1.3515*35 3.6593*04
6.291705 0.0000 6.8534*13
2.6025*351 2.205S*C5t 1 .77dc; *Q5 t 1.29JC*Cf t
2,13H-0E. 3.5857-34* 5.4743-C3* 1.SS31-C2» 3.6589-C2t S.5253-a2f 1.0896-0^, 1.7134-Olt
1 .5192*11. l,6183*lli 1 .713e)*H t 1.8C33*llt 1.33s3fli * 1.9579*i:t 2.3194*11. 2.0619*11.
1.1600*05» 1.3849*?5. 1.5475*',»£» 1.0126*05« 1.67C4*n5, t,5335*05» 1.5153*05. 1.2361*05«
3.2553*13» O.COOC »
-1.138o»l?
19
TABLE 10
EOSTAB INPUT: BERYLLIUM
fl =
T131 =
Y201 =
T3-!lr
Y^Olr
3.00CO«Q2 <.07ii0*03 5.I8«C«03 S.2940*03 7.101C+03 • .5H0O3 9.62«C*03 1.073<I«0M I.1«ICC*0<»
O.OOCC T.2837»05 2.3831*07 1.8«75*Q8 7,2773*08 1.9799*09 4.2973*09 8.0526*09 1.3624*10 1.8031**10
1.8510*00 1.6903*00 1.5123*00 1.5254*00 1.4278*03 1.3164*00 1.1863*00 1.02 72*00 8.3834-01 4.^261-01
O.OOCC 5.14?9*13 7.4915*10 1.0013*11 1.2764*11 1.5846*11 1.9408*11 2.3756*11 2.9781*11 3.3379*11
3.3785*05 7.0014*05 6.4975*05 5.94 96*05 3.3497*05
2.C7E0*03 4.29^0*03 5.4060*03 6.5163*33 7.626O03 8.7363*33 9.8460*03 1.0955*3«
3.4925*03 1.7635*36 3.8666*07 2.5312*38 9.1011*08 2.3454*39 4.9212*09 9.009?*39 1.4990*10
1.7476*33 1.6753*00 1.5957*33 1.5068*00 1.4067*03 1.2921*00 1.1572*33 9.9011-01 7.4667-01
3.3674*10 5.6042*13 7.9797*10 1.0540*11 1.3351*11 1.6514*11 2.0201*11 2.4775*11 3.1472*11
7.3785*35 6.«>037*05 6.3917*35 5.8341*05 5.2228*05
2.2980*03t 2.520C*03t 2.742n*r3t 4,5180*03» 4.7400*03! 4.9620*03, 5,6280*03» ?,8fC0*C3» 6.0720*03, 6.7380*03» 6.9533*33. 7.1823*33. 7.8480*03. 8.07CO03» 8.2920*03. 8.9580*03» 9.1833*33» 9.4020*33. 1.0C68*04» 1.0290*04» 1.0512*04» 1.1178*04»
1.9C72*04» 3.8236*06» 6.0556*07» 3.3199*08» 1.1235*09» 2.7574*09» 5.6C60*09» 1.3342*10» 1.6448*10»
1.7537*03» 1.6EP1*0C» 1.5737*00, 1.4678*00» 1.3351*03. 1.2E69*nr» 1.1270*00» 9.5C44-01* 6.G873-ni#
3.8C?4*10» 6.0569*10» 8.4752*10» 1.1379*11» 1.3951*11» 1.7232*11, 2.1026*11» 2.5862*11, 3.3615*11»
7.2362*05» 6.8L46«C5» 6.2040*05» E.7I64»05. 5.3130*05»
7.9035*04» 7.5545*36» 9.0727*07» 4.4493*33* 1.37Cfi*C9» 3.2133*39» 6.3540*09» 1.1153*1?»
1.7195*33» 1.6445*00, 1 .5513*3?» 1.4683*00» 1.3629*30» 1.2410*00» 1.0954*30» 9.0764-C1,
4.2478*10» 6.5353*10» 8.9785*10» 1.1630*11» 1.4566*11» 1.7912*11» 2.1890*11» 2.7339*11,
7.1927*35, 6.7039*05» 6.1745*35» 5.5966*05» 4.9535*35,
2.6176*05, 1.3853*07, 1.3139*06, 5.7363*33. 1.6549*09! 3.7333*39. 7.1684*09. 1.2t45»lU.
1.7051*30« 1.6235*00. 1.54 35*30. 1.4483*00. 1.3399*00. 1.2141*00. 1.0622*00. 8.6oe9-ri.
4.6948*!n. 7,3l02*in, 9.4 900*10* 1.219X»!!. 1.5198*11, 1.9647*11. 2.2798*11. 2.8333M1,
7.3979*:5, 6,6017*05, 5.3631*35, 5.4743*CE. 4 .8254*35.
20
TABLE 10, EOSTAB INPUT: BERYLLIUM (Continued)
Y50lr
fsir
y7oi =
3.9MB3*0St 3.1J33»05»
3.3333 t 2.5552-05. ä.ii27-a<«f 3.7915-03t 1.2735-02f 3.1«IB9-02. 6.535M-32, 1.2M69-C1. 2.0078-01. M."261-01
3.35«U*11. 3.8339*11. «t.D3ee»ii. «.229<l*llt N.3S93«llt «.5i49*ll* «t.6ie«*i:t «.«11130*11. 3.8379*11
,3710*0<l* 0866*05.
.i|37M*05. 71C5*OE.
,B9G6*r:5. 9856*051
,9037*05. 7980*05.
,3832*05. 71«il*05
«I.5163*35 3.7892*05 2.9269*35 1.9<466*05
1.71113-37 5.7596-05 9.'J75:-3i| 5.C110-G3 1.^608-32 3.6823-02 7.<i33<l-02 l.MlDf-Cl 2.8n:-31
3.677M*11 3.379^*11 «1.0780*11 M.2o5H*n «.«1296*11 «.5565*11 «.62^5*11 «1.3596*11
1.7529*35 2.1628*05 2.«i985*:5 2.755C*05 2.922E*:5 2.9909*05 2.9«3a*35 2.7m5*05 2.2"i31*35
T»3l; 9.6156*06» 1.1997*07 1.7i|33*llf-3.8695*11
«.«319*35 3.6263*05 2.7191*05 1.7E1««05
8.6264-07 1.1671-0« 1.«378-33 6.«901-03 1 .83«9-32 «.2807-02 8.5123-02 1.6C6«-01 3.3781-01
3.7178*11 3.919«*11 «.1167*11 «.3335*11 «.4585*11 «.5753*11 «.6310*11 «.2?C0*11
1.8179*35 2.2259*05 2.5554*05 2.7959*05 2.91«5*35 2.991E*?5 2.9177*05 2.6747*05 2.3;«4*r,5
•3.5«29*ll «.«197*11
4.2545*35. 4.1033*35. 3,4538*05. 3.2865*05. 2.5453*35. 2.3483*35.
3.2515-36. 9.9239-36« 2,1685-04. 3.7511-04. ?.0175-33» 2.8030-03. 8,2582-03» 1,0346-02» 3.:J
,»8-32. i.6745-?2.
4,95^9-02» 5.6991-02» 9,6737-"'?i 1,0993-01. 1,8237-CX. 2.09C6-01.
1.9241*35» 2,3061*05» 2.6111*35» 2,8332*05» 2,962«*35» ?.9873*05» 2,8352*35» 2.5955*05»
2,3372*05» 2,3732*05» 2,5525*35» 2,8668*05» »,9762*05» 2,9782*0«» >,8456*05» 2,6014*05.
3,67n6*n6» 1,9242*11» 1,3538*11» 1.9404*10
21
TABLE 11
EOSTAB INPUT: TITANIUM
M:
mi;
fZDl:
f331r
mi-.
3.0000*02 1.22:10*03 5.4liS0*03 5.»730*03 1.89«0«03 9.1230*03 1.0ü«l5*0«t 1.1570*0<l 1.2550*04
0.0000 2.5386*25 1.8730*06 ».8681*07 3.1176*08 9.J<I51*08 ?.S<i38*C9 4.847i|*C9 7. f«8CC*C3
4.51CC*00 3.9525*00 3.7382*00 1.«9?«I*C0 3.?190*Q0 2.8936*00 7.4966*00 1.9666*03 1.0163*00
0.0000 2.2425*10 3.04 76*10 3.9342*13 4.9357*10 6.1071*13 7.5522*10 4.5353*13 1.322?*ll
4.6947*05 3.8143*05 3.4994*05 3.1552*05 2.7767*05 2.3498*05 1.8572*05 1.2645*35
3.2400*03* 4.4653*03t 5.6900*03* 6.9153*33f 8.1400*03i 9.3650*33* l.O5«Ci*04i 1.1815*04f
2.8880*03* 6.0264*35* 1.3061*07* 9.6826*37* 4.006 5*03* 1.1737*39. 2.7404*09* 5.S23J09.
4.10BC*C0» 3.9115*33* 3.6923*00i 3.4441*33» 3.1504*001 2.8213*03* 2.4047*00. 1.8285*33.
1.6421*10. 2.3932*13. 3.2176*10. 4.1243*13. 5.1540*10. 6.3695*13. T.B911M0. 1.0064*11*
4.0502*05. 3.7532*35* 3.4332*05* 3.0335*35. 2.6957*05. 2.2573*35. 1.7482*05. 1.1299*35.
3.4{5?*03 4*7103*03 5.9350*03 7.1530*03 8.3850*03 9.6133*03 1.0835*04 1.2360*04
1.1178*04 1.2645*08 2.0795*07 1.3116*03 5.0785*08 1.4J37*09 3.1854*09 6.2583*09
4.0703*00 3.85 97*03 3.6452*00 3.3433*03 3.0956*00 2.7)34*33 2.3C71*00 1.6579*00
1.7891*10 2.5334*13 3.3909*10 4.3136*13 5.3797*10 6.6135*10 8.2530*10 1 .3584*1;
3.9924*05 3.6313*35 3.3659*05 3.0393*05 2.6 125*05 2.1621*35 1.6750*05 9.3725»04
?.73C0*C3 4.9553*33 6.18CC*C3 7.4353*33 8.63CC*C3 9.85,i3*33 1.1C8C*C4 1.2335*34
3.6741*04 2.4635*35 3.1934*C7 1.8323*33 6.3590*08 1.6814*39 3.6823*09 7.!:3'!S*39
4.031 e*rc 3.8259*33 3.5969*00 3.3353O0 3.03C7*CC 2.66.>«*33 2.2028*00 1.4652*33
1.9381*10 2.7172*10 3.5680*10 4.5134*13 5.^132*10 o.9139*lC 8.6426*10 1.1473*11
3.9?3e*C5 3.6233*35 3.2973*05 2.9335*35 2.5272*05 2.0633*35 1.5171*01: 3.4343*34
3.975^*03. 5.2003*33. 6.4250*03» 7.8533*03. 8.8750*03. 1.3103*34. 1.1325*04.
1.0452*05. 4.5287*36. 4.7E01*r7. 2.3931»n8. 7.8723*06. 1.9910*39. 4.2350*09.
3.992c*D0. 3.7831*00. 3.5473*rr. 3.2779*30. 2.9634*00. 2.5836*00. 2.0901*00.
2.0892*10. 2.8904*10» 3.749^*:". 4 .7243*10. 5.8E£4*1C. 7.23JCHC. 9.0670*10.
3.8744*05. 3.S544»3S. 3.227<i*C5. 2.3503*05. 2,4397*05. 1.9623*35. 1.3937*0C.
22
TABLE 11, EOSTAB IN1UT: TITANIUM (Continued)
Y5C1 =
Y6nr
Y7C1-
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26
3. RESULTS AND DISCUSSION
The eighteen RIP calculations performed for this research are assigned calculation numbers and described in terms of the equation of state model, the material, and the spectrum in Table 15 . The calculated impulses and peak stresses at the end of the source time for aluminum, beryllium, and titanium are presented in Tables 16« 17, and 18 respectively. Note that in most cases the agreement between the three equa- tion of state models is fairly good. It should be noted that the calculated data reveals no obvious trends concerning the tendency for one equation of state to consistently predict stresses and impulses which lie above or below the predictions of the other two models.
27
TABLE 15
SUMMARY OF RIP CALCULATIONS
Calculation No. Equation of State Material Spectrum
1 EOS1 Aluminum SPEC 1
2 EOS1 Aluminum SPEC 2
3 EOS1 Beryllium SPEC 1
4 EOS1 Beryllium SPEC 2
5 EOS1 Titanium SPEC 1
6 EOS1 Titanium SPEC 2
7 EOS 2 Aluminum SPEC 1
8 EOS 2 Aluminum SPEC 2
9 EOS2 Beryllium SPEC 1
10 EOS 2 Beryllium SPEC 2
11 EOS 2 Titanium SPEC 1
12 EOS 2 Titanium SPEC 2
13 EOS 3 Aluminum SPEC 1
14 EOS 3 Aluminum SPEC 2
15 EOS 3 Beryllium SPEC 1
16 EOS 3 Beryllium SPEC 2
17 EOS 3 Titanium SPEC 1
18 EOS 3 Titanium SPEC 2
28
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31
4. CONCLUSIONS AND RECOMMENDATIONS
The modified PUFF equation of state is easy to use and gives results which are comparable to the more sophisticated models; this feature of course is the primary reason that modified PUFF is so widely applied in hydrodynamic calculations. However, this model has several disadvantages which are inherent due to its simplicity:
1. It is highly empirical and is therefore dependent on UGT or electron beam gen- erated data.
2. The melting and vaporization phase tran- sitions are described by energy levels which have no volume (or pressure) depen- dence .
3. The model contains no explicit caloric equation of state; the temperature, which may be required for some applications, is therefore unknown.
The ?.IP mixed phase equation of state was developed in order to paztially relax the above restrictions. However, we find that additional work is required before that model can be considered a useful calculational tool. The following improvements are recommended:
1. The lowest temperature-volume point on the mixed phase boundary (see Figure 3) should be the boiling point at ambient pressure, i.e., 6 = Ob, the boiling temperature and V = Vb, the specific volume at the boiling temperature.
2, The join function should extend from the ambient volume, V0, to the boiling volume, Vjj (see Figure 3) . Thus the join function will not penetrate the mixed phase region causing the discontinuities which exist in the present RIP mixed phase equation of state.
32
Join function
0) u 0) m 0) u
Mie Gruneisen solid
.Virial expansion
<V vj Mixed phase boundary
Isotherm
o. v0 vb vL vu
Figure 3. Intersection of isotherms with mixed phase boundary,
33
3. The EOSGEN code should be incorporated into the RIP code as a subroutine which is called after the initial call to the EOS2 subroutine; this would eliminate the need to perform a separate calculation with the EOSGEN code and would greatly simplify data input for the RIP mixed phase equation of state.
The GRAY equation of state contains the most sophisti- cated equation of state models available in the RIP code. How- ever, the procedure used to join the Grover liquid equation of state and the Young-Alder model destroys the physical signi- ficance of the Young-Alder model, particularly in the liquid- vapor region. Additionally, since the join volume used in the GRAY equation of state is too large, the Grover model is applied in regions where its accuracy is highly questionable.
The procedure we recommend for correcting these deficien- cies is best explained in terms of a pressure-volume diagram which is presented schematically in Figure 4. The reference or ambient condition is assumed to be
p = P ~ 0 , pressure
T = T , temperature
and v = v0 ' specific volume
The specific volumes indicated on the abscissa are:
V = reference pressure intercept of solidus
V- = reference pressure intercept of liquidus
V, ■ reference pressure intercept of liquid- vapor phase boundary.
V = critical specific volume c
34
ejtnssejd U
•H
35
Isotherms passing through the reference pressure at VQ, Vmo/ Vt0 and Vc are denoted by To, Tmo, Tbo and Tc respectively. The basic problem with the GRAY equation of state is that the join volume lies to the right of Vbo in Figure 4; thus the Young-Alder isotherms in the liquid-vapor region are badly distorted. Consequently, no attempt was made to perform a Maxwell construction in order to identify the liquid-vapor phase boundary; failure to perform this construction results in the appearance of unstable states to the right of the join volume due to loops in the Gibbs free energy along isotherms. We therefore recommend that the Grover and Young-Alder models be joined at some locus between the liquidus and Vb0, the specific volume at the reference pressure and the boiling tem- perature. Joining the two models in this region will leave the Young-Alder model in tact and allow one to perform the Maxwell construction. One of several methods may be used to join the Grover and Young Alder models:
1. The models could be joined at the locus of intersection of corresponding isotherms if that locus does not lie within the mixed phase region.
2. The two models could be joined by a fairing function applied between V and V. .
3. A fairing function may be applied to the entire region between the liquidus and
bo
It should be emphasized that the recommended changes to the GRAY equation of state are very important because they will make it possible bo determine all phase transitions explicitly; this capability is especially desirable for appli- cations involving the x-ray induced response of porous mate- rials. A recent study performed at Lawrence Livermore Labor- atory Hi] shows that impulse production in porous materials occurs on two distinct time scales. The short time contri- bution, which occurs in less than a microsecond, is due to pore closure by thermal expansion which results in high pres- sures and substantial blow-off velocities. The second con- tribution, which occurs over a time period of tens of micro- seconds, is due to the vapor pressure in the mixed liquid/ vapor region. The vapor pressure tends to accelerate the material in front of it (either blowoff material or material in the missile structure which acts as a tamper) over a long time period and can result in a significant contribution to the total impulse.*
It should be mentioned that explicit identification of phase boundaries is also important with respect to laser induced material response because it would enable one to identify regions in which liquid droplets are present. Liquid droplets can drastically affect the optical properties of the medium and these properties are, of course, quite important for pre- dicting laser interactions.
36
REFERENCES
1. R.H. Fisher and H.E. Read, "RIP, A One-Dimensional Material Response Code", Systems, Science and Software Report SSS-R-72-1324, September 1972.
2. D.R. Tuerpe, R.N. Keeler and S.L. McCarthy, "The Volu- metric Properties and Electrical Conductivity of Aluminum at High Temperature and Pressures", Lawrence Livermore Laboratory Report UCID-4683, January 1964.
3. E.B. Royce, "GRAY, A Three-Phase Equation of State for Metals", Lawrence Livermore Laboratory, UCRL-51121, September 1971.
4. R. Grover, "Liquid Metal Equation of State Based on Scaling", J. Chem. Phys., 55, 3435 (1971).
5. R.N. Keeler and E.B. Royce, "Shock Waves in Condensed Media", p. 51 ff. in Physics of High Energy Density, P. Caldirola and N. Knopfel, eds. (Academic Press, New York, 1971); Lawrence Livermore Laboratory Report UCRL-71846 (1969) .
6. D.A. Young and B.J. Alder, "Critical Point of Metals from the van der Waals Model", Phys. Rev. A3, 364, 1971.
7. Private communication with Janet Lacetera, Effects Analysis Branch, Ballistic Research Laboratory, Aberdeen, Maryland, 25 June 1973.
8. Brian J. Kohn, "Compilation of Hugoniot Equations of State", Air Force Weapons Laboratory, AFWL-TR-69-38, April 1969.
9. Robert C. Weast, Handbook of Chemistry and Physics, The Chemical Rubber Company, Cleveland, Ohio, May 1968, pages D-33, D-34, D-97 and D-103.
10. D.R. Stull and G.C. Sinke.- Thermodynamic Properties of the Elements, American Chemical Society, Washington, D.C., 1965.
11. John J. Ruminer, "Liquid/Vapor Contributions to Impulse from Porous Materials", Lawrence Livermore Laboratory, UCRL-51354, February 21, 1973.
37
APPENDIX A
THE EOSGEN CODE
Three tasks are performed by the EOSGEN code: 1) Eval- uation of the seven coefficients in the virial expansion, 2) definition of the mixed phase boundary, 3) evaluation of the seven coefficients in the join function. These tasks are discussed separately in the following sections of this Appen- dix.
A.l THE VIRIAL EXPANSION
The virial expansion employed by the RIP mixed phase equation of state, with density and energy as independent variables, is
\z1P+z2P +z4p |E
z6 ' zllp+z12p2+z13p3
+ z14p4+z15p5 (1)
where
3P V 9 R(ej = ceo
ec
3P V z, = c c
'1 6 c
-3P z = E^
c o
V2+V Iß V +2R(6 )1 i c oL o o o J
-V«9r.fö«V«+2R(ejl _ o c L o o o J +2R(0 )l + 3P V^e^ , _ h. ... J o J c C o z3 e7T~ c o
Preceding page blank 39
24 = cc 0L00 oj rt
c o
v2e [ß v +R(9 )1 - P v3e o c L o o o J ceo c o
z, = c V
Zn = - Z3 Z5
o
'11
'12
'13
s14
'15
- - m - -m - *&) ■ - m - 4$ --m ■
+ z.
In Equation (1), P is the pressure, p is the density and E is the energy. The coefficients, Zi, are expressed in terms of the critical and reference point characteristics presented in Tables 4, 5 and 6. The coefficients zi through z-j are evaluated by the EOSGEN code and must be read into the first
40
seven locations of the Z array in EOSTAB common in the RIP code. The coefficients zu through zis are evaluated by sub- routine EOS2 in the RIP code and need not be input.
A. 2 DEFINITION OF THE MIXED PHASE BOUNDARY
Corresponding values of pressure, density, energy and sound speed on the mixed phase boundary are required as input for the RIP mixed phase equation of state. These data are determined by locating points having equal pressure and Gibbs free energy on a series of isotherms between the reference tem- perature, e0, and the critical temperature 0C. Referring to the schematic pressure/volume diagram presented in Figure A.l, the two points on the mixed phase boundary of an isotherm with temperature, 60 < 6 < 9C, are determined by the following equalities:
Pu(e' ^ = PL(e' PL* ' (2)
Gu(e, pu) = GL(e, pL) , (3)
where G is the Gibbs free energy, P is the pressure and PL and pu are densities on the low and high density sides of the mixed phase boundary respectively. The pressure, as a function of density and temperature, is given by
p(p,e) = z^p + (z2e+z3)p2 + (z4e+z5)p3 , (4)
and the Gibbs free energy is defined as
G = E+pv-es ,
where E is the specific energy and S is entropy. Taking the differential of the Gibbs free energy yields
dG = dE+PdV+VdP - GdS-Sde ,
which reduces to
dG = VdP (5) >
41
(Pc'V
Mixed phase boundary
Isobar, p=p
Isotherm (e<e ) c
Specific volume
Figure A.l Intersection of isotherms with mixed phase boundary. (Note: pL = 1/VL; PU = 1/VU) .
42
for an isothermal process. Therefore, the Gibbs free energy, referenced to the initial density, is
G(e,p) = j^P vdP= |^Pv(|£)dv . (6)
Substituting Equation (4) into the above relation and perform- ing the indicated integration yields
G(6,p) = -z^ *n(^) +(2ez2 +z3)(p-po) + |(Z4e+z5)(p2.p2),7)
Now, Equations (4) and (7) are substituted into Equation (2) and (3) respectively, and the quantities Pu» Pu and PL are calculated by a numerical iteration scheme in the EOSGEN code. The specific energies on the mixed phase boundary are deter- mined by
E = E(e,p ) , u u
and
EL = E(e,pL) ,
where
Z5 2 E = z-p + T" P + zge+z7 •
is the caloric equation of state employed in the expanded region of the RIP mixed phase equation of state. The sound speeds on the mixed phase boundary are given by
=u " y (If), evaluated at 6 and p = p
43
■M Cy = A/ (ä- I evaluated at 6 and p = p
The quantities on the mixed phase boundary are stored in the following arrays in EOSTAB common in the RIP code:
6 + THETA(I)
Pu - PU(I)
Pu - RHOU{I)
Eu + EU(1)
cu * CU(I)
PL - RHOL(I)
EL •*■ EL (I)
CL ■*■ CL(I) •
The number of data points to be stored in each of the above arrays must be input to Z(8) of the Z(I) array which is also in EOSTAB common.
A.3 THE JOIN FUNCTION
The purpose of the join function is to allow pressure and sound speed continuity between the Mie Gruneisen solid equation of state for the compressed region and the virial expansion for the expanded region. The join function is applied over a density range beginning at the ambient density, p0, and ending at a slightly expanded cut-off density of Pco (see Figure A.2) . It was determined that in general it is possible for the join function originally reported in Refer- ence 1 to contain several singularities over this density range; in fact, these singularities appeared when the calcula- tions performed for this report were first attempted. We therefore decided to develop a new join function which would not exhibit singularities under any conditions.
44
I? 01 a a
0)
o c o
•H p
3 cr 0)
0) 03
a
0) x
•H e a*
töjeua
45
The new join function developed for this series of cal culations is:
P2 " ■.-#£)' • -iW' + c1+c2p+c3p +c4p"
(7)
where P2 is the pressure and the subscript, 2, serves to iden- tify the join function. The seven coefficients, cim,,cT, are determined by continuity conditions at the ambient and cut-off densities. Letting the subscripts 1 and 3 represent the Mie Gruneisen solid equation of state aid the virial expansion respectively, we have
Pl = GopoE + K^y2^3) {l-f) . (8)
where
V = t-1
P, ■ Z1P+Z2P +Z4P"
+ z11P+z12
p +z13p +Z14P +z15p" (9)
The appropriate boundary conditions for determining the coeffi- cients in the join function are:
Pl(po'E) " P2(po'E) (10)
3P1 3P, at p = p. (11)
46
3E
3P,
3^ at p = p.
P2(PCO'
E) =P3(Pco'E)
(12)
(13)
3P2 3P.
9p at P = Pco . (14)
3P.
3E"
3P.
3E" at p = p
CO (15)
Substituting Equations (7), (8), and (9) into boundary condi- tions (10) through (15) yields the following set of indepen- dent equations for the undetermined coefficients in the join function.
C5 = N1 (16)
C5+c6+c7 = N4 (17)
po-pco — <=« + — 6 p -p
o oc C7 " -N6
(18)
cl+c2po+c3po+c4po = 0
c2po+2c3po+3c4po ' N2
cl+c2pco+c3pco+c4pco = N3
c2+2c3pco+3c4pco " N5
(19)
(20)
(21)
(22)
where
Nl " poGo
N2 =A1 47
-n
N3 " zllpco+z12pco+z13pco+Z14pco+z15pco
2 3 M
zlpc0+Z2pco+Z4pco N4 Tr
N5 := zll+2z12pco+3zl3pco+4zl4pco+5z15pco
2 z,+220p^ +3z.p M 1 2^co 4 co
6 Zr
Equations (16) through (21) are easily solved by Cramer's rule; the task is performed by the EOSGEN code. Additionally, the cut-off density, pco, is selected automatically by the EOSGEN code based on the following considerations:
1. The coefficient of E in Equation (9) has a value at p - pco which is less than the coefficient of E in Equation (8) .
2. The remaining terms in Equation (9) have a negative value at p = pco and their deriva- tive has a positive value at p = P_ •
The new join function discussed in the above paragraphs is mathematically cleaner (no singularities) and greatly facil- itates application of the RIP Mixed Phase equation of state.
48
APPENDIX B
CORRECTIONS TO EOS2
Two types of corrections to subroutine EOS2 (the RIP mixed phase equation of state) were required in order to per- form the calculations presented in this report. The first set of corrections are related to the incorporation of the new join function (see Appendix A.3) into subroutine E0S2. The second correction is a temporary measure designed to com- pensate for inherent discontinuities in the RIP mixed phase equation of state. These discontinuities exist between the mixed phase region and the virial expansion region below the zero pressure line in Figure B.l; the mixed phase and virial expansion regions in question are denoted as 27 and 26, respectively, in the same figure. The task of correcting the discontinuity in a thermodynamically consistent manner is beyond the scope and intent of this program; however, a tem- porary correction was required in order to avoid numerical convergence problems encountered when a Lagrangian cell attempted to follow a path from the 26 region to the 27 region. Since such a transition is discontinuous and physically meaning- less and because convergence problems were encountered whenever such a transition occurred, it was decided to artificially for- bid it with appropriate modifications to the E0S2 subroutine.
A FORTRAN listing which details the required correc- tions to the EOS 2 subroutine in the RIP code is presented at the end of this appendix.
49
w
i? •H 01 c
A&zeua
0) +J fl ■p CO
M-l O
c 0
•H
<0
<u to fa
•a Q) X
•H e
H
,2 +J
m o c 0
■H -p
a 0) M a) a Q) u u
I m Q)
•H In
50
"-
THE FOLLOWING CHANGES TO E0b2 (RIP MIXED PHASE EQUATION OF STATE) ARE REQUIRED IN THE EXISTING MOUEL IN ORDER TO INCORPORATE THE JOIN FUNCTION AND TO COMPENSATE FOR DISCONTI NU ITES BETWEEN THE MIXED PHASE REGION AND THE VIKIAL EXPANSION REGION BELOW THE ZERO PRESSURE LINE.
1. IN THE CODING WHICH FOLLOWS ••• POINT IS IN MIXED PHASE REGION ••• COMMENT CARD ADD THE FOLLOWING LINES AFTER THE LINE WHICH STATES:
TEMP»1 I)»PU(NMAXJ
ADD LINES.*
IFITEMPU) .GT. TEMP(lC)) GO TO 155 P(J)=TEMP(10) IVFLAG(J)>26 C(J)»ESTC0N»6,N) GO TO 260
AND CHANGE THE STATEMENT WHICH STATES:
IF«TtMP(l) .GT. TEMP(ll)) TEMP<1)«TEMP( 1 I)
TO THE FOLLOWING:
155 IF(TLMP(|) .GT. TEMP(II)) TEMP(I)»TEMPU 1 )
^. AFTER THE LINE OF COOING,
25C IVFLA6(J)»25 THE FOLLOWING LINES HAVE BEEN DELETED AND REPLACED:
TEMP(2)«C0S(ESTC0N(22,N)»TEMP(l)) TEMP(3)«ESTC0N(2I,N)/TEMP(2)♦ESTCON<23|N) TEMP<M)»SIN(TEMP(I») TEMP(5)«ESTC0NC2I,N)»ESTCON(22,N)»TEMP(H)/«(ESTC0N(1,N)-ESTC0N(I6,
N))»TEMP(2)»»2)
THE NEW LINES SHOULD READ:
TEMP(2)«TEMP( 1 )««2 TEMP(3)»ESTC0N(23,N)*TEMP(2)»(ESTC0N(2I,N)*ESTC0N<22,N)*TEMP(1 )) TEMP(5)"-(2.»ESTC0N(21,N)»TEMP(I)♦3.«ESTCONt22.N)»TEMP(2))/
(ESTCON(I,N)-ESTC0N(16.N))
51
APPENDIX C
CORRECTIONS TO EOS3
Two coding errors were discovered In the expression for temperature in the liquid-vapor region of the GRAY equation of state. The FORTRAN changes required to correct these errors are presented in the listing at the end of this appendix.
Due to the lack of continuity between the liquid vapor region and the liquid, melt, and solid regions (see Figure C.l) problems (complex temperatures) were encountered whenever Lagranglan cells attempted to cross the boundary between them; this problem is inherent to the GRAY equation of state because the liquid-vapor region is continuous only with the hot liquid region. The task of providing complete continuity in this equation of state is beyond the scope and intent of this project. Therefore, temporary FORTRAN changes were incorpor- ated into the E0S3 and EOS subroutines which prevent the liquid-vapor region from being entered by any path not leav- ing the hot liquid region. These changes are detailed in the FORTRAN listing which follows.
Preceding page blank 53
1
>1
U 0) a w
Liquid-Vapor
Hot Liquid
Liquid
Melt (2 phase)
Solid
max
J
Density
P-r ■ density at which equations of state are joined
Figure C.l Schematic representation of GRAY EOS.
54
THE FOLLOWING CHANGES TO EOS AND EOS3 (GRAY EQUATION OF STATE) SUBROUTINES ARE RLQUIRED IN THE EXISTING MODELS.
SUBROUTINE EOS
1. IN THE SECTION OF CODING FOLLOWING THE COMMENT STATEMENT ••• LRL EOS ••• ADD TWO LINES AFTER THE ONE WHICH STATES:
30 RSAVfc«RHO(J)
ADDED LINES:
PSAVt2»P<J) ISAVE-IVFLAGU)
ALSO IN THIS SAVE SECTION TWO LINES ARE ADDED IMMEDUTLY FOLLOWING THE LINE STATING:
PSAVt-«P( J)
ADDED LINES:
IVFLAG<JlsISAVE P(J)«PSAVE2
b. SUBROUTINE E0S3
I. IN ThE SECTION OF CODING FOLLOWING THE COMMENT STATEMENT ••• VAPOR REGION ••• ADO SIX LINES OF CODING BETWEEN THE COMMENT STATEMENT AND THE LINE WHICH READS:
90 IVFLMG(J)>3S
90 IF(IVFLAGIJ) tGE* 33) GO TO 85 P(J)"0. GO TO 100
85 IF(P(J) .GTt I.E-12) GO TU 86 P(J)»0. GO TO 100
ANü CHANGE THE LINE OF COOING WHICH READS:
90 IVFLAG(J)>35
TO READ:
86 IVFLAG(J)-36
55
ALSO, IN THE SÄHE SECTION OF CODING CHANGE THE STATLHENT WMlCH STATES;
TEHPt7»»-<E(J)*EC0N(l7,N)«TEMP(|)/ESTC0N(26,N)-EC0N(M,N)«TEMP(6)- ECON(IO.N)
TO THE FOLLOüINfa:
TEMPi7»»-(E(J»*ECON(I7,N>«TEMP( il/ESTCON(2*,N) I -ECONI l,«,N>«TEMP(6>-ECON( I 0 I N > • 2%
ANO ADD THE FOLLüWINd l'j LINES.
( 1 H . N ) i
7 &X,IOHAYPKIME = ,EI2.3,//, 8 5X,5HFF « ,E12.3,//) CALL SPRINT «6HEOS3 ,6HLRRüR9)
92 CONTINUE
56
