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UNCLASSIFIED

cC-1 4 REPORT COUECTION

REPRODUCTION

COPY >

LOS ALAMOS SCIENTIFIC LABORATORY

OF THE UNIVERSITYOF CALIFORNIAo LOSALAMOS NEW MEXICO

_.. ——

,’

. . . . . . . . . . . . .

.. .. . . .. .

.,’

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INVESTIGATIONS OF UNDERGROUND- EXPLOSIONS

,.

. . . .

,,:

IN NEVADA SANDY SOIL

(Title .Unclassified)

● ✎☛☛✍☛☛ ● J- . . ,

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● ✎ ● ✎ ✎

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LEGAL NOTICE

This reportwas preparedas an accountof Government sponsoredwork. Neitherthe UnitedStates,nor the

- Commission, nor any personactingon behalfof the Commission:

A. Makes any warrantyor representation,expressed

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providesaccess to,any informationpursuantto his employmentor contractwiththe Commission, or hisemployment with such centractor.

~

...” .*. ! ..&. -

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..

.J.

7.

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n ● . 9 **iii

● .*... .. . . . ...*,:

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.==3iE=siEiE. ● ● *

UN(XASSlflED +:: ~ ‘H~@~~’Z3

●

● 0● om●●eemmm9*9● 0●m ● **a ● 0 :: c: ● * :,0.0

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LEASABLE ‘

““”** “?$’-’”%i~.”. “:iv

LAF&s-2919

F’SS-10 Date; ~b-9F

ac”1’4Date:&-=q&–

This document consistsof 30 pages

Los ALAMOS scientific LEORATORY

OF THE UNIVERSITYOF CALIFORNIA LOS ALAMOS NEW MEXICO

REPORT WRITTEN: January15,1962

REPORT DISTRIBUTED: August 12, 1963

VERIFIED UNCLASSIF~D A.

By

INVESTIGATIONS OF UNDERGROUND EXPLOSIONS

IN NEVADA SANDY SOIL

(TitleUnclassified)

by

KennethH. Olsen

.

ContractW-7405-ENG. 36 withthe U. S. Atomic Energy Commission

All LAMS reporte are informal documents, ueually prepared for a special purpose and prtromlly preparad for use witbfn tbe Laboratory rather tbcn for

genenil distribution. Tida report bee not been edited, reviewed, or verified

for accuracy. All LAMS reports exprese the views of the authora as of the

time they wem wxltten and do not neoesscrily reflect the opinions of the Loe

Alamoe Scientific Laboratory or the final opinion of the authora on the subjeot.

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c-i b -0 ● 9

● **a* .: ●

● **9 :

. ● , : :0

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AEC Classified Technical Library

Lawrence Radiation Laboratory, Livermore

Manager, ALO (Paul. f%ger)

“Los Alamos Report Library

1-3

4

5

6-40

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I INTRODUCTION

This report presents further information on blast calculations

carried out with the J-IS Lagrangian radiation-diffusion hydrodynamics

code. Here we deal primarily with explosions in Nevada alluvium. The

purpose of these calculations is to determine total bomb yields from

observed shock pressures and arrival times.

In order to make these calculations, we require the initial configuration and the equations of state of the materials involved. The following

simplifying assumptions have been made:

1. The soil is assumed to be homogeneous.

2. Spherical symmetry is assumed. Although the bomb chamber is

certainly not a sphere, the bomb mass and chamber dimensions

are usually small as compared with the mass of the soil and the

shock radii at the ranges of Interest. In this case, the exact

properties of the explosion chamber and materials can be expected

to have only slight effects on the time history of the shock

expansion in the soil.

3. The material is assumed to be fluid with no shear stresses.

‘k. The equation of state of desert alluvium is taken to be of the

Mie-Grkeisen type below the melting point:

UNCLASSIFIED

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APPROVED FOR PUBLIC RELEASE
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7G(V)

P= PH+~ (E -EH).

Here P is the pressure, V is the specific volume,and

energy.For a given ‘phase,the empirical relationship

velocity Us and the particle velocity U is found to

P

us =C + Su .

P

Combining (2) with the Rankine-Hugoniot relations gives

C2(V0 - v)

PH =

[V. - S(vo -V)12

and

(1)

E the specific internal

between the shock

be

PH(VO - v)

EH .

2

All changes after shock

E=-

f

‘PdV.

0

(2)

(3)

. (4)

passage are assumed to be adiabatic giving

(5)

V. Is the specific volume under ambient conditions.

Above the melting point, a gaseous equation of state is used.

This is based on a Thomas-Fermi-Dirac atomic model calculated by R. D. Cowan

with low pressure corrections calculated in J-15 using the Saha ionization

equation.

We shall now proceed to a more detailed discussion of the basic

quantities c, s, and p .,o: ●:. :$ ● ● **

● ● ●°9

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APPROVED FOR PUBLIC RELEASE
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9* ● m ● O

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II EQUATION OF

A. GMX-6 Data.

R. G. McQueen of GMX-6 has

STATE PARAMETERS

measured the Hugoniots of two samples

of Nevada alluvium furnished by J-15 (IAMS-2760). The samples were

very friable, and it was impossible for GMX-6 to machine specimens suitable

for their shots. As an alternative, the samples were broken up, damp

packed into the base plates, and dried before shooting. The two samples

were considerably different as regards particle size distribution, the

coarser aggregate having a dry density of 1.80 gin/ccand the finer

aggregate having a density of 1.54 gin/cc. The results of the experiments

are shown in the Us, Up plane in figure 1. The shock pressures are listed

alongside of the observed Us, U points.

P

We have taken the in situ density in Area Three at NTS to be 1.7

.—

gin/ccfor lack of a more accurate number; the solid line in figure 1

represents the Us versus Up curve for this ambient pre-shot density. For

many solids in certain shock stress ranges, the shock velocity Us may be

less than the sound velocity in the unstressed material. Shocks of these

strengths will therefore travel at less than the ambient sound speed and

will be preceded by

Nevada are somewhat

of 1.5 km/see. The

hence expect to see

a sonic “elastic precursor.” Sonic surveys made in

uncertain, but indicate sonic velocities of the order

GMX-6 data extrapolate to 0.55 km/see, and we would

a two wave structure below shock pressures of about 15

kilobars if the dynamic yield strength of the material is taken to be 1

kilobar. A sonic precursor of this strength Is expected to trigger the

timing indicators during the expansion of the shock so that the more

distant stations will see the elastic precursor rather than the main shock.

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-5

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1

0 1 2 3 4 5

uP

PARTICLE VEWCITY (KUCMWEM /sEC)

Figure 1

UNCLASSIFIED

6 7

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APPROVED FOR PUBLIC RELEASE
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● ●

● : ● 0 ● 9* ::

Preliminary information fro&.{h: J’~sher~e&nt!&~dicatesthat the ambient

sound speed should perhaps be as low as 1 km/see in which case the two

wave structure should develop below about 5 kilobars. Figure 1 shows

oIilythe elastic precursor for an ambient sound velocity of 1.5 km/see.

According to McQueen, the break in the Us versus Up curve between

lM and 200 kilobar6 is a mixed phase region which is almost certainly

associated with a similar transition observed in granite at higher pressures.

Above about 400 kilobars, the slope of the curve is such that the volume

increases with increasing pressure. We have chosen to interpret this as

an indication that the alluvium melts at about 400 kilobars. In any

case, the J-15 numerical integrations are unstable under this type of situation,and we have chosen to extrapolate the phase line between 200 and 400

kilobars until we reach our specified melting temperature at which time we

switch to our gaseous equation of state.

Although the GMX-6 data clearly indicate at least three phase regions

of interest, the best single straight line fit of the data is

Us(km/see) = 1.3 + 1.35UP. (6)

In order to evaluate some of the uncertainties in the calculation, several

problems were run with equation 6 as well as a more complete set of

problems with the multiphase Hugoniot.

In summary, the GMX-6 data define the Hugoniot of Nevada alluvium

reasonably well below 400 kilobars. For yields of the order of 10 kilotons

and the present layout of measuring stations in Area Three, the region

above 400 kilobars Is not of great importance for time of arrival measurements. This is because the mass and volume of soil shocked to such pressures

Is relatively small and would be expected to have only minor effects on time

of arrival at the more distan~ swtipns .... ..O ..

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APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
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● ●9* : ●o, ● ●°:

● 9*

●

9*** :0: 9**

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The most important pressure for the J-15 integrations is below

about 50 kilobarsi test problems indicate that the peak shock pressures

drop below

Since this

GMX-6 type

50 kilobars at ranges of about 10 to 15 meters (for 10KT).

(40-50 kb) is about the minimum pressure obtainable in the

experiments, it is worthwhile to verify that the extrapolation

of the GMX-6 data to zero pressure is reasonable. Ideally, the best

method for doing this would be to make simultaneous measurements of both

time of arrival and peak pressure at every measuring station for the

full scale nuclear shot. Time of arrival measurements would give the

shock speed, Us, at each station; and the pressure measurement, when

combined with the Rankine-Hugoniot condition,

P = pou#p,

could be used to calculate

could then be constructed.

Unfortunately, the

have not been instrumented

(7)

u. From these data, the Us versus Up curves

P

first few shots of the Nougat-Ivanhoe series

for pressure measurements at every range

station, although it is hoped that the later shots will include at least

some of these measurements. It is not now known if these ~ situ pressure

measurements will have the precision necessary to give a well determined Us

versus U

P

curve for each shot, but if they could be made sufficiently

accurate, such a technique would eliminate several of the present uncertainties

in the equation of state. The most obvious advantage of this method would

be the elimination of the present assumption that the data obtained by

GMX-6 is typical of all the material in which the Nougat-Ivanhoe shots are

to be made. The method would also help answer the question of whether the

● e ● O* ● ** ● ● *9

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APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
90 ● O* ● s0 ●:* :M ,0

● **

● ooecoe ::

● O ● e ●:00 ● * ● *

● ● 00 ● S:* :9* ● m

technique of

the material

repacking the semples as done by GMX-6 really approximates

packing at depth in Nevada.

B. CMF-5 Data.

Since the in situ method of measuring the Hugoniot of Nevada ——

alluvium as discussed above has not yet been performed or may not be of

sufficient accuracy in the future, an attempt was made to verify the

extrapolation of the GMX-6 data below 40 kilobars by making static pressurevolume measurements on the samples of alluvium obtained from Nevada. The

work was kindly undertaken for us by Karl Gschneidner of CMF-5 who has a

press capable of attaining pressures up to 10 kilobars.

The idea of the experiment is to obtain the average bulk properties

of the mineral grains, neglecting effects due to voids between the grains

and the fracturing of grain edges and corners. The experimental technique was to measure the grain density and then to introduce a slurry

of grains and glycerin into the pressure cell. The glycerin acts as a

pressure transmitting fluid around the Individual grains and also excludes

the air voids. Corrections were made for the effects of the glycerin

component on the observed compressibilities.

where

(2.16

The results of Gschneidner8s measurements are shown In figure 2,

the reference density is taken as the average observed ~rain density

gin/cc). The sample was first compressed to about 10 kb (Run 1), the

pressure relieved and again compressed to 10 kb (Run 2). Note the divergence

of the two runs at the higher pressures. A possible explanation of this

divergence may be that the sample grains may still be somewhat porous and

that the first application of pressure forced glycerin into these pores

thus making the material “harder” when it was recompressed. The consequences of this hypothesis will be discussed in more detail below.

● ** ● ● 00 ● *9 ● * ●b* ● ● .

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APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
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2

Figure 2

(chfl?-5Data)

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90 :

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9**** 9.* .9

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
9* . . . 80 ●**99.:

● eeeae

:.: .

●

● ** o 0 Obo

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The CMF-5 points must be corrected to a reference density

gin/ccin order to compare them with the extrapolated GMX-6 data.

of 1.7

This was

done by solving for the specific volume using Gschneidner8s observed

grain density of 2.16 gin/ccand then recomputing the relative volume

change with a reference density of 1.7 gin/cc. The points were transferred

to a Us versus U

P

vation equations:

>=1-

S

plot by making use of

U2=PVOV .

s ,. \

(L --V.)

Here V. is the reciprocal density, V. =

these transformations of the CMF-5 data

extrapolated line from the GMX-6 shots.

two of the Rankine-Hugoniot conser-

(8)

(9)

0.588 = l/1.7. The ’resultsof

are shown in figure 3 together with the

It will be noted that the static

measurements reduced in the above manner fall reasonably close to the

extrapolated line, but that the indicated slope is greater than the expected

value of 1.85. On the other hand, the bulk sonic speed calculated from the

initial slope of the CMF-5 P-V curve gives an intercept of about 0.52 km/see

after making the density correction noted above. This is in quite good

agreement with the value of 0.55 km/see obtained from the extrapolatioriof

the dynamic data. If we accept the idea that the mineral grains were somewhat porous and that the effect of the glycerin entering these pores on

compression was to make the material harder to compress, a steeper slope

in the Us versus U would be expected.

P

As a test of this effect, the AV/V

o

points from figure 2 were multiplied by a constant factor of 2 in the range

3 to 10 kb and the reference density correction made to these “adjusted”

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APPROVED FOR PUBLIC RELEASE
2

2

1

1

0

!.5

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APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
● e 8*9 ● 0 ●*O**.: ●am●

9*9** ● **

● oe ● : ● **

● eo 9*O ● ●00:

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values. The results are shown as crosses in figure 3. It is seen that

such an adjustment brings the CMF-’jdata more nearly Into agreement with

the extrapolation of the dynamic GUX-6 data, although the constant factor

of 2 used seems to be an overcorrection, especially for the lower pressures.

In any case, the static experiments indicate that the low pressure (below

100 kb) Hugoniot follows more nearly the relation

Us = 0.55 + 1.85 U

P

(lo)

rather than a curve similar to equation 5 . Although we cannot at

present rule out the possibility that the static measurements may indicate

the existence of different phase region than that given by equation 10,

the indications are that the difference in slope may be explained by the

effects of porous grains in the static measurements. Until further information is available, it seems reasonable to use equation 10 down to

pressures at which the

c. Grheisen Ratio.

In addition to

sonic precursor is effective.

the quntities c and s, another Important parameter in our equation of state Is the Grtieisen ratio} YG” The size of

YG determines how rapidly the pressure falls off behind the shock and

hence is effective In determining the expansion rate of the shock front

as well as the amount of cooling in the adiabatic expansion. There are

several methods for estimating the size of this parameter:

1. For many metals, the Dugdale-MacDonald relation (cf., McQueen

and Marsh J.A.P., & 1253 1960) has been found to be

generally accurate to within about 35$. ‘1’hi.srelation yields

the fogi%~l~”: . . . .

● ** ● ** ● 0

●  

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9 ● e

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APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
●m ● 9*9

● ●’e 9 9** ● ●°:

● ● e* ● **O

● ● *9 ● ::::

9***

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(11)

At zero pressure (V/V. = 1 ) equation 11 reduces to

70 =2s -1, (12)

so that yG may be obtained directly from the observed Us

versus Up curve. For our observed

gives 7G = 2.7 at zero pressure.

2. The value of 7 at room temperature

by the

7m=v

J.

and

c=

a=

K=

thermodynamic identity

(~

E Kc .2 .&

v ‘P

s of 1.85, equation 12

and pressure may be computed

the measured quantities:

adiabatic bulk sound speed

Q

1 IV volume coefficient of expansion - —

V’m

isothermal compressibility = ‘~ (L)

JT

(13)

‘P = specific heat at constant pressure

Cv = specific heat at constant volume

Table 1, compiled by Stan Msrsh from available handbook data

on common igneous rocks gives a comparison of 70 computed

from equation 13 and also from equation 12. Note that for

most rocks, equation 13 gives a considerably smaller 7 than

does equation 12. This tends to cast some doubt on the

applicability of the Dugdale-MacDonald relation to rocks.

9* ● *9 ● oa ● ● a. ● .0

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I

,

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:

(a

ccl

c

al

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* ● O

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9

: ●

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● 9 ● ● ea

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Equation 13 was used to estimate 7 for Nevada alluvium by

assuming the specific heats given by the Dulong-Petit law,

the thermal expansion coefficient of granite and the ob8erved

intercept of the U9 versus U curve which is confirmed by the

P

CMF-5 measurements. This estimate gives a value of 7T of

the order of 0.01. The greatest uncertainty in this computation

is the value of the thermal expansion coefficient a. It may be

argued that instead of the granite value, a more reasonable a

to use would be a value of the order of that for an ideal gas

because of the large volume of air entrapped in the sedimentary

alluvium. Since a for a gas is about two orders of magnitude

larger than that for granite, a value of about unity would

then be obtained for 7.

3. The third Rankine-Hugoniot equation can be written

‘H -

E. =#PH+Po)(Vo - V). (14)

where zero subscripts refer to the ambient state. If the material

is available at two different initial densities and these

are both shocked to the same final density, the pressures and

energies will therefore be different. This fact may be used

to compute Gr~eisenvs ratio directly from the shock measurements

of the two samples

%=<%)=2V

by the equation

.- -1

‘IQ- ‘H1

!J

. (15)

‘pm + ‘02)(V02 - ‘)-(PH1 + ‘O1)(VO1 ‘v

b ● ● *9 ● ee

.** ● ● ● ●*e ● ●*:

● ● 8, ● ● **

● : ● **W ● :: ● 0: 8**

● **O* ● ** ● e

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● m

● *****

:.: .

●

● e@.@ ● ao ●

● ea ● ** ● ***

. . . . . . . . . . . .

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● ● 8

● : :* ● 0 ::

60 :* ●:

●

:: ●:0 : ●:0 :00 ● 0

Such an experimental procedure for iron has recently been

reported in the Russian literature (Alttshuler ~ ~. Soviet

phys JETP~, 606, 1958). Since the GMX-6 shots were made

on two repacked samples of different densities, this procedure

was used to obtain yet another estimate of y .

G

Inspection of

the Hugoniot curves measured by G~-6 (see figure 5 on page 13

of IAMS-2760) showed that the only region In which the

two Hugoniots were simultaneouslywell determined at a single

density was in the vicinity of a density of 3.0 gin/cc.

Application of equation 15 gave a yG of about 1.5 to 1.7.

Equation 11 gives a value of L.25 at this compression. The

sketched in curves

give a 7G of about

yields 0.25. This

the material is in

in IAMS-2760 at a density of 3.8 gin/cc

0.58 whereas application of equation 11

point is of questionable validity because

the mixed phase region at this compression.

In view of these estimates, it seems probable that y lies

somewhere between 0.5 and 2 with a value of about 1 being

a reasonable compromise. A constant Grheisen ratio of 0.5 \

has been used in many of the problems that have been run to

date and gives the slope of the radius time curve for Fisher

shot much better than yG = 1.5. We conclude that equation 11

Is of questionable validity especially when the material

behind the shock has expanded greatly thus giving very large

7G- We are at present inclined to use a constant 7G rather

than the variable form indicated by equation 11.

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APPROVED FOR PUBLIC RELEASE
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. ●=* ● ●°0 ● .*:

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9* ● *

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:: 0000

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80 ● *e 8:* ● ● 9* ●

III NUMERICAL INTEGRATIONS

To start the numerical integrations, the energy of the nuclear

explosion is distributed uniformly into the central zone over a period

of 5 shakes. The central zone is assumed to be a sphere of approximately

the same volume as the bomb chamber. The initial density is taken to be

the mass of material vaporized by the explosion divided by the chamber

volume. It has been estimated that this initial density is approximately

that of the surrounding alluvium (1.7 gin/cc). A gaseous iron equation of

state taken from Cowan~s Thomas-Fermi-Dirac calculations (LA-2124) is

assumed to be pertinent for this zone; all other zones use a combination

of the Gr&m?isen equation

in part II above.

It was found that

and an appropriate TFD equation as discussed

Cowan~s iron tables often gave negative pressures

when extrapolated below temperatures of about 1 volt. The temperature of

this first zone has been initially set at 1 volt to overcome this difficulty;

the error introduced by this Initial temperature should be negligible since

the first zone temperature rises very rapidly as the explosion energy is

integrated into the zone during the first 5 shakes. All the zones representing alluvium are initially set at temperatures of 3 x 10‘2 volts (about 350°K).

Energy transfer by radiation and conduction are assumed to be

negligible in all problems

these processes could have

time curves, especially at

point at a later date when

other J-15 calculations.

run to date. It is possible that the inclusion of

some influence on the computed shock radius versus

very early times. We propose to clarify this

opacities and conductivitieabecome available from

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● ●°:

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● : ● *** ● :: ● 0: ● **

● 0 9 ● * ● 9* ● C

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
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● ****O ●

● a*&* :0: 0

:** 9 9 .** 

. . . . . . . . .

8* ● ✎ ✎ ● ✎✎ ● *9 9.* . .

● *O ● *

● *O* ● * ::

● * :.*. ●:

●

● * ●:0 : ●:0 :00 ● *

The code has provision for including latent heats of fusion

and of vaporization at the melting and vaporization temperatures respectively. Since these quantities are very poorly known for many minerals

and not measured for Nevada alluvium they are usually set to zero in our

calculations. In addition, finite latent heats complicate our integrations by introducing instabilities which damp out slowly in the melted

region. In any case, the effects on the arrival time appear to be only

important at ranges considerably closer to the bomb than the closest

station in the present layout.

Somewhat similar instabilities occur when a zone, omemelted,

cools by expansion below the melting point. If the “resolidified” zone

is allowed to change back from a TFD equation of state to a Gr~e&sen

type equation of state, an instability is set up and the time interval

used in the integration must be temporarily reduced in order to allow the

instability to damp out. If the zone is forced to always use the gaseous

TFD equation after melting on compression and not allowed to change back

to the Gr&misen equation after the expansion, the problems run much more

rapidly and fewer instabilities occur. Since the shock has progressed a

considerable distance in the solid by the time the melted region cools

again to the melting point, the continued use of the gaseous TFD equation

in the imer zones does not have much effect on the radius versus time

curves for the later stages of the expansion, except to make the problem

easier to run on the machine. As en example,two problems were run at 13

kilotons and a constant yG of .01. Melted zones in problem 65 were allowed

to resolidify to the Gr~neiaen equation of state,whereas the same zones

in problem 68 were forced to retain the gaseous equation of state. At

8 milliseconds the shock radius for problem 65 was 23.2 meters and that

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9* ● ** ● ** ● ●

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● *9,* ● ●*oC*: ::0

● * ● bb ● ● ● ● 4

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
● 0 ● ● O*

● ,ee ● .=* ● ●°:

● ● m ● O ● ***

● ● ** c ● ** ● 00090:0 ● 9

● ● . 8.. ● . . . ● *

● 0 ● ma ● o* ● ** ● ** ● *

● 0 9* ●: : .:*O ::00

● O : : :0

● 0 ●0: ●:0 ● coo c

for problem 68 was about 25.5 meters. Comparing problem 63, run at

10 kilotons we have R~ = 22.0 m at 8 ms. Thus the error introduced

by not allowing resoltdificatlon appears to be worth the equivalent of

about 1 kiloton at yields in the 10 to 15 kt range. Since we do not

expect accuracies much better than about ~ for yield measurements

●

by this method,

once a zone has

on expansion.

we have

melted,

chosen to retain the gaseous equation

and not change back to the Grhei6en

of state

equation

The numerical integrations are reasonably insensitive to the size

of the zoning used in setting up the problems. The usual zoning scheme

is to assign the first zone (bomb chamber) a radius of 1 meter; each

successive alluvium zone is assigned a radius such that its mass is about

a factor of 2 above the mass of the preceding zone. This factor of 2 is

applied until the difference in radii of two successive zones is about

2 meters (for yields in the vicinity of 10 kilotons) and thereafter a

constant Z!Rof 2 meters is maintained. For yields of the order of 1 or 2

kt, the maximum M allowed is 1 meter or less. An even finer mesh would

be desirable.in order to more accurately locate the shock front within a

zone but the above

To see the effects

radius at 5 meters

prescription seems to be adequate for most purposes.

of zoning changes, problem 61 run with the first zone

(using alluvium equation of state rather than iron)

may be compared with problem 79 in which the first zone had R. = 1 meter

and used the iron equation of state for that zone. Both problems were

run at 10 kt and used a variable yG (equation 11). Problem 61 gives shock

radii greater than problem 79 by about 0.5 meter for times greater than

3 or 4 milliseconds. Pr6iblem61 also had maximum zone widths of about 3.5

meters whereas the maximum for problem 79 was 2 meters. Such a zoning

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b ● 00

● ●°:

● ● **

● : 99** ● :: ●0: ● *9

● 8mb* ● ** . .

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● ☛ ● e* bo

*O* ●°0 : ●** ●

beobao

00060 h: “o 8 ● :.- 9.* :.* : : ●-a

. .

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:

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difference is extreme

zoning differences to

range.

and we expect the error in our integrations due to

be less than the equivalent of 1 kt in the 10 kt

A variation in the assumed

chamber) has fairly small effects.

density of the first zone (bomb

A change in assumed density of 1.7

gin/cc

radii

and a

(problem 72) to 1.0 gin/cc(problem 73) gives a difference in shock

of about 2$ at ~ milliseconds. Both problems were run at 1 kiloton

constant y

G

= 0.5.

We are at present using smelting point of 0.5 volt (5800°K)

for alluvium. It is very likely that this is too high. However, Cowan’s

TFD equations are not expected to be reliable below temperatures of 1

volt and in fact sometimes give negative pressures below 0.5 volts. Hence

the uncertainties In the

us to make a choice of a

that the break in the Us

is an indication

on being shocked

accept a melting

Cowan tables below

melting point that

versus Up curve at

temperatures of 1 volt force

is not too low. We believe

about 400 kilobars (figure 1)

of melting.

to about 400

point of 0.5

Nuckolls of LRL also states that tuff melts

kllobars (see UCRL 5675, page 12j). If we

volt we find from our listings that alluvium

reaches this temperature by being shocked

and 500 kilobars depending on the assumed

the pressure reaches 340 kb just prior to

to pressures between

Grheisen’$ ratio.

melting and for 7G =

about 340

For yG = 0.5

4.5, a

pressure of 490 kb is present Just below the melting temperature. In any

case, a fairly large change in the assumed melting points does not seem to

affect the computed radius versus time curves. As an example, problem 78

(melting temperature = 1 volt) and problem T9 (melting temperature = .5 volt)

show a negligible difference in the shock radius (20.5 meters) at 5

milliseconds. Both problems were run at 10 kilotons.

● *9 ● ● O* ● ** ● e ●** b ● * .*

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● O ● *9 ● ● ● ● *

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
● e ● * ● e* so ● m***Oe aoa

● ● * boo, :0: ● ● ** :06 ● ● ● ●C: ● am

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● e 90

9*9** ●:09

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se ● *O ● *U 8 ● 00 ●

Although the shock radius versus time curves do not seem to be

very sensitive to the assumed melting points, the computation of the size

of the melted region and the subsequent cavity would show a greater

sensitivity. In view of possible interest in estimating the cavity size,

some further discussion of the proper melting temperatures will be made

here.

AS far as we are aware, no extensive experimental work has been

done on the variation of melting point with pressure for various minerals and

rocks. Various theoretical estimates for the melting points and temperatures

for minerals in the earths deep interior are reviewed by Verhoogen (in

physics and Chemistry of the Earth. Volume 1, 1956) and by Gutenberg (Physics

of the Earths Interior). We shall summarize the theory briefly below. .— —

The increase of melting point T*

of the material is given by the Clapeyron

dT* AV AV

‘===W’W d.p ‘

with pkessure P for a given phase

equation

(16)

where AV/As is the ratio of the volume change to the entropy change at

the melting point, L is the latent heat of fusion. Some information on

latent heats and volume changes for minerals at atmospheric pressure are —

given in the Handbook of Physical Constants (Geological Society of America — —

SpeCial Paper No. 36 - edited by F. Birch). Verhoogen (page 26) ~uote8

the following experimental values for the initial slope of the melting point

curve in degrees per kilobar: Fosterite 4.7; diopside 13; albite, 26.

Jefferies (The .— Earth,

for silicate rocks:”

T: = 1300°K,

4th edition, page 288) adopts as “fairly typical values

L = 100 cal/gm,

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● * ● ** ● ** ● ** ● 9* 9*

● ● 99* 900 ●’0 ● ● b ●**

●

● ●*:

● bm ● ● ● ● ●

● ● 9*** ● :: ●0: ● **

●   ● 00 ● e

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APPROVED FOR PUBLIC RELEASE
● e ● ea ● ** 9*9: ** ●** b

● mmOO ● **

● 00 ● : ● ob o

● 00 ● ** ● ***

.* *.**.. ..**

● ☛ ● 00 ● *9 ● 9* ● ** . .

● ● *O

● :000.. ::

● 9**.. ● :

●

● 8 ●:0 : ●:* :00 ● 0

If ps is taken to be of the order of 3 gin/ccfor these “typical minerals”,

we get

give a

point”

dT*/dp ~ 10°K/kilobar. A value of dT*/dp . 10°K/kilobar would

melting point of x 5500°K at 400 kb if we assume the “melting

of alluvium is approximately 1500°K.

Unfortunately, the slope of the melting point curve Is expected to

decrease appreciably at higher pressures because AV becomes smaller (the

compressibility of the liquid is usually higher than that for the corresponding solid). According to Verhoogen, As Is not likely to be very much

affected

basis of

pressure

by increasing pressure. Thus, melting points estimated on the

the initial slope alone are likely to be too high.

Simon has experimentally studied the change in melting point under

for substances (eg. helium) with very low melting points and which

can be investigated over a wide range of pressure. He finds that these

substances obey the law

()

P T*C

-=

0’ ~ ‘1” (l-#)

Here T* is the melting point at pressure P, T: the melting point at zero

pressure, c is a constant and a is related to the “internal pressure”,

-(bE/hV)T.Simonhas used equation 17 to estimate the melting point of iron

under high pressure (Nature, 172, ‘746,1953).

Combining equations 16 and llwe find

L

at=—.

AV

Simon finds thatc%/(~U/hV)T = 1.5 from the

assumes this to hold for other materials.

(18)

data on alkali metals and

The constant, a, may be estimated

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● amme ● *9

● 9* + ●em ●

● 9 *** ● ● s

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● ✎ ● ☛☛ 9.. .me . . . . .

● 9 ● ● s ● 00:0 ●:**

● :: : : : :0

x ● m 800 ●:0 ● ● 00 ●

uz

c TVo

&v T

=7G~;

where Cv is the specific heat, V the

friend, Grbeisen’s ratio.

Applying Simonts formula to

of the melting point curve we arrive

(19)

specific volume and YG is our old

various estimates of the initial slope

at Table II as estimates of the

melting temperatures at a pressure of 400 kilobars. The melting point

at atmospheric pressure for alluvium was assumed to be 1500°K in these

computations. Case c corresponds to Nuckolls’ estimate of a latent heat

of fusion of 715 calories/ginfor (UCRL 5675, Page 123), an assumed

density of 1.7 gin/ccand a volume change of 10% on melting.

In view of the extreme

into the computations listed in

point as high as 5800°K, but it

uncertainty in the basic data entering

Table II, we cannot rule out a melting

seems more likely that it is not over

about 2500°K at 400 kb. Until our gaseous TFD equation of state is adjusted

to give better reeults at low temperature, we have chosen to retain the

5800° figure.

IV RESULTS AND CONCLUSIONS

Some of the computed shock radius versus time curves are summarized

in figures 4, 5 and 6. They show the sensitivities to various assumed

parameters. The curves are the best smooth curves as determined from the

individual shock radii given by the listings.

Figure 4 shows the difference between equation 6 and the three

phases defined by the constants:

co =

0.55 m/ms, so = 1.85, valid for V> 0.317 m3fgm

c1 = 2.40 m/ms, sl = 0.80, valid for 0.317 > V > 0.242 (20)

APPROVED FOR PUBLIC RELEASE

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40

30

20

P?

10

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1 2 345678910 ~~1314

t (~1.llseconde)

Figure 5

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12 345678910 U~1314

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Figure 6

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These are the first three phases (except for the sonic precursor) as

defined by the GMX-6 data (see solid line in figure 1). Both problems

were run with a variable yG (equation 11). Figure 4 points up the

importance of verifying by static measurements the low pressure extrapolation of the GMX-6 data as was discussed in part II above.

Figure 5 shows the effect of the assumed GrheisenVs ratio.

All of these problems were run with the phase constants (20) plus a son,ic

precursor of 1.> m/ms up to 1 kb as illustrated in figure 1. All these

problems were computed at 13 kt but with constant values of 7G of 0.5, 1.5

and 4.5. At early times the curves are somewhat insensitive to values

of 7G below about 1.5. The curve for yG = 0.01 (not shown) lies just

slightly below that shown for 7G = 0.5. It is interesting to see the

effect of a variable 7G (equation 11) in the curve for problem 69. At

late times when a large fraction of the material behind the shock has

expanded considerably, equation 11 indicates that, on the average, 7G is

quite large behind the shock and this is confirmed by the curve in figure 5.

As has been stated above, the Dugdale-MacDonald relation seems of dubious

merit when applied to rocks,and we have chosen to make our yield analyses

on the basis of a fixed 7G somewhere in the range 0.5 - 1.5. For exemple,

problem 66 with 7G s 0.5 matches the observed

well. If the Dugdale-llac!kmaldrelation were

data analyses, we would have a very difficult

points as well as we do by using

Finally, figure 6 shows

fixed 7G of 0.5. All the curves

a fixed 7G.

the computed

points from Fisher shot very

used for

time matching the observed

yield sensitivities for a

are plotted to times at which the computed

pressures at the shock front were of the

times we would expect to see essentially

is independent of yield.

● ● am ● ● om ● m. ● 9

●-0 0- i ● -O*

● O* ..29 ::0

● m.*

● : ● 0

● 0 ●:0 :90 ● OO :00 ● 0

order of 5 to 10 kb. At later

the sonic precursor whose velocity

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APPROVED FOR PUBLIC RELEASE
● ☛☛✎

● ●:0 ● m. . . . . .°:

● *

::

●

: :0

● 0 ●0: ●:9 ● ●:0 ●

In view of the uncertainties in the ambient density and Grheisents

ratio and the computed sensitivities to melting point

it

to

would appear that the yield accuracy obtainable by

observed times of arrival is of the order of &@.

●la ,90Wl 0!0..! . .*., .,

010108,1

:~o”: { .: ; ‘.:

● 0 ..* . . . . . . . . . . .

and zoning differences,

fitting these curves

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE