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WAVE PROPAGATION IN ANISOTROPIC ROCKS 

Semiannual Technical Report No. 1 

Contract No. H02j|p021 

Sponsored by Advanced Research Projects 
Agency, ARPA Order No. 1579, Amend. 3 

Program Code 62701D 

Principal Investigator: Dr. W. Goldsmith 
Faculty Investigator: Dr. J. L. Sackman 

University of California 
Berkeley, California 94720 

Sept. 25, 1972 


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l O"<0iNA ting ACTUITv (Cufprfmlm mullimt) U*- NIPONT ICCuNi Tv CLASS! FICATION 

. . , I Unclassified 

University of Califo-nia, Berkeley 


J REPORT TITLE 

WAVE PROPAGATION IN ANISOTROPIC ROCKS, III 


4 OCSCMiF TivC noth (Typm ml tmpomi mnd Inelumlvm dmimmj 

Semiannual Technical Report, *tebruary 23, 1972 to August 31, 1972 


• authORHI middim initiml, iff nmm*) 


Werner Goldsmith and Jerome L. Sackaan 


t M404T OATC 


trn. TOTAL NO. or Stall |»*. NO. OF NETS 


September 30, 1972 


S«. CONTRACT OR GRANT NO- 

H0220021 
6. fnojcc t no 

ARPA Order No. 1579, Amend. 3 

C. 

Program Code No. 62701D 


io. oiirmiuTiON itatimint 



•6. other REPORT NOI8) (Any othmt number• thmt mmy M 
ihi» rvpetl) 


Distribution of this document Is unlimited 



11. IPONIORING MILITARY ACTIVITY 


Advanced Research Projects Agency 


■ i. aoitnact report summarizes progress on the analytical and experimental aspects 

of the Investigation concerned with wave propagation resulting from Impact on a 
Yule marble block and an Investigation of Its mechanical properties. The system 
was modeled as a transversely Isotropic halfspuce with the elastic symmetry axis 
lying in the free surface, loaded by a concentrated normal force of arbitrary time 
variation. The analysis for the displacements, strains and stresses in the interior 
Is complete and numerical results have been obtained by integral transform methods. 

A corresponding finite element program has also been completed. Transducer pack¬ 
ages for the measurement of stresses in the interior of a Yule marble block have 
been developed, calibrated and tested. Drilling and embedment techniques for the 
testing of the Yule marble blocks under actual impact conditions have been 
evolved. Static and dynamic tests in tension and compression for various orienta¬ 
tions of samples have been completed as well as cyclic loading, creep and some 
fracture tests. 


,1473 


S/N 0101*807.6801 


iclassifled _ 

Security Cl*t*ific«tion 


(PAGE 1) 















.iiaailicitiofl 









SEMIANNUAL TECHNICAL REPORT NUMBER ONE 


ARPA Order Number: 1579, Amendment 3 
Program Code Number: 62701D 

Contractor: The Regents of the University of California 
Effective Date of Contract: February 23, 1972 
Contract Expiration Date: February 22, 1973 
Amount of Contract: $62,436.00 
Contract Number: H0220021 

Principal Investigator: Professor W. Goldsmith, (415)642-3739 
Project Engineer: Professor J. L. Sackman (415)642-2950 
Title: "Wave Propagation in Anisvtropic Rocks" 

Report Period: February 23, 1972 to August 31, 1972 

Sponsored by 

Advanced Research Projects Agency 
ARPA Order No. 1579 Amend. 3 
Program Code 62701D 

This research was supported by the Advanced Research Projects Agency 
of the Department of Defense and was monitored by the Bureau of Mines 
under Contract No. H0^j0021. 

The views and conclusions contained in this document are those of the 
authors and should not be interpreted as necessarily representing the 
official policies, either expressed or implied, of the Advanced Research 
Projects Agency or the U. S. Government. 



SEMIANNUAL TECHNICAL REPORT NUMBER ONE 


ARPA Order Number: 1579, Amendment 3 
Program Code Number: 62701D 

Contractor: The Regents of the University of California 
Effective Date of Contract: February 23, 1972 
Contract Expiration Date: February 22, 1973 
Amount of Contract: $62,436.00 
Contract Number: H0220021 

Principal Investigator: Professor W. Goldsmith, (415)642-3739 
Project Engineer: Professor J. L. Sackman (415)642-2950 
Title: "Wave Propagation in Anisotropic Rocks" 

Report Period: February 23, 1972 to August 31, 1972 

Sponsored by 

Advanced Research Projects Agency 
ARPA Order No. 1579 Amend. 3 
Program Code 6270ID 

This research was supported by the Advanced Resonrch Projects Agency 
of the Department of Defense and was monitored by the Bureau of Mines 
under Contract No. H0^0021. 

The views and conclusions contained in this document are those of the 
authors and should not be interpreted as necessarily representing the 
official policies, either expressed or implied, of the Advanced Research 
Projects Agency or the U. S. Government. 



SUMMARY 


This report summarizes progress on the analytical and experimental 
aspects of the investigation concerned with wave propagation resulting 
from impact on a Yule marble block and an Investigation of its mechani¬ 
cal properties. The system was modeled as a transversely isotropic 
halfspace with the elastic symmetry axis lying in the free surface, 
loaded by a concentrated normal force of arbitrary time variation. The 
analysis for the displacements, strains and stresses in the interior is 
complete and numerical results have been obtained by integral transform 
methods. A corresponding finite clement program has also been completed. 
Transducer packages for the measurement of stresses in the Interior of a 
Yule marble block have been developed, calibrated and tested. Drilling 
and embedment techniques for the testing of the Yule marble blocks under 
actual impact conditions have been evolved. Static and dynamic tests 
in tension and compression for various orientations of samples have been 
completed as well as cyclic loading, creep and some fracture tests. 



1 


INTRODUCTION 

This report represents the first semi-annual tecinical report 
under contract H02ip021 between the U. S. Bureau of Mines and the 
University of California on the subject entitled "wave Propagation in 
Anisotropic Rocks." The current contract period from February 23, 1972 
to February 22, 1973 represents the second year of the program; the 
present document summarizes progress up to August 31, 1972, with 
emphasis on the work completed since the beginning of the second year 
of the contract. A full account of the activities on this project 

* 

during the first year of its existence may be found in Reference 1. 

The scope of the program is detailed in the proposal identified 
as UCB-Eng 3286 dated April 26, 1971 and submitted on behalf of the 
University of California, Berkeley, by W. Goldsmith as Principal In¬ 
vestigator and is also spelled out in Artile I of the subject contract. 
A program schedule was also submitted at the end of March, 1972; this 
schedule has been faithfully observed with the exception of the work 
on fracture initiation and propagation; in some areas, however, the 

work is progressing ahead of schedule. 

The work has been subdivided into the following categories: 

(a) Theoretical Examination of the Wave Propagation in an 
Anisotropic Solid. 

(b) Experimental Examination of Wave Propagation in an 
Anisotropic Solid. 

^Bibliography appears at the end of the text. 



2 


(c) Experimental Determination of the Physical Properties of 
Yule Marble under Static and Dynamic Loading. 

The following personnel have been employed on the project during the 
report period: (a) Mr. S. L. Suh who is concerned with the develop¬ 
ment and evaluation of the history of the field variables in a trans¬ 
versely isotropic half-space subject to an impact or itu free surface 
using integral transform methods, (b) Mr. M. Katona who completed the 
f. rst pha.?e in the development of a finite-element program for the 
soluMon of the subject problem and who has since departed to resume 
his permanent position with the U. S. Navy, (c) Mr. R. Kenner who 
assisted in various tasks as a Fortran programmer, but departed for 
an industrial position in Anril, (d) Mr. G. Dasgupta who has taken 
over the completion of the subsequent phases of the finite element 
analysis, (e) Mr. K. Krlshnamoorthy who has been concerned with 
category (b) of the investigation, (f) Mr. Tom Jones who has been 
assisting Mr. Krlshnamoorthy in the development of experimental 
equipment, (g) Mr. S. Howe who has been assigned to the pursuit of 
category (c), assisted for a brief period by Mr. J. Thorington, and 
two laboratory assistants, Mr. E. Lin and Mr. G. Wilcox. Messrs. 

Suh , Katona, Dasgupta, Krlshnamoorthy and Howe will use or have used 
part or all of fieir work under this contract as part or all of their 
graduate research requirements. Supervision and active collaboration 
have been provided by Professors W. Goldsmith and J. L. Sackman who 
are responsible for the conduct of the program. In the development 
of the finite element program, the expert resistance of Prof. R. L. 
Taylor of the Department of Civil Engineering, University of Californio, 



3 


Berkeley without compensation is gratefully acknowledged. 

The only technical difficulty that has been encountered during the 
first six months of the present contract that is as yet unresolved 
concerns the evaluation of the surface displacements of a transversely 
isotropic half-space under the action of a concentrated, time-dependent 
surface lead. This quantity cannot be obtained directly from the 
corresponding solution of the interior displacements evaluated at the 
free surface, but rather must be analyzed as an independent problem. 

It appears that the crux of the difficulty may lie in the accuracy of 
the numerical evaluation of the integration that converts the two- 
dimensional solutions to the three-dimensional situation. This phase 
of the investigation will be further scrutinized during the remainder 
of the contract period. 

Another difficulty concerns appropriate staffing of the investi¬ 
gation. Although it was anticipated that the portion of the program 
dealing with fracture iniatiation and propagation could not be com¬ 
pleted during the first two years of the contract, it was hoped to 
make a significant start in this direction by securing the services of 
a graduate student, Mr. J. Thorington, who expressed considerable 
interest in pursuing research in the area of fracture of brittle 
materials. However, after a very brief association with the project, 

Thorington accepted a full-time position in industry for financial 
reasons. Consequently, a replacement must be located and suitably 
trained which will considerably delay the planned execution of this 
phase of the project. 



4 


The remainder of the report will be concerned with a more detailed 
technical description of the progress achieved. 

2. SUMMARY OF PROGRESS ACHIEVED DURING FIRST YEAR OF INVESTIGATION 

The integral transform technique was developed and partially 
tested to obtain the displacement field in the transversely isotropic 
solid with an axis located in the free surface under a Heaviside in¬ 
put. This process constructed a three-dimensional solution from the 
integration of a series of two-dimensional problems associated with 
line loads on the surface of the half-space. It employed a Cagniard- 
de Hoope transformation which simplified the inversion process and 
led to a physical interpretation of the transient wave process in 
terms of well-established concepts of wave and slowness surfaces that 
have been employed in the field of crystal acoustics. 

A finite element program had been written for the three-dimensional 
problem cited above and was in the debugging and test phase. It had 
also been specialized to the simpler case of isotropic behavior where 
other solutions for checking purposes exist. Results obtained at 
that time included the uniaxial wave process in a rod and the surface 
motion of a half-space. 

In the experimental wave propagation phase, the major accomplish-; 
ments achieved at that time included the development of a crystal 
transducer package with a laterally unconstrained crystal employed as 
a sensing element, the calibration of both the crystal and the entire 
package, the development of installation techniques for the embedment 
of the transducer inside cores drilled in rock bars, and some progress 


in the development of a suitable grouting material to fill the core 
holea after inatallation of the tranaducer unit with a minimum of 
dynamic miamatch. 

Crystallographic techniquea were developed for the location of 
the axla of elastic symmetry of the Yule marble specimens. Static 
compreaalve testa, aome with repeated loading, were conducted on 
samples of the material, indicating significant non-linearities and 
the presence of hysteresis. A technique was developed for the genera¬ 
tion of constant strain rates in the intermediate range of 10 to 100 
aec utilizing an adaptation of a Hopklnson-bar procedure. 

3. PROGRESS DURING REPORTING PERIOD 

(a) Theoretical Examination of Wave Processes 
(1) Integral Transform Approach 

Introduction 

It was reported previously [1] that alownesa, velocity, and wave 
curves for the Yule marble characterized by the elastic constants 
obtained by Ricketts [2], have been evaluated. Furthermore, Cagniard- 
de Hoope paths for different angular ranges of the medium were 
generated. From these, displacement fields in the half-apace due to 
a point load representing a Heaviaide function of time denoted as the 
fundamental displacement solution, were constructed. Corresponding 
computer programs were written to perform the required numerical 
evaluations. During the laat aix months, formulations for the dis¬ 
placements, strains and streaaes produced by a realistic input have 
been obtained by uae of convolution integrals in conjunction with the 






6 


fundamental solutions. Also, a computer program for the above with 
an assumed realistic force input given bjr I * 1 sin ^ y has 
been written and corresponding curves for several positions in the 
medium have been obtained. 

During the next six months, it is planned to check the solution 
and to compare numerical results both with experimental data, yet to 
be obtained, and with the results of a corresponding finite element 
analysis. An additional effort will be extended to obtain numerical 
results on the surface of the half-space. Some exploratory efforta 
may also be directed towards obtaining numerical resulta for the 
displacement fields due to a point source in the medium and possibly 
for an orthotropic material 


Formulatlona 
i) Body Waves 

As indicated in Eq. (35) of Ref. ID, the dlaplacement response 
in a transversely isotropic half-space subjected to a force normal to 
the free surface represented by a Heaviaide time function may be 


expressed as 


u (*,t)« 

k h 


TT 


1 

2tt 2 PR 



<t,p,e/5,Q,R>] Pj 





(k * 1,2,3) 


(1) 


in which p (1) = p (1) (t,R,6,'5,Q) is a Cagniard-de Hoop path defined 
by Eqs. (29) and (30) of Ref. [1]. 




7 


6 Is an angle defined In the alowneaa apace, 
t la the tlae, 

x la the poaltlon vector of a response point in a Cartoalan 
coordinate systea In the aedlua 

(R, 1.Q ) 1. tho poaltlon of a response point in spherical 
coordinates, 

W* 1 * are coaplex algebraic functions 

H(t) * o t <0 } iB t *'* Hetvl * i<le f unction * 

ro[ 1 indicates the real part of a coaplex quantity. 


In Eq. (1), k represents the coaponent of the diaplaceacnt field In 
s Cartesian syatea, and J Indicates the nuabering of the roots of 
Eqs . (29) and (30) of Ref. HI. 

Equation (1) was utilized to o'itsln dlsplsceaents resulting froa 
the input of an arbitrary tlae function through s convolution lntogral. 
Let 1 «= l(t) be the tlae history of tho surface force, then the 
corresponding diaplaceacnt u^(x,t) Is given by 

t 

u^x.t) - J i(t-t') u^ (x,t') dt' (2) 


•here 


1 (t) 


dl(t) 

dt 


9 


and t' Is a duaay variable. 

7*0 diaplaceacnt gradients which will be used to compute strain 
fields are obtained analytically by slaply taking spatial derivatives 
of Eq. (2). In the usual indlclsl notation, 



8 


<x,t) 


i< k (x,t) 


au, 


[ i(t-t') » at' 

o J 


(3) 


Though Eq. (3) appears simple, the expresalon la actually quite 

Involved beC4use »; are complicated Implicit functlona of x aa 

H J 

can be aoen by Inspection of Eq. (1). For the purpose of writing and 


testing a computer program which carries out the convolution Integral 
Indicated by Eq. (2), the Input force has been taken in the fora 


I(t) 


ain 


/nt' 


(4) 


where I Q Is the aaplltude, and t* la the duration of the Input pulse. 
It turns out that expression (4) la a good approxlaation for the In¬ 
put force produced when a steel ball Is dropped vertically onto the 
free surface of a half-space. The tlae derivative of Eq. (4) is 


1(0 


S 



(5) 


which Is to be substituted into Eq. (3). It is convenient to Intro¬ 
duce a parameter t * , where R Is the distance from the Impact 

point on the free surface of the half-space to a receiver In the 
medium. Rewriting Eq. (3) by use of Eq. (5) results in the following 

/■* x *o r 

U k (X(t) = - J 8ln 2TT 

0 


dT ' 

H 


( 6 ) 


9 


whore T* “ — . This represents ihe displacement field under the 
input force given by Eq. (4). From the infinitesimal theory of elas¬ 
ticity, the components of the strain tensor are defined as 

'lj *5 ‘“l.j * “j.l’ 1.2.3) (7) 

where u^ are given by Eq. (3), and the components of tho stress 
tensor are expressed as 


°1J “ C lJkX °kjt 


(l.J.k.i = 1,2,3) 


( 8 ) 


For a transversely isotropic medium, c^ ^ consists of 5 Independent 
constants. These were obtained experimentally by Ricketts [2] for 
the Yule marble considered here. 

It is convenient to use a reduced form of Eq. (8) written as 


0 . = c e 
I IJ J 


(I,«I — 1,2,3,... ,6) 


( 8 ) 


where Oj = O n > ^ = 0 22 . ®1 = ®ll' ®2 = ®22 ’ * *’ ’ and 

c jj " c jjjj ( c j 2 = ®U 22 1 "' etc. [2]. The matrix form of Eq. (8) 
for the computer programming may be expressed as 



10 



O) 


In the existing computer program which evaluated fundamental 
displacement solutions given by ( 1 ), .subroutine which 
performs a convolution Integration simultaneously for the three 
Cartesian components of displacement has been incorporated, debugged 
and tested using the input force given by Eq. (4). Furthermore, a 
subroutine which finds displacement gradients given by Eq. ( 3 ), 
strains given by Eq. (7), and stresses given by Eq. (8)' has been 

written and tested. A listing of this program will be included in the 
final report. 


ii) Surface Motion 

For the surface motion, the Cagniard-deHoop paths defined by 
Eqs. (29) and (30) of Ref. [1] collapse into the real axis on the 
p-plane on which the Cagniard-deHoop paths are defined [1]. However, 
a simple pole lies on the real axis which corresponds to the Rayleigh 
pole for vhr, isotropic medium. The presence of the simple pole at 
P = - — , where v R is the phase velocity of the Rayleigh wave, 
requires an indentation of the path of integration to avoid the pole. 
When this is done, the points on a small semicircular indentation 



around the pole no longer corresponds to real tine. Therefore, the 
contribution from the pole waa obtained separately from those of 
ordinary points on the Cagnlard-deHoop path and expressed as 


11 


H pole 


(Xj.OjXg.t) 


■ fl i "(* 



( 10 ) 


The Heaviside function appears In the relation above Instead of the 

( r N 

Dlrac-delta function 611 - -/ which was Incorrectly exhibited 

v (*5, 6) ' 

in Eq. (36) of Ref. [1]. To obtain solutions for a realistic Input 
for the displacements, strains and stresses, a procedure analogous 
to that employed for the body wave solution may be applied. 

Results and Discussions 
1) Body Waves 

A subroutine which finds the roots of the Cagnlard-deHoop path 
by solving a quartlc algebraic equation has been modified so as to 
be faster and more accurate. This Is rather important since most of 
the computing time Is spent In finding the roots of the quartlc equa¬ 
tion defining the path. Examples of the output obtained from the 
computer program for the force Input given by Eq. (4) will be presented. 
Three components of the fundamental displacement u., u and u along 

1 « O 

** _ 

a ray at 0 = 80 , <p = 45 , which lies In the fan of critical angular 
ranges for both *5 and <p , [1], are shown In Fig. 1. As can be seen, 
although the u^ component appears very much like the Heaviside 
function In time, u^ and exhibit a sharp change In shape at 


12 


( l \ 

r )~ 1*0 X 10 ° whlch is •pproximately the arrival time for the 
quasi-shear waves. All three components of the displacement approach 
their static values at approximately t =* 2 X 10 -5 . Curves along 
other rays have also been obtained, although they are not included 
here. The ratios of the amplitudes of the three components become 
quite different depending on the angles'? and <p . Figures 2, 3, 4 
and 5 represent convolved components of displacement of Eq. (6) for 
the same ray as shown in Fig. 1 but with R = 1,2,5,10 inches, re¬ 
spectively. As can be seen from this sequence, the amplitudes of 

the displacements decrease proportionately to | as would be expected 
for the body wave. 

In the evaluation of the convolution integral of Eq.(6), the 
time steps employed were, At = 1 xl0‘ 6 , 2.08 Xlo“ 6 , 8.33 x lo" 7 , 
and **.1x10 for Figs. 2,3,4 and 5 respectively. It can be seen 
that the AT employed for Fig. 3 appears to have been too big resulting 
in curves which are distorted compared to the other results obtained 
with smaller integration steps. Figures 6,7,8,9 and 10 represent 
the fundamental normal strain components e n> e 22 , a^, the convolved 
strain components e n , e^, e^, e^, ^ and e 23 for the input 
force given by Eq. (4), the fundamental normal stress components 

CT 11 * °22’ °33 and the conv °lved normal stresses Ojj, 0 22 and c 33 , 
respectively, along a ray defined by ? = 80°, <p = 45 °. A s can be 
seen, the fundamental strains as well as the fundamental stresses 
vary rapidly for the range of t which lies between 6 X 10 - ^ and 
1.5 x 10 and they appear to approach their static values at a 
time of about t = 2 X 10 5 . The above time interval brackets the 



13 


arrival times of the two quasi-shear waves for this medium. 

Additional stress and strain data have been obtained, but are 
not included in the present report. The total time to compute the 
fundamental as well as the convolved displacements, strains and 
stresses was about 7 seconds on the CDC 7600 computer for a single 
receiver point. 

ii) Surface Motion 

The Rayleigh function for this medium has been checked against 
Buchwald's expression [3l who utilized a plane wave approach. A 
program giving the Rayleigh wave speeds for different directions on 
the free surface of the half-space has been developed. A program 
which computer* fundamental displacements on the surface from a 
Heaviside input force has also been written. The results obtained 
show spurious oscillatory motion for the time intervals from zero 
to the time of the Rayleigh wave arrival, although, at later times 
the solution appears to be reasonable. The cause of this difficulty 
has not yet been determined. For this reason, no further attempt to 
compute convolved quantities has been made at this moment but effort 
will be directed toward resolving this problem during the next 

a 

6 months. 

iii) A computer program which plots the history of a field variable 
at a given receiver point has been written using the GDS-CALCOM 
system [4] developed at the Computer Center of the University of 
California, Berkeley. This program was used to produce the plots 


presented here. 


14 


Conclusions 

A program which yields displacements, strains and stresses for 
an Input of the form of Eq. (4) has been developed. This program 
computes above quantities In the half-space except on the free sur¬ 
face and on the vertical axis given by cp = 90° . These numerical 
results will be checked against experimental data and results derived 
from a finite element program. The program which computes surface 
displacements will be examined and possible modifications will be 
attempted to correct suspected arrors. If time is available an attempt 
will be made to modify the program so as to solve the similar boundary 
value problem for an orthotropic material. Finally a solution to 
the problem of an internal source in the half-space may be examined. 

2. FINITE ELEMENT METHOD 

The annual report [1] contains in detail the formulation of the 
algorithm used In the finite element method of solution of wave 
propagation problems. Mr. M. Katona authored a report [5] that is 
presently employed as a user's guide. This report described the 
solution algorithm and contained a discussion of the stability of the 
explicit method of solution and the choice of Integration tine step. 
Mr. Katona enlarged the Finite Element Assembly Program (FTAP) 
originally written by Professor Robert L. Taylor of the Department of 
Civil Engineering, University of California, Berkeley, adding two 
main subroutines, namely EXPLCT (for explicit time integration) and 
DYNPLT (to give printer plots of stress evolution )• An option 
was provided to input any desired pulse shape either as a prescribed 
function or as a table of values of the impulsive force at each time 


15 


step. 

ne,e ”" M [5] de8crlbea “• ■‘"•'■It. see propagation 
problems used chock the accuracy and efficiency of ,b. coding 

developed. The case o, a column aubjected to a triangular load 
lmpulae .as run. modelling the sy.te. .i th . „ t . ek of th „ e 

slonal elements .1th eight node, per element (the ao-c.lled 
"Brick-8 element”) and .ith Poisson’c r.„„ v = 0 . ,, ... 

in the numerical , h>t , „o„d,.perslvo a,res. pulse propagated 

do.n the column a. predicted by the analytical for 

dimensional ...e propagation p ,„ b le.. Encouraging numorlcal result, 
•ere also obtained upon comparing the aurl.ee displacement lor ...e 
propagation In an laotroplc halt apace modelled by aal.ymmetrlc 
element. .1th the theoretical .elution. * problem involving a ,i„ e _ 
squared Impulse acting „„ the surtace ot an anisotropic hall space 
using the three dimensional general anisotropic element ... also 
evaluated. A study ol the time step, l„ ,„e ,„ dlc , t(?d 

the latter is practically Insenaltlve to their size provided they .re 
smaller than the critical one. 


The report also Include, a dlacus.lon of an efficient .ay of 
modelling half space preblems .1th a finite „„„ b er of elements. I„ 
this procedure, the maklmum dll.t.tien.l .ave speed In the medium ... 
estimated as the bar speed .1th the largest constitutive modulus. 

From the l.rgeat coordinate In the region of Interest and fro. the 
duration of the pulse the total aolutlon time of Interest Is estimated 
at that station so as to Include the effect of the entire Input pulae. 
Then the size of the finite element model Is evaluated such that no 




16 


reflections could arrive from the finite boundary at the point of 
Interest during the solutior period. With an assumed mesh density 
(normally 8 to 10 per pqlse length), the element size for adequate 
discretization was obtained. Calculations for the time step used in 
the explicit integration scheme for the desired time span and for 
blank common storage required to execute the FEAP coding are also 
contained in Ref. [5]. 

Examples of input data for various problems were given and an 
updated version of the user's manual was Included. The input in¬ 
structions were also given on the use of the general isotropic 
solid element, the axlsymmetrie, isotropic solid element and the 
general anisotropic solid element. 

Professor R. L. Taylor introduced many changes in the program 
after March 1972, and made the program more efficient. He Introduced 
a section to estimate the critical time step. For a step larger than 
the estimated one, the program gives an error message and stops 
execution. This is to prevent the user from encountering numerical 
instabilities due to the time of transmission of a disturbance 
through an element being less than the integration time step. Without 
modifying the algorithm some statements in the EXPLCT subroutine 
were changed to increase the computing speed by about ten percent. 

Cases were run for the problem of wave propagation through an 
Isotropic rock block both with a sine-squared impulse and an experi¬ 
mentally-measured load history. Good correspondence with the experi¬ 
mental results was observed as far as the radial stress a „ is 

rr 



17 


concerned; however, the axial stress o did not conpare to a 

zz 

satisfactory degree. This indicates that the PEAP program should be 
reexamined to check the accuracy of the numerical analysis; in addi¬ 
tion, the experimental calibration procedure should be further veri¬ 
fied. 

At present the following modifications in the program are being 
considered; a) the output option in the entire program is to be 
changed radically so as to obtain only those outputs of interest. 

Thus, for the four point quadrilateral element, only the center stress 
will be printed out as the others at the corners are of no significant 
value. (b) The present time step estimate for the stability check is 
very conservative as the eigenvalue is calculated at the element 
level. A section will be added to calculate the largest eigenvalue 
of the full stiffness matrix. This may add to the economy of 
running as it might be possible to use a larger time step than pre¬ 
sently. 

It is Intended to prepare and publish a report containing all 
the details of the FEAP program, with a listing and with elaborate 
user's instructions in the next six months. 

» 

(b) Experimental Examination of the Wave Processes in an 
Anisotropic Rock Material 

The examination of body and surface waves will be conducted 
using the two available blcks of Yule marble whose dimensions are 
24" X 24" X 10" . In order to attain this objective a program of 
experimental work was completed during the first year of the contract 
to develop the necessary body-wave sensors and the technique of 







18 


embedding the -tensors in the Yule marble blocks. The first annual 
report [1] contains a description of the initial experimental efforts 
tc achieve this goal. The present report is a continuation of this 
earlier work. The main accomplishment to be reported at the present 
stage of the overall research project is the completion of th^ 
development work leading to the suitable design, adequate calibration 
methods and successful embedding technique of the body-wave sensors. 
Actual measurements have been made using these body-wave sensors 
embedded in a block of limestone which is nearly isotropic. A method 
of loading the free surface by means of sphere impacts through an 
intermediate loading bar has been used for these measurements which 
was found to be convenient because it permitted repeatability of the 
test, allowed measurement of the input pulse, and permitted close 
control of the impact conditions. The monitoring of the loading pulse 
by means of an input crystal at the contact point has also been 
successfully accomplished. In the sequel further details will be 

given. 

As reported in [ll, at the end of the first year of contract, 
a number of experiments were conducted establishing the design of 
crystal transducers which could be suitably calibrated. Further 
experiments indicated that these crystal transducers could be in¬ 
stalled in deep holes, leads extracted out and the remaining voids 
filled with a suitable grouting material. A mixture of alumina and 
epoxy in appropriate proportions was considered as a possible filler 
material. However, since a metal bar with a crystal transducer 


19 


mounted at one end could be inserted with ease into a deep hole, the 
previously developed quartz crystal transducer was redesigned as 
shown in Fig. 11. Since magnesium has an acoustic impedance closely 
matching the average acoustic impedance of Yule marble, such a trans¬ 
ducer package was constructed using a 3/16" diameter rod of magnesium. 
A block of sandstone 14" X 12" X 9" was obtained and one face of it 
was machined flat. A 3/16" diameter hole was core drilled in the 
center of the block from the bottom. The transducer package was in¬ 
serted in the hole. Smearing the bar with wax before insertion 
enabled the bar to be held firmly in the block with the crystal trans¬ 
ducer anchored securely at about 1*" from the free surface. Simple 
drop tests with 1/2" diameter steel balls indicated that the quartz 
crystal was not sensitive enough. Also, a reexamination of the 
piezoelectric relations of x-cut quartz crystals showed that even 
though x-cut quartz crystals are suitable for one dimensional experi¬ 
ments, because of the coupling between the piezoelectric constants 
of x-cut quartz crystals, they are not suitable for obtaining the 
truly unidirectional stress in a complex stress field. Thus a transi¬ 
tion from x-cut quartz crystals to the more sensitive PZT-5a crystals 
was made. In these ceramic crystals the absence of coupling between 
the piezoelastic constants enables the measurement of a truly uni¬ 
directional stress even in a complex stress field. However, the 
lateral surface of the crystal must still remain free as in the case 
of x-cut quartz. Table 1 shows the comparison of the 
appropriate piezoelectric relations and constants for x-cut quartz 


and PZT-5a crystals. 



20 



OF QUARTZ AND PZT CERAMIC CRYSTALS 




21 


The choice of magnesium a. . suitable material for the transducer 
package was governed by the cloae a.tch In ihe .couatlc impedance of 
magnesium and the average acoustic impedance of Yule marble material. 

A compariaon of the phyaical propertica of Yule marble and the varioua 
materials uted In the PZT-5. cryat.l tranaducer package i. presented 
in Table 2. 

The PZT-5. ceramic cryatala, 1/8" dia. x 1/32" thick were ob¬ 
tained from Valpey Piacher Corporation, Massachusetts. A tranaducer 
packaga was conatructed aa ahown in Fig. ll with the PZT-5a cryatal 
replacing the quartz cryatal. Again the bar wa. smeared with wax and 
the package waa inaerted in the sandstone block to within lj" from the 
free surface. Simple drop testa by 1/4" diameter steel spheres showed 
that the crystals were extremely sensitive. The experimental arrange¬ 
ment is shown in Pig. 12. Typical records obtained by two different 
sphere Impact, directly above the crystal package are shown in Figs. 13 
and 14. In Fig. 13 the record waa obtained by voltage amplification 
•nd in Pig. 14 charge amplification was used 11). The shapes of the 
pulaww .nd the degree of sensitivity indicated that pulses could be 
recorded deep in a block of rock material. However, the crystals and 
the transducer packages had to be calibrated before any quantitative 
results could be established. 

The calibration of x-cut quartz crystals was made using an alumi¬ 
num split Hopklnson bar technique. But magnesium and PZT-5a crystals 
were found to be nci compatible in the sense that they do not have the 
same acoustic impedance in the relevant direction of the crystals. 
However, since the discontinuity is small compared to the pulse 
length, magnesium split Hopklnson bar was used to test the 



22 


crystals. Thus a number of quality control tests were made using a 
1^8 dlan, eter split Hopklnson bar of magnesium. The experimental 
arrangement Is shown In Fig. 15. Since the PZT-5a crystals were 
highly sensitive, a 100x attenuator was used before the output was 
recorded on the oscilloscope. The crystal output was compared with 
the strain gage output. In view of the rather high crystal outputs 
interfering with the strain gage circuit,as shown in Fig. 16 by the 
drastic deviation in the output from the strain gage station upon 
initiation of the crystal signal.which has a peak value of 400V to 500V, 
the output from th-j strain gage station immediately after the crystal 
station was not recorded. However, a comparison of the strain gage 
response and the attenuated crystal response for a number of crystals 
indicated that the crystals were essentially identical, in Fig. 16 
a typical record of the strain gage response and the attenuated 
crystal response obtained on a dual beam oscilloscope is shown. Now 
a number of crystal packages were constructed as shown in Fig. ll. 
However the lead wires were omitted and the packages were tested using 
the experimental arrangement shown in Fig. 17. The calibration tech¬ 
nique was essentially the same as the previously established split 
Hopklnson bar technique reported in [1]. The piezoelectric constant 
was computed from the measurements. A typical record of the strain 
gage reaponse and the attenuated crystal response is shown in Fig. 

18. The interference of the crystal output with the strain gage out¬ 
put was eliminated by attaching the two bars together with epoxy, thus 
Isolating the bar with the strain gage station from the transducer 
package. An average value of 1557 pcoulombs/Lb. was obtained for the 



23 


piezoelectric constant which compared with the published value of 

1650 pcouloirbs/Lb. for the PZT-5a ceramic crystal material, the order 

of agreement being the same as those obtained for quartz [1] and 

within manufacturer's tolerance. A limestone block with dimensions 

15 x 15 X Hi which was also used by Ricketts [2] for surface wave 

measurements was drilled as shown in Fig. 19. Four transducer 

packages were assembled and inserted in the 3/16" diameter holes again 

using wax as the bonding agent between the bars and the rack material. 

The experimental arrangement is shown in Fig. 20. The free surface 

was loaded by shooting 3/16" diameter balls with an air gun through 

an 1/8 diameter loading bar. The loading pulses were measured by 

means of both x-cut quartz crystals and PZT-5a crystals at the contact 

point as shown in Fig. 20. Both voltage amplification and charge 

amplification were adopted to measure the crystal responses inside 

the block. Typical input and output pulse shapes are shown in Fig. 

21(a) for a sphere impact as shown in Fig. 21(b). Similarly a number 

of measurements were made for different impact locations. Each crystal 

station measures the unidirectional stress in the direction of the 

crystal axis i.e. in the direction of the bar. For example, referring 

to Fig. 19, station 1 measures a and stations 2 and 3 measure 

zz 

O xx . The various pulse shapes and the corresponding input pulse 
shapes and Impact locations are shown in Figs. 22 to 29*. A compari¬ 
son of the properties of limestone and magnesium is given in Table 2. 

Presently the experimental results are being compared with the 
results obtained from a finite element program. 

* 

Table 3 shows a summary of stresses measured and approoriate scale 
factors for the various pulse shapes 



TABLE 2 











































Input 1 Output - PZT-5a Crystal Transducers 


•Atp/isd 

JCUOBj 

STDDS 


m a* 

rH Li 

•h *H 3 

mom** 
+* m c 

m o> a> 

a s 6 


Q> 

3* s 

fl o » *- 


10 b£ c 

II) P O 

I* 0) P 

+-<!>+- 
W 73 B 


I c • 

1 o o 

p p 35 
CO P 


>> 
fciP 
(UP 
V > 

3P a» 
ppo be 
ifipp c 
cm a 
a c a 

h 0) 

Hen 


V 


H 

H 

n 

rH 

CO 

CO 

CO 

• 

> 

> 

> 

p 

p 

p 

■o 

73 

■c 

s 


s 

> 

> 

> 

1 CM 

CM 

p 

o' 

6 

o' 


9) P 

be vi c 
a p o 

1 


p p p 
p ft p 

o R a 
> a o 


b 

N K 

N ^ 

0 « 

m 


to 

CM 

CM 

CM 

1 

CM 

CO 



V 

pH 


CO 

pH 

CO 

CO 

<0 


p ft p 
o B a 
> 3 o 


g g 

b* b* 


[ 

1 oo 

H 

» 

CM 

CM 

CM 


(O J TJ 


CM CO 


CM ^ • 

V J3 P 
1C J TJ 


J 

ODUB J 

-to 


m a> 

H C< 

P V 3 

a O w p 




O 

> 

p 

p 

u 

. -a 

Fa 

02 

x>\ 

0) P > 

• 

oe\ 

pH pH 

c > « 

« 

3 B • 
ft p o 


v 

j: O m c 

£ I 


a> p 

be on c 
a p o 
p p p 
p ft p 

o e a 
> a « 


Q> P 

be v c 
a p o 
p p p 
p ft p 
o E a 
> a o 


rH 


a 

■H 

■** 


m 

a> 

>i p 

u 

u 



uoTTBOcn 

JOBdUII 


•os uny 


TABLE 3 













































26 


A plexiglass model of the Yule marble block currently in hand 
was made. The optimum locations of the crystal transducer packages 
in the rock are being determined using this model in conjunction with 
the analytical results. The drilling of the rock and embedding of 
crystals will commence soon. The actual drilling will be done at 
right angles to the surfaces of the block which has been determined 
by the choice of a rectangular coordinate system for the location 
of the transducer packages. Since the PZT-5 crystals are highly 
sensitive,it had been decided to use 1/16" diameter x 0.020" thick 
crystals with 1/8' diameter bars. These transducer packages are being 
constructed. The calibration of these packages will be completed 
shortly. These smaller diameter transducer packages will be used in 
Yule marble blocks thus reducing the possibility of altering the 
wave processes due to the insertion of metal bars in rock like 
material. 

The following is the approximate schedule of endeavors planned 
for the rest of the contract period: (1) The construction and cali¬ 
bration of the 1/8" dla. transducer packages which is expected to 
be completed by the end of September 1972. (2) The drilling of one 

Yule marble block and mounting of these transducer packages in the 
block which will be completed by the middle of November 1972 (3) The 

actual measurements of body waves in the Yule marble block using 
essentially the methods of impact and measurement used already for 
the Limestone block which will be completed by the end of Dec. 1972. 

(4) The repetition of body wave measurements in the second Yule marble 


block,making any necessary changes in the location of the transducers 



27 


and in the methods of measurements,which is planned for completion 
in Jan. 1972. (5) Further investigation of surface wave measurements 
using the second Yule marble block which will be completed by the end 
of February 1972 . (6) The comparison with the finite element method of 
solution and integral transform analysis will be pursued concurrently. 



28 


(e) Experimental Determination of the Physical Properties of Yule 
Marble under Static and Dynamic Loadi ng 

Introduction 

As expressed in the previous technical report, the overall 
objective of this phase of the investigation consists of the determin¬ 
ation of the geometric and mechanical properties of Colorado Yule 
marble, including fracture properties, aa a function of strain rate. 

Up to the time of publication of the prevloua report, the following 
had been accomplished: 

1. the axis of clastic symmetry (AES) of a slab of marble was 
determined using crystallographic techniques 

2. a method of obtaining intermediate strain rates of 10 to 100 
aec * was obtained 

3. the mechanics of rock coring and apecimen preparation were 
mastered 

4. the aet vip and calibration of instrumentation necessary to 
record stresa-atraln relatlonahlns at various strain rates was 
accomplished 

5. some nreliminary mechanical oronerties were obtained. 

Thia report details the progress from that point. 

Objectives 

During the last six months the following objoctives were 
accomplished: 

1. the determination of the AES of a new alab of Yule marble used 
in the extensive testing of the rock, and, the verification of 



29 


the AES in the block used by Ricketts [2]. Both of these studies 
were performed using the ontioal crystallographic technique 
described in the previous technical report. 

2. the determination of the static end dynamic elastic constants 
for Yule marble 

3. th*> investigation of fracture stress versus strain rate and 
specimen orientation for both tension and compression. 

In addition to these objectives, some additional subdivisions 
of the testing program, to be completed in the next 6 month period, 
are described: 

4. the characterization of the nonlinear and/or non elastic behavior 
of the rock 

5. in order to get dynamic fracture strengths, a split Hopkinson 
bar will be used with a reduced and specially contoured rock 
midsection. This procedure should allow for the construction 
of a dynamic stress-strain curve to fracture for both tensile 
and compressive pulses. 

6. the characterization of the failure process in terms of the 
strain rate parameter, including optical observations of the 
phenomena. From this information, a comprehensive model of the 
mechanical response of the substance may be solved. 

Methodology and Procedure 

Previous experiments on the first marble slab obtained from the 
Clervi Marble Co. are described in the last technical report [11• It 
was found using crystallographic methods that the AES of this slab 


was neither in the slab nor perpendicular to it but rather at an 
inconvenient angle. For that reason, a new slab was sought and 
obtained, again from the Clervi Marble Co. of San Francisco. The 
manner in which the test specimens were cut from this slab is shown 
in Fig. 30. The AES of the slab lay in the p-iane of the slab and 
was determined both by visual inspection of the distinct bedding 
planes and by crystallographic technique*. The orientation of the 
optic axis of 100 crystals of a specifically prepared thin section 

is shown graphically in Fig. 31. The thin section was cut so that 

* 

the orientation of the AES as deduced from vitual inspection of 

the bedding planes (assuming the AES is perpendicular to these planes) 

is normal to the slide surface. 

Compression specimens measuring 1.05 inches in diameter and of 
lengths between 1.4 and 2.2 inches were cored using an oil cooled 
diamond bit. The emit; of these specimens were milled flat using a 
special Jig and an oil cooled milling machine with a diamond abrasive 
surface. These surfaces were flat and parallel to 1/1000 and the 
lateral dimensions of each specimen at the ends of two mutually 
orthogonal diameters as well as the dimeter of the specimen was 
recorded. 

Specimens used for tensile testing and in the Hopkinson bar 
experiments were cored using a 7/8" O.D. diamond bit and cut on 
a water cooled gravity feed drill press especially converted for the 
coring of rock specimens. These cores measured ~ 3/4" diameter by 
nominally 6" in length and were cut nearly perpendicular at the ends 
using a diamond cut off saw at very slow feed rates (2"/hour). 



31 


Ten compression specimens were cut from each of three mutually 
orthogonal directions, X,Y,Z. The Z axis lies in the plane and the 
X axis is normal to the slab. Y and Z direction specimens were 
nominally 2.0 inches long, while the X direction specimens, owing to 
the thickness of the slab, measured roughly 1.4 Inches. 

These specimens were mounted with Baldwin-Lima-Hamilton 
FAE-12-12S9L epoxy backed strain gages of 120 Q resistance having a 
gage factor of 2.01. These 1/8" gages were mounted longitudinally 
and diametrically opposed at the midsection of the specimen using 
EPY-150 epoxy. They were wired in series to eliminate the measure¬ 
ment of any bending component. 

The gages were Incorporated in an AC-excited bridge curcuit in¬ 
cluding an amplifier and the output was recorded on a plotter. The 
records were calibrated by means of kno*n shunt resistances across 
the gage. The applied compressive force was determined by means of 
a calibrated 0-20,000 lb range Instror. load call. Strain-time and 
stress-time records were recorded on a strip chart recorder. For the 

fastest of these quasistatic loading tests (less than 2 seconds 
duration) an oscilloscope was used to record both stress-time and 
strain-time data. 

An Instron testing machine was used to load the specimens as 
shown in Fig. 32. A ball and socket loading member with a 3" diameter 
bearing plate and a 2" dia. ball assured uniform contact with the 
loading surfaces. A specimen of 7075 aluminum was mounted with 
longitudinal and transverse gages and tested. The data from these 
tests were reduced in a manner identical to that used for the rock 
specimens. The elastic constants obtained from this data were 



32 


g 

E = 11.6x10 psi and v = .34 as compared with the accepted values 
0 

of E = 10.6 X 10 psi and v B .33. Although there Is some error In 
Young's modulus relative to the accepted value (< 10%), the value 
for Poisson's ratio agrees closely (<3%) , and there is no hysteresis 
present. 

In order to determine the static elastic constants of Yule 
marble, three compression specimens were prepared as above. The 
constitutive relation for a transversely isotropic material may be 
written in terms of well-known constants as follows: 

Let the z direction be the AES and E', v 7 be Young's modulus 
and Poisson's ratio in that direction. Let x and y denote coordi¬ 
nates lying in the plane of isotropy (PI). Let E and v be the 
elastic constants in this plane. Let G / denote the shear modulus 
in any plane perpendicular to the PI. Then the constitutive equa¬ 
tion is: 


xx 


yy 


e(°xx VC yy') E 1 
l( 0 yy ^xx) E 7 


zz 


zz 


v .1 

€ *= - - (o + o ) + -/ c 

'’zz E xx yy E zz 

G 

r vz = •<£ 

yz G 


xz 


xz 

G' 


c = JSZ = 0 

xy G E xy 


E 


33 


Using the specimens mounted with strsin gages as shown in Fig. 33 ( 
all the elastic constants can be determined using these three 
specimens. They were tested in compression from 0-1200 psi at about 
100 psi/sec loading rate for seven cycles and elastic constants were 
measured on this seventh cycle. The specimens were then tested at 
0-6000 psi at 300 psi/sec for three more cycles and stress and 
strain data as a function of time was recorded. 

An attempt has been made to obtain the same type of information 
as above for the rock in tension. Different geometries have been 
employed but none have been very successful. One series of tests 
were performed on rock rods glued to aluminum endcaps as shown in 
Fig. 45. The rock cores were glued to the aluminum endpleces in a 
V block using high strength Scotchweld Structural Adhesive. Strain 
gages were mounted at the midsection and tensile tests to fracture 
were performed at various strain rates for specimens oriented both 
in the Y and the Z directions. The results of these tests are given 
in the next section. 

In order to measure the dynamic elastic constants for Yule 
marble, rock cores 3/4" <t> x 6" l were cut as desc~ibed above. The 
ends of these rods were cut perpendicular using a diamond cutoff 
saw. Three rods, all with the same orientation, were glued together 
and mounted with strain gagos as shown in Fig. 34. These strain 
gages ’vere monitored by an oscilloscope through a dynamic bridge 
balance and calibrated using a known shunt resistance. Aluminum 
endcaps 1/2 "l x 3/4" ♦ were attached with wax to the impact end of 
the rod to prevent local fracturing. These specimens were held 





34 


vertically in a drop teat aet up aa shown in Fig. 35. A 1/2" diameter 
ball was dropped from 22) through an aligning tube onto the rock rod. 
Strain pulses were recorded st two stations, oach using the same time 
bsse. The scope was triggered using a piezoelectric cryatal taped 
to the side of the rod. This apparatus was to measure strsin histories 
at three gage locations: two longitudinal atralns and one transverao 
strain. Peak strain arrival times were used to calculate waveapeeds 
and these dynamic elastic constants using E * Pc^ for both tensile 
and compressive pulses. The ratio of tranaverae to longitudinal atraln 
gave Poisson's ratio in those tests. In order to ascertain the error 
in thla aetup, the dynamic elaatlc constants of a rod of 7075 alumi¬ 
num with the same dlmenalons as the rock rod was used. For this rod, 
wsvespeeds of 200,000 in/sec implying E - 10.2 x10 6 psl were obtained 
for both tensile and compressive pulses. Poisson's ratio obtained wss 
.335. 

Results 

Ten compressive specimens in each of three mutually orthogonal 
directions X,Y and Z were cut and instrumented as described in the 
previous section. Two specimens from each direction were tested at 
*ach of five strain rates and both stress-tine and strain-time dsts 
to fracture were gathered. From these data streas-atraln curves 
were constructed. The genersl shape of the atress-strain curve in 
oach direction does not seem to depend on strain rate. Figures 36, 

37, 38 show the two most deviant curvea from esch of the three 


directions. 





The shape of the stress-strain curve la both character utlc 
with respect to apeclaen orientation and reproducsble. Initially, 
there ia a region of increasing stiffneas fro* an initial tr.ntent 


35 


Modulus Ej to a region of constant Mcdulus E_ . This at iffaring 
d 2 c 

region where —r >0 porsl.ita until roughly one third the raxlmua 
dc 

stress o la reached. The Material exhibits constant Modulus E 
* 2 
2 a. 

between — and — and with increasing stress enters a region 

of decreasing tangent Modulus, —| <0. Directly proceeding failure, 

dc 

the Modulus becomes negative and the material falls at a value of 
fracture stress c f lower than . Figure 39 indicates the values 
of Ej snd E ? for the apeclaena tested. Due to low saplifIcstlons 
of both stress and strain signals near the origin, error in the 
detenalnstlon of Ej is estlasted at around ±20% of the actual 
value. This Modulus ia obtained with greater accuracy lr subsequent 


tests. Sirsin rates for these tests were calculated on the baals of 
strsin 2t %i 

• Strains become large as failure approaches giving 


increasing strain rates near fracture. 

In the region near fracture, strain recorded by the gages no 
longer is an accurate Measurement of average cross-sectional specimen 
strsin. Frscture paths initiate, slmoat without exception, at the 
corners of the specimens and travel diagonally through producing one 
plsne of failure. In X and Y oriented specimens, thne planes were 


invariably oriented such that their normal lay in the plane produced 
by the test axis and a point lying on the AES or Z direction. Gages 
mounted such that this plsne of failure traveled through them pro¬ 
duced larger strsin readings than those located away from the fracture 



36 


path. This conclusion was reached after «ounting four gages at 90 
intervals around both Y and Z direction specimens and comparing o- e 
curves produced by gages through which the fracture path occurred 
with those produced by gages distant from this path. The results of 
this test are shown in Fig. 40. Specimens in the X direction were 
unintentionally mounted so that the fracture path was as far as 
possible from the gage while specimens in the Y were mounted such 
that the fracture path always passed through the gages. Thus, the 
apparent difference in material properties near failure. X direction 
specimen, appearing brittle while Y direction specimens showing 
remarkable amounts of seemingly near plastic strain, is explained. 
Figure 40 clearly indicates how the structure of the material is 
such that the assumption of one-dimensionality breaks down near 

failure. 

Specimen. In the Z direction show no preference In fracture 
pUne orl.nt.tlon, although the, . 1.0 .pllt dl.gon.lljr, .ometlmo. 
accompanied by accond.r, cr.ck. Initiating .t the be.rlng pl.te. .nd 
running p.r.11.1 to the .peclmen ..1., In the four g.ge tent, there 
... very little difference In .tr.ln between two .et. of gage, even 

though the fracture path lay clearly through one .et. 

Maximum .tie.. «r .. • function of at.tic .train rate end dlrec 
tlon 1. ahown In rig. 41. He atrong .train rate dependence la 
evident for either the X or Y direction.. Z aho*. a greater depen¬ 
dence of maximum atrea. .1th log 1Q C . The difference In 0„ for 
X and Y direction, may be attributable to .hotter (1.4" va. 2.0') 

X direction specimens. 



37 


The results of the compression tests used to obtain the static 
elastic constants are shown in Fig. 42. As is evident from these 
figures, more of these "constants" are truly constant. Previous 
tests on this material have shown a permanent deformation accompanying 
each virgin stress level. On repeated loading to this stress level, 
the stress-strain curve exhibits hysteresis but no appreciable 
permanent set after this first cycle. In order to avoid measuring 
any "first cycle" effects, the specimens were all loaded through 
six cycles before the stress-strain relationships were recorded on 
the seventh. The following elastic constants were determined: 

g 

Ej = 4.7 X 10 psi 
= .13 

=2.3 X 10 6 psi 
uj = .05 

g' =2.8 X 10 6 psi 

These same specimens were then loaded through three additional 
cycles at 300 psi/sec to 6000 psi. A typical result of these tests 
is shown in Figs. 43, 44. As can be observed from the figure, an 
initial permanent deformation occurs after the first loading cycle. 
Subsequent loading cycles show only minor amounts of permanent defor¬ 
mation after each cycle, while retaining a nonlinear form at low 
stresses. Hysteresis in the second and third cycles is greatly 
reduced. Again, there exists a region between C m /3 and 2c m /3 
of very nearly constant modulus. 


38 


Several preliminary teata have been performed to try to indicate 

the viscoelastic nature of the rock, if any. Creep tests at stresses 
2 a 

m 

up to seem to yield little viscoelastic phenomena in the 1-1800 

sec range. Some definite creep is observed for Y compression specimens 
at about 7/8 m but this data is still preliminary in nature. 

Dynamic elastic constants were obtained by measuring strain-time 
histories at two longitudinal and one transverse station in base 
oriented in each of the three directions Y, Z and 43° . These tests 
produced the following dynamic elastic constants: 


v 

C' 


compressive pulse 
11.3 x 10 6 psi 
.29 

5.0 x 10 6 psi 
.13 


tensile pulse 
11.3 x 10 6 psi 
.24 

3.2 x 10 6 psi 
.09 


Although the value of g' has not been calculated as yet, the pro¬ 
cedure is straightforward and enough data is available in terms of 
five dependent tests to obtain five elastic constants. This method 
of measuring the elastic constants is identical with the method 
used in the static compression tests in so far as location of recording 
gages with respect to X, Y, and Z directions is concerned. The 
following relations were used 



39 


E ' - PC 2 \) - - —— Z-axis bar 

€ 

Z zz 

2 ^vv 

E = PC v = - —— Y-axis bar 

°Y £ xx 

Wavespeed and a Poisson's ratio have been obtained for the 45° 
specimen and thus a dynamic G ' can be determined. These elastic 
constants were measured at peak longitudinal strains of less than 
200 p, in/ in and thus fall into the initial elastic region as 
determined by the static tests. As indicated by Ricketts’ [2], 
conversion of these technical constants to corresponding velocity 
constants is straightforward. 

Some problems were encountered in the development of a satisfac¬ 
tory method of testing the rock in tension. An initial tensile spec¬ 
imen design was patterned after standard metal test specimen geo¬ 
metries and loaded by means of a friction hold with end clamps as 
shown in Fig. 45. These specimens broke as they were being attached 
to the testing machine. 

A second geometry was considered. Rock slabs measuring 6' xl$ X1/4 
were attached to aluminum end pieces 3" X 1$" equipped with a \ 
diameter loading hole using Scotchweld Structural Adhesive. Load was 
applied through a chain so that if the specimen and endcaps arrangement 
was symmetrical with respect to a line between the loading holes, no 
bending moment could be introduced. These ypecimens proved unsatis¬ 
factory as they consistently broke at the rock-aluminum interface. 

In addition, these specimens were extremely difficult to fabricate 
and align properly. In order to get some indication of the strain 


40 


.n th, rock, . brittle c„.ti„g, stress Co.,, ... . pplied to the 
rock surface. At roo. temperature this ,. tert .i cr „ cks ne , r 500 

although ..log special techniques to cool the coating, this Unit 
can be lowered to 200 Even In the regi p„ directly 

to the tensile fracture path, only very few cracks in the Stress 
Co., .ere observed. Maximum t e„. lle f . llur e y „ le 

is about the same as that required for crack initiation of the Stress 
Co., and thus this brittle coating is no. very useful in obtaining 
.n overall Prefecture strain a,.,, necessary in order to detect 
stress concentrations. A third specimen geometry was decided on, 
orim.rlly because of ease of fabrication. This tensile test con¬ 
sisted of 6" long 3/4" diameter rock rods glued with Scotch.eld 
Structural Adhesive to threaded aluminum end cap, as shown in p,g. 

45. The ends of the rods were cut nearly perpendicular to the axis 
using a diamond cu, off a... The endcap, ..re wrapped with trans¬ 
parent tape until they measured within .002 of the diameter of the 
rock rods. The endcaps were glued to the rock by placing the pieces 
in an aluminum L member. Weights were applied to the rod and ond- 
C.ps to assure tha, they remained securely in the bottom of the L as 


shown in Fig. 46. 

These specimens were mounted with diametrically opposed longi¬ 
tudinal strain gages located at the midsection of the rock. fbur 
specimens in both the Y and Z directions were prepared in this manner. 

The rock tensile specimens were tested at four different loading 
rates to fracture and the resultant stress-strain curves are shown 
in Figs. 47 and 48. All of the Y direction specimens broke at the 



41 


rock-spec men interface. The stress-strain curves obtained from 
these tests are probably in error in three ways. First, the strain 
measured by the gages is probably much smaller than that at the rock- 
aluminum interface where failure eventually occurred. Second, 
fracture stress is underestimated since a stress concentration 
apparently exists at the interface. Third, the apparent increase in 
fracture stress with respect to strain rate may be partially due to 
a strain rate property of the glue and thus a strain rate sensitivity 
of the stress concentration. Specimens oriented in the Z direction 
are also subject to these criticisms but to a lesser extent as two 
of the four specimens broke relatively near the gages. 

The specimen geometry for tension is not considered satisfactory. 
Now in progress is an attempt to fabricate a reduced section contour 
from 6 "l X 3/4" 0 rods. This configuration should eliminate distal 
fractures. However, affects of additional machining of the rock 


must be investigated. 


42 


CONCLUSIONS 

After one and one-half years of theoretical and experimental investi¬ 
gation involving the propagation of body and surface waves in a trans¬ 
versely isotropic natural rock, namely Yule marble, the following results 
and conclusions can be summarized: 

(1) An analysis has been developed based on integral transform 
procedures utilizing the Cagniard de-Hoop technique of inversion for the 
propagation of waves in a transversely isotropic half-space with the axis 
of symmetry lying in the free surface when subjected to an arbitrary 
concentrated normal force. This model simulates the normal impact of a 
sphere on a large block of Yule marble as represented by corresponding 
experimental conditions. The displacements, strains and stresses pro¬ 
duced by body waves in the interior of the medium have been completely 
delineated, and numerical results have been obtained specifically for a 
step input and a sine-squared input in time; the latter is one of the best 
simple analytical representations of an actual impact by a snhere. The 
formulation for the surface response has been completed and the procedure 
for obtaining numerical results has been initiated; however, computational 
difficulties have been encountered at singular points of the solution and 
attempts are currently under way to resolve this oroblem. 

(2) A finite-element program for the identical problem was evolved 
approximately five months ago, but has since been refined and improved, 
although requiring further checking. Typical numerical difficulties are 
encountered when results are desired close to the source, requiring for 
its improvement very fine mesh sizes leading to extensive computational 


efforts. 


43 


(3) Complete transducer packages for the direct measurement of 
dynamic stress components in the interior of rock and ceramic-type 
materials have been developed and externally calibrated both by means of 
split Hopkinson bars and embedment within rock bars. Drilling 

and insertion techniques have been evolved and applied to rods and blocks 
composed of nearly isotropic rock. Techniques are currently being 
constructed for in situ calibration of these packages in the interior 
of anisotropic rock masses. 

(4) In the area of the determination of mechanical properties of 
the marble, the axes of elastic symmetry of static and dynamic test 
specimens as well as of the blocks to be used in the body wave experi¬ 
ments have been crystallographically determined. A large number of 
quasi-static and some dynamic tests have been performed both in tension 
and compression for various orientations of the material. Creep, 
cyclic loading and fracture studies have also been executed, and 
dynamic elastic constants have been determined for the material. 

During the remainder of the contract period, it is expected to 
complete the analytical and experimental studies of body wave response 
and to compare the integral transform, finite-element and experimental 
results as a test of the validity of the model and the theoretica 1 and 
test techniques employed. The characterization of selected mechanical 
properties of the Yule marble both under static and dynamic conditions 
is anticipated to be largely complete with the exception of fracture 
initiation and propagation studies that will require further effort. 



44 


The analytical procedures developed will permit ready extension to 
other classes of substances, such as orthotropic materials, while 

the techniques Involving the construction and embedment of transducer 
packages will find wide applicability in the study of dynamic processes 
in the interior of large blocks or structural elements. 


MMTVi 




45 


BIBLIOGRAPHY 


1. Goldsmith, W., and J. L. Sackman, "Wave Propagation in Aniso¬ 
tropic Rocks," Annual Technical Report Number One, Feb., 1972. 
Contract H0210022 Bureau of Mines. University of California, 
Berkeley, Feb. 1972. 

2. Ricketts, E., "Spherical Impact on an Anisotropic Half- 
Space," Ph.D. thesis, University of California, Berkeley, 197C. 

3. Buchwald, V. T., Rayleigh Waves in Transversely Isotropic 
Media," Quart. J. Mech. Appl. Math., Vol. 14, pt. 3, 1961. 

4. Paradis, A. R., and Hussey, D. F., "Graphical Display System," 
Computer Center, Univ. of Calif., Berkeley, Anr., 1969. 

5. Lion, K. S., Instrumentation in Scientific Research; Electrical 
Input Transducers , New York, McGraw-Hill. 1959. 

6. Mason, Warren P. , Physical Acoustics, Principles and Methods, 

Volume I - Part A , New York, Academic Press, 1964. ” 





47 



1 Cartesian Displacements Due to a Heaviside Input 
for 0 = 85°, 9 = 45° 

2 Cartesian Displacements Due to the’Input Force 

I(t) = I sin 2 [jE , at R»l", 0^80°, m=45° 

° \t* / 

with Time Step At® 1 usec/in, t*=50 usee 

3 Cartesianjlisplacements Due to the I(t) above 
at R=2", e=80°,9*45°, with At= 2.08 usec/in 


Page 

50 

51 

52 


4 Cartesian Displacements Due to the I(t) above 

at R=5", UnJ0°, 9 = 45 °, sith At = .833 usec/in 53 

5 Cartesian Displacements Due to the I(t) above 

at R=10", $=80°, 9 * 45 °, with At = .41 usec/in 54 

6 Cartesian Normal Strains Due to a Heaviside Input 

for 0= 85°, 9 = 45° 55 

7 Cartesian Normal Strains Due to the Input Force of 

Fig. 2 at R=l" ( e^O 0 , 9 = 45 ° with At® lusec/in. 56 

8 Cartesian Shear Strains Due to the Input Force of 

Fig. 2 at R=l", 0^=80°, 9 = 45 ° with At® 1 usec/in 57 

9 Cartesian Normal Stresses Due to a Heaviside Input 

for 0^5°, 9 = 45 ° 58 

10 Cartesian Normal^Stresses Due to the Input Force of 

Fig. 2 at R=l", 0^=80°, 9 * 45° with At® 1 usec/in 59 

11 Transducer Package Employing Magnesium Bar and 

1/8" Dia. Crystal 60 

12 Experimental Arrangement for Drop Tests on Sand¬ 
stone Block with 1/4" Dia. Steel Balls 61 

13 Signal Produced by the Drop Test on Sandstone 

Block Measured by Voltage Amplification 62 

14 Signal Produced by the Drop Test on Sandstone 

Block Measured by Charge Amplification 62 


15 Experimental Arrangement for Testing PZT-5a 

Crystals Using 1/8" Dia. Magnesium Split Hopkinson 
Bar 


63 



48 


List of Figures 


Figure 

Title 

Page 

16 

Crystal Record (inverted) and Strain Gage Signal 
for 1/8 Dia. Magnesium Snlit Hopkinson Bar Teat 

63 

17 

Experimental Arrangement for Testii.g Transducer 
Packages with 1/8" Dia. PZT-5a Crystaia 

64 

18 

Calibration Data for Transducer Package with 

PZT-5a Crystals in a Split Hopkinson Bar Test 

64 

19 

Sketch Showing the Drilling of Liuestone Block 

65 

20 

Experimental Arrangement of Impact Test of Lime¬ 
stone Block for Body Wove Measurements 

66 

21-29 

Typical Input and Output Records for the Impact 

Tests on the Limestone Block and Correaponding 

Impact Locations (Se Table 3 for details) 

67-72 

30 

Orientation of Tensile and Compresaive Specimens 

Cut from Rock Slab 

73 

31 

Crystallographic Plot o’ the Orientation of Optical 
Axis of 100 Crystals of Yule Marble Slab 

74 

32a 

Instron Machine with Tensile Specimen 

75 

32b 

Instron Mschlne with Compression Snecimen 

75 

33 

Geometry of Compression Specimens Showing Gsge 
Locations 

76 

34 

Geometry of Hopkinson Bars and Tensile Specimens 
Showing Gage Locations 

77 

35 

Hopkinson Bar Drop Test Arrangement 

78 

36 

X-Direction Compression Tests 

79 

37 

Y-Direction Compression Tests 

80 

38 

Z-Direction Compression Tests 

81 

39 

Table of Static Elastic Moduli 

82 

40 

Variations in Stress-Strain Curve with Respect 
to Gage Location for Y and Z Direction 

Specimens 

84 



Llat of Figures 


49 


Figure 

Title 

Page 

41 

Maximum Fracture Stresa aa a Function of Strain 

Rate for X, Y, and Z Directiona 

85 

42 

Static Elastic Constanta Plot 

86 

43 

Typical Y-Direction 0-6000 PSI Compreaaion Testa - 
3 Cycles 

87 

44 

Longitudinal versus Transverse Strain for Compression 
Test of Figure 43 

88 

45 

Three Prototype Tensile Geometries 

88 

46 

Fabrication Jig for Rod Tensile Geometry 

89 

47 

Y-Direction Tensile Test Results 

90 

48 

Z-Direction Tensile Test Results 

91 



FUNDRHENTRL. 

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FUNDAMENTAL 

TIME-STRAINS 



11 CDMP 
22 CDMP 
33 CDMP 








11 CDHP. 
22 CDHP. 
33 CDHP. 







59 


S3SS3aiS ££9IS‘U9IS 



• » • 

CL CL CL 
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61 



Fig. 12 


Reproduced from 
best available copy. 




5° ^ sec/div 



5v/div 

Transducer 
sensltiivty: 
10 neb 

Range: 
21b/volt 


Fig. 14 





63 




Fig. 16 



































64 



fig;, 17 




Crystal response 
2v/div. 

Strain gage response 
2mv/div. 


Fig. 18 














Fig. 19 



66 



Fig. 20 



67 


20 |i, sec/div. 



Run No. 1 

x-cut quartz 
Input crystal 
response by 
charge amplification 

0.5 v/div. 

Output at’Station No. 1 
by charge amplification 

0.5 'f/div. 























69 


Run No. 5 

PZT-5a Input 
Crystal response 
5v/div. ; 100 x 

Output crystal 
response at 
Station No. 2 
0.2v/div. 
















70 


: 


20 |i sec./'div. 



Run No. 5 

Output Crystal 
response at 
Station No. 1 
0.2v/div. 


Fig. 25 

20 p, sec. /’div. 



Run No. 5 

Output Crystal 
response at 
Station No. 3 
0.lv/div. 



i 

* 



















20 |i sec . /div. 



Run No. 7 
PZT-5a 

Input crystal 
response 
5v/div.; 100X 

Output crystal 
response at 
Station No. 2 
O.lv/div. 


Fig. 27 






« 




72 


20 p sec . /div . 



Run No. 7 

Output crystal 
response at 
Station No. 1 
O.lv/div. 


Fig. 28 


20 p, sec./div. 



Run No. 7 

Output crystal 
response at 
Station No. 3 
O.lv/div. 














































Fig. 31 


75 


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Fig. 32b 













A, OonO E 
Series 

X 


I Y d>— k* 

AX- _ 


Y 




EY- 


Z 




oz- 


Seriaft 





■ D Strain Goge 
Stations 


Z is Axis of Eiostic Symmetry 

















78 











Fig. 38 






SPECIMEN I.D. 


sec" 1 Xl0° nsi xlO 6 nsi 


DZ-1 

1.8 xlO -6 

1.7 

5.3 

UZ-2 

2.0xl0 -6 

2.9 

6.2 

DZ-3 

1.6 xlO~ 6 

2.8 

6.2 

DZ-4 

1.8xl0 _5 

3.0 

6.3 

DZ-5 

1.7X10" 4 

2.4 

6.2 

DZ-6 

2.0X10" 4 

1.9 

5.7 

DZ-7 

- 

- 

- 

DZ-8 

5.7X10" 3 

2.9 

5.0 

DZ-9 

- 

- 

- 

DZ-10 

-3 

5.5x10 

2.4 

5.2 

X ave 

- 

5.6 

8.9 

Y ave 

- 

6.6 

8.1 


Z ave 


8.6 


5.8 



84 



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Fig. 41 



86 







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90 



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