: in shock compression of condensed matter-1997, edited by...
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WAVE GENERATIONS FROM
CONFINED EXPLOSIONS IN ROCKS
C� L� Liu and Thomas J� Ahrens
Seismological Laboratory� California Institute of Technology� Pasadena� CA �����
In order to record P� and S�waves generated from con�ned explosions in rocks in the labo�
ratory� a method is developed based on the interactions between incident P� and SV�waves
and free�surfaces of rocks� The relations between particle displacements of incident P� and
SV�waves� and the strains measured using strain gauges attached on free�surfaces of rocks are
analytically derived� P� and SV�waves generated from con�ned explosions in Bedford limestone
are recorded�
INTRODUCTION
Virtually all the methods that have been pro�posed for discrimination of underground explo�sions � mb � �� from earthquakes and mining ex�plosions�� and �� are based on various P�to�S am�plitude ratios� Although there are many previousstudies of seismic radiation patterns from decou�pled explosions�� � �� it is still unclear what con�trols the radiation pattern of S�waves in tampedand decoupled explosions ��� Therefore studyof P� and S�wave generation from con�ned explo�sions is important for discrimination purposes� Inorder to investigate waves generated from small�scale laboratory explosions in rocks a measure�ment method is required to monitor both P� andS�waves� Conventional seismic recording systemsand methods for laboratory scale high strain rateexperiments��� can not be utilized� Based onthe analysis of the interactions between P� andSV�waves and free surfaces we have developed amethod to monitor P� and SV�wave pro�les usingtwo perpendicular strain gauges attached to thefree�surfaces of samples� The method and someinitial experimental data are presented below�
MEASUREMENT METHOD
When elastic P� and S� waves generated fromexplosions in rocks re ect at free surfaces thesegenerate di�erent displacement�time history� We
determine P� and SV�wave displacements usingthe strains measured along two perpendicular di�rections at a series of stations on free�surfaces ofthe rock samples�
Data reduction method
The strain gauges are attached at positionsalong the intersection of the plane containing theaxis of the spherical wave front and the sam�ple free�surface as shown in Fig��� The strainsrecorded by the gauges include the contributionsfrom incident P� or S�waves and re ected P� andS�waves� The relation between the measuredstrains and incident P� and S�wave particle dis�placements are derived as follows�
P�wave re�ections at free surfaces
The displacement re ection coe�cients forincident planar P�waves at free surfaces ���are PP � �B � A���B � A� PS ���� sin���� cos��j���A � B�� where PP and PSare re ection coe�cients of P� and SV�wave dis�placements due to incident P�waves and � and� are P� and S�wave velocities respectively� A �cos���j� and B � ��� �
� sin��j� sin����� where �and j are P�wave incident angle and S�wave re� ection angle respectively �Fig����The displacements of particles on free surfaces
after P�wave re ections are
uppar � u
Ip��� � PP � sin � � PS cos j� � Hparu
Ip� ���
: in Shock Compression of Condensed Matter-1997, edited by S.C. Schmidt, D.P. Dandekar, and J.W. Forbes, pp. 859-862, AIP Press, Woodbury, NY, 1998 859-862.
upper � u
Ip���� PP � cos � � PS sin j� � Hperu
Ip� ���
where uIp is the particle displacement of in�cident P�waves� uppar and upper are the parti�cle displacements along the directions shown inFig�� respectively� Having substituted PP andPS into Eqs���� and ��� the two coe�cientsare Hpar � � cos � sin��j���A�B� and Hper �� cos � cos��j���A�B��
SV�wave re�ections at free surfaces
For incident SV�waves the re ection coe��cients for P�waves �SP � and SV�waves�SS����are SP � �
� sin������As � Bs�� SS � �As �
Bs���As � Bs�� where As � cos������ Bs ���� �
� sin���� sin��j�� � and j are SV�wave incidentangle and P�wave re ected angle respectively�The displacements of particles on free surfaces
after SV�wave re ections are
usvpar � u
Isv ����SS� cos ��SP sin j� � Gparu
Isv� ���
usvper � u
Isv��SS� �� sin ��SP cos j� � Gperu
Isv� ��
where usvpar and usvper are particle displacements af�
ter re ection along the directions shown in Fig��and uIsv is the particle displacement of incidentSV�waves� The two coe�cients are determinedto be Gpar � � cos���� cos ���As � Bs�� Gper �
���� cos j sin������As �Bs��
Particle displacements of incident P�waves
Because upper is perpendicular to free surfacesand incident waves are assumed to be sphericalthe strain due to upper along direction � �per� issimply expressed as
�per� � HperuIp�r�� ��
where r� is the distance between the center ofexplosive source and the free surface of samplesat � � ��uppar does not result in any strains along direc�
tion � so the total strain induced by the incidentP�waves is determined to be �p� � H�u
Ip�r�� where
H� � Hper� Therefore the strain along direction� yields the particle displacement of incident P�waves as
uIp � r��p��H�� ���
Since both upper and uppar have contributions to
strains along direction � we need to consider the
resultant displacements� The length of the gauge
after re ection �s is equal to �r�n � ��rn�� ����
�
���
where �� � ls cos �rn
ls is the initial length of thestrain gauge rn is the distance between the ex�plosive source center to the position of a gaugeupon P�wave re ection� Here rn can be expressedas rn � r�u cos������where r is the distance be�tween the explosive source center and the gaugebefore P�wave re ection u is the resultant dis�placement of the point at � on the free surfaceand � is the angle between u and upper� u and �
are given by u � uIp� cos�����A�B� and � � �j�From the expressions above �s is given as
�s � �rn ��
�rn�rn�
��
���� ���
BecauseuIpr � � �s is rewritten as
�s � �r���tan����
���uIp�W ������
tan����
���tan���
dW
d������
���
where W ��� � � cos��� cos�� � ����A�B��From the de�nition of strains
�p� ��s� ls
ls�
H�uIp
r�� ���
where H� � cos����W �� � tan������� �tan � dWd� ���� � tan��������The particle displacement of an incident P�
wave determined from a gauge along direction �is
uIp � r��p��H�� ����
Particle displacements of incident SV�waves
Using the same above formulation the rela�tions between incident SV�wave particle displace�ments and strains along the two directions areobtained�The particle displacements of incident SV�
waves from the strains along direction � and �are
uIsv � r��sv� �G�� ����
anduIsv � r��
sv� �G�� ����
where G� � Gper
: in Shock Compression of Condensed Matter-1997, edited by S.C. Schmidt, D.P. Dandekar, and J.W. Forbes, pp. 859-862, AIP Press, Woodbury, NY, 1998 859-862.
Rock sample
θ
θj
Incident P- or SV-wave
Reflected SV- or P-wave
Reflected P- or SV-wave
12
3
4
56
7
8
Figure 1. Schematic arrangement of experiments
Rock joint
Explosives
Spherical cavity
Gauge
u per
par
Direct
ion 2
(u
)
Direction 1
Gauges (7, 8)
(5,6)
(3, 4)
(1,2)
G� � cos����Ws���tan����
�� � tan �
dWs
d����� �
tan����
���
Ws �� cos�� � �� cos���
As �Bs
�cos����� � ��
���
cos� j sin�����
�
�
�
and tan � � �� cos ���� cos j�� tan�����
Characteristics of strains in di�erent directions
Figure � shows the dependence of H�� H�� G�
and G� on incident angle that were calculatedusing the equations derived above� For incidentP�waves the constant H� is relatively insensi�tive to � and H� changes from ��� at � � �o
to ��� at � � ��o for Bedford limestone� Theconstant H� is very sensitive to � and it variesfrom � to ���� when � varies from � to ��o� Itcan be seen that the strains induced by compres�sional P�waves along direction � are always posi�tive but the strains along direction � are positivewhen � is less than ��o and negative when � islarger than ��o� This change in polarity is con�trolled by the ratio of the projection of P�wavedisplacements along direction � to that along thedirection that is perpendicular to the free sur�face� If the strain induced by the displacementalong direction � is less than that due to the dis�placement along the perpendicular direction thestrain is positive otherwise the strain is nega�tive�
From the above calculation the gauges alongdirection � are not sensitive to an incident SV�wave however the gauges along direction � arevery sensitive to an incident SV�wave� The po�larities of the strains along direction � are alwaysnegative and the polarities of the strains alongdirection � are determined by the direction of SV�wave particle motion �Eqs���� ���� ���� and ���� give the relations
between strains along the two directions and in�cident P� and SV�wave particle displacements� Ifstrains along the two directions can be recordedthe P� and SV�wave amplitudes can be deter�mined experimentally�
EXPERIMENTS AND RESULTS
The method described above was used to mon�itor P� and SV�waves generated from con�ned ex�plosions in rocks� The rock sample �Bedford lime�stone� was assembled with two blocks as shownin Fig��� The rock sample with strain gauges wasplaced inside a tank that was pressurized to ��bars�The recorded strains for one of the experiments
are shown in Figs�� and �� The characteristics ofthe strains recorded by the gauges are the sameas predicted from our derived equations� Thestrains along direction � induced by incident P�waves are always positive while the strains alongdirection � change polarities as P�wave incidentangle increases �Figs�� and ��� The strains in�
: in Shock Compression of Condensed Matter-1997, edited by S.C. Schmidt, D.P. Dandekar, and J.W. Forbes, pp. 859-862, AIP Press, Woodbury, NY, 1998 859-862.
duced by incident SV�waves along direction � arenegative and are much larger than that along di�rection ��From the records it is straightforward to de�
termine P and SV�wave amplitudes for the ex�periment using the expression given above� Fromthe P� and SV�wave velocities of Bedford lime�stone the expected S�wave arrivals are labeledon the records� The time di�erence between theexpected and the recorded is less than �s for theexperiment�
CONCLUSIONS
In this work a method has been developed formeasuring P� and SV�wave amplitudes generatedfrom explosions in rocks� The relations betweenthe strains given by gauges placed on the free sur�faces of rocks and incident particle displacementsare derived analytically� The experimental re�sults showed that the characteristics of recordedstrains along the two directions are in good agree�ment with the predictions�
ACKNOWLEDGMENTS
Research was sponsored by Air Force Techni�cal Applications Center� Contribution ���� Di�vision of Geological and Planetary Science Cali�fornia Institute of Technology�
REFERENCE
�� Blandford� R� R�� AFTAC�TR�������� �����
�� Helmberger� D� V� and Woods� B�� Monitoring a
Comprehensive Test Ban Treaty� edited by Husebye�E�S� and Dainty� A�M�� Kluwer Academic Publishers�the Netherlands� ����� ����
� Glenn L� and Goldstein P�� J� Geophys� Res�� �������� � ��������� ��
� Glenn L�� Ladd A�� Moran B�� and Wilson K�� Geo�phys� J� R� astr� Soc�� ��� ��� � �������
�� Murphy J� � op� cit in Ref����� pp � �����
�� Sykes L�� op� cit in Ref����� pp ����� ��
�� Kim S�� Clifton R� and Kumar P�� J� Appl� Phys��
��� ��� ���������
� Aki� K� and Richards� P�� Quantitative Seismology
Theory and Methods� W� H� Freeman and Company����� ch��� ���
-4.0
-2.0
0.0
2.0
4.0
0 15 30 45 60
H1
H2
G1
G2
Figure 2. Incident angle (degree)
H o
r G
Gauge-2
4
68
Stra
in(1
00µε
)
Time ( µ s)
Figure 3. Strains induced by P-wave along direction 1
20 40 60 100800
2
4
3Gauge-1
5
7
Time( µs)
Str
ain
(100
µε)
" " ------ Indicates SV-wave arrival
Figure 4. Strains induced by incident SV-waves along direction 2
20 40 60 80 100
0
-2
: in Shock Compression of Condensed Matter-1997, edited by S.C. Schmidt, D.P. Dandekar, and J.W. Forbes, pp. 859-862, AIP Press, Woodbury, NY, 1998 859-862.
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