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Internal stress wave measurements in solids subjected
to lithotripter pulses
S. M. Gracewski, Girish Dahake, and Zhong Ding
Department of Mechanical Engineering, University of Rochester, Rochester, New York 14627
S. J. Burns
Materials Science Program, Department of Mechanical Engineering, University of Rochester, Rochester,
New York 14627
E. Cart Everbach
Electrical Engineering Department, University of Rochester, Rochester, New York 14627
(Received:l August 1992; accepted for publication 20 April 1993)
Semiconductor strain gauges were used to measure the internal strain along the axes of spherical
and disk plaster specimens when subjected to lithotripter shock pulses. The pulses were
produced by one of two lithotripters. The first source generates spherically diverging shock
waves of peak pressure approximately 1 MPa at the surface of the specimen. For this source, the
incident and first reflected pressure (P) waves in both sphere and disk specimens were identified.
In addition, waves reflected by the disk circumference were found to contribute significantly to
the strain fields along the disk axis. Experimental results compared favorably to a ray theory
analysis of a spherically diverging shock wave striking either concretion. For the sphere,
pressure contours for the incident P wave and caustic lines were determined theoretically for an
incident spherical shock wave. These caustic lines indicate the location of the highest stresses
within the sphere and therefore the areas where damage may occur. Results were also presented
for a second source that uses an ellipsoidal reflector to generate a 30-MPa focused shock wave,
more closely approximating the wave fields of a clinical extracorporeal lithotripter.
PACS numbers: 43.80.Ev, 43.80.Sh
INTRODUCTION
Both cavitation and direct stress wave effects have
been proposed as mechanisms that cause kidney stone and
gallstone fragmentation during extracorporeal shock wave
lithotripsy (ESWL). •-4 A typical clinical lithotripter
source produces a focused shock wave incident on the sur-
face of a stone with a peak positive pressure of duration
• 1 ps and magnitude • 100 MPa, followed or preceded
by a tensile stress of lower magnitude but typically of
longer duration. In addition, larger tensile stresses can de-
velop within the stone as the compressive stress wave prop-
agating in the stone material is reflected at the posterior
stone surface. The cavitation hypothesis predicts that the
rarefactional portion of the incident shock wave causes
microbubbles present in the liquid surrounding the stone to
expand and collapse violently near the stone surface. 2
Upon collapse, high localized stresses are produced that
result in surface pitting and crack propagation that may
ultimately fragment the stone. Alternatively, the direct
stress wave hypothesis predicts damage on the anterior
surface of a stone caused by compressive stresses and spal-
ling of the posterior surface caused by tensile stresses gen-
erated within the stone. •
In stones with nonplanar boundaries, internal focusing
effects may also occur. For example, a plane wave striking
a sphere 5 or an explosion or point impact on the surface of
a sphere 6'7 will result in caustics: lines along which linear
theory predicts infinite stresses for a shock wave front.
These caustics are for two extreme cases: the first is for a
plane wave interacting with a sphere, that is, the source is
far from the sphere; the second is for a diverging wave with
the source on the surface of the sphere. The experiments
studied in this paper are between these limits. The spheri-
cally diverging acoustic source is near enough to the tar-
gets so that curvature of the wave fronts is important. The
location of the caustics is of interest since experimental
studies have shown that fractures in Plexiglas spheres sub-
jected to point explosive loading at the sphere surface oc-
cur initially at points where the caustics cross the symme-
try axis. 6
In vitro lithotripsy experiments have presented evi-
dence for both the cavitation and direct stress wave
mechanisms. 2-4'8-•ø However, the relative importance of
these mechanisms for stone fragmentation in vivo is still
not well understood. A better understanding of the mech-
anisms involved in stone fragmentation and their depen-
dence on lithotripter parameters such as pulse shape, du-
ration, rise time, and peak positive or peak negative
pressures may lead to improvements in lithotripter design
or in more effective clinical procedures.
Knowledge of the evolution of stress fields inside con-
cretions subjected to lithotripter pulses, along with mea-
surements of the mechanical properties of the stones, may
clarify failure processes. In this paper, we describe a tech-
nique of implanting silicon strain gauges within plaster
samples to obtain information about these internal stress
fields. Test specimens of simple geometries (i.e., disks and
spheres) have been chosen for this study so that the reflec-
652 J. Acoust. Soc. Am. 94 (2), Pt. 1, August 1993 0001-4966/93/94(2)/652/10/$6.00 ¸ 1993 Acoustical Society of America 652
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tions from the boundaries can be more easily identified and
compared with theoretical models. Experimental results
are obtained for both spherically diverging and focused
shock wave sources. Theoretical models based on geomet-
rical acoustics are presented for a spherical wave front
incident on a disk and on a sphere. The caustic surfaces for
a spherically diverging wave incident on a sphere are pre-
sented. Finally, predictions from these models are com-
pared with the experimental results.
I. EXPERIMENTAL METHODS
A. Strain gauge characteristics and sample
preparation
Monocrystalline silicon semiconductor strain gauges
(UFP-500-060, Kulite Semiconductor Products, Inc., Le-
onia, NJ) were chosen because of their high sensitivity and
small size. Their gauge factor (140+5% at 24 ø C) is two
orders of magnitude larger than conventional metallic wire
strain gauges. The silicon gauges are U shaped with a half-
width of 0.2 mm and an effective gauge length of 0.8 mm.
These silicon gauges, like conventional wire gauges, are
intended to be long and slender, so that corrections for
transverse strains are small. The longitudinal and shear
wave speeds of the gauge material are •=8930 m/s and
•=5320 m/s, respectively, as calculated from published
data for the density (2331 kg?m3), Young's modulus ( 161
GPa), and Poisson's ratio (0.225) of silicon. • The fre-
quency response of the gauge is primarily limited by the
relationship between the gauge length and the wavelength
of the lithotripter pulse and by the amplifying circuit band-
width. For instance, the pulse length in the gauge for a
longitudinal wave of 1-bts duration is 9 mm, so the pulse
length is nine times the effective length of the gauge. In the
surrounding plaster, the pulse length would be 3 mm. Be-
cause the total change in resistance of the gauge will be
proportional to the average strain along the length of the
gauge, the shock rise time and peak value will not be rep-
resented accurately. However, information about the ar-
rival times, relative magnitudes, and polarity of the strain
wave pulses can be obtained from the strain gauges.
A strain gauge produces a change in resistance when
subjected to a stress or strain. The lead wires of the gauge
are attached to an electronic circuit that converts the
change in resistance to a change in voltage and amplifies
the signal. The output signal is displayed on a LeCroy
model 9400 digital oscilloscope and transferred to a com-
puter for plotting and analysis. Electromagnetic noise from
the spark sources is minimized by using short coaxial ca-
bles and by electrically shielding the electronics and the
strain gauge.
Spherical and disk-shaped specimens were fabricated
by pouting a. plaster mixture, 100 parts Ultracal 30
(United States Gypsum Company, Chicago, IL) to 36
parts water by weight, into Plexiglas molds and allowing
the plaster to set. These molds were designed so that a
strain gauge, glued to a 10-btm-diam. fiber of Kevlar TM
(DuPont Corp., Wilmington, DE), could be positioned
within the mold before the plaster was poured. Minimal
amounts of glue were used to bond the gauges to the fibers
and unbacked gauges were used to enhance the bonding
between each gauge and the plaster. For the experiments
described in this paper, a gauge was positioned at the cen-
ter of each of the specimens, and in the disk, the gauge was
aligned with the axis.
The plaster spheres were 2.25 cm in diameter. The
plaster disks were 10.2 cm in diameter and approximately
3 cm thick. Before each experiment, the plaster samples
were submerged in degassed water and degassed under a
24-in. Hg vacuum for at least 1 h. Samples were then pres-
surized at 10 atm for approximately 45 min to drive any
remaining air bubbles into solution.
B. Wave speed and density measurements of plaster
The longitudinal and shear wave speeds and the den-
sity of the plaster are required for the theoretical analysis.
The wave speeds were calculated from the propagation
times in 1- and 0.5-cm-thick plaster samples that were de-
gassed as described above. The samples were cut with a
diamond saw to obtain parallel faces. Half-inch diameter
2.25-MHz longitudinal wave and 1-MHz shear wave trans-
ducer pairs were used (Panametrics, Inc., Waltham, MA).
Transmit and receive transducers were coupled directly to
the sample's surfaces. The measurements on eight samples
were averaged. The measured value for the longitudinal
wave speed ca is 3290+40 m/s and for the shear wave
speed Cs is 1750 4- 90 m/s (mean 4- standard deviation).
The specific gravity Sg for three degassed plaster sam-
ples was calculated from measurements of their weight in
air We and their weight in water Ww as
Sg= We/( We-- Ww). Their specific gravity was found to
be 1.88 4-0.01.
C. Experimental setup for strain measurements
Two lithotripters were used in this study. The first, a
Wolf model 2137.50 Electrohydraulic Lithotripter (Rich-
ard Wolf GMBH, Postfach 4D, D-7134 Knittlingen, Ger-
many), generates spherically diverging shock waves by
producing an underwater spark at the end of a 3-mm co-
axial cable (9F probe). With each spark, a bubble is cre-
ated that expands and collapses twice before it shatters into
microbubbles. A shock wave is generated with the creation
of the initial bubble and upon each collapse.
Campbell et al. 12 describe the details of the bubble behav-
ior and characterize the resulting shock waves fields. They
report that the standard deviation of the spark-to-spark
peak positive pressure variation is on the order of 20%.
There is negligible rarefactional pressure. The positive
pressure amplitude of each shock wave pulse as it propa-
gates in degasseal water is inversely proportional to the
radial distance from the shock source. The time history of
the pressure, measured at 9 cm from the source by a Mar-
coni (Marconi Research Center, Chelmsford, England) bi-
laminar PVDF membrane hydrophone with a 1.0-mm-
diam. sensitive element, is shown in Fig. 1. At this distance
from the shock source, the amplitude of the first shock
wave pulse (see insert) is about 1 MPa. The second pulse
653 J. Acoust. Soc. Am., Vol. 94, No. 2, Pt. 1, August 1993 Gracewski et at' Stress measurements of lithotripter pulses 653
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2OO
200
'•'• 100 -2 5
133 -100
-200 t ß
o.o
Time (msec)
FIG. 1. Signal from a membrane hydrophone positioned 9 cm from a
Wolf electrohydraulic lithotripter. The first two acoustic shock waves
arriving at the hydrophone, indicated by the arrows, are of comparable
amplitude. Each acoustic pulse is followed by noise generated as the
waves reflect off the hydrophone mount. 12 The insert, with time axis in/•s,
shows the first acoustic pulse in more detail.
arrives approximately 1 ms later. For the experiments pre-
sented here, times were chosen so that only the effects of
this first shock wave striking the sample as measured by
the strain gauges were recorded. Similar results were also
obtained for the second shock wave. The Wolf source was
used in most of our experiments because of the simplicity
in modeling the spherically diverging waves.
The second lithotripter used in this study was designed
by Coleman 13 as a low-cost experimental lithotripter that
would facilitate the examination of acoustic fields in
ESWL. The shock wave source and measured waveforms
are similar to those of the clinical Dornier HM3 litho-
tripter. The shock waves, generated by spark discharge
under water, are focused by a brass ellipsoidal reflector.
The focal diameter is 1 cm with a peak positive pressure of
about 30 MPa. The Coleman lithotripter was used to ob-
serve the strain fields in plaster disks subjected to fields
more typical of clinical ESWL.
Samples were always positioned so that the strain
gauges were along the axis of symmetry of the lithotripters,
thereby aligning each strain gauge with a principal direc-
tion of strain. For the Wolf lithotripter, the samples were
aligned using a three-way positioner so that their front
surface was either 9 or 15 cm from the shock source. For
the Coleman lithotriper, they were positioned so that their
front surface was at the lithotripter focus.
Two masks could be placed between the Wolf source
and a disk sample to partially block the incident wave.
Mask A consisted of a 5-mm-thick aluminum plate with a
5.1-cm-diam. hole at its center, covered with a 1.4-mm-
thick corprene layer containing a 4.2-cm-diam. hole, con-
centric with that of the plate. Corprene is a corklike ma-
terial that will prevent the acoustic waves that strike it
from propagating to the sample. This mask was positioned
so that the direct wave could reach the strain gauge. The
mask was large enough, however, to block any waves that
would have reached the outer rim of the disk. Mask B
consisted of a 1.3-cm-diam., 1-cm-thick piece of Plexiglas
covered with a 0.5-mm-thick layer of corprene. This mask
was positioned to block the direct wave incident on the
strain gauge, but allow waves to reach the outer rim of the
disk. Masks were not aligned with the strain gauge axis so
that edge diffraction effects would be minimized. No masks
were used with the Coleman lithotripter because, for this
focused wave, the energy reaching the outer rim of the disk
is negligible.
II. GEOMETRICAL ACOUSTICS
Ray theory is used to approximate the wave propaga-
tion and reflections within the samples for the shock waves
generated by the Wolf lithotripter. These shock waves are
only weakly nonlinear, so linear theory is likely to provide
a good approximation over short propagation distances.
The plaster material is assumed to be homogeneous, iso-
tropic, and linear elastic. The coupling water is assumed to
be inviscid and to extend to infinity so acoustic reflections
from the liquid boundary can be neglected. Absorption and
scattering will contribute to additional attentuation of the
waves within the samples, but are neglected since experi-
mental values are not available. The models, discussed be-
low, are axisymmetric and geometrical acoustics is used to
predict the arrival times and approximate amplitudes of
internally reflected waves arriving at the strain gauges. The
location of the caustics, pressure contours of the incident
pressure wave, and wave fronts at various times are also
determined for the sphere. Because the number of reflected
waves arriving at the strain gauge rapidly becomes large as
time increases, the analysis for the disk is carded out for 50
/•s after the arrival of the direct pulse. The reflection coef-
ficients, needed to calculate the amplitudes and phases of
reflected and refracted waves, are summarized in the Ap-
pendix.
For the following analyses, the displacement vector u
in the solid is expressed in terms of the scalar potential •
and the vector potential • in the form
u= V•b-I- VX•,, (1)
where • and • satisfy the wave equations
;t+2
V2•--C• at2, C•-- P ,
1021P 2 •
(2)
with ;t and/• denoting the Lam• constants and p the den-
sity of the solid. In the liquid, the displacement vector u• is
expressed in terms of the scalar potential • that satisfies
the wave equation
(3)
1 O2•l k l
with k• and Pt denoting the bulk modulus and the density
of the liquid, respectively.
654 J. Acoust. Soc. Am., Vol. 94, No. 2, Pt. 1, August 1993 Gracewski et al.: Stress measurements of lithotripter pulses 654
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B
• C •
Shock sourc%.....-- •
Strain Gauge
FIG. 2. Propagation path of a ray generated at a point shock source that
reflects off the circumference of a solid disk and strikes the center of a
strain gauge aligned with the disk axis.
A. Disk
A diagram of a disk of radius R and thickness B, at a
distance L from the shock source is shown in Fig. 2. A
strain gauge of length 2D and width W is aligned with the
central axis and positioned at the center of the disk. In
addition to the wave that propagates from the source di-
rectly to the strain gauge, oblique waves can also be re-
flected by the circumference of the disk so that they strike
the gauge. The ray path at an angle a from the axis is
shown for a wave incident on the front of the disk. The
path b, c, g ultimately strikes the center of the gauge. Only
the pressure (P) waves generated at each reflection or re-
fraction are shown. However, shear (S) waves are also
produced resulting in 2 •v+2 wave fronts, where N is the
number of reflections by either face of the disk. For the ray
path in the figure, N= 1.
We have considered all possible combinations for N<3
for which the wave front arrives at the strain gauge less
than 50/•s after the direct wave does. These combinations
can be represented in terms of the five cases presented in
Table I. The case label represents the wave type. For ex-
ample, pPSnpms represents a P wave striking the disk cir-
cumference generating a reflected P wave followed by n ,5'
waves and rn P waves reflected from either the front or
back surfaces and then an ,5' wave that ultimately strikes
the strain gauge. In Table I, the time t for the wave to
reach the strain gauge is given in terms of the incident
angle a and the angles fia and fis that the P and ,5' waves,
respectively, make with the normal to the front surface of
the disk. These angles are determined from Snell's law and
the geometric relation given in Table I for each case. The
plus or minus signs ( 4- ) correspond to waves striking the
edges of the gauge that are farthest or nearest to the source,
respectively.
The change in amplitude of a wave propagating along
a ray can be estimated by assuming that in a narrow tube
of rays the energy remains constant, unless a boundary is
encountered. At a boundary, the amplitude is modified by
the appropriate reflection or transmission coefficient. Ig-
noring the curvature of the wave front at the strain gauge
and making use of the assertion that the reflection and
transmission coefficients at an interface are independent of
TABLE I. Relations for waves reflecting from the circumference of the
disk.
Case
Arrival time and geometric relation
PSS n
ppsnpmp
ppsnpms
sspnsms
sspnsmp
L R R-- L tan a
t--
ct cos a cs sin fs + ca sin fa
R (cot fluq. cot rs) = L tan a cot faq' ( n q- «) B 4- D
L (mq.l.5)B4-D nB
c/cos a ca cos fa cs cos fs
2R = L tan a q.nB tan fsq. [(rn q. 1.5)B4- D]tan fa
L (mq. 1)B (nq.O.5)B4-D
t-- -4
c/cos a ca cos fa cs cos fs
2R = L tan a +[(n q.0.5) Bm D]tan fs
+ (m+ 1 ) B tan flu
t--
L nB (m+I.5)B4-D
q.
c/cos a ca cos fa cs cos fs
2R= L tan ot+nB tan fa+[(m+ 1.5) B4- D]tan fs
L (mq.1)B (nq.O.5)B4-D
t= +
c/cos a q cos fs ca cos fa
2R= L tan a+[(n+0.5) B4- D]tan fa
+ (m+ 1 ) B tan fs
the curvatures of the incident wave front and the
interface, 5 the energy density Eg at the strain gauge can be
expressed as
N
Eg=A*To(a)R c .
Eo 7-fi, 1-[
k=l
(4)
In this expression, T o is the power transmission coefficient
for the wave incident from the liquid into the solid, R c is
the power reflection coefficient for the wave incident on the
circumference, R•, is the power reflection coefficient of the
kth reflection on the disk front or back face. The power
reflection (transmission) coefficients are obtained by
squaring the appropriate potential amplitude coefficient
listed in the Appendix and multiplying by the ratio of the
reflected (transmitted) wave speed to the incident wave
speed. Also, fi•, is defined as the reflected angle for the kth
interaction with the front or back surfaces, with/30=/3½.
The value of E0-.• 5 X 10 -3 MPa m3/s was obtained by fit-
ting measurements of the energy density of the spherically
diverging source wave to the form
E= Eo/ 2, (5)
where ? is the distance from the source.
The geometric factor in Eq. (4) is given by
A*= 7-• •--•g r•sk' (6)
k=l
,,
where r and s are the two principle radii of curvature and
the unprimed and primed quantities correspond to radii of
curvature before and after reflection or transmission, re-
spectively. The subscripts c and g indicate the locations
where the ray intersects the circumference and strain
gauge, respectively, and subscript b indicates the location
655 d. Acoust. Soc. Am., Vol. 94, No. 2, Pt. 1, August 1993 Gracewski et al.: Stress measurements of lithotripter pulses 655
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