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Acta Mechanica Sinica (2014) 30(2):125132DOI
10.1007/s10409-014-0042-9
RESEARCH PAPER
Dynamic tensile characterization of pig skin
H. Khatam Q. Liu K. Ravi-Chandar
Received: 23 January 2014 / Accepted: 20 March 2014The Chinese
Society of Theoretical and Applied Mechanics and Springer-Verlag
Berlin Heidelberg 2014
Abstract The strain-rate dependent response of porcineskin
oriented in the fiber direction is explored under ten-sile loading.
Quasi-static response was obtained at strainrates in the range of
103 s1 to 25 s1. Characterization ofthe response at even greater
strain rates is accomplished bymeasuring the spatio-temporal
evolution of the particle ve-locity and strain in a thin strip
subjected to high speed impactloading that generates uniaxial
stress conditions. These ex-periments indicate the formation of
shock waves; the shockHugoniot that relates particle velocity to
the shock velocityand the dynamic stress to dynamic strain is
obtained directlythrough experimental measurements, without any
assump-tions regarding the constitutive properties of the
material.
Keywords Nonlinear waves Impact tests Digital imagecorrelation
Shocks Hugoniot
1 Introduction
The dynamic mechanical behavior of soft materials such
asrubbers, elastomers, gels, and biological tissues has
attractedmuch attention in recent years; this is driven by the need
fordetermining accurate strain-rate dependent constitutive mod-els
that are used in modeling the response to impact, penetra-tion and
other modes of loading in structural applications aswell as
biomechanical applications. In this article, we focuson pig skin
and its dynamic response. The split-Hopkinsonbar apparatus has
evolved into the most common method fordynamic material
characterization (see for example, the re-view of the technique by
Subhash and Ravichandran [1]); itsuse has been well-established for
a range of materials. Themain advantage of this technique is that
it does not use adetailed analysis of wave propagation through the
specimenmaterial and therefore a priori knowledge of the material
be-
H. Khatam Q. Liu K. Ravi-Chandar ()Center for Mechanics of
Solids,Structures and Materials,The University of Texas at Austin,
USAe-mail: [email protected]
havior is not required. However, the assumptions of unifor-mity
of the stress and deformation within the specimenwhich are needed
for interpretation of the experimentsthrough elementary
analysisplace rather severe restric-tions on the specimen size,
strain rates, and strain levels thatcan be obtained and limit the
applicability of this techniqueto a certain class of materials and
certain range of strain rates.For tension testing, the specimens
need to be quite small(for example on the order of one or two
millimeters); whilethe strain rates obtained are typically in the
range of about102103 s1, the duration of loading is small, and
hence thestrain levels achieved are quite small. The technique is
bet-ter suited for compression characterization, with strain
ratesreaching nearly 104 s1. The measurement of strain-rate
de-pendent tensile behavior of soft materials with a Hopkinsonbar,
particularly for large stretch levels, is fraught with
diffi-culties; in addition to the problems arising from
impedancemismatch with the loading bars that causes a very low
signalto noise ratio, lateral inertia effects in the specimen and
thegeneral inhomogeneity of the stress and deformation in thistest
scheme provide very restrictive conditions under whichthe split
Hopkinson bar may be used in tension. Furthermore,for applications
in many soft materials, very large stretch lev-els are encountered;
this necessitates long duration pulses for example, to reach a
stretch ratio of two at a strain rate of103 s1, a pulse duration of
2 ms is required! In order to over-come such limitations, we have
used an experimental methodfor dynamic tensile characterization of
materials using tran-sient wave propagation (Niemczura and
Ravi-Chandar [24]and Albrecht et al. [5]) and shock wave analysis
(Niemczuraand Ravi-Chandar [3]). In the present work, we apply
thesetechniques, along with some variants that allow lower
strain-rate ranges to be achieved, to examine the dynamic
responseof pig skin.
This paper is organized as follows: the quasi-static re-sponse
of pig skin is discussed in Sect. 2 to provide the un-derlying
characterization that is used as the basis for com-parison of the
dynamic response. This is followed in Sect. 3by a description of
one dimensional wave propagation as the
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126 H. Khatam, et al.
basis for interpreting the high strain-rate experiments.
Inparticular, the theoretical background that is necessary forthe
interpretation of the experiments is discussed. The re-sults of
these experiments are examined in Sect. 4 to explorethe shock
response and to construct the shock Hugoniot. Asummary of
conclusions and a perspective of soft materialcharacterization are
provided in Sect. 5.
2 Quasi-static mechanical response of pig skin
Pig skin is the material considered in the present
investi-gation of the strain-rate dependence of the constitutive
re-sponse. Skin is a connective tissue, but with one
significantdifference from other connective tissues: it is a
protectivebarrier material, for which one side is always subjected
tothe ambient environment. The general practice in tissue test-ing
is to place the specimen inside a hydration bath in orderto
maintain the specimen at physiological temperature andfurthermore,
to avoid excessive drying (dehydration). How-ever, the compromise
in this process is that the sample willabsorb some water and may
exhibit a non-native response,mediated by the swelling. Obviously,
skin in physiologi-cal condition is not fully immersed inside a
liquid and thein-vivo mechanical properties may vary from those
obtainedfrom an overhydrated skin sample [6]. Another
importantfactor in determining the mechanical response of soft
tis-sues is the strain-rate dependence. Although the responseis
expected to become less sensitive to strain rate at very lowstrain
rates [7], this requires very long loading and immers-ing time
which inevitably leads to an overhydrated sample.Finally, studies
(for example, Ref. [8]) have demonstratedthat in normal conditions
and under clothing, the tempera-ture of human skin in most parts of
the body is considerablybelow the typically imposed physiological
body temperature(37C) and is in the range of 28C to 35C with an
averagetemperature of 33C.
For the tests performed in this work, a large cut of pork-belly
was obtained from a local abattoir, and after removingthe muscle
and subcutaneous layer, the skin was preparedand tested within 48
hours post-mortem. Samples from theabdominal area were cut along
the longitudinal and trans-verse axes to uniform width (56 mm);
only the results ofspecimens oriented along the longitudinal or
fiber directionare reported in this article. The measured density
was be-tween 1.100 and 1.160 g/cm3, and skin thickness varied
be-tween 2 and 3 mm. Three different experimental arrange-ments are
considered in this work in order to obtain the re-sponse of pig
skin. All tests were conducted in room tem-perature (25C). We
demonstrated, in previous work [9],the dramatic effect of water
content on the quasi-static re-sponse of pig skin. For the current
tests, samples were keptin a closed container in the presence of
saline-soaked papertowels until testing. After removing a sample
from this mois-ture chamber, the mounting and test were performed
within5 minutes.
The first test was performed in an Instron universal test-
ing machine at strain rates in the range of 103 s1 and theother
two involve one dimensional impact loading. In thefirst type of
impact experimental arrangement, strain rates inthe range of 10 s1
are achieved; in these experiments, theforce on the specimen is
measured directly with a piezoelec-tric load cell. In the second
type of impact experiment, shockwaves are generated necessitating
direct measurements ofparticle velocities and strains in order to
enable determina-tion of the forces.
2.1 Low strain-rate experiments
Specimens of width 5 mm, thickness 2 to 3 mm, and length4 cm
were pulled in uniaxial tension in an Instron test-ing machine at a
cross-head speed of 0.3 mm/s resultingin an average strain rate of
0.007 s1. In order to providea good gripping boundary condition,
self-tightening gripswere used; even in this case, some portion of
the specimenoutside the gage length of the specimen experiences
strainsand therefore, reliance on global displacement
measurementwould overestimate the actual local strains experienced
bythe specimen. Therefore, the digital image correlation
(DIC)technique was used to measure the local strains and
thencorrelated to the measured average stress in order to
extractthe nominal stress vs. nominal strain variation. The
result-ing stressstrain curve is shown in Fig. 1 as the solid
blackline. At about a strain level of 0.4, the specimen began
toslip in the grips and the test could not be continued
further.However, the characteristic response of pigskin is
capturedin this test. It is easy to see that the initial stiffness,
cor-responding to unkinking the crimped collagen structure isquite
small. The initial tangent elastic modulus (for strainlevels below
about 0.2) is around 1 MPa. However, uponstraightening of the
collagen kinks, the linear part of the re-sponse has a considerably
higher tangential elastic modulusaround 100 MPa beyond a strain
level of about 0.4. Althoughnot shown here, the primary difference
between the fiber di-rection and the transverse direction lies in
the fact that thestiffening response is seen at much larger strain
level whentested in the transverse direction. Exact comparison of
theseresults to previous work is difficult for two reasons: first,
theidentification of the reference configuration contains
signif-icant error; therefore matching the strain level at which
sig-nificant stiffening occurs is difficult. Second, handling,
mois-ture content, and temperature influence the response
signif-icantly. Therefore, comparisons can only be in terms of
or-der of magnitude of the quantities of interest. Ankersen etal.
[10] used 30 mm wide pig skin specimens with a dumb-bell shaped
ends and reported a stressstrain response similarto that shown in
Fig. 1. Maximum stress levels at failure is inthe order of 1030 MPa
and strain levels at failure of about0.240.29 are reported; the
present results are comparable tothese results, although the
failure strain levels are somewhatlarger in the present work. In
contrast, Lim et al. [11], whoalso tested pigskin under low strain
rates, reported uniaxialstress strain curves that were
significantly different; in factat a strain level of about 0.35,
the stress indicated by Lim et
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Dynamic tensile characterization of pig skin 127
al. [11] is around 30 kPa, whereas the present results indi-cate
about 10 MPa. The likely reasons for these differencesare not
known, but for consistency and minimization of sam-ple to sample
variability, we shall compare the stressstraincurves obtained at
different strain rates from the same ani-mal, prepared and tested
as indicated above.
Fig. 1 Nominal strain vs. nominal stress variation for pig skin.
Thesolid line indicates the response of a test performed in an
Instrontest machine at a strain rate of 0.007 s1. The two other
sets ofsymbols (, ) correspond to the response from low-speed
impacttests at two different intermediate strain rate tests (17 and
23 s1)discussed in Sect. 2.2
2.2 Intermediate strain-rate experiments
The specimens used in the intermediate strain rate experi-ments
are about 175200 mm long and 56 mm wide; afterremoval of the
subcutaneous fat layer with a surgical scalpel,the specimens were
about 23 mm thick. The specimen iswrapped around a slider
illustrated in Fig. 2, with the sliderinserted into the slotted
muzzle of an air-gun, and the twoends of the specimen are clamped
to a piezoelectric loadcell, providing a minimum gage length of
about 20 mm. Thespecimen is decorated with a random pattern of dots
with anindelible ink to facilitate determination of the
deformationusing digital image correlation. One dimensional loading
isapplied by impacting the specimen holder with an 880 mmlong
projectile driven from the air-gun at a speed of about1 m/s. A
Photron SA1 high-speed video camera, operatedat 5 000 frames per
second, is used to capture the deforma-tion of the specimen upon
impact. The entire process is syn-chronized by a controller: upon
sending a command signal,this system provides a trigger to the
solenoid valve for theair gun, and a trigger with suitable delays
to begin record-ing the high speed camera images and the
oscilloscope forthe load cell data. The details of two tests in the
fiber direc-tion are provided here. The strain field was obtained
from
the images using the commercial DIC program ARAMIS.Stress waves
propagate back and forth between the impactpoint and the fixed grip
at the load cell, and establish uni-form strain conditions in the
central 10 mm of the specimen.Figure 3 shows the time variation of
the strain at a point inthe center of the specimen in two different
tests. In the firsttest, the specimen was held tight in the loading
fixture andresulted in a constant strain rate of 23 s1, while in
the sec-ond specimen, an initial slack in the specimen holder
delayedthe start of straining, and resulted in a small variation in
thestrain rate, ending up at about 17 s1. The correlation of
thestrain measurement with the load measured with the
piezo-electric load cell yields the dynamic stressstrain curve
forstrain rates in the range of 1723 s1; this is shown in Fig. 1as
the filled and open circular symbols. The comparison tothe
quasi-static stress strain curve indicates that the shape ofthe
stressstrain curve does not vary significantly with thechange in
strain rate over four orders of magnitude, but that
Fig. 2 Diagram of the slider; the specimen is wrapped around
theslider in the groove on the cylindrical surface of the
half-disk. Theslider is then inserted into the slotted muzzle of an
air gun. The twoends of the specimen are then clamped either to the
piezoelectricload cell or to the exterior of the barrel itself
Fig. 3 Time variation of the nominal strain in the gage section
forthe intermediate strain-rate experiments. The difference
betweenthe two tests arises from the initial slack or tension in
the specimenwhen placed in the impact apparatus
In fact, translating the stressstrain curve at the intermediate
strain rate along the strain axis, we observed nearly perfect
overlap of thequasi-static and intermediate strain rate
responses.
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128 H. Khatam, et al.
the stiffening response is observed at smaller overall
strainlevels than in the lower strain rate experiments. It should
benoted that there is some uncertainty in the reference
configu-ration which could result in a strain uncertainty, and
henceshifting of the stress strain curve along the strain axis isa
possibility that can not be completely eliminated. How-ever, the
initial stiffness in the intermediate impact range(evaluated at
strain levels below 0.2) suggests an increaseto about 2.5 MPa.
Next, we consider even higher impactspeeds, where wave propagation
effects need to be investi-gated.
3 Shock wave analysis and Hugoniot construction
Consider a semi-infinite specimen of width D and thicknessh
occupying x > 0; at t = 0, the end x = 0 is impacted by
aprojectile traveling at a speed Vp in the negative x direction,and
imposes a particle velocity V Vp; this generates aone dimensional
wave that propagates into the specimen. Ifthe transverse
deformation is small, inertia effects associatedwith the transverse
motion may be neglected and one may as-sume one-dimensional motion
of material points; the presentexperiments provide an opportunity
to explore this aspect.Under such conditions, the subsequent motion
of materialpoints in the specimen is represented completely by a
singlekinematic quantity, u(x, t), the displacement in the x
direc-tion; therefore, the current position of the material point x
atany time t is given by y(x, t) = x+u(x, t). The
correspondingstrain and particle velocity are given by (x, t) = u/x
andV(x, t) = u/t, respectively. Soft materials such as elas-tomers
and rubbers are usually assumed to be incompress-ible; for
biological tissues, the presence of moisture and theresulting pore
pressure may need to be taken into account,but this will not be
explored in the present work. The equa-tions governing the balance
of mass and momentum can bewritten down as
Vx=
t,
x= 0Vt, (1)
where is the nominal stress and 0 is the mass densityper unit
volume. These equations are obtained from basicbalance laws and are
therefore applicable to any material.Therefore, in order to
complete the formulation for a specificmaterial, we must specify
the material behavior; this is thefield of constitutive theory.
This system of equations is com-pleted by the addition of an
equation of state; for example,for an elastic material this can be
represented by the specificinternal energy U() in terms of the
strain . This bringsthe connection between the stress and strain: =
0U/.Note that thermal effects have been neglected in this
formu-lation, but this is adequate for the experiments
consideredhere since temperature changes due to deformation may
beconsidered to be negligible. Solutions that are continuousand
differentiable can be obtained for some constitutive mod-els; there
are numerous investigations of compressive wavepropagation in
nonlinear solids [1216]. There has been rel-atively little work on
tensile waves [2, 5, 1721] that examine
propagation of finite amplitude waves.In order to explore the
wave propagation in pig skin in
view of the stressstrain relations, () obtained in the pre-vious
section, we express Eq. (1) in terms of the particle dis-placement
to obtain the nonlinear wave equation in familiarform
[c()]22ux2=2ut2, (2)
where c() =()/, is the speed (in the reference con-
figuration) of incremental waves propagating in a
specimenstrained to a level with the prime indicating a
derivativewith respect to the argument. Suitable initial conditions
needto be specified; for example, the initial strain and particle
ve-locity along the specimen can be prescribed: (x, 0) = g(x),v(x,
0) = h(x). For a given constitutive response, this corre-sponds to
specifying the initial stress state as well. The gen-eral solution
to this boundary-initial value problem can beobtained analytically
for some forms of the stress strain re-sponse () (see Ref. [20]),
while numerical schemes suchas the method of characteristics must
be used in other cases(see for example, Ref. [2]).
Bethe [22] examined conditions under which discontin-uous
solutions are possible for materials with arbitrary equa-tion of
state and showed that stable shocks are possible when
02
2> 0. (3)
Such discontinuities are called shocks, and a vast,
classicalliterature exists that explores the generation and
propaga-tion of shocks, characterizes the response of materials
undershock conditions, and examines the development of modelsfor
the equation of state. The classical book by Zeldovichand Raiser
[23] and the more recent book by Davison [24]provide a summary of
shock propagation in solids. In orderto account for such
discontinuities where the derivatives arenot defined, the governing
differential equations in Eq. (1)have to be rewritten in terms of
the jumps across the discon-tinuities or shocks. Let the shock
occur at x = s(t) and letus denote the Lagrangian shock speed as s;
then these jumpconditions can be written as
s[[]] + [[V]] = 0, 0 s[[V]] + [[]] = 0, (4)
where [[g]] = g(s+(t), t) g(s(t), t) is the jump operator,which
is defined as the relevant parameter ahead of theshock (superscript
+) minus that same parameter behindthe shock (subscript ). If we
consider that the shock ispropagating into an undisturbed medium,
the + states arequiescent and Eq. (3) represent two equations
(representingmass and momentum balance) that govern four unknowns(,
, V, and the shock speed s). Typically, one of thequantities behind
the shockeither the particle velocity V
or the stress is imposed as a boundary condition. Theother must
be provided either through another measurementor through an
appropriate constitutive law: (); herein liesthe crux of the
problem! How does one get the appropri-
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Dynamic tensile characterization of pig skin 129
ate constitutive law? There are two approaches: the
experi-mental approach is to use a direct measurement of the
shockspeed and construct diagrams of the variation of the
shockspeed with impact speed. The diagram of the shock speedvs.
impact speed is called a shock Hugoniot. Such a shockHugoniot curve
must be characterized for each initial state;the Hugoniot
corresponding to the quiescent initial state iscalled the Principal
Hugoniot. It is not a complete mate-rial constitutive
characterization, but is sufficient to performcalculations of shock
effects. The second approach is to de-termine an equation of state
through lower scale models ofdynamic material response, and then to
establish the Hugo-niot curve analytically. In the present work, we
focus ondetermining the principal Hugoniot for pig skin
experimen-tally.
For the case of pig skin, we use the experimentally de-termined
stress strain curve shown in Fig. 1 for the interme-diate strain
rate tests, and obtain the variation of the wavespeed c() =
()/; the result is shown in Fig. 4. We
note that for small strain levels, < 0.14, the wave speedis
nearly constant at 43 m/s (to within experimental accu-racy in
determining the stressstrain curve in the low stiff-ness region).
For larger strain levels > 0.14, the wavespeed increases rapidly
with strain. The interpretation of thisstrain-dependence of the
wave speed is as follows: any straindisturbance with magnitude <
0.14 will propagate into thespecimen with a speed of 43 m/s. For
> 0.14, greater strainlevels will move faster than lower strain
levels indicating theformation of a shock wave. In other words, the
Bethe condi-tion in Eq. (3) is satisfied for > 0.14, and shock
waves canform.
Fig. 4 Dependence of the wave speed on strain level. In the
ini-tial region ( < 0.14), the wave speed is nearly independent
of thestrain, but beyond 0.14, there is a significant increase
causedby the stiffening response of the material
4 High strain-rate experiments
Higher strain-rates than that indicated in Sect. 2 were
ob-tained by impacting the specimen with a shorter projectile(2.5
cm long) driven to higher speeds (in the range of 8 to
37 m/s); the main difference from the experiments describedin
Sect. 2.3 is that the two ends of the specimen are clampeddirectly
to the gun-barrel without the piezoelectric load cell,because the
frequency response of the piezoelectric load cellis insufficient to
provide force measurement in this experi-ment. However, this
inability to measure the force directlyis not a serious impediment
since we may use shock jumpanalysis in Sect. 3 to extract the
stress in the specimen. Sincethe impact speeds are high, and the
response times short,the high speed camera was set to capture
images at 50 000frames per second or intervals of 20 s. Other
elements ofthe experiment are as indicated above. The details of
onetestcorresponding to an impact speed of 16.6 m/swiththe specimen
oriented in the fiber direction are providedhere.
(1) Selected frames from the high speed image se-quence with an
overlay of the strain field as determined byDIC are shown in Fig.
5; the time interval between the im-ages is 80 s. These results
provide a number of importantobservations regarding the
deformation. In the first two im-ages (the first 80 s), the
projectile has impacted the flangeand begins to move, but there is
no indication of deformationin the specimen; in the third image
(160 s), the strain pulseis seen to move into the field of view. In
the fourth image(at 240 s), while there is a significant amount of
strain onthe impact side, the right side of the specimen is
completelyunstrained. There appears an abrupt jump in the strain
from = 0 to over a small spatial extent (because of er-rors in the
strain measurement that arise due to the unknowninitial
configuration, we will determine the strain level fromconservation
of mass, rather than from direct kinematic mea-surements). With
time, the location of the strain jump movesto the right along the
specimen. This indicates that in re-sponse to high speed impact, a
wave is developed, acrosswhich the specimen experiences a jump in
strain as well asparticle velocity.
(2) The time variations of the displacement, particle ve-locity
and strain at four selected points along the specimen,labeled as
stage point n (n = 0, 1, 2, 3) in Fig. 5, are shown inFig. 6. Prior
to the arrival of the wave, the specimen is in anunloaded state: (+
= 0, V+ = 0). Upon arrival of the waveat any point, the particle
velocity and strain increase rapidlyand attain a constant value: V
= 16.6 m/s and ,respectively.
(3) The position of the strain jump moves along thespecimen with
a speed s which can be estimated from prop-agation of a constant
level of strain into the material; in thepresent experiment, the
speed of propagation of the strainlevel of 0.80 was used to find s
= 105.2 m/s. Notethat this is larger than the elastic wave speed of
43 m/s cor-responding to strain levels < 0.14.
(4) Since the material is initially at rest, the stress,
strainand particle velocity ahead of the shock are zero and thejump
conditions in Eq. (4) can be used to write the strain andstress
behind the shock as follows
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130 H. Khatam, et al.
Fig. 5 Sequence of high speed images (0.08 ms time interval) of
the pig skin specimen subjected to one-dimensional impact at a
speedof approximately 16.6 m/s. The red dashed line indicates the
motion of the front end of the specimen holder. The yellow line
indicates apoint on the specimen, just outside of the holding
flange; the position variation of this point is used to calculate
the particle velocity in thespecimen. The strain field calculated
from digital image correlation is overlaid on the physical image of
the specimen decorated with aspeckle pattern. The strain propagates
into the specimen, towards the fixed end at a speed of
approximately 105 m/s, significantly greaterthan the small-stretch
elastic wave speed of 43 m/s. For scale, the specimen initial width
is 6.5 mm
Fig. 6 Time variation of the particle displacement, particle
velocity,and strain at four points identified in Fig. 5 as stage
points 0 through3. The coordinates of the stage points are: x3 = 0,
x2 = 6.3 mm,x1 = 12.6 mm, x0 = 18.9 mm
= V/s, = 0 sV. (5)
The particle velocity, V = 16.6 m/s, is imposed by theprojectile
impact and has been measured. The speed of prop-agation of the
strain jump, s = 105.2 m/s, has also been mea-sured directly.
Therefore, the strain and stress behind thejump can be calculated
from Eq. (5). For this experiment,we get ( = 0.16, = 1.92 MPa).
Similar experiments were performed on other speci-mens obtained
from the same animal from neighboring lo-cations and different
impact speeds. The measured values ofV and s are shown in Table 1;
these are used subsequentlyin Eq. (5) to determine the strain and
stress levels behind theshock that are indicated in the 3rd and 4th
columns of Table1. These values of strain and stress form the
principal Hugo-niot for pig skin. From Eq. (5), it is clear that it
is sufficientto identify the pairs (V, s) in order to identify the
Hugo-niot; this relationship is shown in Fig. 7 and appears to
benearly linear over the range of impact speeds considered. A
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Dynamic tensile characterization of pig skin 131
linear V s Hugoniot of this type has been observed formost
solids (see Zeldovich and Raiser, 2002 [23]) and is ex-pressed
as
s = C0 + S V. (6)
A least-squares fit to the data yields C0 = 50 m/s and S =
4.3(with R2 = 0.82) and is also shown in Fig. 7. The constantsC0 is
usually given physical interpretation (see Zeldovichand Raiser,
2002 [23]) based on the limit as V 0. Thelimit for s as V 0 should
be the wave speed correspond-ing to small strain levels; it is
observed that, indeed, the valueof C0 = 50 m/s is close to the wave
speed of 43 m/s estimatedin Sect. 3 for < 0.14. It is also
instructive to consider theprincipal Hugoniot in terms of the
stress and stain: (, );these are the end states achieved through
impact and are plot-ted in Fig. 8; for comparison, the quasi-static
and intermedi-ate strain-rate stressstrain curves are also shown.
It is clearthat the stressstrain states reached through impact are
sig-nificantly different from those attained through
quasi-staticloading. The line connecting the quiescent initial
state andthe shock end state, illustrated with one example in Fig.
8, iscalled the Rayleigh line. The slope of this line is
proportionalto the square of the shock speed, and the area
underneath thisline gives an estimate of the energy expended in
straining tothe level behind the shock.
Table 1 Impact test results indicating impact speed, shock
speedand the corresponding stress and strain levels behind the
shock
Test Impact speed Shock speedStrain Stress/MPa
ID V/(ms1) s/(ms1)F 20 8.1 73.3 0.11 0.65
F 40 16.6 105.2 0.16 1.92
F 60 20.0 171.4 0.12 3.77
F 80 20.9 147.6 0.14 3.39
F 100 37.3 199.8 0.19 8.20
Fig. 7 The V s Hugoniot for pig skin. The dashed line
corre-sponds to a linear fit to the experimental data: s = 50 +
4.3V withR2 = 0.82
Fig. 8 The principal Hugoniot (pentagram symbols) for pig skinin
the fiber orientation is shown in comparison to the intermediate(,
symbols) and static stressstrain response (solid line). TheRayleigh
line for one of the impact tests is shown by the arrowwhich
indicates the shock jump from a quiescent initial state to anend
state that is quite far away from the low and intermediate
re-sponse. For the same strain level, the stress attained under
shockconditions is substantially greater
In closing, we note a few points for further considera-tion:
first, the Hugoniot determined in the present work cor-responds to
the fiber direction orientation of the specimen;similar
characterization must be performed in the transversedirection.
Second, we have only examined the initial pulsepropagation into the
material; at longer times, the continuedmovement of the projectile
and the finite dimensions of thespecimen result in further
straining and eventual failure ofthe specimen at high strain-rates;
these have not been ana-lyzed. Third, the maximum particle speed
considered was37 m/s; extension of the Hugoniot to higher impact
speedsshould be considered. Finally, while we have interpreted
themeasurements through the jump equations, and constructedthe
shock Hugoniot, the finite rise time of the strain pulseseen in
Fig. 6 suggests that it may be possible to postulate aviscoelastic
or viscoplastic constitutive model for the mate-rial and analyze
the results using the propagation of a steady-wave (see Ref.
[24]).
5 Conclusion
The quasi-static and dynamic mechanical response of pigskin
specimens were determined through uniaxial tensionand
one-dimensional wave propagation experiments. Thequasi-static
mechanical response of pigskin in the fiber ori-entation, when
strained at rates of about 103 s1 was ob-tained in an Instron test
machine. Higher strain rates (in therange of 10 to 25 s1) were
obtained through projectile im-pact on a short strip of specimen at
speeds of about 1 m/s.High speed photography and digital image
correlation wereused to determine the strain variation over a few
millisec-onds; a piezoelectric load cell, with adequate frequency
re-sponse, was used to identify the force. The resulting
charac-terization of the stressstrain response indicated that the
ef-
-
132 H. Khatam, et al.
fect of the strain-rate is to bring the stiffening portion of
thestressstrain curve to smaller strain levels. Characterizationof
the response at even greater strain rates is accomplished
bymeasuring the spatio-temporal evolution of the particle ve-locity
and strain in a thin strip subjected to high speed impactloading
that generates uniaxial stress conditions. These ex-periments
indicate the formation of shock waves; the shockHugoniot that
relates particle velocity to the shock velocity,and the dynamic
stress to dynamic strain is obtained directlythrough experimental
measurements, without any assump-tions regarding the constitutive
properties of the material.Interestingly, a linear relation between
the impact speed andthe shock speed was observed, as is typical for
many materi-als.
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