Download Measurements of the compressive properties of scleral tissue.

Measurements of the Compressive Properties
of Sclerol Tissue
J. L. Barraglioli and R. D. Kamm
Tests were conducted on small samples of sclera taken from cattle eyes and human eyes to determine
the mechanical properties of the tissue in compression. It was found that the elastic modulus for radial
compressive stress of human sclera was more than a factor of 100 less than the modulus for circumferential
tensile stress. The stress-strain relationship was found to be mildly nonlinear and exhibited hysteresis
when cycled through a pattern of increasing-then-decreasing stress. Given sufficient time, however, the
sample returned to its original state and, therefore, exhibited no permanent deformation. Measurements
of Poisson's ratio yielded values of 0.46 to 0.50, indicating that these samples were essentially incompressible. Invest Ophthalmol Vis Sci 25:59-65, 1984
rection. Since the collagen fiber bundles run primarily
in the circumferential direction, they are better able
to resist changes in globe circumference than changes
in radial thickness. This is also evident in the tendency
for the stroma7 and, to a lesser extent, the sclera8 to
imbibe fluid when placed in water or an aqueous solution, and swell in thickness while maintaining relatively constant dimensions in the circumferential
plane.9 Aside from these measures of the swelling
properties of cornea and sclera, the literature contains
little information concerning the scleral elastic modulus
when the tissue is subjected to compressive stresses in
the radial direction (hereinafter referred to as the radial
compressive modulus), even though the sclera is normally subjected to radial compressive loads due to the
intraocular pressure. (See Appendix A for expressions
describing the stresses in the wall of spherical shell.)
Knowledge of scleral elasticity is essential for predicting the response of the globe to changes in intraocular pressure,10 to externally applied forces during
tonometry10 and oculoplethysmography, and to contraction of the extraocular muscles.'' In each of these
problems, a structural model of the globe can be used
to determine the distribution of stresses in the sclera
and the resulting deformations. Although these models
are of varying complexity, it is generally assumed that
the sclera is isotropic—that its structural properties
are the same in all directions. While assuming isotropic
structural properties, some previous studies of scleral
deformation do take account of the nonlinear stressstrain relationship for the sclera (see Woo and coworkers10). These assumptions can, in certain cases,
lead to erroneous conclusions and must therefore be
carefully assessed in each situation. Such errors can
occur, for example, when one examines the potential
collapse of vessels passing through the sclera, due to
Measurements of the tensile stiffness (the ability of
a material to retain its original dimension when subjected to forces that tend to stretch the sample) of
scleral tissue have previously been reported by several
investigators.1"3 Results from such tests are usually
reported in terms of a relationship between the applied
force per unit area (stress) and the corresponding fractional change in length (strain) of the sample. These
tests have shown that the sclera exhibits a nonlinear
relationship between stress (<r) and strain (e). The elastic
modulus, or Young's modulus (E), is the ratio of change
in stress to the associated change in strain and, therefore, represents a measure of the "strength" of a material. When the elastic modulus is high, little deformation occurs with a small increment in stress. The
same stress increment will produce large deformations
when the elastic modulus is small. Many materials
exhibit values for the elastic modulus that change as
the material is stretched. The sclera is one such material
exhibiting values of the elastic modulus (for tensile
stresses acting to stretch strips of sclera cut from the
globe), which typically range from 107 to 109 dynes/
cm 2 with a tendency for the sclera to become less compliant (higher modulus) as the applied stress increases.
This behavior can be attributed to the tendency for
the collagen fibrils to be wavy at low levels of stress,
but increasingly taut as the applied stress increases.4"6
As has been recognized for some time, the elastic
properties of the sclera and corneal stroma are different
in the radial direction than in the circumferential diFrom the Massachusetts Institute of Technology, Department of
Mechanical Engineering, Cambridge, Massachusetts.
Supported by the NIH, grant # EY 03141.
Submitted for publication February 11, 1982.
Reprint requests: Prof. Roger D. Kamm, MIT Rm. 3-260, 77
Massachusetts Avenue, Cambridge, MA 02139.
59
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INVESTIGATIVE OPHTHALMOLOGY & VISUAL SCIENCE / January 1984
Vol. 25
that run in the circumferential direction can be shown
to collapse with increases in intraocular pressure. This
phenomenon has obvious implications both in the flow
of aqueous humor from the eye and in the regulation
of blood flow at the point where veins pass through
the sclera and is the primary motivation for this study.
We have conducted experiments to examine the
compressive modulus of the sclera in the radial direction. Ourfindingsshow (1) that the radial compressive
modulus is two orders of magnitude less than previously reported values of the tensile modulus, and (2)
that Poisson's ratio for the sclera falls between the
values of 0.46 and 0.50 for changes in stress that occur
over a time scale of less than 1 hour where the latter
value corresponds to a perfectly incompressible material.
Materials and Methods
Apparatus (Fig. 1)
OPTICAL
DISPLACEMENT
PROBE
TRANSPARENT.
HOLDER. \ \
SAMPLE
DISH
ISOTONIC
SALINE
SAMPLE
FORCE FROM
AIR PISTON
Fig. 1. Top, Photograph of experimental apparatus. Bottom, Crosssectional view of sample holder and optical measurement probe.
changes in intraocular pressure.12 If one assumes that
the sclera is isotropic and linearly elastic, then it can
be shown that all vessels will increase in cross-sectional
area as intraocular pressure begins to increase, regardless of the orientation of the vessel within the sclera.
If, however, the tissue actually is more compliant in
compression than in tension (which we will proceed
to demonstrate by our experimental results), vessels
Tests were performed on small cylindrical sections
of scleral tissue (1-mm thick by 9-mm diameter), which
were placed in a sample dish and covered with an
isotonic saline solution. A compressive stress is produced by pressurizing the air chamber beneath a lowfriction graphite piston contained within a 0.627-cm
pyrex cylinder. Forces are transmitted-to the sample
by way of a connecting rod, which isattached to the
sample dish through a ball joint to ensure that the
stresses are applied normal to the sample surface-Pressure inside the pyrex cylinder is regulated, and measured with a water manometer to an accuracy of 50
dynes/cm2. The sample dish and piston assembly are
enclosed within a plexiglass chamber and kept at a
temperature of 37 ± 1 °C.
Since the sample thickness was typically about 1
mm, the displacements during compression were extremely small. These were measured using an optical
displacement probe (Fotonic Sensor, Mechanical
Technology, Inc.), which not only had excellent resolution (0.25 um) but also precluded the need for direct
mechanical contact with the sample dish assembly.
Sample Preparation
The first set of experiments was conducted using
cattle eyes (obtained from the Joseph T. Trelagan Co.,
Cambridge, MA). The eyes were placed in an isotonic
solution shortly after enucleation and stored at 4°C
until used. All experiments were conducted within 48
hours of enucleation.
For the second series of tests, three pairs of human
eyes were obtained from the Massachusetts Eye and
Ear Infirmary Eye Bank. The ages of the donors were
93, 74, and 63 years. These eyes were also stored at
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No. 1
COMPRESSIVE PROPERTIES OF SCLERAL TISSUE / Dorroglioli er ol.
4°C, and the experiments were conducted within 72
hours of enucleation.
To prepare a test sample, the conjunctiva was removed and, for the cattle eyes only, the stubs of the
rectus muscle were excised. The globe was then bisected
at the equator, and the lens, iris, and choroid were
removed.
Cylindrical samples were taken from the sclera just
posterior to the cornea (in the region of but not including the trabecular meshwork) using a 9-mm trephine and stored in isotonic saline at 4°C until tested,
always within 3 hours of sample preparation. Some
swelling of the tissue is likely to have occurred while
the sample was stored in saline solution, but this is
estimated to be no more than 15% of the initial
volume.13
61
mations are encountered (see reference 14). However,
even at the maximum values of strain, we observed
(0.15), the difference between equation 1 and the engineering strain is less than 10%.
In order to determine Poisson's ratio (j>),t the sample
was photographed from above, under magnification,
with different loading conditions. The cross-sectional
area of the sample was thus determined both before
(As0) and after (As) compression. These areas, in conjunction with the measured compressive strain, yield
a value of Poisson's ratio:
v = (VAs/As0 - l)/6
(2)
as derived in Appendix B. The measured areas As0
and As correspond to compressive stresses of 2 X 103
and 4 X 104 dynes/cm 2 , respectively. All stresses are
computed based on the area As0.
Test Procedures
The initial thickness of the scleral sample was measured to an accuracy of 10 /im using a spring-loaded
thickness gauge and then placed into the sample dish
filled with saline. A compressive stress of 2 X 103 dynes/
cm2* was then applied to ensure firm contact between
the sample and holder, and 30 minutes was allowed
for stabilization. This point was used as a reference
(strain = e = 0) from which all subsequent strains were
measured. At this level of stress we could see, by observing the sample through the transparent holder, that
the entire upper surface of the sample was in contact
with the sample holder and, therefore, that the initial
curvature had been eliminated. This observation is
consistent with that of Hedbys and Dohlman 8 who,
using a sample holder of similar design, found that
changing the diameter of their sample had no effect
on their results.
The optical displacement probe was then brought
into position so that sample thickness (t) could be
measured after deformation under a certain level of
stress (stress was computed as F/AJQ where F is the
force applied by the sample holder on the specimen
and As0 is the initial specimen area). Given the initial
sample thickness (to), compressive strain can be defined
using the relationship:
e = In (to/t)
(1)
Here we depart from the more common definition of
compressive strain [e = (to - t)/to], which is applicable
for small deformations and apply instead the expression
for natural strain generally used when large defor* An eye with an intraocular pressure of 15 mmHg would have
a radial compressive stress on the inner layers of the sclera of about
2 X 104 dynes/cm2 as calculated from the expressions given in Appendix A.
Results
Cattle Sclera
The first tests were conducted using cattle eyes, essentially for the purpose of perfecting the experimental
technique. However, the experiments exhibited several
interesting features. First, it was found that cattle sclera
continues to deform for some time after the load has
been applied. This characteristic is common to many
physiologic tissues15 and is often referred to as viscoelastic creep. Appreciable creep was observed in these
experiments during the first 10-20 minutes following
the application of pressure, corresponding approximately to the period of primary creep observed in
rabbit sclera by Greene and McMahon. 16 Consequently, we found it necessary to wait 30 minutes
between pressure increments to ensure a reasonably
stable measurement. Second, we observed that the
stress-strain relationship is somewhat nonlinear in that
the tissue becomes less compliant with increasing
compressive loads.
A linear regression of the stress-strain data produced
values for the compressive modulus (using Equation
1) ranging from 1.21 X 105 to 1.88 X 105 dynes/cm 2 ,
for compressive stresses between 2.0 X 103 and 2.6
X 104 dynes/cm2. Although the behavior was somewhat
nonlinear, the linear regression coefficients for the individual sets of data ranged from 0.975 to 0.985.
Human Sclera
As with cattle sclera, human sclera exhibited considerable creep over a period of up to 20 minutes folt Poisson's ratio is found by dividing the induced lateral strain
by the axial strain. If v = 0.5, then the material is perfectly incompressible.
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Vol. 25
INVESTIGATIVE OPHTHALMOLOGY & VISUAL SCIENCE / January 1984
• INCREASING STRESS
o DECREASING STRESS
CVJ
O
0
0.04
Q08
0.12
0.12 °0
0.16
0.0-4
0.08
0.12
0.08
0.12
STRAIN U)
Fig. 2. Compressive stress-strain data for all human eyes. Test 1 shows the hysteresis loop formed by first loading, then unloading, the
sample. Numbers refer to identifications in Table 1.
which has previously been employed by others.17 19
Note that we continue to use the expression e = In
(to/t), whereas the others referenced here use e = (to
- t)/to. The parameter often used to describe the extent
of nonlinearity in the resulting stress-strain relationship
is the "stiffening coefficient"; a in equation 3. Low
values of a are indicative of a lesser degree of nonlinearity. Values of a for the tensile elastic modulus of
the sclera have been found to be over 40. l7 Our measurements of the radial compressive modulus yield
values ranging from 3-10, with a mean of 6.1. Thus,
due to the high degree of linearity in the data, we can
express our results either by assuming a constant modulus of elasticity or an exponential form of the type
in equation 3. The differences between the two would
be slight.
The procedure to determine Poisson's ratio was followed in six of the eight eyes, for compressive stresses
between 2 X 102 and 4 X 104 dynes/cm 2 , with the
results given in Table 1. The values of Poisson's ratio
lowing a change in the applied stress level. The stressstrain relationship was also found to be mildly nonlinear, becoming stiffer for higher strains, consistent
with measurements of the tensile modulus. 1 ' 217 Sample
1 was cycled through a pattern of increasing-then-decreasing stress with increments every 30 minutes and
exhibited some hysteresis (Fig. 2), but returned to its
original thickness when allowed to stabilize at the reference stress level. Thus the sample experienced no
permanent set.
All data for the eight eyes tested are shown in Figure
2. A linear regression yields values for the compressive
modulus ranging from 2.7 X 105 to 4.1 X 105 dynes/
cm2 (0.982 < r2 < 0.992) with errors of less than 13%
(Table 1).
The degree of nonlinearity exhibited by these results
can be compared to that of the tensile modulus by
expressing our results in the general form
a = A(e<" -
1)
(3)
Table 1. Measured properties of human scleral samples
Sample
Sample no.
Radial compressive
modulus (dynes/cm2)*
Correlation coefficient
(linear regression)
Poisson 's
ratiof
[i
4.12
4.00
3.31
3.27
105
105
105
105
0.982
0.993
0.991
0.988
0.50
Age: 74 years
One sample from each eye
II
2.69 X 105
2.89 X 105
0.992
0.982
0.48
0.47
Age: 63 years
One sample from each eye
{I
3.18 X 105
3.24 X 105
0.989
0.982
0.46
0.47
Age: 93 years
Two samples from each eye
* Errors less than 13%.
t Errors less than 6%.
I 4
X
X
X
X
X Poisson's ratio not measured.
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—t
0.49
-t
No, 1
COMPRESSIVE PROPERTIES OF SCLERAL TISSUE / Darroglioli er al.
all are approaching 0.5, indicating that the sample
remains nearly at constant volume when undergoing
deformation. This finding is consistent with previous
estimates.10
63
(a
Discussion
The values we found for the radial compressive
modulus of the sclera are roughly a factor of 100 lower
than values previously reported for the tensile modulus
in the circumferential direction. This apparent discrepancy can be explained by reference to the structural
arrangement of the collagen bundles, which provide
the globe with its considerable stiffness. In the anterior
segment of the eye, the collagen bundles form a ring
around the cornea, which blends into the woven pattern
that exists over most of the remainder of the eye. This
structure provides for optimal strength in the circumferential direction, which is needed in order to maintain
the structural integrity of the optical components of
the eye. Since few of the collagen fibers are oriented
in the radial direction, the tissue is much less capable
of resisting deformation when subjected to radial
stresses. In addition, because the stiffness of most collagen containing tissues increases with increasing strain
(as do arteries, for example15), compression of these
same tissues will tend to reduce the elastic modulus
due to the release of tension and a consequent increase
in the waviness of the collagen fibrils.4"615 For these
two reasons, we would expect the elastic modulus for
radial compressive stresses to be considerably less than
that found when the tissue is subjected to tensile stresses
in the circumferential direction, consistent with our
observations.
While it is relatively straightforward to compare our
measurements of compressive modulus to the values
of tensile modulus reported in the literature, it is perhaps more appropriate to consider a different collection
of experiments—those in which swelling pressure of
the corneal stroma or sclera have been measured (see
references 9 and 13). In the experimental arrangement
of Hedbys and Dohlman, 8 for example, the swelling
pressure is that pressure exerted by a sample when
confined between two rigid plates and exposed to water
or an aqueous solution. Due to the tendency of the
sample to imbibe fluid, it expands against the rigid
plates, thereby producing an expansion force that can
be measured by a force transducer. Scleral samples
similar to those we have tested will exhibit swelling
pressures strongly dependent upon sample hydration.8
By relating hydration to sample volume, and assuming that circumferential dimensions remain relatively constant in experiments of the type described
above,13 one can, in principle, determine the compressive modulus of the sample from the experiments
(b)
c)
Fig. 3. A schematic representation of the deformations produced
by these experiments, (a), cross-sectional view of the scleral sample
prior to flattening. The wavy lines represent collagen fibrils oriented
in the circumferential direction, (b), same sample shown after flattening, (c), same sample shown after compressive stresses have been
applied above that required for flattening.
of Hedbys and Dohlman. While it is difficult to determine the elastic modulus precisely, a rough calculation based on an estimate of the slope at the point
of normal hydration of the swelling pressure-hydration
relationship for the sclera (Fig. 3 in reference 8) suggests
values for the modulus somewhat lower than ours,
possibly as high as 1 X 105 dynes/cm 2 . This apparent
discrepency can be directly attributed to a subtle difference in the experimental systems. While similar in
most respects, the two systems differ in that the upper
and lower plates were porous in the experiments of
Hedbys and Dohlman, but impermeable in ours. In
addition, their samples remained at nearly constant
volume since they were placed between two plates constrained to prevent relative motion. What little volume
change occurs is facilitated by the close proximity of
a permeable boundary. In order for our samples to
change volume, fluid must pass to the peripheral edge
at the outer radius of the sample. The resistance to
flow by this pathway is enormous due to the low permeability of scleral tissue21 and would, as a result,
% The time required to reach equilibrium in an experiment such
as this is determined by the dimensions of the sample (the distance
the fluid has to travel before leaving the sample), sample stiffness
and porosity, and the permeability of the sample to flow. According
to consolidation theory, which describes the rate at which a compressible, porous material is compacted by surface loads (see reference
22), the time for a sample to reach equilibrium once subjected to a
new load will vary as the square of the distance from the center of
the sample to the nearest permeable boundary. Therefore, for a disk
1 mm thick and 9 mm in diameter, the equilibration time for a disk
with permeable flat surfaces would be approximately 81 times less
than one with only the outer circumference permeable.
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64
INVESTIGATIVE OPHTHALMOLOGY & VISUAL SCIENCE / January 1984
cause the time scale for changes in fluid volume to be
extremely long in our experimental set-up.f
Since we found values for Poisson's ratio of nearly
0.5, the time for significant changes in fluid volume
must therefore be long compared to the duration of
an experiment. As Bert and Fatt21 point out, swelling
pressure curves are influenced only by the swelling
components of the tissue (the mucopolysaccharide gel),
while other structures are relatively unimportant. In
contrast, since the hydration of our samples remained
constant (as evidenced by values of Poisson's ratio
close to 0.5), our results reflect the structural properties
of the entire tissue matrix, including both the mucopolysaccharide gel and the collagen fibrils. It is impossible, however, to distinguish from these measurements, the relative importance of these two constituents.
To illustrate, consider the sample shown schematically in Figure 3. Before flattening (a), all the collagen
fibrils are relaxed and are assumed to have roughly
the same initial degree of waviness. When the upper
and lower plates are bought into contact with the sample, the globe segment is flattened (b). This flattening
produces an increased waviness of the outermost (top)
fibrils due to the local compressive stresses associated
with bending, and a straightening of the fibrils near
the inner wall due to the local tensile stresses. With
subsequent compression (c), all the fibrils are lengthened, and the tissue matrix is deformed so as to maintain the total volume nearly constant. The strength of
the tissue (or its ability to resist deformation) is, therefore, due to at least three factors: 1) stretching of the
circumferential collagen fibrils, 2) compression of the
radially oriented fibrils (not shown in the schematic
representation of Figure 3), and 3) the general deformation of the tissue matrix. Measurements of swelling
pressure reflect the effects of (2) and (3) but not (1),
since the circumferential dimensions are observed to
remain constant.
Based on these arguments, it would seem most appropriate to use our results in situations where the
tissue hydration remains relatively constant. This
would presumably include situations in which the stress
level in the eye changes rapidly as, for example, in the
application of an applanation device to the ocular surface or when stresses are imposed by the extraocular
muscles. We would expect even greater deformations
(and a lower apparent modulus of elasticity) given a
particular compressive stress if the sample were allowed
to come to complete equilibrium and if the hydration
were to change during this process.
Key words: mechanical properties of scleral tissue, elastic
modulus, structural modals of the globe
Vol. 25
Acknowledgments
We thank the members of the Howe Laboratory of Ophthalmology at the Massachusetts Eye and Ear Infirmary for
their many helpful suggestions, and Mark Johnson who
helped in the data analysis.
References
1. Gloster J, Perkins ES, and Pomier ML: Extensibility of strips
of sclera and cornea. Br J Ophthalmol 41:103, 1957.
2. Curtin BJ: Physiopathologic aspects of scleral stress-strain. Trans
Am Ophthalmol Soc 67:417, 1969.
3. Yamada H: Strength of Biological Materials, Krieger, Huntingdon,
New York Press, 1973, p. 238.
4. McCally RL and Farrell RA: Effect of transcorneal pressure on
small-angle light scattering from rabbit cornea. Polymer 18:444,
1977.
5. Ku DN and Greene PR: Scleral creep in vitro resulting from
cyclic pressure pulses: applications to myopia. Am J Optom
Physiol Opt 58:528, 1981.
6. McCally RL and Farrell RA: Structural implications of smallangle light scattering from cornea. Exp Eye Res 34:99, 1982.
7. Cogan DG and Kinsey VE: The cornea. V: Physiologic aspects.
Arch Ophthalmol 28:661, 1942.
8. Hedbys BO and Dohlman CH: A new method for the determination of the swelling pressure of the corneal stroma in vitro.
Exp Eye Res 2:122, 1963.
9. Fatt I: Physiology of the Eye—An Introduction to the Vegetative
Functions, Boston and London, Butterworth Inc., 1978.
10. Woo SL-Y, Kobayashi AS, Lawrence C, and Schlegel WA:
Mathematical model of the corneo-scleral shell as applied to
intraocular pressure-volume relations and applanation tonometry. Ann Biomed Eng 1:87, 1972.
11. Greene PR: Mechanical considerations in myopia: relative effects
of accommodation, convergence, intraocular pressure, and the
extraocular muscles. Am J Optom Physiol Opt 57:902, 1980.
12. Battaglioli JL and Kamm RD: The collapse of small vessels in
the wall of a spherical cavity. Proc. 34th ACEMB, p. 364, 1981.
13. Maurice DM: The cornea and sclera. In The Eye Vol I: Vegetative
Physiology and Biochemistry, Davson, H, editor. New York,
Academic Press, 1962, pp. 289-368.
14. Popov EP: Introduction to Mechanics of Solids, Englewood Cliffs,
NJ, Prentice-Hall, 1968, p. 94.
15. Fung YC: Biomechanics: Mechanical Properties of Living Tissues,
New York, Springer-Verlag, 1981, pp. 267-276.
16. Greene PR and McMahon TA: Scleral creep versus temperature
and pressure in vitro. Exp Eye Res 29:527, 1979.
17. Woo SL-Y, Kobayashi AS, Schegel WA, and Lawrence C: Nonlinear material properties of intact cornea and sclera. Exp Eye
Res 14:29, 1972.
18. Graebel WP and van Alphen GWHM: The elasticity of sclera
and choroid of the human eye, and its implications on scleral
rigidity and accommodation. J Biomech Eng 99:203, 1977.
19. Nash IS, Greene PR, and Foster CS: Comparison of mechanical
properties of keratoconus and normal corneas. Exp Eye Res
35:413, 1982.
20. Fatt I and Hedbys BO: Flow of water in the sclera. Exp Eye
Res 10:243, 1970.
21. Bert JL and Fatt I: Relation of water transport to water content
in swelling membranes. In Surface Chemistry of Biological Systems, Blank M, editor. New York, Plenum Press, 1970, pp. 287294.
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COMPRESSIVE PROPERTIES OF SCLERAL TISSUE / Dorroglioli er ol.
22. Yong RN and Warkentin BP: Introduction to Soil Behavior,
New York, Macmillan, 1966, pp. 192-198.
23. Timoshenko S: Theory of Elasticity, New York, McGraw-Hill,
1934, pp. 323-326.
Appendix A
For a spherical shell of uniform thickness subjected to a
uniform intraocular pressure (IOP), the radial stress (trr) and
circumferential stress (<rc) have the following distributions
(23):
3
3
<rr = IOP[l - (b/r) ]/[(b/a) - 1]
3
3
<rc = IOP[2 + (b/r) ]/[2 - 2(b/a) ]
(Al)
(A2)
where a and b are the inner and outer radii of the shell,
respectively, and r is the radial position measured from the
center of the sphere. Notice that the radial stress is negative
for positive values of intraocular pressure and ranges from
- I O P at the inner surface (r = a) and zero at the outer surface
(r = b). The circumferential stress is always positive and is
typically about five times as large as the intraocular pressure.
65
ness Xo to a final thickness x as a result of compression
between two flat, rigid surfaces, the other dimensions of the
sample will also generally change. If the initial dimensions
in the two orthogonal directions perpendicular to the direction
of applied stress are y0 and Zo, respectively, then the final
dimensions are expressible in terms of the Poisson's ratio as
given below:
y = yoO + «)
(Bi)
z = Zo(l +
where e is the strain in the direction of stress application and
the deformations are assumed to be small. Accordingly, the
initial volume of the sample is XoyoZo and the final volume
is
xyz = XoyoZoU + e)(l + vef
(B2)
Similarly, the sample area in the plane perpendicular to the
x-direction is
As = A s O ( l + ^ ) 2
(B3)
where A^ = yozo. Rearranging A3 one obtains
Appendix B
If a sample of an elastic material, such as the scleral disks
used in these experiments, is deformed from an initial thick-
v = (As/Aso - l)/e
which is used to determine Poisson's ratio.
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(B4)