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Development of a method for quantitative stress analysis in bones by three-dimensional photoelasticity.

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THE ANATOMICAL RECORD 206:227-237 (1983)
Development of a Method for Quantitative Stress Analysis in
Bones by Three-Dimensional Photoelasticity
Istituto di Anatomia Umana Normale, Uniuersita di Milano (R.B., L. K,
A.M.), and Dipartimento di Meccanica, Politecnico di Milano (f? C.), Italy
Comparative critical examination of methods suitable for
studying stress in bones have shown that the three-dimensional photoelastic
method is one of the most reliable. Described herein is the method for obtaining, by fusion, full-scale models in epoxy resin, that are exactly equivalent to
external shape of the prototypes.
This technique offers the advantages of being applicable without variation
to any bone segment and of enabling a large number of additional resin
castings to be made from the same mould. Hence it is possible to produce a
very large number of copies of the same bone segment that will be suitable for
comparative studies of different load situations.
As a n example, quantitative data expressing both surface and internal tension trends in the proximal third of a normal human femur are given.
It is well known that a bone exposed to
mechanical stress becomes elastically deformed and develops corresponding internal
tensions which, influencing its biodynamics,
improve the bone structure, making it mechanically better suited for the stresses applied. The internal structure of bones thus is
in direct relation to the mechanical stresses
exerted on them.
The correlation between mechanical requirements and bone structure, with particular reference to the arrangement and
orientation of cancellous bone trabecules, has
been known since early in the last century.
Ward (1838) was the first to provide detailed
description of cancellous bone in the femoral
neck, by identifying the three main trabecular systems; furthermore, comparing the construction of femur epiphysis to that of a
crane, he identified a zone subjected to
compression along the medial cortical bone
and a zone of tension along the lateral cortical bone. These results were later confirmed
by Wyman (1857). Humphry (1858) observed
that in the femoral neck the main trabecular
systems cross each other at right angles, and
that the trabeculae are arranged perpendicularly in relation t o the articular surface of
the femur head. In 1867 von Meyer published, according to data of Culmann (1866),
0 1983 ALAN R. LISS, INC.
his theory regarding cancellous bone architecture, with a definition of trabecular trajectories arranged along the principal stress
On the basis of Meyer’s and Culmann’s
data, the close relationship between the function and architecture of cancellous bone was
demonstrated by Wolff (1892), and developed
and confirmed by Roux (1893) in his theory
of functional adaptation.
More recently the now undisputed link between shape, structure, and mechanical
function of bone has been verified in a series
of studies which have used more sophisticated and accurate methods of investigation.
Among these, the most reliable are: finite
elements (theoretical calculation method),
electric strain gauges, brittle coatings, holographic interferometry, and photoelasticity
methods (all experimental methods).
The finite elements method applied to bones
(Zinkiewicz, 1971; Scholten, 1976, on the human femur; Thresher and Saito, 1973; Farah
et al., 1973; Lavernia et al., 1981, on the
human tooth; Chand et al., 1976, on the human knee joint; Knoell 1977, on the human
Received December 9,1982; accepted March 11, 1983.
mandible; Hayes et al., 1978, on the human
tibia) makes it possible to determine the
stresses produced by applied loads, but only
after having determined the law that governs the relations between stresses and
strains (the law of constitution) and geometrical shape of the bone. The simplest law of
constitution assumes that stresses are proportional to strains, but in the case of bones,
proportionality may be different among different sectors of the same bone, such as, for
example, between cortical and cancellous
bone. These factors are important as regards
calculations complexity.
The method also involves a description of
the geometrical shape of the bone by means
of a model formed by solid two- or threedimensional elements (representing the bone
structure more accurately). Generally these
are closer together in instance of more irregular surfaces. This is, in fact, the case for
With the finite elements method it is also
possible to consider bone material anisotropy, but in this case calculation costs would
be very high.
For all these reasons the reliability of results may need experimental verification.
On the other hand, electric strain gauge
analysis determines directly the strains on
the bone surface (Evans, 1953, on the canine
tibia; Hirsch and Brodetti, 1956a,b; Indong
and Harris, 1978; Jacob and Huggler, 1980,
on the human femur; Fisher et al., 1976, on
the swine skull; Wright and Hayes, 1979, on
the bovine tibia and metatarsals). These
measurements have also been carried out “in
vivo” on various animals (Lanyon, 1972, on
the sheep vertebrae; Lanyon, 1973, on the
sheep calcaneus; Baggott and Lanyon, 1977,
on the sheep and goat radius; Carter et al.,
1980, on the canine radius and ulna; Caler et
al., 1981, on the canine radius and femur).
This method is based on variation in the
electric signal of transducers (electric strain
gauges) secured to the bones, this variation
being proportional to the strains which the
transducers undergo when the bone is loaded.
This technique gives very accurate data on
the state of stress but is limited by the fact
that the strain gauge, being of rather small
dimensions, reveals only what happens on a
relatively small surface, and thus provides
only local information.
The brittle coatings are lacquers which are
sprayed or spread onto the bone surface
(Gurdjian and Lissner, 1945, 1946, 1947, on
the human skull; Evans and Lissner, 1948;
Evans et al., 1953; Pedersen et al., 1949; Kalen, 1961, on the human femur) and break
orthogonally in the direction of maximum
strain when the local deformation limit-value
is exceeded. This method gives a total view
only of the surface stress state and does not
specify by how much the deformation threshold value is exceeded.
Holography is a method that is not as well
documented in the literature on bone biomechanics (Fuchs and Schott, 1973, on the human skull; Hewitt, 1977, on the primate
skull; Kragt, 1979, on the human skull). It is
based on interferometry and optical diffraction. This technique makes it possible to register clearly only the displacements normal
to the surface, while the total displacement,
and thus the stresses, can be obtained only
by very difficult experimental processes.
Photoelasticity is based on the properties of
certain plastic materials which are transparent to light and become birefringent when
loaded and observed in a polarized light. It is
possible to see (with a polariscope) lines (or
fringes) of differing optical intensity which
express the trajectories and the entity of the
stress in the element. With this method, photoelastic coatings are applied to the bone (in
which case the results are essentially quantitative, particularly in the most irregular
areas) or models (two- or three-dimensional)
are made from special plastic resin.
We think that of all these experimental
methods, photoelasticity applied to models is
overall the most satisfactory. In fact, this
method provides a quite accurate overall
view of the stresses, both on the surface and
inside the bone, while other methods either
yield only local results (electric strain
gauges), even if they are very accurate, or
overall results (photoelastic coatings, brittle
coatings), which are only superficial and of
low accuracy, or overall results of high accuracy (holography) but that are restricted to
the external surface.
One of the first applications of the photoelasticity technique to bones was that of Milch
(19401, who utilised it to study bone shape on
two-dimensional models. Photoelastic studies were later carried out on biological
models, both with two-dimensional models
(Pauwels, 1951, 1955, 1965, 1973; Kummer,
1956; Fessler, 1957, on the human femur;
Maquet et al., 1966; Maquet and Pelzer, 1977,
on the human knee) and with three-dimensional models (Knief, 1967, on the human
femur; Johnson et al., 1968; Farah et al.,
1973, on the human tooth; Yoshizawa, 1969,
on the human spine; Chand et al., 1976, on
the human knee).
The most common photoelastic method reported in the literature employs two-dimensional models obtained directly from a plate
of plastic resin. This method is, therefore,
simple, rapid, and inexpensive. Fundamental to this field are the studies carried out by
Pauwels that have extended and elaborated
what was already known about bone biomechanics. However, the two-dimensional photoelastic method has a significant limitation,
for real structures (such as bones) are usually
three-dimensional and are loaded spatially.
Quantitative studies of stresses on two-dimensional models may produce results not
fully applicable to real biological situations.
Thus it appears that photoelastic analysis
with three-dimensional models is the most
reliable method, because it provides a generally accurate picture of stress distribution a t
both the external surface and inside the bone.
It is, of course, necessary that the real conditions of the actual bone be simulated on the
model, and that the different characteristics
of resin and bone tissue be considered.
Three methods of analysis by three-dimensional photoelasticity are available: the
sandwich method, the diffused-light method,
and the stress-freezing method. In the sandwich method, a transparent sheet of photoelastically active resin is inserted in a
transparent three-dimensional body of very
low photoelastic activity. Thus the study is
of a two-dimensional type as only the inserted sheet is photoelastically active. It is
possible to apply successive and different load
systems to the “sandwich,” which is useful,
but the study of internal stresses is limited
to the region where the material is photoelastically active. Problems of adhesion among
the parts and the duration of usefulness
sometimes make this method rather difficult.
Furthermore, the model as a whole requires
final processing after the glueing of the parts
and so it is practically impossible to reproduce very irregularly shaped elements with
a high degree of accuracy.
The diffused-light method is based on the
principle that, in a transparent field (specimen) crossed by a beam of light, a light is
emitted, plane polarized, perpendicular to the
incident beam. Thus, the specimen acts as
analyser and polarizer. The optical fringes
that appear (oriented perpendicular to the
incident beam) are due to the sum of the
photoelastic effects along the path of the
same incident beam; to obtain the single local stress values, it is necessary to perform
some analytical and trigonometrical operations, which sometimes makes the method
difficult and inaccurate.
The stress-freezing method, described below, makes it possible to visualize the state
of stress of the model point by point, both on
the external surface and inside the bone, even
taking into account that it is practically impossible to faithfully reproduce the organization of cancellous bone.
The last technique (stress-freezing) has
been chosen, for our study, because in our
judgment it is the best of the experimental
methods. It also has certain advantages over
the finite elements method, which, it will be
recalled, also takes three-dimensionality into
account. In fact, as already pointed out, the
finite elements method does make it possible
to provide a good representation of a structure which is as spatial and irregular in
shape as a bone, especially if the elements
are three-dimensional and locally very dense.
However, it presents disadvantages of preparation time, and construction costs are considerable, and the division into elements
must be begun again with each shift in bone
segments studied.
Once techniques have been perfected for
the preparation of three-dimensional photoelastic models to be frozen, the stress-freezing
method presents no such difficulties when
one moves from one bone to another, and
models can have the same external shape a s
the actual bone segments. It has one disadvantage: Each model allows the study of only
one loading situation, while the finite elements method permits a variety of loads to
be applied.
Thus both methods, in our view, are of comparable value; the choice of the experimental
method was prompted by the desire to have
a n accurate method that, once refined, might
become almost routine for subsequent comparative studies and any kind of bone.
In this work, therefore, our aim has been
1)to perfect a technique for the preparation
of three-dimensional models of any bone, not
by manual shaping from blocks of resin
(which may require considerable geometrical
approximations) but with a technique which
reproduces the original external shape faithfully, in full scale; and 2) to carry out a n
accurate three-dimensional quantitative
analysis of stress distribution in the proximal third of the normal human femur. The
subject was chosen because this particular
bone segment has been thoroughly discussed
in the literature and so our data can be compared with those of earlier workers. We see
the technique as suitable for further biomechanical investigations on other bones.
The full-size model of a normal human femur was obtained by placing the bone in a
suitable aluminum container into which was
poured a n elastomer formed from two liquid
components (RTV-M 533 with T 35 Wacker
hardener) which, when mixed in appropriate
proportions, solidify. This produced a negative surface shape or mould of the bone, having a very high level of precision. Later, a
thermosetting epoxy resin cast (two components: CIBA Araldit B 46 + HT 903) was
made in the same mould.
The femur model so obtained may be seen
in Figure 1. It should be noticed that the
elastomer must have a thermal expansion
coefficient very close to that of epoxy resin,
so as to avoid separation phenomena between the two materials during cooling.
The method followed for the study of the
stresses in the loaded model is known as the
“stress freezing process” (Avril, 1974). The
model was heated in a oven at a temperature
of 140°C; a suitable load was applied; then it
was slowly cooled, still under load, to room
temperature. In such a way, the mechanical
stresses typical of the loaded condition were
locked (“frozen”) in the model, to obtain a
permanent effect of optical birefrangence
(fringes) proportional to such mechanical
As the elastic and optical characteristics of
epoxy resin vary with temperature, the same
thermal cycle was applied to a calibration
disc so as to,obtain the two fundamental parameters for the material of the model:
Young’s elastic modulus (E) and the fringe
value (F).
A very significant static condition was experimentally reconstructed-i.e., that corresponding to erect position with the body
weight on one limb only. The situation is the
most stressful, for in these conditions a force
F higher than the weight P of the body acts
on the femur head (Fig. 2). This is true because the action line of the weight P has a
lever arm relative to the rotation center of
the ilium: the moment due to the weight P is
balanced by the action T of pelvis-trochanter
abductor muscles.
Figure 3 shows the model of femur with
the load applied and the tie rods schematizing the abductor muscles.
The choice of the load applied is dependent
on the mechanical theory of the models (Wilbur and Norris, 1950); in fact, as the bone
and the epoxy resin have different elastic
moduli, the loads applied are different, if we
wish to have the same deformation on each.
The following condition must be respected:
- 1. a) Normal human femur. b) Femur model in
epoxy resin.
T- P h
Fig. 2. Schematization of the load acting on the femur model
P, = load applied to the model,
P, = load applied to the prototype (bone),
Em = Young’s modulus of the model,
E, = Young’s modulus of the prototype,
L, = linear dimension of the model,
L, = linear dimension of the prototype.
The stresses in the bone and in the model
are in this ratio:
Of course, in our case, the ratio LmL, is
equal to 1because, as previously stated, the
model is a full-size model of the prototype.
From the calibration of the disc, the elastic
modulus of the resin is equal to: Em = 19 N/
mm2, and as the bone elastic modulus corresponds to (10 + 20) lo3 N/mm2, in order to
obtain the same deformations it is necessary
to apply to the model a weight P approxiof the corremately equal to (1 + 2 )
sponding real body weight.
The optical response of the loaded model is
proportional to the mechanical stress according to the relationship:
n = -u . s
a, = mechanical stress in the model,
a, = mechanical stress in the prototype.
in which n is the number of fringes and s
Fig. 3. Model of the bone under load and Calibration disc placed in the oven.
represents the model thickness. It can be seen
that n also depends on the fringe value F of
the material, so that with the same ( T . s the
number of fringes is inversely proportional
to the value F. The calibration of the resin
used gave a value F = 0.275 N/mm fr; with
this value of F, the application of a weight P
corresponding to the average body weight of
a man would have produced too low an optical response, so we preferred to use a slightly
higher load. Thus, we applied a load of 2.45
N (0.250 kg).
Figure 4 shows the proximal third of the
frozen model, inserted in the polariscope, in
which appear the fringes caused by the
stresses. Such fringes represent the average
of the local stresses, which usually vary along
the thickness of the model crossed by the
single optical beam; so it is necessary to obtain rather thin “slices” (sections) along
which the stresses may be held sufficiently
constant. It is important to mention that such
cuts, if carefully made with a machine tool,
will not introduce optical disturbance signals. In this way the slices will maintain,
completely unchanged inside them, the mechanical stresses caused by the load applied
on the three-dimensional model.
The analysis was carried out on two different frontal sections, one median and the other
one displaced with respect to the median by
10 mm. The choice of frontal sections was
made on the basis of the literature indicating
them to be the most highly stressed.
The more constant the state of stress (in
both value and direction) along the thickness, the more accurate and reliable are the
data obtained from the fringe readings, and
thus the tension values. This is usually all
the more so in thinner sections. Unfortunately, however, as we have seen before, the
optical response (the number of fringes) is
higher at the same stress in thicker sections.
Thus there are two contrasting requirements. Therefore we considered the following
as the best way to operate: obtaining, a t the
beginning, slices of 8.2-mm thickness, noting
the data, and then decreasing the thickness
to 5.8 mm, noting further data, and finally
decreasing it again to a thickness of only 3.6
mm. The differences in data collected between the first slice and the subsequent ones
express the stress values in the removed
The method allows us to obtain two kinds
of information. If the slices are observed in
circular polarized light, the observation of
fringes gives the point-by-point difference between the maximum and minimum stress
which acts on the plane of the slice. On the
borders, moreover, one of the two stresses is
usually known, after determining the position of the loads applied; thus it is possible to
obtain the value of the single stress:
(u = n
from the fringe order without any difficulty.
If the slices are observed in polarized plane
light, it is possible to see not only the preceding fringes but also some lines (isoclinics)
which give the direction point of such maximum and minimum stresses in the plane.
The results obtained from the fringe readings provide a variety of information: the
Fig. 5. Isochromatic fringes in the median frontal section of the femur model.
~ i 4, ~Interference
fringes in the
three-dimensional model.
toto,, femur
isochromatic fringes (Fig. 5) on the borders of
the sections show that negative stresses
(compressions), in agreement with the qualitative data in the literature, are in modulus
greater than the positive ones (tensions) in
all the observed sections. This is valid not
only in median sections but also in the posterior ones. Figure 6 shows the stress values
both on the borders and inside the median
sections; as can be seen, their numerical values in the corresponding geometrical points,
according to the different thicknesses of the
median section, are not constant even if the
qualitative trends are still of the same kind.
In particular, the maximum values, in the
median slice, increase when moving from the
greatest thickness to the intermediate and
then to the minimum one. This means that
the state of stress is different in the subsequent points examined at the same level.
The maximum variations are of about 15%.
In the median slice it is possible to see that
stresses tend toward zero in the trochanter
zone except for the muscle insertion points.
+ 3.7 ‘ 10-2
+2,1 .10-2
+ 5,3.10-2
Fig. 6. Stresses on the borders and inside the median frontal section of the femur model for decreasing thicknesses (a-c).
- 3.8. lo-2
+ d,4.16'
- 1,6 * lo-'
flow in the central slice, surveyed by 10" in
10" per increasing angle clockwise.
The critical analysis of the results obtained
with three-dimensional photoelastic method
shows that the values of stresses obtained
vary in relation to slice thickness; this indicates that the state of stress is variable along
the thickness, which implies that schematization with a two-dimensional model is not
completely reliable, if not only qualitative
but also quantitative results are needed.
The isoclinic fringes show that stress trajectories in the three-dimensional model coincide with the orientation of trabeculae, as
is shown by the extensive data in the literature on this subject.
Similar and equally valid results can be
obtained also by applying the previously
mentioned finite elements method, as underlined earlier, which requires a calculation
schematization necessarily different for each
bone. Even though the technique for the
preparation of the three-dimensional model
Fig. 7. Stresses on the borders and inside a frontal
section displaced by 10 mm with respect to the median
where the stresses, purely at a local level,
reach values corresponding to about half the
maximum values found in the diaphysis tensile area.
It is also interesting to note that, in the
loading situation studied, the neutral axis is
closer to the tensile area than to the one
compressed in the diaphysis, while going up
the femur head it continually moves toward
the compressed side, as is shown in Figures
6 and 7.
The ratio between the surveyed maximum
compression, in modulus, and maximum tension is about 1.3 for the slices of maximum
thickness, which then decreases toward 1.1
as the slice thickness decreases.
The second kind of information which was
obtained from the fringe reading indicated
the direction of maximum and minimum
stresses (principal stresses) which operate in
the plane of the sections studied; Figure 8
shows a n example of a 20" isoclinic of the
median slice of the 8.2-mm-thick model. On
the other hand, Figure 9 shows the isoclinics
Fig. 8. 20" Isoclinic fringes in the frontal median
section of the femur model.
Fig. 9. Complete course of isoclinics in the frontal
median section of the femur model.
requires high technical competence, it makes
possible the obtaining of models identical in
external shape to the prototypes and, for this
reason, we consider it the most valid at this
stage of our knowledge of materials.
Among the main advantages that this experimental methodology offers, one should
mention the possibility of its application to
practically any bone segment without the
need for any methodological variation, and
the fact that the elastomer from which the
mould is made can withstand many further
resin casts which harden both when cold and
when hot, so that it is possible to reproduce
a very large number of copies of the same
bone segment which can be used for comparative studies in different loading situations.
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development, dimensions, photoelasticity, method, three, analysis, stress, bones, quantitative
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