Biophysics Department, Faculty of Science, Cairo University
[4]
Measurement of Young's modulus of a bone-equivalent
material
Aim:
The purpose of the experiment is to measure the Young's modulus by depression of a boneequivalent material.
Theory:
All types of deformation in a solid result in the relative displacement of particles making up
the body. Such displacements create forces opposing the deformation. Those forces termed
elastic forces act both between parts of the strained body and between other bodies causing
the deformation. When these forces vanished or deleted, the body tend to reestablish its initial
shape and volume. The property of strained bodies to reestablish their initial shape and
volume after the external forces have ceased to act is termed elasticity. Deformations which
vanish with lifting of the external stresses are termed elastic.
Experience shows that the deformed body cannot be able to reestablish its original shape after
the external forces (stresses) have been lifted. This deformation is termed plastic
deformation. Internal forces appearing in the body in the course of plastic deformation are
also termed plastic forces. So, the strained body in this case is said to be plastic.
Hook established a relation between the plastic deformation and the internal forces acting in a
material. Hook's law states that: "For any elastic material the stretching stress is directly
proportional to the longitudinal strain, i.e.
Stress α Strain
F/A
α ∆L / L
F/A = E (∆L/L)
(1)
where F/A is the stretching (tensile) stress in N/m2, ∆L/L is the longitudinal strain, and E is a
constant called Young's modulus of elasticity (or Elastic modulus).
Hook's law (equation1) is valid also for the case of compression and can be illustrated in
Figure (1).
Tension or Compression
(F/A)
C
Stress
B
Figure (1)
A
Strain
23
First Level, Practical Biophysics
Biophysics Department, Faculty of Science, Cairo University
Bone as well as some other materials shows an elastic behavior. Bone consists of two quite
different materials plus water:
(1) Collagen: It is the major organic fraction, which is about 40% of the weight of solid
bone and 60% of its volume.
(2) Bone mineral: It is the inorganic component of bone, which is about 60% of the weight
of the bone and 40% of its volume.
Collagen is flexible like rubber, while bone mineral is very fragile. Bone mineral crystals are
believed to be made of calcium hydroxyapatite [Ca10(PO4)6(OH)2] and are rod shaped with
diameters of 20 -70 Ǻ and length of 50 -100 Ǻ.
There are two types of bone:
(1) Solid or compact bone which has constant density throughout the life (ρ = 1.9 g/cm3).
(2) Spongy or trabecular bone.
Young's modulus for compact bone is equal to 1.8 x 1010 N/m2, while it is only 8 x 107 N/m2
for trabecular bone. This means that compact bone has better elastic properties.
Bone material is as strong as granite in compression and 25 times stronger than granite under
tension. Healthy compact bone is able to withstand a compressive stress of about 170 N/m2
before it fractures. The midshaft of the femur has a cross-section area of about 3.3 cm2, it
would support a force of about 5.7 x 104 N (6 tons).
Generally, bones are not as strong under tension as they are in compression; a tension stress
of about 120 N/mm2 will cause a bone to break. However, bone is stronger under tension than
many common materials such as porcelain, oak wood and concrete.
In this experiment, loads are fixed horizontally at one end (i.e. a cantilever) is used to
determine Young's modulus of bone (Figure 2)). If a cantilever has a load W suspended at the
extreme end, the depression y of the end is given by the equation
4 L3
W
Y=
3
Ebd
(2)
where L, b and d are the length, the breadth and the thickness of the cantilever respectively,
and E is the Young's modulus for the material of the beam and W is the weight (= mg).
So,
4 L3 mg
Y=
Ebd 3
(3)
Therefore, the Young's modulus (E) can be calculated as:
4 L3mg
E=
Ybd 3
(4)
24
First Level, Practical Biophysics
Biophysics Department, Faculty of Science, Cairo University
L
y
Cantilever
d
b
mg
L
Figure (2)
Procedure:
1.
Take the zero reading (without putting the hook in the cantilever).
2.
Apply various loads on the hook. Then note the corresponding position of the extreme
end of the cantilever and find the depression (Y) in each case. Increasing the load by
equal amounts (50 grams each). Taking six or seven readings both with increasing and
with decreasing loads.
Note: Remove the load from the hook after taking each reading.
3.
Tabulate the results in a table.
4.
Draw the relation between the applied load (m) and the corresponding mean depression
(Y).
5.
From the straight line obtained, determine the slope of equation (3) and the intercept
(Figure 3).
6.
From the slope determine the Young's modulus (E) as:
4 L3g
Slope =
Ebd 3
(5)
25
First Level, Practical Biophysics
Biophysics Department, Faculty of Science, Cairo University
Therefore,
4 L3g
E=
7.
(6)
slope . bd 3
The straight line intersects the negative mass axis at a value corresponding to the mass of
the pan (hook), mo.
Y (cm)
Slope = Y/m
m (g)
mo
Figure (3)
26
First Level, Practical Biophysics
Biophysics Department, Faculty of Science, Cairo University
Results:
Zero Reading =
Depression (Y) (mm)
Load (m)
(gram)
Load increasing
Load decreasing
Mean depression
(Y)
(cm)
50
100
150
200
250
L=
cm
b=
cm
d=
cm
Slope = Y/m
cm/g
4 L3g
E=
slope . bd 3
E=
dyne/cm2
27
First Level, Practical Biophysics