Biomedical Engineering Reference
In-Depth Information
Fig. 5.15  A body such as a wall structure subjected to a force load. a The initial length is L o and
the deflection caused by the force load is d. For elastic materials b the stress-strain relationship is
governed by Hooke's Law producing a linear line that is the Young's modulus. c For non-linear
elastic deformation the Young's modulus is found by taking the incremental gradient
the force, the body deforms and the amount of deformation, d as a ratio of its initial
shape, e.g. length, L o is defined as the strain, ε . The amount of force applied per unit
area, A is the stress loading, δ . During linear elastic deformation the relationship
between stress and strain, known as Hooke's Law is shown in Fig. 5.15b where the
slope of the linear line is the material's Young's modulus, E . These relationships,
stress, strain, and Hooke's Law are given below:
d
F
σ
(5.25)
ε
=
σ
=
E
=
L
A
ε
o
Young's modulus, E describes how stiff a material is; a higher E produces a steeper
slope, which means that a greater force loading is needed to deform the material.
Arteries exhibit non-linear elastic behaviour, where the material does not follow
Hooke's Law (Fig. 5.15 ). The relationship is non-linear showing that as the stress
in an artery increases, the material becomes stiffer and resists strain. In such cases
Young's modulus is defined as the slope of the curve at a given stress-strain point,
which is termed the incremental Young's modulus, defined as
d
σ
ε
(5.26)
E
=
inc
d
If we consider the strain in all three directions, then Eq. (5.25) which refers to defor-
mation in one direction can be rewritten in its separate coordinates as
σ
σ
σ
y
x
z
(5.27)
ε
=
ε
=
ε
=
x
y
z
E
E
E
During material deformation, axial strain occurs in the direction of the force load. In
addition there is deformation in the other two directions, laterally and perpendicular
to the force direction (Fig. 5.16 ).
 
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