Biomedical Engineering Reference
In-Depth Information
the equivalence of the last equality for the bulk modulus k may be seen from the last
column of Table 6.2 . The isotropic strain-stress relations are obtained from ( 6.24 ),
thus
1
E
E
¼
þ nÞ
T
trT
Þ
g:
1
1
(6.26)
Reverting briefly to the case of orthotropic symmetry, the strain-stress relations
may be written in the form
2
4
3
5
1
E 1
n 21
E 2
n 31
E 3
2
4
3
5 ¼
2
4
3
5
E 11
E 22
E 33
T 11
T 22
T 33
n 12
E 1
n 32
E 3
1
E 2
(6.27)
n 13
E 1
n 23
E 2
1
E 3
T 23
2 G 23 ;
T 13
2 G 13 ;
T 12
2 G 12 ;
E 23 ¼
E 13 ¼
E 12 ¼
where E 1 , E 2 , E 3 represent the Young's moduli in the x 1 , x 2 , x 3 directions; G 23 , G 31 ,
G 12 represent the shear moduli about the x 1 ,x 2 , x 3 axes and
n 23 ,
n 31 ,
n 12 ,
n 32 ,
n 13, and
n 21 are Poisson's ratios. The Poisson ratio
n 21 represents the strain in the x 1 direction
due to the normal strain in the x 2 direction and where symmetry of the compliance
tensor requires that
n 13
E 1 ¼ n 31
; n 13
E 1 ¼ n 31
; n 23
E 2 ¼ n 32
:
(6.28)
E 3
E 3
E 3
Biomedical Historical Note: Thomas Young (1773-1829) was a child prodigy, a well
educated physician, a physicist and a student of languages who attempted to decipher
Egyptian hieroglyphics and who translated the Rosetta Stone. Although he is well known
for his concept of the modulus of elasticity, he did significant work in explaining how the
eye functioned. He argued that the lens of an eye changed shape to focus light as necessary.
He suggested that the retina responded to three principal colors that combined to form all
the other colors. More generally he considered the nature of light and discovered the
principle of interference of light.
As noted in the introductory paragraph of this section, the system of 15 equations
in 15 unknowns can be reduced to a set of three equations in three unknowns or,
equivalently, to a single vector equation in three dimensions. The resulting
equations are known as the Navier equations of elasticity and they are similar in
form to the Navier-Stokes equations of viscous fluid theory developed in the
following section. To obtain these equations for an isotropic material one
substitutes (2.49) into ( 6.24 ) and then places the modified ( 6.24 ) for the stresses
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