142
3 Continuum Mechanics and Nonlinear Elasticity
Orthotropy
In the undeformed state, an orthotropic material contains three axes of symmetry
at each point. For example, tissue composed of three mutually orthogonal fiber families embedded in an isotropic matrix is orthotropic (Fig. 3.25d). Before deformation,
let the unit vectors N f and M f define the orientations of two of these fiber families;
the third is given by N f × M f . Such a material has a strain-energy density function
of the form (Spencer 1984)
W = W m (I 1 , I 2 , I 3 ) + W f (I 4 , I 5 , I 6 , I 7 ).
(3.231)
The first five invariants are the same as those already defined; similar to Eq. (3.228),
the others are
I 6 = M f · C · M f = M f · (I + 2E) · M f = λ
2
f
I 7 = M f · C
2
· M f = M f · (I + 2E)
2
· M f .
(3.232)
As shown in the previous example, including both I 5 and I 7 captures all possible
combinations of shear.
For an incompressible material, one choice for W is obtained simply by adding
another fiber term to Eq. (3.230), i.e.,
W =
μ
2
(I 1 − 3) +
c 1
2c 2
e
c 2 (I 4 −1) 2 − 1
+
c 3
2c 4
e
c 4 (I 6 −1) 2 − 1
.
(3.233)
Blood vessels provide a specific example. Arteries are often treated as circular
cylinders composed of incompressible orthotropic material with properties that
differ in the radial (R), circumferential (), and axial (Z) directions. Variations
of the form
W = c
e
Q
− 1
Q = c 1 E
2
RR + c 2 E
2
+ c 3 E
2
ZZ + 2c 4 E RR E + 2c 5 E E ZZ
+ 2c 6 E ZZ E RR + c 7
E
2
RR + E
2
R
+ c 8
E
2
Z + E
2
ZZ
+ c 9
E
2
ZR + E
2
RZ
(3.234)
are commonly used in studies of artery mechanics (Humphrey 2002).
Here, we emphasize the following. First, all the forms for W given above are
just examples. An infinite number of other possibilities exist, so long as they satisfy
the fundamental principles listed at the beginning of this section and are consistent
with experimental data. In addition, any nonlinear form for W should reduce to the
appropriate linear relations in the limit of small strain. In the linear regime, a general
3 Continuum Mechanics and Nonlinear Elasticity
Orthotropy
In the undeformed state, an orthotropic material contains three axes of symmetry
at each point. For example, tissue composed of three mutually orthogonal fiber families embedded in an isotropic matrix is orthotropic (Fig. 3.25d). Before deformation,
let the unit vectors N f and M f define the orientations of two of these fiber families;
the third is given by N f × M f . Such a material has a strain-energy density function
of the form (Spencer 1984)
W = W m (I 1 , I 2 , I 3 ) + W f (I 4 , I 5 , I 6 , I 7 ).
(3.231)
The first five invariants are the same as those already defined; similar to Eq. (3.228),
the others are
I 6 = M f · C · M f = M f · (I + 2E) · M f = λ
2
f
I 7 = M f · C
2
· M f = M f · (I + 2E)
2
· M f .
(3.232)
As shown in the previous example, including both I 5 and I 7 captures all possible
combinations of shear.
For an incompressible material, one choice for W is obtained simply by adding
another fiber term to Eq. (3.230), i.e.,
W =
μ
2
(I 1 − 3) +
c 1
2c 2
e
c 2 (I 4 −1) 2 − 1
+
c 3
2c 4
e
c 4 (I 6 −1) 2 − 1
.
(3.233)
Blood vessels provide a specific example. Arteries are often treated as circular
cylinders composed of incompressible orthotropic material with properties that
differ in the radial (R), circumferential (), and axial (Z) directions. Variations
of the form
W = c
e
Q
− 1
Q = c 1 E
2
RR + c 2 E
2
+ c 3 E
2
ZZ + 2c 4 E RR E + 2c 5 E E ZZ
+ 2c 6 E ZZ E RR + c 7
E
2
RR + E
2
R
+ c 8
E
2
Z + E
2
ZZ
+ c 9
E
2
ZR + E
2
RZ
(3.234)
are commonly used in studies of artery mechanics (Humphrey 2002).
Here, we emphasize the following. First, all the forms for W given above are
just examples. An infinite number of other possibilities exist, so long as they satisfy
the fundamental principles listed at the beginning of this section and are consistent
with experimental data. In addition, any nonlinear form for W should reduce to the
appropriate linear relations in the limit of small strain. In the linear regime, a general