3.6 Constitutive Relations
141
where C is the right Cauchy-Green deformation tensor, and the unit vector N f
defines the undeformed fiber direction. As shown by Eq. (3.76), I 4 is the square
of the fiber stretch ratio λ f , whereas I 5 includes shear terms relative to the fiber
direction (see example below). These are invariants because they are geometric
quantities computed relative to a specific direction.
For example, the form
W =
n
i=0
n
j =0
c ij (I 1 − 3)
i (λ f − 1)
j
(3.229)
has been used successfully to describe the properties of passive heart muscle
(Humphrey 2002), taken as incompressible. Holzapfel et al. (2000) proposed the
relatively simple expression
W =
μ
2
(I 1 − 3) +
c 1
2c 2
e
c 2 (I 4 −1) 2 − 1
,
(3.230)
which provides a reasonably accurate approximation for many incompressible soft
tissues using only three parameters. To include compressibility in these relations, a
term involving I 3 can be added.
Example 3.21 Consider a transversely isotropic material with fibers oriented along
the X 1 -direction. Compute the invariant I 5 in terms of the components of C.
Solution
Setting C = C ij e i e j gives
C
2
= C · C = (C ij e i e j ) · (C kl e k e l ) = C ij C kl e i e l δ jk
= C ij C jl e i e l = C ij C jk e i e k .
With N f = e 1 , Eq. (3.228) 2 yields
I 5 = e 1 · C
2
· e 1 = e 1 · (C ij C jk e i e k ) · e 1
= C ij C jk δ 1i δ k1 = C 1j C j 1
= C
2
11 + C 12 C 21 + C 13 C 31 ,
which includes shear terms involving the fiber direction.