Fundamentals - Preventive Biomechanics - page 76

Expansion of the four-fold scalar product in ( 3.221 ) and considering the 105

ð 8 Þ

independent isotropic tensors of rank eight I j outlined in Silber (1986) leads to the

following strain energy function composed of the nine ''mixed'' basic invariants I j
(j = 1, 2,…,9)

Þ¼ X

9

w ¼ w G ; M 1 ; M 2 ; M 3 ð c k I k G ; M 1 ; M 2 ; M 3 ð Þ

k ¼ 1

with I 1 : ¼ tr 2 M 1 G ; I 2 : ¼ tr 2 M 2 G ; I 3 : ¼ tr 2 M 3 G
I 4 : ¼ trM 1 G M 2 G ; I 5 : ¼ trM 1 G M 3 G
I 6 : ¼ trM 2 G M 3 G ; I 7 : ¼ trM 1 G

ð 3 : 222 Þ

ð Þ trM 2 G ð Þ

I 8 : ¼ trM 1 G ð Þ trM 3 G ð Þ; I 9 : ¼ trM 2 G ð Þ trM 3 G ð Þ

with nine independent material parameters c k in the orthotropic case.
In the case of transversal isotropy, the material isotropy is characterized by
only one direction tensor

K 1 M 3 : ¼ M ¼ e 3 e 3 ð 3 : 223 Þ

(WLOG, the e 3 -direction has been chosen) such that ( 3.218 ) transforms to the
following expression

ð 4 Þ ð 4 Þ

ð 8 Þ

w ¼ w G ; ð Þ¼ C

C

G ð ; : ¼ J M ð Þ: ð 3 : 224 Þ

Expansion of the fourfold scalar product in ( 3.224 ) leads to a strain energy
function composed of five ''mixed'' basic invariants I j (j = 1, 2, 3, 4, 5) with five
independent material parameters c k in the transversal isotropic case:

w ¼ w G ; ð Þ¼ X

5

c k I k G ; ð Þ

k ¼ 1

ð 3 : 225 Þ

with I 1 : ¼ tr 2 G ; I 2 : ¼ trG 2 ; I 3 : ¼ tr ðÞ trM G

ð Þ

I 4 : ¼ tr 2 M G ; I 5 : ¼ trM G 2 :

Generalized F UNG Model. Fung et al. (1979) proposed the following strain
energy function for biological soft tissue

1

2 ð J 2 1 Þ ln J

f ð J Þ : ¼ 1
D

w ð C ; J Þ¼ c 2 ð e Q 1 Þþ f ð J Þ with

ð 3 : 226 Þ

where the exponent Q

ð 4 Þ ð 4 Þ

Q : ¼ G A

G A G

G ¼ 2 C ð Þ

ðÞ with

ð 3 : 227 Þ