Fundamentals - Preventive Biomechanics - page 87

2G I G I þ I G I ¼ 2 X

3

X

3

M i G M j þ M j G M i

i ¼ 1 j ¼ 1

Þ I ¼ X

X

Þ I ¼ X

3 3 3

tr ð I ¼ trI G ð ð trM i G Þ M j ; ð trM i G ð trM i G Þ M j

i ¼ 1 j ¼ 1 j ¼ 1

Þ M i ¼ X

3

M i ; M i G M i ¼ trM i G

tr ð M i ¼ trI G ð trM j G ð Þ M i

j ¼ 1

M i G ¼ M i G I ¼ X M i G M j ; G M i ¼ I G M i ¼ X

3 3

M j G M i

j ¼ 1 j ¼ 1

ð 3 : 280 Þ

whereby ( 3.279 ) can be transformed in the following form with nine independent
material coefficients u i

P II ¼ u 1 trM 1 G þ u 2 trM 2 G þ u 3 trM 3 ð Þ M 1
þ u 2 trM 1 G þ u 4 trM 2 G þ u 5 trM 3 G

ð Þ M 2

þ u 3 trM 1 G þ u 5 trM 2 G þ u 6 trM 3 G ð Þ M 3

þ u 7 M 1 G M 2 þ M 2 G M 1 ð Þþ u 8 M 1 G M 3 þ M 3 G M 1 ð Þ

þ u 9 M 2 G M 3 þ M 3 G M 2 ð Þ:

ð 3 : 281 Þ

The material equation ( 3.281 ) differs from the form presented in Boehler (1975,
1979) by the latter three terms in parentheses, which are identified by M i G þ
G M i (i = 1, 2, 3). They can, however, be transformed into each other employing
( 3.280 ) 4 -( 3.280 ) 7 .
For transversal isotropic materials, the material tensor ( 3.277 ) converts to the
following fourth order tensor function of the single direction tensor M using
( 3.224 ) 2

þ l 3 IM þ MI

ð 4 Þ

ð 4 Þ ð 4 Þ

C : ¼ l 1 II þ l 2 I 1 þ I 2 ð Þþ l 4 MM

ð 3 : 282 Þ

M :

ð 6 Þ ð 6 Þ ð 6 Þ ð 6 Þ

þ l 5 I 2 þ I 3 þ I 11 þ I 12

In ( 3.282 ), the underlined term represents the particular case of isotropy.
Substituting ( 3.282 )in( 3.276 ) leads to the second P IOLA -K IRCHHOFF stress tensor
with five independent material coefficients u i (the isotropic part is underlined)

I

P II ¼ u 1 trG þ u 2 trM G

ð 3 : 283 Þ

þ u 2 trG þ u 3 trM G ð Þ M þ u 4 G þ u 5 M G þ G M ð Þ: