Fundamentals - Preventive Biomechanics - page 136

strain-energy function must be consistent with the linear theory of isotropic
elasticity, appropriate for small deformations.

Uniaxial Deformation: According to ( 3.253 ) 2 the coordinates of the first
P IOLA -K IRCHHOFF stress tensor read P ii ¼ ow = ok i . Using ( 3.96 ) and ( 3.99 ) one finds
P I ¼ F 1 s and further with regard to ( 3.188 ) and ( 3.274 ) the following
expressions for the coordinates of the first P IOLA -K IRCHHOFF stress for high com-
pressible materials (O GDEN -H ILL model) in the case of uniaxial loading in

1-direction ð P I 22 ¼ P I 33 ¼ 0 ; k 2 ¼ k 3 and J ¼ k 1 k 2 ;Þ are obtained:

¼ 0

;

P I 11 ¼ 2 X

N

l j
a j

P I 22 ¼ f k 1 ; k ð Þ P

N

l j
a j

k a j 1
1

k a j 1
2

k 1
1

k 2 J a j b j

J a j b j

j ¼ 1

j ¼ 1

ð 3 : 408 Þ

According to (cf. Attard and Hunt 2004) the initial Y OUNG modulus E 0 is
defined as

k 1 ¼ 1 ; k 2 ¼ 1

E 0 : ¼ o P I 11
ok 1

ð 3 : 409 Þ

so that with regard to ( 3.408 ) 1 E 0 in the case of the O GDEN -H ILL model reads:

k 1 ¼ 1 ; k 2 ¼ 1

k 1 ¼ 1 ; k 2 ¼ 1

¼ 2k 1 X

N

k a j 1 þ J a j b j

E 0 ¼ o P I 11
ok 1

l j
a j

k 1 þ a j b j J 1 o J a j 1

ok 1

j ¼ 1

!

k 1 ¼ 1 ; k 2 ¼ 1

¼ 2 X

N

1 þ b j o J

ð 3 : 410 Þ

l j

ok 1

j ¼ 1

On the basis of ( 3.189 ) and with regard to k 2 ¼ k 3 , which implies J ¼ k 1 k 2 , the
derivation oJ = ok 1 reads

k 1 ¼ 1 ; k 2 ¼ 1

k 1 ¼ 1 ; k 2 ¼ 1

k 1 ¼ 1 ; k 2 ¼ 1

oJ
ok 1

¼ o
ok 1

¼ k 2 þ 2k 1 k 2 ok 2 k 1 k 2

ok 1

k 1 ¼ 1 ; k 2 ¼ 1

¼ 1 þ 2 o k 2
ok 1

: ð 3 : 411 Þ

It is to note that the derivation ok 2 = ok 1 in ( 3.411 ) represents the derivation of
the material specific dependence (implicit relation) of both stretches k 1 and k 2
resulting from ( 3.408 ) 2 which can be constituted from ( 3.408 ) 2 as follows

k 1 ¼ 1 ; k 2 ¼ 1 k 1 ¼ 1 ; k 2 ¼ 1 ¼ 2k 2 X

N

o P I 22
ok 1

l j
a j

k a j 1
2

o k 2
ok 1 þ J a j b j k 1

o k 2
ok 1 þ a j b j J 1

1 þ ð Þ o k 2
ok 1

a j 1

2

j ¼ 1

" # ok 2

ok 1

" # þ X

k 1 ¼ 1 ; k 2 ¼ 1 k 1 ¼ 1 ; k 2 ¼ 1

¼ X

N

X

N N

o k 2
ok 1

l j 1 þ 2b j þ b j l j 1 þ 2b j l j b j ¼ 0

J ¼ 1 j ¼ 1 j ¼ 1

ð 3 : 412 Þ