Fundamentals - Preventive Biomechanics - page 135

solved analytically, and root finding regarding ( 3.404 ) must thus be performed
numerically. In this context A BAQUS utilizes a N EWTON ' S method to find corre-
sponding k 2 values for given values of k 1 .
Similarly to the presented approach, the principal K IRCHHOFF stresses based on
the O GDEN strain-energy form for slightly compressible materials ( 3.398 ), as used
to characterize human soft tissue material in Chap. 5 ,( 3.272 ) 2 , read

" ! #

s i ¼ 2 X

X

N 3

l j

k a j i 1
3

þ j

a j J a 3

k a j
k

D j JJ ð Þ 2j 1

ð 3 : 405 Þ

j ¼ 1 k ¼ 1

Considering the total differential of ( 3.405 ) the relation between changes in the
K IRCHHOFF stress and changes in logarithmic strain can be described by the matrix
equation ( 3.400 ) whereby the stiffness matrix is defined by

2
4

3
5

A 1 þ A p A o þ A p A n þ A p

X

N

W : ¼ 2
3

l j A o þ A p A 2 þ A p A m þ A p ð 3 : 406 Þ

A n þ A p A m þ A p A 3 þ A p

j ¼ 1

using the abbreviations

3 H ð 3k a j i þ X

3

A i : ¼ 1 k a k Þ ð i ¼ 1 ; 2 ; 3 Þ

k ¼ 1 ;

A m : ¼ 1

3 H 2 ð k a j 2 þ k a 3 Þ k a j

1

A n : ¼ 1

3 H 2 ð k a j 1 þ k a 3 Þ k a j

ð 3 : 407 Þ

2

A o : ¼ 1

3 H 2 ð k a j 1 þ k a 2 Þ k a j

3

A p : ¼ 2j
D j

ð k 1 þ k 2 k 3 Þ; H : ¼ J 3 a j

k 1 : ¼ð J 1 Þ 2j 1 ; k 2 : ¼ð J 1 Þ 2j 2 ; k 3 : ¼ð 2j 1 Þ J 2

Check for material stability is performed analogue to the Hyperfoam model
( 3.274 ).

3.4.10 Restrictions Based on Classical Linear Theory

Irrespective of the D RUCKER criterion, conclusions on the material parameters of
the O GDEN -H ILL model and the O GDEN model for highly compressible and slightly
compressible isotropic materials, respectively, can be drawn from the relations
governing the linear elastic regime. At the initial stress-free reference state, the