Fundamentals - Preventive Biomechanics - page 133

This assumption represents a possible approach for developing a criterion for
the material stability in terms of material parameter restrictions. Using the
example of the modified O GDEN -H ILL strain-energy potential (Ogden 1972a, b;
Hill 1978), (a.k.a. Ogden foam model) which is referred to as Hyperfoam in
A BAQUS , and used to characterize soft polymeric foam materials in Sect. 4.3 , the
concept of D RUCKER stability is briefly summarized as follows.
Regarding ( 3.96 ), ( 3.99 ), ( 3.274 ) and ( 3.352 ) and also using spectral repre-
sentation for G H the spectral forms of the K IRCHHOFF stress tensor and the H ENCKY
strain tensor read

s ¼ F P I ¼ X

3

¼ 2 X

N

l j
a j

ð 3 : 395 Þ

s i k i P ii k i o w
ok i

k a j i J a j b j

s i n i n i ;

i ¼ 1 j ¼ 1

and

G H ¼ X

3

G i G i ¼ lnk i n i n i with ð 3 : 396 Þ

i ¼ 1

with the identical eigenvectors n i ; and s i and G i are the principal values of tensor
s and G H ; respectively. Substituting ( 3.395 )in( 3.394 ) leads to the following
expression

X

3

ds i dG i [ 0 : ð 3 : 397 Þ

i ¼ 1

With regard to ( 3.396 ), the differential logarithmic strain and the volume ratio
follow:

dG j ¼ d ln k j ¼ dk j

ð j ¼ 1 ; 2 ; 3 Þ

k j

ð 3 : 398 Þ

Þ J X

3

¼ J X

3

dk j
k j

dG j dJ ¼ d k 1 k 2 k 3 ð

j ¼ 1 j ¼ 1

respectively, and therefore the total differential of ( 3.395 ) 2 reads

ds i ¼ 2 X

N

l j ½ð k a j i þ A i Þ dG i þ A j ð dG 1 þ dG 2 þ dG 3 dG i Þ ð i ¼ 1 ; 2 ; 3 Þ:

j ¼ 1

ð 3 : 399 Þ

whereby the abbreviation A j : ¼ b j J a j b j is introduced.
Considering equation ( 3.399 ), the relation between changes in the K IRCHHOFF
stress and changes in logarithmic strain are described by the matrix equation
(underlined symbols denote column vectors and matrices, respectively).