Fundamentals - Preventive Biomechanics - page 73

with

; w 2 Cof ð Þ : ¼ X h i

w 1 ðÞ : ¼ X

N M

b j tr Cof ð Þ bj = 2 3 trC a i = 2 3 a i

i ¼ 1 j ¼ 1

ð 3 : 207 Þ

T

w 3 ðÞ : ¼ f ðÞ; CofC : ¼ C III C T adj ð Þ T ¼ o C III
oC

; C ¼ F T F

where w is subdivided into three parts. The three argument functions of the
deformation gradient in ( 3.206 ) represent the transformation of a surface- and
volume element of the ICFG into the CCFG, cf. ( 3.54 ) and ( 3.55 ).

Slightly and Highly Compressible Elastomers. The following strain energy
function was proposed by Ogden (1972a) for slightly compressible materials

f ð J Þ¼ X

N

1
D k

w ¼ w ð k 1 ; k 2 ; k 3 Þþ f ð J Þ with ð J 1 Þ 2k

k ¼ 1

w ð k 1 ; k 2 ; k 3 Þ : ¼ 2 X ð k a 1 þ k a k 2 þ k a k 3 3 Þ 2 X

X

N N 3

l k
a k

l k
a k

ð k a k i 1 Þ: and

k ¼ 1 k ¼ 1 i ¼ 1

ð 3 : 208 Þ

For N = 1 and a 1 = 2 and for N = 2 and a 1 = 2 and a 2 =-2 and based on
( 3.208 ) the Neo-H OOKE and the M OONEY -R IVLIN model results, which however,
can also be generated by substituting ( 3.184 )in( 3.200 ) and ( 3.201 ), respectively,
considering ( 3.191 ). For the description of highly compressible materials, Hill
proposed the following form (Hill 1978)

w ¼ 2 X

N

l k
a k

f k ð J Þ¼ 1
b k

½ k a 1 þ k a k 2 þ k a k 3 3 þ f k ð J Þ mit ð J a k b k 1 Þð 3 : 209 Þ

k ¼ 1

which was employed in an experimental work by Storåkers (1986). This model
was found to provide good correspondence with test data for both highly com-
pressible and slightly compressible elastomers.

3.2.6.3 Anisotropic Representations of Strain Energy Functions

Polynomial Form. One possible approach to generate constitutive equations for
anisotropic hyperelastic materials based on the isotropy of space (Boehler 1975,
1979) is considered in Silber (1988, 1990). Accordingly, the strain energy function
w in form of a (scalar-valued) tensor function of the G REEN strain tensor G and
N additional direction tensors K i (i = 1, 2,…., N) based on ( 3.69 )

w ¼ w G ; K 1 ; K 2 ; :::: K N ; ð Þ ð 3 : 210 Þ

must satisfy an extended isotropy condition according to ( 3.179 )