Biomedical Engineering Reference
In-Depth Information
where, according to ( 3.195 ), the following properties must be satisfied
C ¼ I
C ¼ I ; C ¼ I
f 0 J ¼ ð Þ¼ 0 ; w 0
ð
Þ 0
w d
ð
Þ¼ 0 :
ð 3 : 306 Þ
In ( 3.305 ), w 0 ð C ; J Þ is the initial strain energy function with the deviatoric (or
isochoric) elastic part w 0
ðÞ and the volumetric elastic part f 0 ð J Þ in the equilib-
rium state of the material and w d ð C ; C Þ is the dissipative part of the strain energy
function w, extended for viscoelastic materials. Considering ( 3.305 ) 1 , equation
( 3.304 ) transforms into
P II ¼ 2 o w C ; ð Þ
oC
and o w C ; ð Þ
oC
C 0 :
ð 3 : 307 Þ
Note: In particular the split into deviatoric and volumetric parts is analogous to
that regarding hyperelastic materials, starting with equation ( 3.254 ). The entities
regarding equilibrium elasticity introduced in the 'hyperelastic materials-' section
are additionally indexed with ''0''!
Considering ( 3.305 ) and following the outlines for the one-dimensional form
( 3.301 ) 1 , the stress tensor ( 3.307 ) 1 can be divided into an equilibrium elasticity
part P I 0 and a dissipative part P I d where P I 0 can be further divided into a volumetric
and isochoric part P I 0J and P I 0 such that employing the chain rule
ow 0 ð C ; J Þ= oC ¼ o w 0 ð C Þ= oC þ of 0 ð J Þ= o w 0 ð C Þ
¼½ ow 0 ð C Þ= oC ½ oC = oC þ½ of 0 ð J Þ= oJ ð oJ = oC Þ
all parts can be written as follows
P II ¼ P I 0 þ P I d ¼ P I 0J þ P I 0 þ P I d
ð 3 : 308 Þ
with
P I 0 ¼ 2 ow 0 C ; ð Þ
P I d ¼ 2 ow d C ; ð Þ
: ¼ P I 0J þ P I 0
ð 3 : 309 Þ
and
oC
oC
and
P I 0 ¼ 2 ow 0
o C o C
P I 0J ¼ 2 o f 0 ðÞ
oC ¼ 2 o f 0 ðÞ
oJ
oC ;
ðÞ
oC
¼ 2 ow 0
ðÞ
oJ
oC
ð 3 : 310 Þ
P I d ¼ o C C ð Þ
¼ o C C ð Þ
o C
½
½
oC :
oC
oC
With the expression
o C C ð Þ
½
¼ C
ð 3 : 311 Þ
o C
as well as ( 3.256 ) and ( 3.257 ), the relations ( 3.310 ) and thus all three terms of
constitutive equation ( 3.308 ), transform to
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