Fundamentals - Preventive Biomechanics - page 96

Constitutive Equation for the K IRCHHOFF Stress Tensor. Based on the form
( 3.316 ), the constitutive equation for the Kirchhoff stress tensor is derived. Ana-
logue to ( 3.260 ) 2 and ( 3.264 ) 4 , the following relation holds between the deviatoric
parts of the stress tensors s 0 , P I 0 and S 0 and the fictitious stress tensors s 0 P I 0 and S:
s 0 ¼ F P I 0 F T ¼ J S 0

P I 0 ¼ F 1 s 0 F T ¼ JF 1 S 0 F T

respectively

s 0 ¼ F P I 0 F T ¼ JS 0

P I 0 ¼ F 1 s 0 F T ¼ J F 1 S 0 F T
ð 3 : 321 Þ

respectively

respectively.

For the first term in ( 3.316 ) 3 of the second P IOLA -K IRCHHOFF stress, the modified
push-forward operation yields

¼ F ðÞ F T J 3 F ðÞ F T ¼ J 3 F ðÞ J 3 F T

ð 3 : 322 Þ

|{z}
F

|{z}
F T

as well as the following identical transformation using ( 3.190 ) 2 ,( 3.257 ), ( 3.260 ),
( 3.261 ), ( 3.262 ), and ( 3.321 ) 4 :

F T ¼ p

h i ¼

ð 4 Þ

ð 4 Þ

ðÞ P I 0 ðÞ F P I 0 ðÞ F T

J 2 = 3 F P

2 3

5 ¼ ð 4 Þ s 0 ðÞ s 0 ðÞ

ð 3 : 323 Þ

¼ ð 4 Þ F F 1

s 0 ðÞ F T F T

4

|{z}
I

|{z}
I

For the second term in ( 3.316 ) 3 of the second P IOLA -K IRCHHOFF stress consid-
ering the definition of the relative deformation gradient F t , it can be found

F t t ðÞ : ¼ F t ðÞ F 1

ðÞ:

ð 3 : 324 Þ

In addition, the relation between the dyadic product of the right C AUCHY strain
tensor and its inverse and its modified versions and based on ( 3.192 ), i.e.

1

C 1 C ¼ J 2 = 3 C

J 2 = 3 C ¼ J 2 = 3 J 2 = 3 C 1 C ¼ C 1 C ð 3 : 325 Þ

together with trC A ¼ trF T F A trF A F T ; yield the following identical
transformations: