Fundamentals - Preventive Biomechanics - page 49

It is to note that even though S is symmetric (according to ( 3.128 )), P I

in

general, is not symmetric since

P I T ¼½ JF 1 S T ¼ J ð F 1 S Þ T ¼ JS T F T ¼ JS F T 6¼ P I : ð 3 : 97 Þ

Furthermore, the second P IOLA -K IRCHHOFF stress tensor which is symmetric and is
obtained and is related to P I and S (via pull-back operation F 1 ðÞ F T ) yields

P II ¼ JF 1 S F T ¼ P I F T ð CofF Þ T S F T
¼ F 1 S ð CofF Þ¼ J 1 P I ð CofF Þ:

ð 3 : 98 Þ

The latter tensor strictly speaking, is a pseudo stress measure which has no
physical meaning. It however, represents an important stress measure, especially in
the context of finite element methods. The K IRCHHOFF stress tensor s which is used
in various applications is introduced in the following. It is related to the stress
tensors introduced previously. In ( 3.99 ) 1 the term F ðÞ F T

is referred to as

push-forward operation)

s ¼ F P II F T ¼ JS P II ¼ F 1 s F T ¼ JF 1 S F T :

ð 3 : 99 Þ

respectively

3.2.4.4 Principle Stresses (Eigen-Value Problem)

To show constitutive equations as a function of eigen-values (spectral form) in the
following, the eigen-value problem is demonstrated using the example of the stress
tensor. The method is applicable for arbitrary (second order) tensors.
The stress measures (stress coordinates change in magnitude and direction in
every (material) point of a loaded body. To exclude structural failure of a body due
to high mechanical stress, the stress limits must be known. According to ( 3.89 ) the
general state of stress at a point is given by three direct and three shear stress
coordinates, i.e. r ii and r ij (i = j). By specifically rotating the coordinate system
on which the r ij are projected, a stress state can be found where the shear stress
vanishes and the remaining direct stress components have their maximum
(extreme) value, cf. Fig. 3.19 . This process is referred to as principal axis trans-
formation and the direct stress is referred to as principal stress.
The stress tensor S can be formulated in diagonal form equivalent to ( 3.89 ),
containing only the principal stress components (state of principal stress). Hence,
the following relation applies (note that on the right-hand side of ( 3.100 ) the
capital sigma notation must be used since more that two equal indices occur)

S ¼ r ij e i e j ¼ ! X

3

r Hi n Hi n Hi ð 3 : 100 Þ

i ¼ 1