1-2 Stretch or length distortion 15
In order to visualize the eigenspace of the left and right Cauchy–Green deformation tensors C l
and C r relative to the left and right metric tensors G l and G r , we are forced to compute in addition
the eigenvectors, in particular, the eigencolumns (also called eigendirections) of the pairs {C l , G l } and
{C r , G r }, respectively. Compare with Lemma 1.6.
Lemma 1.6 (Left and right general eigenvectors, left and right principal stretch directions).
For the pair of positive-definite symmetric matrices {C l , G l } and {C r , G r }, an explicit form of the left
eigencolumns (also called left principal stretch directions) and of the right eigencolumns (also called
right principal stretch directions) is
1st left eigencolumn, Λ 1 :
F 11
F 21
=
1
c 22 − Λ 2
1 G 22
2 G 11 − 2
c 12 − Λ 2
1 G 12
c 22 − Λ 2
1 G 22
G 12 +
c 12 − Λ 2
1 G 12
2 G 22
×
×
c 22 − Λ
2
1 G 22
−
c 12 − Λ
2
1 G 12
,
2nd left eigencolumn, Λ 2 :
F 12
F 22
=
1
c 11 − Λ 2
2 G 11
2 G 22 − 2
c 11 − Λ 2
2 G 11
c 12 − Λ 2
2 G 12
G 12 +
c 12 − Λ 2
2 G 12
2 G 11
×
×
−
c 12 − Λ
2
2 G 12
c 11 − Λ
2
2 G 11
,
(1.60)
1st right eigencolumn, λ 1 :
f 11
f 21
=
1
C 22 − λ 2
1 g 22
2 g 11 − 2
C 12 − λ 2
1 g 12
C 22 − λ 2
1 g 22
g 12 +
C 12 − λ 2
1 g 12
2 g 22
×
×
C 22 − λ
2
1 g 22
−
C 12 − λ
2
1 g 12
,
2nd right eigencolumn, λ 2 :
f 12
f 22
=
1
C 11 − λ 2
2 g 11
2 g 22 − 2
C 11 − λ 2
2 g 11
C 12 − λ 2
2 g 12
g 12 +
C 12 − λ 2
2 g 12
2 g 11
×
×
−
C 12 − λ
2
2 g 12
C 11 − λ
2
2 g 11
.
(1.61)
End of Lemma.