38
1 From Riemann manifolds to Riemann manifolds
Table 1.3. Various deformation tensors of the first kind.
Definitions
Author
Comments
E 1 = C l = S l R
∗ G r RS l = J
∗
l G r J l
A. Cauchy (1889,1890)
(“left Cauchy–Green”)
if G r = I, then C l = S
2
l = J
∗
l J l
E 2 = C r = S r R
∗ G l RS r = J
∗
r G l J r
G. Green (1839)
(“right Cauchy–Green”)
if G l = I, then C r = S
2
r = J
∗
r J r
E 3 = C l G
−1
l
E. Grafarend (1995)
(“left-right Cauchy–Green”)
if G l = I, then E 3 = C l
E 4 = C r G
−1
r
E. Grafarend (1995)
(“right-left Cauchy–Green”)
if G r = I, then E 4 = C r
E 5 = G l C
−1
l
E. Grafarend (1995)
(“inverse left-right Cauchy–Green”)
J. Finger (1894a)
if G l = I, then E 5 = E
−1
3
E 6 = G r C
−1
r
E. Grafarend (1995)
(“inverse right-left Cauchy–Green”)
G. Piola (1836),
if G r = I, then E 6 = E
−1
4
E 7 = C
m/2
l
∼ {Λ
m
1 , Λ
m
2 }
B. R. Seth (1964a,b)
(m ∈ Z, m = 0)
m = 2 : E 7 = E 1
E 8 = ln C l ∼ {ln Λ 1 , ln Λ 2 }
H. Hencky (1928)
–
E 9 = C
m/2
r
∼ {λ
m
1 , λ
m
2 }
B. R. Seth (1964a,b)
m = 2 : E 9 = E 2
E 10 = ln C r ∼ {ln λ 1 , ln λ 2 }
H. Hencky (1928)
–
E 11 = E l =
1
2
(C l − G l )
A. Cauchy (1889,1890)
(“left Euler–Lagrange”)
if G l = I, then E l =
1
2
(C l − I)
E 12 = E r =
1
2
(G r − C r )
E. Almansi (1911)
(“right Euler–Lagrange”)
if G r = I, then E r =
1
2
(I − C r )
E 13 = E l G
−1
l
=
1
2
(C l G
−1
l
− I)
E. Grafarend (1995)
(“left-right Euler–Lagrange”)
if G l = I, then E 13 = E l
E 14 = E r G
−1
r
=
1
2
(I − C r G
−1
r )
E. Grafarend (1995)
(“right-left Euler–Lagrange”)
if G r = I, then E 14 = E r
E 15 =
1
2
(C
−1
l
− G
−1
l )
Z. Karni and M. Reiner (1960)
if G l = I
then E 15 =
1
2
(C
−1
l
− I)
E 16 =
1
2
(G
−1
r
− C
−1
r )
Z. Karni and M. Reiner (1960)
if G r = I
then E 16 =
1
2
(I − C
−1
r )
E 17 = G l E
−1
l
E. Grafarend (1995)
(“inverse left-right Euler–Lagrange”)
if G l = I, then E 17 = E
−1
l
E 18 = G r E
−1
r
E. Grafarend (1995)
(“inverse right-left Euler–Lagrange”)
if G r = I, then E 18 = E
−1
r
E 19 = E
m/2
l
∼ {K
m/2
1
, K
m/2
2
}
B. R. Seth (1964a,b)
(m ∈ Z, m = 0)
m = 2 : E 19 = E 11
E 20 =
1
2
ln E l
H. Hencky (1928)
–
E 21 = E
m/2
r
∼ {κ
m/2
1
, κ
m/2
2
}
B. R. Seth (1964a,b)
m = 2 : E 21 = E 12
E 22 =
1
2
ln E r
H. Hencky (1928)
–