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1
2
I
1
=
2
(
Λ
1
+
Λ
2
)=
2
tr[C
l
G
−
l
]
,
1
(1.306)
2
i
1
=
2
(
λ
1
+
λ
2
)=
2
tr[C
r
G
−
r
]
represent the
average Cauchy-Green deformation, distortion energy density
of the first kind, also
called
Cauchy-Green dilatation
. In contrast,
ln
√
I
2
=
2
(ln
Λ
1
+ln
Λ
2
)=ln
det[C
l
G
−
l
]
,
(1.307)
ln
√
i
2
=
2
(ln
λ
1
+ln
λ
2
)=ln
det[C
r
G
−
r
]
are the
geometric mean of Cauchy-Green deformation
or
distortion energy density
of the sec-
ond kind. Note that similar
Hilbert invariants
can be formulated and interpreted for the Euler-
Lagrange deformation tensor.
Alternative measures of distortion energy density are intro-
duced in continuum mechanics. By means of the
weighted
Frobenius matrix norm
of Box
1.48
, we have given quadratic
forms of Cauchy-Green and Euler-Lagrange deforma-
tion density. The weight matrices W
l
and W
r
are
Hooke
matrices
, also called
direct and inverse stiffness matrices
.
Boxes
1.47
and
1.48
have reviewed local scalar-valued defor-
mation measures, namely distortion densities of the first
and the second kind. As soon as we have to map a certain
part of the left surface as well as the right surface, we should
consequently introduce global invariant distortion measures
as summarized in Box
1.49
, which constitute Cauchy-
Green and Euler-Lagrange deformation energy. d
S
l
denotes
the left surface element, while d
S
r
den
otes the
right sur-
face elem
ent, fo
r instance, d
S
l
=
det[G
l
]d
U
d
V
and
d
S
r
=
det[G
r
]d
u
d
v
, respectively. The vec operator is a
mapping of a matrix as a two-dimensional array to a column
as a one-dimensional array: under the operation vec[A], the
columns of the matrix A are stapled vertically one-by-one.
An example is
A
2
×
2
,vec[
A
]=(
a
11
,a
21
,a
12
,a
22
).
∈
R
Box 1.48 (Weighted matrix norms of Cauchy-Green and Euler-Lagrange deformations).
Cauchy-Green deformation:
C
l
G
−
1
2
W
l
C
r
G
−
1
2
W
r
:=
:=
l
r
versus
:= tr[(C
l
G
−
l
)
T
W
l
(C
l
G
−
l
)]
:= tr[(C
r
G
−
r
)
T
W
r
(C
r
G
−
r
)]
,
(1.308)
C
l
G
−
1
2
W
l
=
C
r
G
−
1
2
W
r
=
l
r
versus
=(vec[C
l
G
−
l
])
T
W
l
(vec [C
l
G
−
l
])
= (vec[C
r
G
−
r
])
T
W
r
(vec[C
r
G
−
r
])
.
Euler-Lagrange deformation:
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