Geography Reference
In-Depth Information
1-6 Review: The Deformation Measures
Review: the family of 22 different deformation measures, compatibility conditions, integra-
bility conditions, differential forms.
By means of Table
1.3
, let us introduce a collection of various deformation measures, i.e. defor-
mation tensors of the first kind based upon the reviews by
Macvean
(
1968
),
Morman
(
1986
), and
Grafarend
(
1995
). For the classification scheme various representation theorems of
Ting
(
1985
)
are most useful.
Compatibility conditions
for Cauchy-Green deformation fields have been formu-
lated by
Duda and Martins
(
1995
). They are needed for the problem to determine the mapping
equations
U
K
=
f
K
(
u
k
)or
u
k
=
f
k
(
U
K
) from prescribed left or right Cauchy-Green deformation
fields as tensor-valued functions. In the context of
exterior calculus
, these compatibility condi-
tions are classified as
integrability conditions
. The various deformation measures honor the works
of
Almansi
(
1911
),
Cauchy
(
1889
,
1890
),
Finger
(
1894a
),
Green
(
1839
),
Hencky
(
1928
),
Karni
and Reiner
(
1960
),
Piola
(
1836
), and
Seth
(
1964a
,
b
). The inverse deformation matrices, namely
E
5
,
E
6
,
E
15
,
E
16
,
E
17
,andE
18
, appear in the various forms of distortion energy. Logarithmic
and root measures of deformation appear in special stress-strain relations, which very often are
called
constitutive equations
. The measures E
3
and E
4
as well as E
13
and E
14
build up the special
eigenvalue problems. They correspond to definitions of the curvature matrix K =
HG
−
1
,in
−
(d
g
)
2
=
g
μν
d
u
μ
d
u
ν
as
surface geometry built on the matrices of the
first differential form I
∼
(d
h
)
2
=
h
μν
d
u
μ
d
u
ν
, which is also called the
Hesse
well as on the
second differential form II
∼
form
. Indeed, they establish the matrix pair
{
H
,
G
}
, where G is positive definite.
1-7 Angular Shear
A second additive measure of deformation: angular shear (also called angular distortion), left
and right surfaces, parameterized curves.
An alternative
additive measure
of deformation is
angular shear
, also called
angular distortion
.
Assume that two parameterized curves in
2
2
M
l
as well as their images in
M
r
intersect at the point
U
M
U
N
2
as well as
u
1
and
u
2
being elements of
U
0
as well as
u
0
, respectively. Two vectors
1
and
the corresponding local tangent spaces
T
U
0
M
l
as well as
T
U
0
M
r
,
U
1
U
2
∈
l
,
l
versus
u
1
∈
r
,
u
2
∈
r
,
∈
T
U
0
M
T
U
0
M
T
U
0
M
T
U
0
M
(1.132)
include the angles
Ψ
l
and
Ψ
r
. (Note that prime differentiation is understood as differentiation
with respect to arc length. In contrast, dot differentiation is understood as differentiation with
respect to an arbitrary curve parameter, called “
t
l
”and“
t
r
”, respectively.) As it is illustrated by
Fig.
1.19
, the left angle
Ψ
l
as well as the right angle
Ψ
r
are represented by the inner products
Search WWH ::
Custom Search