[4, 5]. The use of the modified theory of shell dynamics, which takes into account the inertia of rotation and
transverse shift of a normal element, is due to the fact that polymer and composite materials, widely used in
modern technology, are characterized by weak resistance to shear deformation, which are not taken into
consideration by the classical theory of shells, and within the framework of this approach take nonzero
values.
Key words: two-dimensional thermoviscoelastic waves, stability of a difference scheme,
convergence of a solution of a difference problem, indenter, deformation, stress tensor.
Introduction
Let us consider a thin shell, the middle surface of which is formed by rotation of a smooth
curve
s
R
around the axis
Oz
(Fig. 1 ). The radius-vector
,
s
r
of an arbitrary point on the
median surface is set as follows
Figure 1
s
z
s
R
s
R
s
r
,
,
cos
,
.
(1)
Directing vectors of the orthogonal local coordinate system in the point are entered as
follows:
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;
,
sin
,
cos
z
R
R
s
r
r
s
;
0
,
sin
,
cos
R
R
r
r
(2)
;
;
ds
s
dz
z
ds
s
dR
R
.
s
R
R
Using these vectors, we define the components of the metric tensor
;
1
:
,
,
2
2
z
R
r
r
g
r
g
s
s
ss
,
;
0
;
2
2
R
g
g
g
g
R
g
s
s
(3)
as well as Lamé coefficients
.
1
,
1
:
1
R
H
H
g
H
s
The normal vector
n
will be set as
;
,
sin
,
cos
1
R
z
z
r
r
g
n
s
;
,
sin
,
cos
R
z
z
s
n
n
s
;
0
,
cos
,
sin
z
z
n
n
(4)
;
2
2
ds
s
R
d
R
;
2
2
ds
s
z
d
z
and the components of the second metric tensor
b
will be calculated as
:
r
n
b
b
R
z
z
R
r
n
b
s
s
ss
or, considering the relations between
;
1
2
R
z
,
1
2
R
R
R
z
get
;
1
1
0
2
s
ss
R
R
R
b
;
0
2
R
R
z
R
r
n
b
(5)
.
0
s
s
b
b
Here
,
1
2
0
R
R
R
s
2
0
1
R
R
R
- radii of curvature in directions
s
and
respectively.
Mixed components of the second metric tensor are calculated as follows:
;
1
0
s
s
s
R
b
;
1
0
R
b
.
0
s
s
b
b
(6)
Christophele's characters in the selected coordinate system are as follows
;
0
s
s
s
s
ss
s
ss
r
r
r
r
r
.
;
R
R
r
r
R
R
r
s
s
s
(7)
Now let us proceed to the finding of the basic ratios of the shell dynamics. The determining
equations in tensor form for the shell of arbitrary curvature are given in [6]. Let us write the
equations of motion, Hooke's law and Cauchy's relations, keeping only their linear part.
Motion equation:
;
p
Q
b
N
h
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;
q
N
b
Q
w
h
(8)
;
q
N
b
Q
w
h
Here, the second derivative in time
t
is marked by two dots above the letters. For recording
the expressions in the right part (8), the rule of summing up by two repetitive indexes is used.
Symbols
,
denote variables
,
,
,
,
w
s
− the counter-variant components of the
displacement vector and the normal angle of rotation, respectively.
M
Q
N
,
,
− counter-variant
components of force and moment tensors;
m
q
p
,
,
− intensities of forces (tangential and
transverse), as well as moments distributed on the shell surface.
− covariant derivative,
−
density of material,
h
− thickness of the shell,
12
3
h
I
Hook's Law:
;
h
N
;
I
M
(9)
;
3
3
2
w
hC
k
Q
;
3333
33
33
C
C
C
C
The coefficients
C
are Hook's elastic coefficient matrix:
.
2
,
1
;
3
,
2
,
1
,
,
,
s
l
k
j
i
C
kl
ijkl
ij
6
5
2
k
− the shear coefficient in the S.P.Timoshenko theory. The tensors included in the
right parts (9) are determined from Cauchy ratios:
;
2
1
l
l
;
2
1
;
2
1
w
r
;
w
b
l
(10)
;
.
b
w
w
The representation of relations (8) − (10) in terms of physical components of
displacements, forces and moments is illustrated by the example of the first equation in (8) at
s
. By definition of the covariant derivative from the invariant tensor is obtained as follows
l
l
l
l
N
N
x
N
N
or at
s
RN
R
N
R
R
N
s
N
N
N
N
N
N
s
N
N
ss
s
ss
s
s
s
s
s
s
s
s
s
ss
ss
s
ss
s
3
2
Here we used formulas (7). Further from (6) we have
.
1
0
s
s
s
s
s
s
s
Q
R
Q
b
Q
b
Q
b
Then the first equation in (8) takes the form
.
1
0
s
s
s
ss
s
ss
s
p
Q
R
N
R
R
N
R
R
N
s
N
h
(11a)
Similar calculations lead to the following system of equations:
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;
1
3
0
p
Q
R
N
R
R
N
s
N
h
s
s
;
1
0
2
0
q
N
R
R
N
R
Q
R
R
Q
s
Q
w
h
s
ss
s
s
s
(11b)
;
s
s
ss
s
ss
s
m
Q
M
R
R
M
R
R
M
s
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