Volume 03 Issue 11-2023
226
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
11
Pages:
226-231
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
A
BSTRACT
In this work, in a general three-dimensional formulation, the problem of vibration of two-layer piecewise
homogeneous viscoelastic plates of constant thickness is formulated. General equations of vibration are
derived, expressions are given for displacements and stresses at the internal points of the plate through
functions that describe the displacements and deformations of the points of the contact plane.
K
EYWORDS
Three-dimensional, two-layer, plate, equation, vibrations, displacement, deformation.
I
NTRODUCTION
Plates are one of the main elements of many
technical and building structures.
In many cases, the plates are non-uniform in
thickness, in particular, piecewise homogeneous
(two-layer, etc.) [1-4].
At present, there is practically no theory of
vibration of piecewise homogeneous plates, and
therefore the development of the theory and
methods for calculating such plates is an urgent
problem in structural mechanics [5-9].
Let us consider a piecewise-homogeneous
viscoelastic plate of constant thickness, as a
piecewise-homogeneous layer of the same
geometry, with the thickness of the upper
component being equal to
0
h
, and the thickness
of the lower component
1
h
. The plate occupies the
Journal
Website:
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Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Research Article
EXACT THREE-DIMENSIONAL BOUNDARY VALUE PROBLEM
FOR PIECEWISE HOMOGENEOUS PLATES
Submission Date:
November 13, 2023,
Accepted Date:
November 18, 2023,
Published Date:
November 23, 2023
Crossref doi:
https://doi.org/10.37547/ijasr-03-11-37
M.L. Djalilov
Fergana Branch Of The Tashkent University Of Information Technologies Named After Muhammad Al-
Khorazmiy, Fergana, Republic Of Uzbekistan
Volume 03 Issue 11-2023
227
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
11
Pages:
226-231
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
area
-
)
y
,
x
(
;
0
1
h
z
h
−
, while the
homogeneity interface coincides with the plane
.
z
0
=
T
HE MAIN PART
The movement of the material of the constituent layers of the plate in Cartesian coordinates
)
,
,
(
z
y
x
is
described by the equations of motion in stresses [10-16].
;
t
u
z
y
x
)
k
(
k
)
k
(
xz
)
k
(
xy
)
k
(
xx
2
2
=
+
+
;
t
z
y
x
)
k
(
k
)
k
(
yz
)
k
(
yy
)
k
(
xy
2
2
=
+
+
(1)
;
t
w
z
y
x
)
k
(
k
)
k
(
zz
)
k
(
zy
)
k
(
xz
2
2
=
+
+
Where
)
(
к
ij
are the components of the stress tensor;
)
k
(
u
,
)
к
(
,
)
к
(
w
are the components of the
displacement vector.
In this case, stresses, displacements, and density in each of the layers will be denoted by the corresponding
index “0” or “1”, i.e.,
k
takes the values “0” and “1”.
The dependences of stress
)
k
(
ij
on deformations
)
k
(
ij
at points of the plate are described by linear
operator equations, that is, we will assume that they are specified in the form of Boltzmann relations:
);
(
M
)
(
L
)
k
(
ij
k
)
k
(
k
)
k
(
ij
2
+
=
);
(
M
)
k
(
ij
k
)
k
(
ij
=
).
z
,
y
,
x
j
,
i
;
j
i
(
=
(2)
Where are the viscoelastic operators
k
L
and
−
k
M
linear integral operators of the form
−
−
=
d
)
(
)
t
(
f
)
t
(
)
(
L
)
k
(
t
k
k
1
0
;
−
−
=
d
)
(
)
t
(
f
)
t
(
)
(
M
)
k
(
2
t
0
k
k
; (3)
where
)
t
(
f
)
k
(
j
are the kernels of viscous operators,
k
k
,
−
elastic constants or Lamé coefficients.
Volume 03 Issue 11-2023
228
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
11
Pages:
226-231
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
Let us introduce the potentials
)
k
(
and
)
k
(
longitudinal and transverse waves according to the
formula:
)
k
(
)
k
(
)
k
(
rot
gradФ
U
+
=
).
w
,
v
,
(u
(k)
(k)
(k)
)
k
(
)
k
(
U
U
=
(4)
Where
U
k
)
(
- vector of movement of plate points.
Then, instead of equations (1), we obtain integro-differential equations:
;
t
)
(
M
;
t
Ф
)
Ф
(
N
)
k
(
k
)
k
(
k
)
k
(
k
)
k
(
k
2
2
2
2
=
=
(5)
where the operator
k
N
is equal
k
k
k
M
L
N
2
+
=
−
three-dimensional Laplace operator
2
2
2
2
2
2
z
y
x
+
+
=
.
By virtue of the Helmholtz theorem /4/, in the absence of internal sources, the vector potential
)
k
(
of
transverse waves must satisfy the condition
;
di
)
k
(
0
=
).
,
,
(
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
3
2
1
=
(6)
Condition (6) for the vector components
)
k
(
takes the form
,
z
y
x
)
k
(
)
k
(
)
k
(
0
3
2
1
=
+
+
(7)
That is, we obtain a closing equation for determining the vector potential
)
k
(
.
Equations (5) and condition (6) are sufficient to find general solutions for the scalar and vector potentials
(k)
and
)
k
(
.
Vibrations of a viscoelastic piecewise homogeneous plate are caused by external forces applied to the
surfaces of the plate. Therefore, the boundary conditions take the form:
at
0
h
z
=
(on the upper surface of the plate)
);
t
,
y
,
x
(
f
);
t
,
y
,
x
(
f
)
(
xz
)
(
xz
)
(
zz
)
(
zz
0
0
0
0
=
=
);
t
,
y
,
x
(
f
)
(
yz
)
(
yz
0
0
=
(8)
at
0
=
z
(contact plane)
;
)
(
zz
)
(
zz
1
0
=
;
)
(
xz
)
(
xz
1
0
=
;
)
(
yz
)
(
yz
1
0
=
Volume 03 Issue 11-2023
229
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
11
Pages:
226-231
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
;
u
u
)
(
)
(
1
0
=
;
)
(
)
(
1
0
=
)
(
)
(
w
w
1
0
=
; (9)
at
1
h
z
−
=
(on the bottom surface of the record)
);
t
,
y
,
x
(
f
)
(
z
)
(
zz
1
1
=
);
t
,
y
,
x
(
f
)
(
xz
)
(
xz
1
1
=
)
t
,
y
,
x
(
f
)
(
yz
)
(
yz
1
1
=
. (10)
The initial conditions of the problem are zero, that is
.
t
Ф
t
Ф
)
k
(
)
k
(
)
k
(
)
k
(
0
=
=
=
=
(11)
Displacements
)
k
(
u
,
)
k
(
strains
)
k
(
ij
and stresses
)
k
(
ij
in Cartesian coordinates through the potentials
(k)
of
)
k
(
both longitudinal and transverse waves are determined by the following formulas /5/ .
For travel:
;
z
y
x
Ф
u
)
k
(
)
k
(
)
k
(
−
+
=
2
3
k
;
x
z
y
Ф
)
k
(
)
k
(
)
k
(
−
+
=
3
1
k
;
y
x
z
Ф
w
)
k
(
)
k
(
)
k
(
−
+
=
1
2
k
(12)
For deformations:
;
z
y
y
x
x
)
k
(
)
k
(
)
k
(
)
k
(
xx
−
+
=
2
2
3
2
2
2
;
x
y
z
y
y
)
k
(
)
k
(
)
k
(
)
k
(
yy
−
+
=
3
2
1
2
2
2
(13)
;
z
y
z
x
z
)
k
(
)
k
(
)
k
(
)
k
(
zz
−
+
=
1
2
2
2
2
2
;
x
y
z
y
z
x
y
x
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
xy
2
3
2
2
3
2
2
2
1
2
2
2
−
+
−
+
=
(13)
;
y
z
z
x
y
x
z
y
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
yz
2
1
2
2
1
2
2
2
2
2
2
2
−
+
−
+
=
;
y
x
x
z
z
y
z
x
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
xz
−
+
−
+
=
1
2
2
2
2
2
2
2
3
2
2
2
For voltages:
Volume 03 Issue 11-2023
230
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
11
Pages:
226-231
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
);
z
x
y
x
x
(
M
)
(
L
)
k
(
)
k
(
)
k
(
k
)
k
(
k
)
k
(
xx
−
+
+
=
2
2
3
2
2
2
2
);
(
)
(
)
(
)
(
)
(
)
(
)
(
y
x
z
y
y
M
2
L
k
3
2
k
1
2
2
k
2
k
k
k
k
yy
−
+
+
=
(14)
);
z
y
z
x
z
(
M
)
(
L
)
k
(
)
k
(
)
k
(
k
)
k
(
k
)
k
(
zz
−
+
+
=
1
2
2
2
2
2
2
);
z
y
z
x
y
x
y
x
(
M
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
k
)
k
(
xy
−
+
+
−
=
2
2
1
2
2
3
2
2
3
2
2
2
);
z
x
y
x
z
y
z
y
(
M
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
k
)
k
(
yz
−
+
+
−
=
3
2
2
2
2
1
2
2
1
2
2
2
);
z
y
y
x
z
x
y
x
(
M
)
k
(
)
k
(
)
k
(
)
k
(
)
k
(
k
)
k
(
xz
+
−
−
+
=
3
2
1
2
2
2
2
2
2
2
2
2
Thus, the exact three-dimensional problem of
vibration
of
a
viscoelastic
piecewise
homogeneous plate of constant thickness is
reduced to solving the vibration equations (5) in
potentials
)
k
(
and
)
k
(
under boundary
conditions (8), (9), (10) and zero initial
conditions (11).
C
ONCLUSIONS
1. The study of vibrations of piecewise
homogeneous plates in an exact three-
dimensional
formulation
allows,
without
involving any hypotheses, to derive the general
and approximate equations of vibration of such
plates based on them.
2. For piecewise homogeneous plates there is
neither a purely transverse nor a purely
longitudinal vibration; as shown, this is an
equation of the sixth order in derivatives, which
for a homogeneous plate of constant thickness
becomes the product of two integro-differential
operators describing longitudinal and transverse
vibrations.
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Volume 03 Issue 11-2023
231
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
11
Pages:
226-231
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
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