Моделювання контактної взаємодії нагрітого жорсткого еліптичного штампа з трансверсально-ізотропним пружним півпростором
On the base of a rigorous mathematical model, the problem of the contact interaction of a heated flat punch of an elliptical section with a transversely isotropic elastic half-space is investigated. It is assumed that the half-space surface is the isotropy plane of a transversely isotropic material,...
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System research and information technologies| _version_ | 1867334409407954944 |
|---|---|
| author | Kirilyuk, Vitaly S. Levchuk, Olga I. Gavrilenko, Olena V. Viter, Mykhailo B. |
| author_facet | Kirilyuk, Vitaly S. Levchuk, Olga I. Gavrilenko, Olena V. Viter, Mykhailo B. |
| author_institution_txt_mv | [
{
"author": "Vitaly S. Kirilyuk",
"institution": "The Department of Mechanics of Stochastically Inhomogeneous Media of S.P. Timoshenko Institute of Mechanics of National Academy of Sciences of Ukraine, Kyiv"
},
{
"author": "Olga I. Levchuk",
"institution": "The Department of Mechanics of Stochastically Inhomogeneous Media of S.P. Timoshenko Institute of Mechanics of National Academy of Sciences of Ukraine, Kyiv"
},
{
"author": "Olena V. Gavrilenko",
"institution": "The Department of Computer-Aided Management and Data Processing Systems of the Faculty of Informatics and Computer Science of the National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Mykhailo B. Viter",
"institution": "The Department of Information Systems and Technologies of the Faculty of Transport and Information Technologies of the National Transport University, Kyiv"
}
] |
| author_sort | Kirilyuk, Vitaly S. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2021-01-19T12:18:25Z |
| description | On the base of a rigorous mathematical model, the problem of the contact interaction of a heated flat punch of an elliptical section with a transversely isotropic elastic half-space is investigated. It is assumed that the half-space surface is the isotropy plane of a transversely isotropic material, and also that there is a smooth (without friction) contact. Expressions of contact stresses and displacements of a heated flat elliptical punch are found explicitly. In the form of a simple inequality, a condition for separating the elastic material from the surface of a flat elliptical punch is obtained. Numerical calculations are carried out. Contact interaction of a heated flat punch is studied taking into account the separation of material from the punch. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2020.3.10 |
| first_indexed | 2025-07-17T10:27:01Z |
| format | Article |
| fulltext |
V.S. Kirilyuk, O.I.Levchuk, V.V. Gavrilenko, M.B. Viter, 2020
138 ISSN 1681–6048 System Research & Information Technologies, 2020, № 3
UDC 539.3
DOI: 10.20535/SRIT.2308-8893.2020.3.10
MODELING OF CONTACT INTERACTION OF A HEATED
PLANE RIGID ELLIPTICAL PUNCH WITH A TRANSVERSALLY
ISOTROPIC ELASTIC HALF-SPACE
V.S. KIRILYUK, O.I. LEVCHUK, V.V. GAVRILENKO, M.B. VITER
Abstact. On the base of a rigorous mathematical model, the problem of the contact
interaction of a heated flat punch of an elliptical section with a transversely isotropic
elastic half-space is investigated. It is assumed that the half-space surface is the
isotropy plane of a transversely isotropic material, and also that there is a smooth
(without friction) contact. Expressions of contact stresses and displacements of a
heated flat elliptical punch are found explicitly. In the form of a simple inequality, a
condition for separating the elastic material from the surface of a flat elliptical punch
is obtained. Numerical calculations are carried out. Contact interaction of a heated
flat punch is studied taking into account the separation of material from the punch.
Keywords: mathematical model, contact interaction, elastic half-space, transver-
sally-isotropic material, plane elliptical punch, heating, stress distribution, domain of
material separation.
INTRODUCTION
Currently, methods for solving spatial problems of contact interaction for iso-
tropic elastic bodies are quite well developed. Among the papers on this topic,
classical monographs [1–5], as well as articles [6–8], can be noted. However, the
solution of spatial contact problems for transversely isotropic bodies is associated
with significant mathematical difficulties, since the initial system of equations for
determining the stress state has a more complex structure. Contact problems of
thermoelasticity for a transversely isotropic half-space were studied in [8–10]
and others. An approach was used in [9, 10] that allows one to investigate prob-
lems only for a circular contact region. In [8], the contact problem of thermoelas-
ticity is studied with a special distribution of the temperature field on the surface
of the punch, which is proportional to the contact pressure under the paraboloidal
punch. In the papers [11–15] and [16–21], spatial problems for transversely-
isotropic elastic and electroelastic bodies respectively were considered. At the
same time, analytical solutions of spatial contact problems for transversely
isotropic elastic bodies were not obtained when hard punch heated in an arbitrary
manner.
In this paper, the problem of thermoelasticity on the indentation of a heated
plane hard punch of elliptical cross-section into a transversely isotropic elastic
half-space is considered. Expressions of contact stresses and displacements of a
heated flat elliptical punch are found explicitly. In the form of inequality, the rela-
tionship between the values of the indentation force, the heating temperature, and
the thermoelastic properties of a transversely isotropic material is obtained, which
makes it possible to predict the appearance of a material separation zone under a
Modeling of contact interaction of a heated plane rigid elliptical punch with a transversally
Системні дослідження та інформаційні технології, 2020, № 3 139
flat elliptical punch (for given force and temperature influences). The influence of
material properties, heating temperature, and indentation force on the distribution
of contact pressure is investigated. It is shown that the appearance of separation
(peeling) of the material significantly affects the type of the distribution of contact
stresses under the punch.
Formulation of the problem. Let us consider a transversely isotropic half-
space that occupies a region 0z and into which a heated flat hard punch of el-
liptical section is pressed without friction. We assume that the axis z0 coincides
with the axis of symmetry of the transversely isotropic material. The boundary
conditions on the surface of the half-space have the following form:
0 yzxz , 0z ; 0zz , ),( yx ;
;0),(;),(),,()0,,( |00 yxTyxyxTyxT
;\),(,0)0,,( 2 RyxyxT
)0,,( yxuz , ),( yx , (1)
where 1//: 2222 byax ; 0)0,,( yxT — punch heating temperature; —
unknown displacement value. The indentation force applied in the center of the
punch is related to the contact pressure by the ratio
dxdyyxpP ),( , where
),( yxp is the unknown contact pressure.
Basic relations. The equations of stationary thermoelasticity for an elastic
transversely isotropic medium in the absence of body forces and heat sources in
the body according to [8] can be written as
2
2
442
2
12112
2
11 )(
2
1
z
u
c
y
u
cc
x
u
c xxx
x
T
z
u
cc
y
u
cc
x
zy
)()(
2
1
44131211 ;
2
2
442
2
112
2
1211 )(
2
1
z
u
c
y
u
c
x
u
cc yyy
y
T
z
u
cc
x
u
cc
y
zx
)()(
2
1
44131211 ;
z
T
y
u
x
u
z
cc
z
u
c
x
u
y
u
c yxzzz
144132
2
332
2
2
2
44 )( ; (2)
0/// 22
4
2222 zTnyTxT .
In the above expressions, ijc are elastic constants; 41,, n — constants
depending on the thermophysical properties (thermal conductivity and thermal
linear expansion coefficients) of the material. The solution of the system of equa-
V.S. Kirilyuk, O.I.Levchuk, V.V. Gavrilenko, M.B. Viter
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 140
tions (2) can be represented by means of four potential functions i ( i = 1, 2, 3, 4)
in according to [8] in this way:
,///
;////
;////
442211
4321
4321
zmzmzmu
yxyyu
xyxxu
z
y
x
(3)
where 321 ,, are functions satisfying the equations
,0)///( 222222 jj znyx
also 211211443 ,);/(2 nncccn are the roots of the quadratic equation
0])([ 4433
2
44131133
2
44
2
4411 ccncccccncc ; (4)
)2,1(
)(
4433
4413
4413
4411
j
cnc
ccn
cc
cnc
m
j
jj
j .
The function 4 simultaneously satisfies two equations
.;0 32
4
2
42
2
42
2
2
2
Tm
zz
n
yx
We use the notation )4,3,2,1(2/1 jznz jj . Functions ,),,( 11 zyx
),,,( 22 zyx ),,( 33 zyx , ),,( 44 zyx will be harmonic functions in the corre-
sponding coordinate system. The constants 43,mm included in relations (3) de-
pend on the elastic and thermophysical properties of a transversely isotropic me-
dium and are written as follows:
.
)()(
)()(
;
)( 4413144433
44134114441
4
4114441344
3 cccnc
ccncnc
m
ncmccc
m
Solution method. We write the temperature field in the form of the har-
monic potential of the double layer
2
4
22
0
4
4
)()(
),(
2
1
),,(
zyx
ddT
z
zyxT .
It follows from the properties of the derivative of the potential of a simple
layer [3] that
.),(,0
;),(),,(
|),,( 0
04 4 yx
yxyxT
zyxT zz
Harmonic function is a solution to the Dirichlet problem for the stationary
heat conduction equation for a half-space (for a given distribution of the tempera-
ture field inside a flat region and zero temperature outside this region on the sur-
face of the half-space [3]). Note that contact problems of thermoelasticity with a
known temperature distribution in the contact area and the absence of a tempera-
ture field outside it (in the contact plane) were also considered in [8, 24].
Modeling of contact interaction of a heated plane rigid elliptical punch with a transversally
Системні дослідження та інформаційні технології, 2020, № 3 141
Next, we present the solution of problem in the form of a superposition of
states, for the first of which we take the function 4 in the form of one of the
Boussinesq potentials [2]
),,(),,( 44
)1(
4 zyxFzyx
ddzzyxT 4
2
4
22
0 )()(ln),(
2
1
.
For the first state, we also set
)2,1(),,(),,()1( izyxFzyx iiii ; 0)1(
3 ,
where 21, are the unknown constants.
Constants 1a , 2a , we define by means of this way:
)(
)(
12
42
2/1
4
2/1
1
1 mm
mm
n
n
a
;
)(
)(
12
41
2/1
4
2/1
2
2 mm
mm
n
n
a
.
As a result, for the first state we get
0)1()1( yzxz , 0z ;
0| 0
)1( zzu , ),( yx ;
.),(,0
;),(),(
| 0
Trans
0
)1(
yx
yxyxT
zzz
We find the value Trans in the form
)( 13443331
Trans cnmcm
;)1(
)(
)(
)1(
)(
)( 2/1
11
21
422/1
22
21
412/1
4344
nm
mm
mm
nm
mm
mm
nmc (5)
133131113121114 2;)(;/ ccccckkn ,
where kk /1 is the ratio of the coefficient of thermal conductivity in the direction
0z to the coefficient of thermal conductivity in the direction 0x (or 0y); 1, are
the coefficients of linear thermal expansion of the material in the direction of 0x
(or 0y) and 0z. In the transition from a transversely isotropic material to the iso-
tropic material, we obtain )1/()1(Trans ( is the Poisson's ratio,
is the shear modulus), which fully corresponds to the result [5] for an isotropic
material. Note that expression (5) for Trans it was also used in the papers
[22, 23] to find the thermo-stressed state of a transversely isotropic material with
an elliptical crack.
For the second state of superposition, we choose the functions j
)4,3,2,1( j as follows:
V.S. Kirilyuk, O.I.Levchuk, V.V. Gavrilenko, M.B. Viter
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 142
ddzp
nncm
n
zyx )ln(),(
)()1(2
1
),,( 112/1
2
2/1
1441
2/1
1
1
)2(
1 ;
ddzp
nncm
n
zyx )ln(),(
)()1(2
1
),,( 222/1
2
2/1
1442
2/1
2
2
)2(
2 ,
where
222 )()( ii zyx )2,1( j .
In this case we obtain
222
)2(
)()(
),(
2
1
),,(
i
ii
i zyx
ddyp
zyx
z
)2,1( j .
Take also 0)2(
3
)2(
4 .
As a result of superposition of states, we obtain
0)2()1()2()1( yzyzxzxz for 0z ;
22
Trans
0
)2()1(
)()(
),(
2
1
|)(
yx
ddp
Auu zzz ;
.),(,0
;),(),,(),(
|)( 0
Trans
0
)2()1(
yx
yxyxTyxp
zzzzz
The value TransA is determined as
)1()1()(
1
21
12
2/1
2
2/1
144
Trans
mm
mm
nnc
A
)()(
)()(
1321113111
4413
2/1
2
2/1
1
44
11
cnccnc
ccnn
c
c
. (6)
From the obtained expression (6) for a transversely isotropic material, one
can easily obtain the case of an isotropic material. Let's put
121 nn ; 211c ; 13c ; 44c .
Then from formula (6) it follows
v
A
1
)(2
2Trans .
Thus, for an isotropic material we obtain a coincidence of the results with
the known data [5].
Note that expressions TransA (6) can be converted to a more convenient
form. Using Vieta's theorem for the roots of the quadratic equation, from (6) we
obtain
3311441344
2
133311
44
2
133311
11Trans 22
)(
cccccccc
cccc
c
A .
Modeling of contact interaction of a heated plane rigid elliptical punch with a transversally
Системні дослідження та інформаційні технології, 2020, № 3 143
The obtained expression allows to find the desired value by directly substi-
tuting the elastic constants of the material into it without first determining the
roots of the quadratic equation (4), as in the case of formulas (6).
Correspondence between solutions of contact problems for isotropic and
transversely isotropic elastic half-spaces (in contact with heated plane rigid
punch of elliptical section). According to the results of [5], the solution of the
contact problem of thermoelasticity for an isotropic elastic half-space with
boundary conditions (1) can lead to a search for an unknown potential density of a
simple layer. It remains to satisfy the boundary condition
22 )()(
),(
2
)1(
)0,,(
yx
ddpv
yxuz . (7)
Stress distribution has such form under the plane punch
),(
)1(
)1(
),(| 00 yxT
v
v
yxpzzz
, ),( yx . (8)
As a result of a superposition of states for a transversely isotropic half-space,
we obtain
22
Trans
)()(
),(
2
1
)0,,(
yx
ddp
Ayxuz . (9)
The normal stresses under the plane punch in this case have the form
),(),(| 0
Trans
0 yxTyxpzzz , ),( yx . (10)
All other boundary conditions (1) are satisfied. Comparing expressions (7),
(8) and (9), (10), we conclude that such contact characteristics as contact pres-
sure and displacement under the plane punch for a transversely isotropic half-
space can be calculated from the corresponding expressions for an isotropic
half-space by replacing the values /)1( v with TransA and
)1/()1( vv by Trans .
Solutions of new contact problems. When pressing a flat elliptical punch
(in the absence of the rotations around the axes 0x and 0y) according to the found
correspondence of expressions (7), (8) and (9), (10) and results [4, 5] we obtain
the values of contact pressure and displacement under the plane punch:
),(1
2
),( 0
Trans
2/1
2
2
2
2
1 yxT
b
y
a
x
ab
QP
yxp
;
)(
2
Trans1 eKA
a
QP
, (11)
where a is the semimajor axis of the ellipse, e is its eccentricity;
Э
dxdyyxTQP ),(0
Trans
1 . (12)
When inequality (12) is fulfilled, the contact stress under the flat punch are
compressive and have a root singularity when approaching the punch boundary,
which is determined by the first term in formulas (11).
V.S. Kirilyuk, O.I.Levchuk, V.V. Gavrilenko, M.B. Viter
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 144
Consider the distribution of the temperature field under the plane punch in
the form
q
q byaxTyxT )//1(),( 2222
0 , 0q .
Then, for a heated plane hard elliptical punch, the values of stresses and dis-
placements under the punch we obtain in the form
;)//1()//1(
2
),( 2222Trans2/122221 q
qzz byaxTbyax
ab
QP
yx
)(
2
Piezo
1
1 eKB
a
QP
;
dydxbyaxTQ q
q )//1( 2222Trans
1 ; 1QP . (13)
Note that formulas (13) have the following physical meaning. When the ine-
quality 1QP is fulfilled, the plane punch is pressed to the material over the en-
tire contact area and under it there is no area of separation (delamination) of the
material. After integration, we obtain the inequality
abT
q
QP q
Trans
1 1
1
. (14)
If the opposite inequality holds
abT
q
P q
Trans
1
1
tensile stresses arise when apprsoaching the punch edge (due to the first term in
the stress components in formulas (13)), i.e. a material separation zone appears.
In [7], for the problem of the contact of a heated plane circular punch with
an elastic isotropic half-space, it was proposed to search for a new contact zone,
which is smaller than the size of the punch itself, from the problem for a non-
planar punch, directing R . The two-dimensional contact problem of ther-
moelasticity was considered in a similar way in [24]. Using this approach, and
considering for this the problem of a heated paraboloidal punch of elliptical cross
section, which is pressed into an elastic transversely isotropic half-space without
friction, we obtain
3/1
2
3/1Trans
1
1
3 )()(
2
3
])[(
e
eEeK
AQP
R
a
,
where e is the eccentricity of the elliptical base of the punch. Directing 1R , we get
**
Trans
1 1
1
baT
q
QP q
,
where **,ba are the semi-axes of the new contact area under the punch. They are smaller
than the corresponding semi-axes of the plane punch. However, contact zone remains el-
liptical and the relation remains abab // ** Using the expression (14) we find further
2Trans
*
1
)1(
eT
qP
a
q
; 2
** 1 eab .
Modeling of contact interaction of a heated plane rigid elliptical punch with a transversally
Системні дослідження та інформаційні технології, 2020, № 3 145
Contact stress under a heated plane punch in the case of separation of mate-
rial near the edge of the punch takes the form
q
qzz byaxTyx )//1(),( 2
*
22
*
2Trans , (15)
since the singular term disappears in formulas (13) at 1QP . Therefore, an in-
crease in punch heating when a certain threshold value is exceeded, which de-
pends on the strength P and thermoelastic properties of the transversely isotropic
material, leads to the appearance of separation zone of the material under the
plane punch.
Note that for a plane circular punch ( 0e ), the radius of the contact area
when the material is separated from the punch according to formulas (15) takes
the form
qT
qP
a
Trans*
)1(
, ** ab .
Analysis of the results of numerical investigations. Consider the case of
the distribution of the temperature field under a plane circular punch of radius a
in the form 4/12222
0 )//1( ayaxT , where 00 T . We investigate next
three cases of punch heating: 1)
2Trans0
2,1
a
P
T
; 2)
2Trans0
3,1
a
P
T
and
3)
2Trans0
5,1
a
P
T
. First, we verify the fulfillment of inequality (14) to find out
whether the material is peeling off. As a result, we obtain that for the first case
inequality (14) is satisfied, i.e. material separation under the punch does not
occur. At the same time, with increasing heating (cases 2 and 3), such a separation
of the material takes place.
After simple calculations for the first case of heating, the pressure expression
under the punch takes the form
4/1222/122
Trans
0
)/1()/1(
60
1
arar
T
zz
.
For the second and third cases, the contact pressure has the same expression
4/12
*
2
Trans
0
)/1( ar
T
zz
.
However, the radii of the new contact area for these cases are different. For
the second case of heating 26/5* aa , at the same time for the third case we
find that 6/5* aa .
Fig. 1 shows the change in contact pressure under the punch, while the
pressure curve for case 1 (without separating the material under the punch) is
shown by line 1, and for the second and third cases – by means of the lines with
corresponding numbers. It is seen that with increasing heating of the flat punch,
the contact area with the half-space decreases.
Consider a flat elliptical punch with the distribution of the temperature
field under the punch in the form 4/12222
0 )//1( byaxT of 00 T . Put
])/[( *Trans
0 abPT . We study cases of punch heating, assuming that it *
V.S. Kirilyuk, O.I.Levchuk, V.V. Gavrilenko, M.B. Viter
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 146
takes the following values: 1) * 1000; 2) * 4; 3) * 2; 4) * 1. Since the
value 0T is inversely proportional to the value * , in the latter case we get the
highest punch heating. For the selected parameter values * , the material does not
separate under the punch, and the stress expression under the punch takes the form:
4/122222/12222*
*
)//1()//1(
5
4
2
1
)]/([
bxaxbxax
abP
zz
.
In fig. 2 shows the distribution of the contact pressure under a flat punch
with an elliptical section at different values of the heating of the punch.
Lines 1–4 in Fig. 2 correspond to punch heating options noted above. It can
be seen an increase of the temperature heating of a plane punch lead to the
stresses increase in the center of the punch and decrease when approaching its
boundary.
Thus, in the paper expressions of contact stresses and displacements of
a heated flat elliptical punch are found explicitly. By means of the inequality, the
condition for the occurrence of material separation under a flat heated elliptical
punch is obtained, which is pressed without friction into the transversely isotropic
elastic half-space. This inequality includes the values of the indentation force,
temperature heating, and thermoelastic properties of a transversely isotropic
Fig. 2. Stress distribution under a plane elliptical punch
1
2
3
x/a
)/( abP
zz
4
0 0,25 0,5 0,75
1,5
1,3
0,5
Fig. 1. The stress distribution under a heated flat punch, taking into account the separation
of the material
1
2
3
r/a
0
TransT
zz
0 0,2 0,4 0,6 0,8
0,8
0,4
Modeling of contact interaction of a heated plane rigid elliptical punch with a transversally
Системні дослідження та інформаційні технології, 2020, № 3 147
material. It is shown that with increasing heating, the region of complete contact
(with separation of the material) decreases. The famous results for an elastic
isotropic material follow from the obtained data as a special case. The influence
of heating on the distribution of contact stresses, as well as the appearance of
a region of separation (delamination) of a transversely isotropic material under a
punch, is investigated.
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V.S. Kirilyuk, O.I.Levchuk, V.V. Gavrilenko, M.B. Viter
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 148
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Received 11.09.2020
INFORMATION ON THE ARTICLE
Vitaly S. Kirilyuk, ORCID: 0000-0002-8513-0378, S.P. Timoshenko Insitute of me-
chanics of NAS of Ukraine, e-mail: kirilyuk_v@ukr.net.
Olga I. Levchuk, ORCID: 0000-0002-6514-6225, S.P. Timoshenko Institute of me-
chanics of NAS of Ukraine, e-mail: 2013levchuk@gmail.com.
Olena V. Gavrilenko, ORCID: 0000-0003-0413-6274, National Technical University
of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail:
iem.gavrilenko@meta.ua.
Mykhailo B. Viter, ORCID: 0000-0003-4109-005X, National Transport University of
Ukraine, e-mail: mbviter@gmail.com
МОДЕЛЮВАННЯ КОНТАКТНОЇ ВЗАЄМОДІЇ НАГРІТОГО ЖОРСТКОГО
ЕЛІПТИЧНОГО ШТАМПА З ТРАНСВЕРСАЛЬНО-ІЗОТРОПНИМ ПРУЖНИМ
ПІВПРОСТОРОМ / В.С. Кирилюк, О.І. Левчук, В.В. Гавриленко, М.Б. Вітер
Анотація. На основі строгої математичної моделі досліджено задачу контакт-
ної взаємодії нагрітого плоского штампа еліптичного перерізу з трансверсаль-
но-ізотропним пружним півпростором. Припускається, що поверхня півпрос-
тору є площиною ізотропії трансверсально-ізотропного матеріалу, а також має
гладкий (без тертя) контакт. У явному вигляді знайдено вирази контактних на-
пружень і переміщення нагрітого плоского еліптичного штампа. У вигляді
простої нерівності отримано умову відділення пружного матеріалу від
поверхні плоского еліптичного штампа. Виконано числові розрахунки. Вивче-
но контактну взаємодію нагрітого плоского штампа з урахуванням відділення
матеріалу від штампа.
Ключові слова: математична модель, контактна взаємодія, пружний
півпростір, трансверсально-ізотропний матеріал, плоский еліптичний штамп,
нагрівання, розподіл напружень, ділянка відділення матеріалу.
МОДЕЛИРОВАНИЕ КОНТАКТНОГО ВЗАИМОДЕЙСТВИЯ НАГРЕТОГО ПЛОСКОГО
ЖЕСТКОГО ЭЛЛИПТИЧЕСКОГО ШТАМПА С ТРАНСВЕРСАЛЬНО-ИЗОТРОПНЫМ
УПРУГИМ ПОЛУПРОСТРАНСТВОМ / В.С. Кирилюк, О.И. Левчук, В.В. Гавриленко,
М.Б. Витер
Аннотация. На основе строгой математической модели исследована задача
контактного взаимодействия нагретого плоского штампа эллиптического
сечения с трансверсально-изотропным упругим полупространством. Предпо-
лагается, что поверхность полупространства является плоскостью изотропии
трансверсально-изотропного материала, а также имеет гладкий (без трения)
контакт. В явном виде найдены выражения контактных напряжений и пере-
мещения нагретого плоского эллиптического штампа. В виде простого нера-
венства получено условие отделения упругого материала от поверхности
плоского эллиптического штампа. Выполнены числовые расчеты. Изучено
контактное взаимодействие нагретого плоского штампа с учетом отделения
материала от штампа.
Ключевые слова: математическая модель, контактное взаимодействие, упру-
гое полупространство, трансверсально-изотропный материал, плоский эл-
липтический штамп, нагрев, распределение напряжений, область отделения
материала.
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| id | journaliasakpiua-article-221386 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:27:01Z |
| publishDate | 2020 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/e3/33fe8ae51a2f47cb67d1358dd79609e3.pdf |
| spelling | journaliasakpiua-article-2213862021-01-19T12:18:25Z Modeling of contact interaction of a heated plane rigid elliptical punch with a transversally isotropic elastic half-space Моделирование контактного взаимодействия нагретого плоского жесткого эллиптического штампа с трансверсально-изотропным упругим полупространством Моделювання контактної взаємодії нагрітого жорсткого еліптичного штампа з трансверсально-ізотропним пружним півпростором Kirilyuk, Vitaly S. Levchuk, Olga I. Gavrilenko, Olena V. Viter, Mykhailo B. mathematical model contact interaction elastic half-space transversally-isotropic material plane elliptical punch heating stress distribution domain of material separation математична модель контактна взаємодія пружний півпростір трансверсально-ізотропний матеріал плоский еліптичний штамп нагрівання розподіл напружень ділянка відділення матеріалу математическая модель контактное взаимодействие упругое полупространство трансверсально-изотропный материал плоский эллиптический штамп нагрев распределение напряжений область отделения материала On the base of a rigorous mathematical model, the problem of the contact interaction of a heated flat punch of an elliptical section with a transversely isotropic elastic half-space is investigated. It is assumed that the half-space surface is the isotropy plane of a transversely isotropic material, and also that there is a smooth (without friction) contact. Expressions of contact stresses and displacements of a heated flat elliptical punch are found explicitly. In the form of a simple inequality, a condition for separating the elastic material from the surface of a flat elliptical punch is obtained. Numerical calculations are carried out. Contact interaction of a heated flat punch is studied taking into account the separation of material from the punch. На основе строгой математической модели исследована задача контактного взаимодействия нагретого плоского штампа эллиптического сечения с трансверсально-изотропным упругим полупространством. Предполагается, что поверхность полупространства является плоскостью изотропии трансверсально-изотропного материала, а также имеет гладкий (без трения) контакт. В явном виде найдены выражения контактных напряжений и перемещения нагретого плоского эллиптического штампа. В виде простого неравенства получено условие отделения упругого материала от поверхности плоского эллиптического штампа. Выполнены числовые расчеты. Изучено контактное взаимодействие нагретого плоского штампа с учетом отделения материала от штампа. На основі строгої математичної моделі досліджено задачу контактної взаємодії нагрітого плоского штампа еліптичного перерізу з трансверсально-ізотропним пружним півпростором. Припускається, що поверхня півпростору є площиною ізотропії трансверсально-ізотропного матеріалу, а також має гладкий (без тертя) контакт. У явному вигляді знайдено вирази контактних напружень і переміщення нагрітого плоского еліптичного штампа. У вигляді простої нерівності отримано умову відділення пружного матеріалу від поверхні плоского еліптичного штампа. Виконано числові розрахунки. Вивчено контактну взаємодію нагрітого плоского штампа з урахуванням відділення матеріалу від штампа. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2020-12-07 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/221386 10.20535/SRIT.2308-8893.2020.3.10 System research and information technologies; No. 3 (2020); 138-149 Системные исследования и информационные технологии; № 3 (2020); 138-149 Системні дослідження та інформаційні технології; № 3 (2020); 138-149 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/221386/223562 Copyright (c) 2021 System research and information technologies |
| spellingShingle | математична модель контактна взаємодія пружний півпростір трансверсально-ізотропний матеріал плоский еліптичний штамп нагрівання розподіл напружень ділянка відділення матеріалу Kirilyuk, Vitaly S. Levchuk, Olga I. Gavrilenko, Olena V. Viter, Mykhailo B. Моделювання контактної взаємодії нагрітого жорсткого еліптичного штампа з трансверсально-ізотропним пружним півпростором |
| title | Моделювання контактної взаємодії нагрітого жорсткого еліптичного штампа з трансверсально-ізотропним пружним півпростором |
| title_alt | Modeling of contact interaction of a heated plane rigid elliptical punch with a transversally isotropic elastic half-space Моделирование контактного взаимодействия нагретого плоского жесткого эллиптического штампа с трансверсально-изотропным упругим полупространством |
| title_full | Моделювання контактної взаємодії нагрітого жорсткого еліптичного штампа з трансверсально-ізотропним пружним півпростором |
| title_fullStr | Моделювання контактної взаємодії нагрітого жорсткого еліптичного штампа з трансверсально-ізотропним пружним півпростором |
| title_full_unstemmed | Моделювання контактної взаємодії нагрітого жорсткого еліптичного штампа з трансверсально-ізотропним пружним півпростором |
| title_short | Моделювання контактної взаємодії нагрітого жорсткого еліптичного штампа з трансверсально-ізотропним пружним півпростором |
| title_sort | моделювання контактної взаємодії нагрітого жорсткого еліптичного штампа з трансверсально-ізотропним пружним півпростором |
| topic | математична модель контактна взаємодія пружний півпростір трансверсально-ізотропний матеріал плоский еліптичний штамп нагрівання розподіл напружень ділянка відділення матеріалу |
| topic_facet | mathematical model contact interaction elastic half-space transversally-isotropic material plane elliptical punch heating stress distribution domain of material separation математична модель контактна взаємодія пружний півпростір трансверсально-ізотропний матеріал плоский еліптичний штамп нагрівання розподіл напружень ділянка відділення матеріалу математическая модель контактное взаимодействие упругое полупространство трансверсально-изотропный материал плоский эллиптический штамп нагрев распределение напряжений область отделения материала |
| url | https://journal.iasa.kpi.ua/article/view/221386 |
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