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Research ArticleThree-Dimensional Unsteady State Temperature Distribution ofThin Rectangular Plate with Moving Point Heat Source
Yogita M Ahire1 and Kirtiwant P Ghadle2
1Department of Applied Science PVGrsquoS College of Engineering Nashik Maharashtra India2Department of Mathematics Dr Babasaheb Ambedkar Marathwada University Aurangabad 431004 India
Correspondence should be addressed to Kirtiwant P Ghadle drkpghadlegmailcom
Received 30 March 2016 Accepted 20 July 2016
Academic Editor Dnyaneshwar S Patil
Copyright copy 2016 Y M Ahire and K P GhadleThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
This paper deals with the study of thermal stresses in thin rectangular plate subjected to point heat source which changes its placealong 119909-axis Governing heat conduction equation has been solved by using integral transform technique Results are obtainedin the form of infinite series As a special case aluminum plate has been considered and results for thermal stresses have beencomputed numerically and graphically
1 Introduction
Material properties are dependent on change in temperatureThe properties like elasticity and stresses at various temper-atures have been studied These nonisothermal problems oftheory of elasticity have attracted the attention of many Thetemperature dependent properties are focused on variousfields like aerodynamics heating which produces intensethermal stresses reducing the strength of structure of highvelocity aircraft [1] Steady state thermal stresses with axissymmetric temperature distribution in a circular plate sub-jected to the upper surface with respect to zero temperatureon the lower surface and thermally insulated circular edgehave been determined by [2] On fixed and simply supportededges [3] has calculated thermal deflection of associated axissymmetrically heated circular plate Reference [4] has consid-ered quasi-static thermal stresses in a thin circular plate dueto transient temperature applied along the edge of a circle onthe upper face with respect to lower face at zero temperatureand a thermally insulated fixed circular edge Reference [5]studied an inverse unsteady state thermoelastic problem ofa thin rectangular plate Quasi-static thermoelastic problemof an infinitely long circular cylinder has been calculatedby [6] Temperature distribution thermal functions anddisplacement at any point of semi-infinite rectangular slabwith internal heat source using integral transform technique
are solved by [7] Using integral transform technique andGreenrsquos theorem [8] has determined temperature distribu-tion and thermal stresses by taking second kind boundarycondition in thin rectangular plate with moving line heatsource Reference [9] has determined thermal stresses on thinrectangular plate by integral transform with internal movingpoint heat source Reference [10] determines temperaturedistribution displacement and thermal stresses of a thincircular plate due to uniform internal energy generation usingHankel transform technique graphically
Integral transform technique is a powerful tool to solvevarious new general purpose numerical methods and can beapplied to any multidimensional problem to get an approxi-mate solution This is the easiest way to find parameters likevariation of temperature and so forth This method is betterthan other methods
Attempt ismade to determine effective solution and studyof thermal stresses in a thin rectangular plate with internallymoving heat point source
Present paper elaborates on determination of tempera-ture and thermal stresses in a thin rectangular plate definedas 0 le 119909 le 119886 0 le 119910 le 119887 and minusℎ le 119911 le ℎ where ℎ lt 119887 lt 119886 andℎ is thickness which is very small Using integral transformtechnique the governing heat conduction equation is solvedResults are obtained in the form of infinite series It has beencomputed numerically and graphically
Hindawi Publishing CorporationIndian Journal of Materials ScienceVolume 2016 Article ID 7563215 7 pageshttpdxdoiorg10115520167563215
2 Indian Journal of Materials Science
2 Formulation of the Problem
We consider three-dimensional thin rectangular plate understeady state temperature defined in region 119877 0 le 119909 le 1198860 le 119910 le 119887 and minusℎ le 119911 le ℎ where ℎ lt 119887 lt 119886 and ℎ isthickness which is very small The plate is subjected to themotion of moving point heat source at the point (1199091015840 0 0)Under these realistic prescribed conditions temperature andthermal stresses in a thin rectangular plate are required to bedetermined
The temperature distribution of the rectangular platedefined in [11] is given by
1205972119879
1205971199092+
1205972119879
1205971199102+
1205972119879
1205971199112+
119892
119896
=
1
120572
120597119879
120597119905
(1)
where 119896 is thermal conductivity and 120572 is thermal diffusivityof the material of the plate
Consider an instantaneous moving heat source at point(1199091015840 0 0) and release its heat spontaneously at time 1199051015840 Such
volumetric moving heat source in rectangular coordinates isgiven by
119892 (119909 119910 119911 119905) = 119892119894
119901120575 (119909minus119909
1015840) 120575 (119910) 120575 (119911) 120575 (119905minus119905
1015840) (2)
where 119892119894119901is instantaneous point heat source
Hence (1) becomes1205972119879
1205971199092+
1205972119879
1205971199102+
1205972119879
1205971199112+
1
119896
119892119894
119901120575 (119909minus119909
1015840) 120575 (119910) 120575 (119911)
=
1
120572
120597119879
120597119905
(3)
Initial and boundary conditions are given by[119879]119905=0= 0
[119879]119909=0= 1198651(119910 119911 119905)
[119879]119909=119886= 1198652(119910 119911 119905)
[
120597119879
120597119910
]
119910=minus119887
= 1198653(119909 119911 119905)
[
120597119879
120597119910
]
119910=119887
= 1198654(119909 119911 119905)
[119879 + 1198961
120597119879
120597119911
]
119911=minusℎ
= 1198655(119909 119910 119905)
[119879 + 1198962
120597119879
120597119911
]
119911=ℎ
= 1198656(119909 119910 119905)
(4)
Thermal stress function 120594 is 120594 = 120594119888+ 120594119901 where 120594
119888
complementary function is and 120594119901is particular integral 120594
119888
and 120594119901are governed by equations
(
1205972
1205971199092+
1205972
1205971199102)
2
120594119888= 0
(
1205972
1205971199092+
1205972
1205971199102)
2
120594119901= minus120572119864D
(5)
Since plate is thin 119911 is negligible and D = 119879 minus 1198790 where 119879
0is
initial temperature Components of stress functions [12] aregiven by
120590119909119909=
1205972120594
1205971199102 (6)
120590119910119910=
1205972120594
1205971199092 (7)
120590119909119910= minus
1205972120594
120597119909120597119910
(8)
with boundary conditions 120590119910119910= 0 and 120590
119909119910= 0 at 119910 = 119887
Equations (1) to (8) represent the statement of the prob-lem
3 Solution of the Problem
Applying finite Fourier cosine transform finite Fourier sinetransform [13] and Marchi-Fasulo transform [14] usingboundary conditions (4) we get
119889119879
lowast
119889119905
+ 120572119876119879
lowast
= 1205720(9)
where 119876 = 119898212058721198862 + 119899212058721198862 + 119886119897
2
0 = [
119898120587
119886
[(minus1)119898+11198652+ 1198651] + (minus1)
1198991198653minus 1198654
+
119901119897(ℎ)
1205721
1198655minus
119901119897(minusℎ)
1205722
1198656
+
119892119894
119901
119896
sin(119898120587119909119868
119886
)119901119897(0) 120575 (119905 minus 119905
119868)]
119879
lowast
= 119890minus120572119876119905
(int 119890120572119876119905+ 1205720 119889119905 minus int1205720 119889119905)
(10)
Taking inverse Marchi-Fasulo transform [14] finiteFourier sine transform and finite Fourier cosine transform[13]
119879 =
4
119886119887
sdot ∬
infin
sum
119897119898119899=1
119901119897(119911)
120582119897
[119890minus120572119876119905
(int 119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(11)
Indian Journal of Materials Science 3
And D = 119879 minus 1198790
D =4
119886119887
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(12)
120594119888=
infin
sum
119898=1
119910 [1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
] cos(119898120587119909119886
) + 119910 [1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
] sin(119898120587119909119886
)
(13)
120594119901=
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(14)
120594 =
infin
sum
119898=1
119910 [1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
] cos(119898120587119909119886
) + 119910 [1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
] sin(119898120587119909119886
) +
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(15)
0 = [
119898120587
119886
[(minus1)119898+11198652+ 1198651] + (minus1)
1198991198653minus 1198654minus
119901119897(ℎ)
1205721
1198655
minus
119901119897(minusℎ)
1205722
1198656+
119892119894
119901
119896
sin(119898120587119909119868
119886
)119901119897(0) 120575 (119905 minus 119905
119868)]
(16)
119879 =
4
119886119887
sdot ∬
infin
sum
119897119898119899=1
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
D = 119879 minus 1198790
D =4
119886119887
sdot ∬
infin
sum
119897119898119899=1
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(17)
As we change the values of 119897 119898 and 119899 from 1 toinfinwe getinfinite terms of this solution which is nothing but infiniteseries
4 Determination of Stress Function
Using (15) in (6)ndash(8) we get
120590119909119909=
infin
sum
119898=1
[2 (
119898120587
119886
1198881119890119898120587119910119886
minus
119898120587
119886
1198882119890minus119898120587119910119886
)
+ 119910 (1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
)] cos(119898120587119909119886
)
+ [2 (
119898120587
119886
1198883119890119898120587119910119886
minus
119898120587
119886
1198884119890minus119898120587119910119886
)
+ 119910 (1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
)] sin(119898120587119909119886
)
+
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)(
minus11989921205872
1198872)
120590119910119910=
minus11989821205872119910
1198862
[1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
] cos(119898120587119909119886
)
+ [1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
] sin(119898120587119909119886
) + (
11989821205872
1198862)
sdot
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
120590119909119910= [(119888
1119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
) +
119898120587119910
119886
(1198881119890119898120587119910119886
minus 1198882119890minus119898120587119910119886
)] (minus
119898120587
119886
) sin(119898120587119909119886
) minus [(1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
) +
119898120587119910
119886
(1198883119890119898120587119910119886
minus 1198884119890minus119898120587119910119886
)]
sdot (minus
119898120587
119886
) cos(119898120587119909119886
) +
1198991198981205872
119886119887
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin(119899120587119910
119887
) cos(119898120587119909119886
)
(18)
4 Indian Journal of Materials Science
Using the boundary conditions 120590119910119910= 0 and 120590
119909119910= 0 at 119910 = 119887
we get
1198881= 0
1198882= 0
1198883=
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890minus119898120587119887119886
1198884=
minus2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890119898120587119887119886
120590119909119909=
infin
sum
119898=1
[(
2119898120587
119886
+ 119910)
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890minus119898120587119887119886
119890119898120587119910119886
+ (
2119898120587
119886
minus 119910)
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890119898120587119887119886
119890minus119898120587119910119886
] sin(119898120587119909119886
) +
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)(
11989921205872
1198872)
120590119910119910=
minus11989821205872119910
1198862
[
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890minus119898120587119887119886
119890119898120587119910119886
minus
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890119898120587119887119886
119890minus119898120587119910119886
] sin(119898120587119909119886
)
minus
11989821205872
1198862
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
120590119909119910= [minus(1 +
119898120587119910
119886
)
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890minus119898120587119887119886
119890119898120587119910119886
+ (1 minus
119898120587119910
119886
)
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890119898120587119887119886
119890minus119898120587119910119886
](minus
119898120587
119886
) cos(119898120587119909119886
) +
1198991198981205872
119886119887
sdot
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin(119899120587119910
119887
) cos(119898120587119909119886
)
(19)
5 Numerical Results
Let 119896 = 05330 120572 = 238times10minus6 119864 = 0675 times 1011 and 119886 =5 cm 119887 = 1 cm ℎ = 02 cm and
120590119909119909=
infin
sum
119898=1
[(
2120587
5
+ 119910)
sdot
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
119890minus12058751198901205871199105
+ (
2120587
5
minus 119910)
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
] sin(1205871199095
) +
20 times 238 times 10minus6times 0675 times 10
11
261205872
sdot
infin
sum
119897119898119899=0
119901119897(02)
120582119897
[119890minus238times10
minus6
119876119905int (119890
120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos (120587119910) sin(120587119909119886
)(
11989921205872
1198872)
120590119910119910=
minus1205872119910
25
[
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
sdot 119890minus12058751198901205871199105
minus
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
] sin(1205871199095
) minus
1205872
25
sdot
20 times 238 times 10minus6times 0675 times 10
11
261205872
23562
155485
cos (120587119910)
sdot sin(1205871199095
)
120590119909119910= [(minus(1 +
120587119910
5
)
sdot
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
119890minus12058751198901205871199105
minus (1 minus
120587119910
5
)
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
)](minus
120587
5
) cos(1205871199095
) +
1205872
5
sdot
20 times 238 times 10minus6times 0675 times 10
11
261205872
sdot
119901119897(02)
120582119897
[119890minus238times10
minus6
119876119905int (119890
120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin (120587119910) cos(1205871199095
)
120593 =
23562
155485
119890minus238times10
minus6
times124912119905int (119890
238times10minus6
times124912119905+ 238
times10minus6
(minus09721 minus 12639119905)) 119889119905 minus int 238
times10minus6
((minus09721 minus 12639119905)) 119889119905
(20)
Indian Journal of Materials Science 5
minus35 minus30 minus25 minus20 minus15 minus10 minus5 00
05
1
15
2
25
3
Time(
t)
Temperature (T)
Figure 1 Temperature versus time
where 119909119868 = 15 119892119894119901= 1 119905 = 1 119905119868 = 15 120582
119897= 155485 119901
119897(02) =
23562 120593 = 02056 119876 = 124912 0 = minus45920 and
119879 = 293083 [119890minus29729119890minus04119905
4420611989044594119890minus04
] cos (120587119910)
sdot sin(1205871199095
)
119879 = 404676 [119890minus07369119905
] cos (120587119910) sin(1205871199095
)
120590119910119910= 119910[102767 times 02056119890
1205871199105+ 450219
times 02056119890minus1205871199105
] minus 59124 cos (120587119910) sin(1205871199095
)
120590119909119910= [(minus2318346 + 1456616119910) 119890
1205871199105+ (716544
minus 450205119910) 119890minus1205871199105
+ 555858 sin (120587119910)] cos(1205871199095
)
120590119909119909= [(minus4637527 + minus3690536119910) 119890
1205871199105
+ (1433047 minus 1140416119910) 119890minus1205871199105
+ 2779289]
sdot sin(1205871199095
)
(21)
6 Graphical Interpretation
See Figures 1ndash7
7 Discussion
In this article the three-dimensional nonhomogeneous heatconduction issue in a thin rectangular plate is studiedWe didnumerical computations for a thin rectangular plate made upof aluminumThe heat source 119892(119909 119910 119911 119905) is an instantaneouspoint heat source of strength 119892
119894 The thermoelastic behavior
is examined such as temperature and thermal stresses
0 05 1 15 2 25 30
05
1
15
2
25
3
35
4
45
120590yy
x-axis
Figure 2 120590119910119910
versus 119909
0 05 1 15 2 25 3minus150
minus100
minus50
0
50120590xy
x-axis
Figure 3 119909 versus 120590119909119910
0 05 1 15 2 25 3minus90
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
120590xx
x-axis
Figure 4 119909 versus 120590119909119909
6 Indian Journal of Materials Science
minus5 0 5 10 15 20 25 30 35 40 450
05
1
15
2
25
3
120590yy
y-axis
Figure 5 120590119910119910
versus 119910
minus100 0 100 200 300 400 500 600 700 8000
05
1
15
2
25
3
120590xy
y-axis
Figure 6 120590119909119910
versus 119910
minus9000 minus8000 minus7000 minus6000 minus5000 minus4000 minus3000 minus2000 minus1000 00
05
1
15
2
25
3
120590xx
y
Figure 7 120590119909119909
versus 119910
From Figure 1 it is found that at first when time iszero temperature is shrinking But as time increasestemperature develops as much as precise restrictionand it turns out to be regularFrom Figure 2 interatomic distance grew to beextensive up to precise value of119909 after specific value of119909 interatomic distance in a plate takes its function as itis When we provide temperature to aluminium plateinitially atoms in a plate get disturbed that is shortstress increases along 119909-axisThermal stress increasesinitially but it is observed that it remains constant andagain slightly decreasesFrom Figure 3 as temperature rises molecule beginsto vibrate more rapidly and push away from oneanother to increase separation between the atoms thatcause expansion in atoms In a plate position of atomsgets separated along 119909-axisThermal stress ofmaterialchanges from minimum of 119909 to maximum that isvariation observed along both 119909-axis and 119910-axisFrom Figure 4 initially when 119909 is zero stress is zerothat is in a plate interatomic constitution is constantbut when we change the worth of 119909 atomic distancegets compressed and at 119909 = 25 it turns into extracompression and again interatomic distance slowlyseparated Then it gets its customary positionFrom Figure 5 at first interatomic distance could bemuch closed as altering the worth of 119910 that distancegrew to be vast that is stress risesFrom Figure 6 stress alongside 119910-axis atomic struc-ture in a plate is rapidly changing its positions as wechange the value of 119910From Figure 7 it shows that originally at 119910 = 0 stressis incredibly minimum suggesting that interatomicdistance is compressed as value of 119910 changes it comesto its original interatomic distance so that it acquiresits original position
8 Conclusion
In this paper we carried out the nonhomogeneous thermoe-lastic problem solved using integral transform techniquesnumerically Results are obtained dependent on values of 119897119898 and 119899 which vary from 1 to infin Hence variation of heatby moving heat sources in a body changes infinitely Fromgraphical study when a body is provided with heat it affectsit in all directions Hence material shows expansion along 119909-axis 119910-axis and 119911-axis respectively We conclude that if timeincreases temperature will also increase Interatomic distancebecamewide along119910-axis As thin rectangular plate subjectedto point heat source which changes its place along 119909-axisinteratomic distance became narrow
The outcomes got here basically applicable in engineeringproblems especially for industrial machines subjected tothe heating such as the main shaft of a machine turbinesthe roll of rolling mill and practical applications in aircraftstructures
Indian Journal of Materials Science 7
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Nowinski Theory of Thermoelasticity with ApplicationSijthoff amp Noordhoff Alphen Aan Den Rijn The Netherlands1978
[2] W Nowacki ldquoThe state of stresses in a thick circular plate dueto temperature fieldrdquo Bulletin of the Polish Academy of SciencesTechnical Science vol 5 article 227 1957
[3] B A Boley and J H Weiner Theory of Thermal Stresses JohnWiley amp Sons New York NY USA 1960
[4] S K Roy Choudhary ldquoA note of quasi static stress in a thincircular plate due to transient temperature applied along the cir-cumference of a circle over the upper facerdquo Bulletin LrsquoAcademiePolonaise des Science Serie des Sciences Mathematiques vol 20pp 20ndash21 1972
[5] N W Khobragade and P C Wankhede ldquoAn inverse unsteadystate thermoelastic problem of a thin rectangular platerdquo TheJournal of Indian Academy of Mathematics vol 25 no 2 2003
[6] K R Gaikwad and K P Ghadle ldquoQuasi-static thermoelasticproblem of an infinitely long circular cylinderrdquo Journal of theKorean Society for Industrial and Applied Mathematics vol 14no 3 pp 141ndash149 2010
[7] V B Patil B R Ahirrao and N W Khobragade ldquoThermalstresses of semi infinite Rectangular slab with internal heatsourcerdquo IOSR Journals of Mathematics vol 8 no 6 pp 57ndash612013
[8] D T Solanke and M H Durge ldquoQuasi static thermal stressesin thin rectangular plate with internal moving line heat sourcerdquoScience Park Research Journal vol 1 no 44 pp 1ndash5 2014
[9] M S Thakare C S Sutar and N W Khobragade ldquoThermalstresses of a thin rectangular plate with internal moving heatsourcerdquo International Journal of Engineering and InnovativeTechnology vol 4 no 9 2015
[10] K R Gaikwad ldquoTwo-dimensional steady-state temperaturedistribution of a thin circular plate due to uniform internalenergy generationrdquo Cogent Mathematics vol 3 no 1 Article ID1135720 2016
[11] N M Ozisik Boundary Value Problem of Heat ConductionDover Mineola NY USA 1968
[12] N Noda R B Hetnarski and Y Tanigawa Thermal StressesLastran 2nd edition 2002
[13] I N SneddonTheUse of Integral Transform McGrawHill NewYork NY USA 1972
[14] E Marchi and A Fasulo ldquoHeat conduction in sectors of hollowcylinders with radiationrdquo Atti della Accademia delle Scienze diTorino no 1 pp 373ndash382 1967
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Nano
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Journal ofNanomaterials
2 Indian Journal of Materials Science
2 Formulation of the Problem
We consider three-dimensional thin rectangular plate understeady state temperature defined in region 119877 0 le 119909 le 1198860 le 119910 le 119887 and minusℎ le 119911 le ℎ where ℎ lt 119887 lt 119886 and ℎ isthickness which is very small The plate is subjected to themotion of moving point heat source at the point (1199091015840 0 0)Under these realistic prescribed conditions temperature andthermal stresses in a thin rectangular plate are required to bedetermined
The temperature distribution of the rectangular platedefined in [11] is given by
1205972119879
1205971199092+
1205972119879
1205971199102+
1205972119879
1205971199112+
119892
119896
=
1
120572
120597119879
120597119905
(1)
where 119896 is thermal conductivity and 120572 is thermal diffusivityof the material of the plate
Consider an instantaneous moving heat source at point(1199091015840 0 0) and release its heat spontaneously at time 1199051015840 Such
volumetric moving heat source in rectangular coordinates isgiven by
119892 (119909 119910 119911 119905) = 119892119894
119901120575 (119909minus119909
1015840) 120575 (119910) 120575 (119911) 120575 (119905minus119905
1015840) (2)
where 119892119894119901is instantaneous point heat source
Hence (1) becomes1205972119879
1205971199092+
1205972119879
1205971199102+
1205972119879
1205971199112+
1
119896
119892119894
119901120575 (119909minus119909
1015840) 120575 (119910) 120575 (119911)
=
1
120572
120597119879
120597119905
(3)
Initial and boundary conditions are given by[119879]119905=0= 0
[119879]119909=0= 1198651(119910 119911 119905)
[119879]119909=119886= 1198652(119910 119911 119905)
[
120597119879
120597119910
]
119910=minus119887
= 1198653(119909 119911 119905)
[
120597119879
120597119910
]
119910=119887
= 1198654(119909 119911 119905)
[119879 + 1198961
120597119879
120597119911
]
119911=minusℎ
= 1198655(119909 119910 119905)
[119879 + 1198962
120597119879
120597119911
]
119911=ℎ
= 1198656(119909 119910 119905)
(4)
Thermal stress function 120594 is 120594 = 120594119888+ 120594119901 where 120594
119888
complementary function is and 120594119901is particular integral 120594
119888
and 120594119901are governed by equations
(
1205972
1205971199092+
1205972
1205971199102)
2
120594119888= 0
(
1205972
1205971199092+
1205972
1205971199102)
2
120594119901= minus120572119864D
(5)
Since plate is thin 119911 is negligible and D = 119879 minus 1198790 where 119879
0is
initial temperature Components of stress functions [12] aregiven by
120590119909119909=
1205972120594
1205971199102 (6)
120590119910119910=
1205972120594
1205971199092 (7)
120590119909119910= minus
1205972120594
120597119909120597119910
(8)
with boundary conditions 120590119910119910= 0 and 120590
119909119910= 0 at 119910 = 119887
Equations (1) to (8) represent the statement of the prob-lem
3 Solution of the Problem
Applying finite Fourier cosine transform finite Fourier sinetransform [13] and Marchi-Fasulo transform [14] usingboundary conditions (4) we get
119889119879
lowast
119889119905
+ 120572119876119879
lowast
= 1205720(9)
where 119876 = 119898212058721198862 + 119899212058721198862 + 119886119897
2
0 = [
119898120587
119886
[(minus1)119898+11198652+ 1198651] + (minus1)
1198991198653minus 1198654
+
119901119897(ℎ)
1205721
1198655minus
119901119897(minusℎ)
1205722
1198656
+
119892119894
119901
119896
sin(119898120587119909119868
119886
)119901119897(0) 120575 (119905 minus 119905
119868)]
119879
lowast
= 119890minus120572119876119905
(int 119890120572119876119905+ 1205720 119889119905 minus int1205720 119889119905)
(10)
Taking inverse Marchi-Fasulo transform [14] finiteFourier sine transform and finite Fourier cosine transform[13]
119879 =
4
119886119887
sdot ∬
infin
sum
119897119898119899=1
119901119897(119911)
120582119897
[119890minus120572119876119905
(int 119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(11)
Indian Journal of Materials Science 3
And D = 119879 minus 1198790
D =4
119886119887
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(12)
120594119888=
infin
sum
119898=1
119910 [1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
] cos(119898120587119909119886
) + 119910 [1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
] sin(119898120587119909119886
)
(13)
120594119901=
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(14)
120594 =
infin
sum
119898=1
119910 [1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
] cos(119898120587119909119886
) + 119910 [1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
] sin(119898120587119909119886
) +
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(15)
0 = [
119898120587
119886
[(minus1)119898+11198652+ 1198651] + (minus1)
1198991198653minus 1198654minus
119901119897(ℎ)
1205721
1198655
minus
119901119897(minusℎ)
1205722
1198656+
119892119894
119901
119896
sin(119898120587119909119868
119886
)119901119897(0) 120575 (119905 minus 119905
119868)]
(16)
119879 =
4
119886119887
sdot ∬
infin
sum
119897119898119899=1
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
D = 119879 minus 1198790
D =4
119886119887
sdot ∬
infin
sum
119897119898119899=1
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(17)
As we change the values of 119897 119898 and 119899 from 1 toinfinwe getinfinite terms of this solution which is nothing but infiniteseries
4 Determination of Stress Function
Using (15) in (6)ndash(8) we get
120590119909119909=
infin
sum
119898=1
[2 (
119898120587
119886
1198881119890119898120587119910119886
minus
119898120587
119886
1198882119890minus119898120587119910119886
)
+ 119910 (1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
)] cos(119898120587119909119886
)
+ [2 (
119898120587
119886
1198883119890119898120587119910119886
minus
119898120587
119886
1198884119890minus119898120587119910119886
)
+ 119910 (1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
)] sin(119898120587119909119886
)
+
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)(
minus11989921205872
1198872)
120590119910119910=
minus11989821205872119910
1198862
[1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
] cos(119898120587119909119886
)
+ [1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
] sin(119898120587119909119886
) + (
11989821205872
1198862)
sdot
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
120590119909119910= [(119888
1119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
) +
119898120587119910
119886
(1198881119890119898120587119910119886
minus 1198882119890minus119898120587119910119886
)] (minus
119898120587
119886
) sin(119898120587119909119886
) minus [(1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
) +
119898120587119910
119886
(1198883119890119898120587119910119886
minus 1198884119890minus119898120587119910119886
)]
sdot (minus
119898120587
119886
) cos(119898120587119909119886
) +
1198991198981205872
119886119887
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin(119899120587119910
119887
) cos(119898120587119909119886
)
(18)
4 Indian Journal of Materials Science
Using the boundary conditions 120590119910119910= 0 and 120590
119909119910= 0 at 119910 = 119887
we get
1198881= 0
1198882= 0
1198883=
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890minus119898120587119887119886
1198884=
minus2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890119898120587119887119886
120590119909119909=
infin
sum
119898=1
[(
2119898120587
119886
+ 119910)
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890minus119898120587119887119886
119890119898120587119910119886
+ (
2119898120587
119886
minus 119910)
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890119898120587119887119886
119890minus119898120587119910119886
] sin(119898120587119909119886
) +
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)(
11989921205872
1198872)
120590119910119910=
minus11989821205872119910
1198862
[
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890minus119898120587119887119886
119890119898120587119910119886
minus
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890119898120587119887119886
119890minus119898120587119910119886
] sin(119898120587119909119886
)
minus
11989821205872
1198862
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
120590119909119910= [minus(1 +
119898120587119910
119886
)
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890minus119898120587119887119886
119890119898120587119910119886
+ (1 minus
119898120587119910
119886
)
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890119898120587119887119886
119890minus119898120587119910119886
](minus
119898120587
119886
) cos(119898120587119909119886
) +
1198991198981205872
119886119887
sdot
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin(119899120587119910
119887
) cos(119898120587119909119886
)
(19)
5 Numerical Results
Let 119896 = 05330 120572 = 238times10minus6 119864 = 0675 times 1011 and 119886 =5 cm 119887 = 1 cm ℎ = 02 cm and
120590119909119909=
infin
sum
119898=1
[(
2120587
5
+ 119910)
sdot
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
119890minus12058751198901205871199105
+ (
2120587
5
minus 119910)
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
] sin(1205871199095
) +
20 times 238 times 10minus6times 0675 times 10
11
261205872
sdot
infin
sum
119897119898119899=0
119901119897(02)
120582119897
[119890minus238times10
minus6
119876119905int (119890
120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos (120587119910) sin(120587119909119886
)(
11989921205872
1198872)
120590119910119910=
minus1205872119910
25
[
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
sdot 119890minus12058751198901205871199105
minus
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
] sin(1205871199095
) minus
1205872
25
sdot
20 times 238 times 10minus6times 0675 times 10
11
261205872
23562
155485
cos (120587119910)
sdot sin(1205871199095
)
120590119909119910= [(minus(1 +
120587119910
5
)
sdot
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
119890minus12058751198901205871199105
minus (1 minus
120587119910
5
)
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
)](minus
120587
5
) cos(1205871199095
) +
1205872
5
sdot
20 times 238 times 10minus6times 0675 times 10
11
261205872
sdot
119901119897(02)
120582119897
[119890minus238times10
minus6
119876119905int (119890
120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin (120587119910) cos(1205871199095
)
120593 =
23562
155485
119890minus238times10
minus6
times124912119905int (119890
238times10minus6
times124912119905+ 238
times10minus6
(minus09721 minus 12639119905)) 119889119905 minus int 238
times10minus6
((minus09721 minus 12639119905)) 119889119905
(20)
Indian Journal of Materials Science 5
minus35 minus30 minus25 minus20 minus15 minus10 minus5 00
05
1
15
2
25
3
Time(
t)
Temperature (T)
Figure 1 Temperature versus time
where 119909119868 = 15 119892119894119901= 1 119905 = 1 119905119868 = 15 120582
119897= 155485 119901
119897(02) =
23562 120593 = 02056 119876 = 124912 0 = minus45920 and
119879 = 293083 [119890minus29729119890minus04119905
4420611989044594119890minus04
] cos (120587119910)
sdot sin(1205871199095
)
119879 = 404676 [119890minus07369119905
] cos (120587119910) sin(1205871199095
)
120590119910119910= 119910[102767 times 02056119890
1205871199105+ 450219
times 02056119890minus1205871199105
] minus 59124 cos (120587119910) sin(1205871199095
)
120590119909119910= [(minus2318346 + 1456616119910) 119890
1205871199105+ (716544
minus 450205119910) 119890minus1205871199105
+ 555858 sin (120587119910)] cos(1205871199095
)
120590119909119909= [(minus4637527 + minus3690536119910) 119890
1205871199105
+ (1433047 minus 1140416119910) 119890minus1205871199105
+ 2779289]
sdot sin(1205871199095
)
(21)
6 Graphical Interpretation
See Figures 1ndash7
7 Discussion
In this article the three-dimensional nonhomogeneous heatconduction issue in a thin rectangular plate is studiedWe didnumerical computations for a thin rectangular plate made upof aluminumThe heat source 119892(119909 119910 119911 119905) is an instantaneouspoint heat source of strength 119892
119894 The thermoelastic behavior
is examined such as temperature and thermal stresses
0 05 1 15 2 25 30
05
1
15
2
25
3
35
4
45
120590yy
x-axis
Figure 2 120590119910119910
versus 119909
0 05 1 15 2 25 3minus150
minus100
minus50
0
50120590xy
x-axis
Figure 3 119909 versus 120590119909119910
0 05 1 15 2 25 3minus90
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
120590xx
x-axis
Figure 4 119909 versus 120590119909119909
6 Indian Journal of Materials Science
minus5 0 5 10 15 20 25 30 35 40 450
05
1
15
2
25
3
120590yy
y-axis
Figure 5 120590119910119910
versus 119910
minus100 0 100 200 300 400 500 600 700 8000
05
1
15
2
25
3
120590xy
y-axis
Figure 6 120590119909119910
versus 119910
minus9000 minus8000 minus7000 minus6000 minus5000 minus4000 minus3000 minus2000 minus1000 00
05
1
15
2
25
3
120590xx
y
Figure 7 120590119909119909
versus 119910
From Figure 1 it is found that at first when time iszero temperature is shrinking But as time increasestemperature develops as much as precise restrictionand it turns out to be regularFrom Figure 2 interatomic distance grew to beextensive up to precise value of119909 after specific value of119909 interatomic distance in a plate takes its function as itis When we provide temperature to aluminium plateinitially atoms in a plate get disturbed that is shortstress increases along 119909-axisThermal stress increasesinitially but it is observed that it remains constant andagain slightly decreasesFrom Figure 3 as temperature rises molecule beginsto vibrate more rapidly and push away from oneanother to increase separation between the atoms thatcause expansion in atoms In a plate position of atomsgets separated along 119909-axisThermal stress ofmaterialchanges from minimum of 119909 to maximum that isvariation observed along both 119909-axis and 119910-axisFrom Figure 4 initially when 119909 is zero stress is zerothat is in a plate interatomic constitution is constantbut when we change the worth of 119909 atomic distancegets compressed and at 119909 = 25 it turns into extracompression and again interatomic distance slowlyseparated Then it gets its customary positionFrom Figure 5 at first interatomic distance could bemuch closed as altering the worth of 119910 that distancegrew to be vast that is stress risesFrom Figure 6 stress alongside 119910-axis atomic struc-ture in a plate is rapidly changing its positions as wechange the value of 119910From Figure 7 it shows that originally at 119910 = 0 stressis incredibly minimum suggesting that interatomicdistance is compressed as value of 119910 changes it comesto its original interatomic distance so that it acquiresits original position
8 Conclusion
In this paper we carried out the nonhomogeneous thermoe-lastic problem solved using integral transform techniquesnumerically Results are obtained dependent on values of 119897119898 and 119899 which vary from 1 to infin Hence variation of heatby moving heat sources in a body changes infinitely Fromgraphical study when a body is provided with heat it affectsit in all directions Hence material shows expansion along 119909-axis 119910-axis and 119911-axis respectively We conclude that if timeincreases temperature will also increase Interatomic distancebecamewide along119910-axis As thin rectangular plate subjectedto point heat source which changes its place along 119909-axisinteratomic distance became narrow
The outcomes got here basically applicable in engineeringproblems especially for industrial machines subjected tothe heating such as the main shaft of a machine turbinesthe roll of rolling mill and practical applications in aircraftstructures
Indian Journal of Materials Science 7
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Nowinski Theory of Thermoelasticity with ApplicationSijthoff amp Noordhoff Alphen Aan Den Rijn The Netherlands1978
[2] W Nowacki ldquoThe state of stresses in a thick circular plate dueto temperature fieldrdquo Bulletin of the Polish Academy of SciencesTechnical Science vol 5 article 227 1957
[3] B A Boley and J H Weiner Theory of Thermal Stresses JohnWiley amp Sons New York NY USA 1960
[4] S K Roy Choudhary ldquoA note of quasi static stress in a thincircular plate due to transient temperature applied along the cir-cumference of a circle over the upper facerdquo Bulletin LrsquoAcademiePolonaise des Science Serie des Sciences Mathematiques vol 20pp 20ndash21 1972
[5] N W Khobragade and P C Wankhede ldquoAn inverse unsteadystate thermoelastic problem of a thin rectangular platerdquo TheJournal of Indian Academy of Mathematics vol 25 no 2 2003
[6] K R Gaikwad and K P Ghadle ldquoQuasi-static thermoelasticproblem of an infinitely long circular cylinderrdquo Journal of theKorean Society for Industrial and Applied Mathematics vol 14no 3 pp 141ndash149 2010
[7] V B Patil B R Ahirrao and N W Khobragade ldquoThermalstresses of semi infinite Rectangular slab with internal heatsourcerdquo IOSR Journals of Mathematics vol 8 no 6 pp 57ndash612013
[8] D T Solanke and M H Durge ldquoQuasi static thermal stressesin thin rectangular plate with internal moving line heat sourcerdquoScience Park Research Journal vol 1 no 44 pp 1ndash5 2014
[9] M S Thakare C S Sutar and N W Khobragade ldquoThermalstresses of a thin rectangular plate with internal moving heatsourcerdquo International Journal of Engineering and InnovativeTechnology vol 4 no 9 2015
[10] K R Gaikwad ldquoTwo-dimensional steady-state temperaturedistribution of a thin circular plate due to uniform internalenergy generationrdquo Cogent Mathematics vol 3 no 1 Article ID1135720 2016
[11] N M Ozisik Boundary Value Problem of Heat ConductionDover Mineola NY USA 1968
[12] N Noda R B Hetnarski and Y Tanigawa Thermal StressesLastran 2nd edition 2002
[13] I N SneddonTheUse of Integral Transform McGrawHill NewYork NY USA 1972
[14] E Marchi and A Fasulo ldquoHeat conduction in sectors of hollowcylinders with radiationrdquo Atti della Accademia delle Scienze diTorino no 1 pp 373ndash382 1967
Submit your manuscripts athttpwwwhindawicom
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CompositesJournal of
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Biomaterials
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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Nano
materials
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Journal ofNanomaterials
Indian Journal of Materials Science 3
And D = 119879 minus 1198790
D =4
119886119887
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(12)
120594119888=
infin
sum
119898=1
119910 [1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
] cos(119898120587119909119886
) + 119910 [1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
] sin(119898120587119909119886
)
(13)
120594119901=
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(14)
120594 =
infin
sum
119898=1
119910 [1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
] cos(119898120587119909119886
) + 119910 [1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
] sin(119898120587119909119886
) +
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(15)
0 = [
119898120587
119886
[(minus1)119898+11198652+ 1198651] + (minus1)
1198991198653minus 1198654minus
119901119897(ℎ)
1205721
1198655
minus
119901119897(minusℎ)
1205722
1198656+
119892119894
119901
119896
sin(119898120587119909119868
119886
)119901119897(0) 120575 (119905 minus 119905
119868)]
(16)
119879 =
4
119886119887
sdot ∬
infin
sum
119897119898119899=1
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
D = 119879 minus 1198790
D =4
119886119887
sdot ∬
infin
sum
119897119898119899=1
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
(17)
As we change the values of 119897 119898 and 119899 from 1 toinfinwe getinfinite terms of this solution which is nothing but infiniteseries
4 Determination of Stress Function
Using (15) in (6)ndash(8) we get
120590119909119909=
infin
sum
119898=1
[2 (
119898120587
119886
1198881119890119898120587119910119886
minus
119898120587
119886
1198882119890minus119898120587119910119886
)
+ 119910 (1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
)] cos(119898120587119909119886
)
+ [2 (
119898120587
119886
1198883119890119898120587119910119886
minus
119898120587
119886
1198884119890minus119898120587119910119886
)
+ 119910 (1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
)] sin(119898120587119909119886
)
+
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)(
minus11989921205872
1198872)
120590119910119910=
minus11989821205872119910
1198862
[1198881119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
] cos(119898120587119909119886
)
+ [1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
] sin(119898120587119909119886
) + (
11989821205872
1198862)
sdot
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
120590119909119910= [(119888
1119890119898120587119910119886
+ 1198882119890minus119898120587119910119886
) +
119898120587119910
119886
(1198881119890119898120587119910119886
minus 1198882119890minus119898120587119910119886
)] (minus
119898120587
119886
) sin(119898120587119909119886
) minus [(1198883119890119898120587119910119886
+ 1198884119890minus119898120587119910119886
) +
119898120587119910
119886
(1198883119890119898120587119910119886
minus 1198884119890minus119898120587119910119886
)]
sdot (minus
119898120587
119886
) cos(119898120587119909119886
) +
1198991198981205872
119886119887
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin(119899120587119910
119887
) cos(119898120587119909119886
)
(18)
4 Indian Journal of Materials Science
Using the boundary conditions 120590119910119910= 0 and 120590
119909119910= 0 at 119910 = 119887
we get
1198881= 0
1198882= 0
1198883=
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890minus119898120587119887119886
1198884=
minus2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890119898120587119887119886
120590119909119909=
infin
sum
119898=1
[(
2119898120587
119886
+ 119910)
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890minus119898120587119887119886
119890119898120587119910119886
+ (
2119898120587
119886
minus 119910)
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890119898120587119887119886
119890minus119898120587119910119886
] sin(119898120587119909119886
) +
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)(
11989921205872
1198872)
120590119910119910=
minus11989821205872119910
1198862
[
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890minus119898120587119887119886
119890119898120587119910119886
minus
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890119898120587119887119886
119890minus119898120587119910119886
] sin(119898120587119909119886
)
minus
11989821205872
1198862
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
120590119909119910= [minus(1 +
119898120587119910
119886
)
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890minus119898120587119887119886
119890119898120587119910119886
+ (1 minus
119898120587119910
119886
)
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890119898120587119887119886
119890minus119898120587119910119886
](minus
119898120587
119886
) cos(119898120587119909119886
) +
1198991198981205872
119886119887
sdot
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin(119899120587119910
119887
) cos(119898120587119909119886
)
(19)
5 Numerical Results
Let 119896 = 05330 120572 = 238times10minus6 119864 = 0675 times 1011 and 119886 =5 cm 119887 = 1 cm ℎ = 02 cm and
120590119909119909=
infin
sum
119898=1
[(
2120587
5
+ 119910)
sdot
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
119890minus12058751198901205871199105
+ (
2120587
5
minus 119910)
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
] sin(1205871199095
) +
20 times 238 times 10minus6times 0675 times 10
11
261205872
sdot
infin
sum
119897119898119899=0
119901119897(02)
120582119897
[119890minus238times10
minus6
119876119905int (119890
120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos (120587119910) sin(120587119909119886
)(
11989921205872
1198872)
120590119910119910=
minus1205872119910
25
[
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
sdot 119890minus12058751198901205871199105
minus
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
] sin(1205871199095
) minus
1205872
25
sdot
20 times 238 times 10minus6times 0675 times 10
11
261205872
23562
155485
cos (120587119910)
sdot sin(1205871199095
)
120590119909119910= [(minus(1 +
120587119910
5
)
sdot
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
119890minus12058751198901205871199105
minus (1 minus
120587119910
5
)
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
)](minus
120587
5
) cos(1205871199095
) +
1205872
5
sdot
20 times 238 times 10minus6times 0675 times 10
11
261205872
sdot
119901119897(02)
120582119897
[119890minus238times10
minus6
119876119905int (119890
120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin (120587119910) cos(1205871199095
)
120593 =
23562
155485
119890minus238times10
minus6
times124912119905int (119890
238times10minus6
times124912119905+ 238
times10minus6
(minus09721 minus 12639119905)) 119889119905 minus int 238
times10minus6
((minus09721 minus 12639119905)) 119889119905
(20)
Indian Journal of Materials Science 5
minus35 minus30 minus25 minus20 minus15 minus10 minus5 00
05
1
15
2
25
3
Time(
t)
Temperature (T)
Figure 1 Temperature versus time
where 119909119868 = 15 119892119894119901= 1 119905 = 1 119905119868 = 15 120582
119897= 155485 119901
119897(02) =
23562 120593 = 02056 119876 = 124912 0 = minus45920 and
119879 = 293083 [119890minus29729119890minus04119905
4420611989044594119890minus04
] cos (120587119910)
sdot sin(1205871199095
)
119879 = 404676 [119890minus07369119905
] cos (120587119910) sin(1205871199095
)
120590119910119910= 119910[102767 times 02056119890
1205871199105+ 450219
times 02056119890minus1205871199105
] minus 59124 cos (120587119910) sin(1205871199095
)
120590119909119910= [(minus2318346 + 1456616119910) 119890
1205871199105+ (716544
minus 450205119910) 119890minus1205871199105
+ 555858 sin (120587119910)] cos(1205871199095
)
120590119909119909= [(minus4637527 + minus3690536119910) 119890
1205871199105
+ (1433047 minus 1140416119910) 119890minus1205871199105
+ 2779289]
sdot sin(1205871199095
)
(21)
6 Graphical Interpretation
See Figures 1ndash7
7 Discussion
In this article the three-dimensional nonhomogeneous heatconduction issue in a thin rectangular plate is studiedWe didnumerical computations for a thin rectangular plate made upof aluminumThe heat source 119892(119909 119910 119911 119905) is an instantaneouspoint heat source of strength 119892
119894 The thermoelastic behavior
is examined such as temperature and thermal stresses
0 05 1 15 2 25 30
05
1
15
2
25
3
35
4
45
120590yy
x-axis
Figure 2 120590119910119910
versus 119909
0 05 1 15 2 25 3minus150
minus100
minus50
0
50120590xy
x-axis
Figure 3 119909 versus 120590119909119910
0 05 1 15 2 25 3minus90
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
120590xx
x-axis
Figure 4 119909 versus 120590119909119909
6 Indian Journal of Materials Science
minus5 0 5 10 15 20 25 30 35 40 450
05
1
15
2
25
3
120590yy
y-axis
Figure 5 120590119910119910
versus 119910
minus100 0 100 200 300 400 500 600 700 8000
05
1
15
2
25
3
120590xy
y-axis
Figure 6 120590119909119910
versus 119910
minus9000 minus8000 minus7000 minus6000 minus5000 minus4000 minus3000 minus2000 minus1000 00
05
1
15
2
25
3
120590xx
y
Figure 7 120590119909119909
versus 119910
From Figure 1 it is found that at first when time iszero temperature is shrinking But as time increasestemperature develops as much as precise restrictionand it turns out to be regularFrom Figure 2 interatomic distance grew to beextensive up to precise value of119909 after specific value of119909 interatomic distance in a plate takes its function as itis When we provide temperature to aluminium plateinitially atoms in a plate get disturbed that is shortstress increases along 119909-axisThermal stress increasesinitially but it is observed that it remains constant andagain slightly decreasesFrom Figure 3 as temperature rises molecule beginsto vibrate more rapidly and push away from oneanother to increase separation between the atoms thatcause expansion in atoms In a plate position of atomsgets separated along 119909-axisThermal stress ofmaterialchanges from minimum of 119909 to maximum that isvariation observed along both 119909-axis and 119910-axisFrom Figure 4 initially when 119909 is zero stress is zerothat is in a plate interatomic constitution is constantbut when we change the worth of 119909 atomic distancegets compressed and at 119909 = 25 it turns into extracompression and again interatomic distance slowlyseparated Then it gets its customary positionFrom Figure 5 at first interatomic distance could bemuch closed as altering the worth of 119910 that distancegrew to be vast that is stress risesFrom Figure 6 stress alongside 119910-axis atomic struc-ture in a plate is rapidly changing its positions as wechange the value of 119910From Figure 7 it shows that originally at 119910 = 0 stressis incredibly minimum suggesting that interatomicdistance is compressed as value of 119910 changes it comesto its original interatomic distance so that it acquiresits original position
8 Conclusion
In this paper we carried out the nonhomogeneous thermoe-lastic problem solved using integral transform techniquesnumerically Results are obtained dependent on values of 119897119898 and 119899 which vary from 1 to infin Hence variation of heatby moving heat sources in a body changes infinitely Fromgraphical study when a body is provided with heat it affectsit in all directions Hence material shows expansion along 119909-axis 119910-axis and 119911-axis respectively We conclude that if timeincreases temperature will also increase Interatomic distancebecamewide along119910-axis As thin rectangular plate subjectedto point heat source which changes its place along 119909-axisinteratomic distance became narrow
The outcomes got here basically applicable in engineeringproblems especially for industrial machines subjected tothe heating such as the main shaft of a machine turbinesthe roll of rolling mill and practical applications in aircraftstructures
Indian Journal of Materials Science 7
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Nowinski Theory of Thermoelasticity with ApplicationSijthoff amp Noordhoff Alphen Aan Den Rijn The Netherlands1978
[2] W Nowacki ldquoThe state of stresses in a thick circular plate dueto temperature fieldrdquo Bulletin of the Polish Academy of SciencesTechnical Science vol 5 article 227 1957
[3] B A Boley and J H Weiner Theory of Thermal Stresses JohnWiley amp Sons New York NY USA 1960
[4] S K Roy Choudhary ldquoA note of quasi static stress in a thincircular plate due to transient temperature applied along the cir-cumference of a circle over the upper facerdquo Bulletin LrsquoAcademiePolonaise des Science Serie des Sciences Mathematiques vol 20pp 20ndash21 1972
[5] N W Khobragade and P C Wankhede ldquoAn inverse unsteadystate thermoelastic problem of a thin rectangular platerdquo TheJournal of Indian Academy of Mathematics vol 25 no 2 2003
[6] K R Gaikwad and K P Ghadle ldquoQuasi-static thermoelasticproblem of an infinitely long circular cylinderrdquo Journal of theKorean Society for Industrial and Applied Mathematics vol 14no 3 pp 141ndash149 2010
[7] V B Patil B R Ahirrao and N W Khobragade ldquoThermalstresses of semi infinite Rectangular slab with internal heatsourcerdquo IOSR Journals of Mathematics vol 8 no 6 pp 57ndash612013
[8] D T Solanke and M H Durge ldquoQuasi static thermal stressesin thin rectangular plate with internal moving line heat sourcerdquoScience Park Research Journal vol 1 no 44 pp 1ndash5 2014
[9] M S Thakare C S Sutar and N W Khobragade ldquoThermalstresses of a thin rectangular plate with internal moving heatsourcerdquo International Journal of Engineering and InnovativeTechnology vol 4 no 9 2015
[10] K R Gaikwad ldquoTwo-dimensional steady-state temperaturedistribution of a thin circular plate due to uniform internalenergy generationrdquo Cogent Mathematics vol 3 no 1 Article ID1135720 2016
[11] N M Ozisik Boundary Value Problem of Heat ConductionDover Mineola NY USA 1968
[12] N Noda R B Hetnarski and Y Tanigawa Thermal StressesLastran 2nd edition 2002
[13] I N SneddonTheUse of Integral Transform McGrawHill NewYork NY USA 1972
[14] E Marchi and A Fasulo ldquoHeat conduction in sectors of hollowcylinders with radiationrdquo Atti della Accademia delle Scienze diTorino no 1 pp 373ndash382 1967
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
4 Indian Journal of Materials Science
Using the boundary conditions 120590119910119910= 0 and 120590
119909119910= 0 at 119910 = 119887
we get
1198881= 0
1198882= 0
1198883=
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890minus119898120587119887119886
1198884=
minus2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890119898120587119887119886
120590119909119909=
infin
sum
119898=1
[(
2119898120587
119886
+ 119910)
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890minus119898120587119887119886
119890119898120587119910119886
+ (
2119898120587
119886
minus 119910)
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890119898120587119887119886
119890minus119898120587119910119886
] sin(119898120587119909119886
) +
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)(
11989921205872
1198872)
120590119910119910=
minus11989821205872119910
1198862
[
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890minus119898120587119887119886
119890119898120587119910119886
minus
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
119890119898120587119887119886
119890minus119898120587119910119886
] sin(119898120587119909119886
)
minus
11989821205872
1198862
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720119889119905]
sdot cos(119899120587119910
119887
) sin(119898120587119909119886
)
120590119909119910= [minus(1 +
119898120587119910
119886
)
2120572119864119887120593 (minus119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890minus119898120587119887119886
119890119898120587119910119886
+ (1 minus
119898120587119910
119886
)
2120572119864119887120593 (119886 + 119898120587119887)
1205872(11988621198992+ 11988721198982)
sdot 119890119898120587119887119886
119890minus119898120587119910119886
](minus
119898120587
119886
) cos(119898120587119909119886
) +
1198991198981205872
119886119887
sdot
4120572119864119886119887
1205872(11988621198992+ 11988721198982)
sdot
infin
sum
119897119898119899=0
119901119897(119911)
120582119897
[119890minus120572119876119905
int (119890120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin(119899120587119910
119887
) cos(119898120587119909119886
)
(19)
5 Numerical Results
Let 119896 = 05330 120572 = 238times10minus6 119864 = 0675 times 1011 and 119886 =5 cm 119887 = 1 cm ℎ = 02 cm and
120590119909119909=
infin
sum
119898=1
[(
2120587
5
+ 119910)
sdot
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
119890minus12058751198901205871199105
+ (
2120587
5
minus 119910)
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
] sin(1205871199095
) +
20 times 238 times 10minus6times 0675 times 10
11
261205872
sdot
infin
sum
119897119898119899=0
119901119897(02)
120582119897
[119890minus238times10
minus6
119876119905int (119890
120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot cos (120587119910) sin(120587119909119886
)(
11989921205872
1198872)
120590119910119910=
minus1205872119910
25
[
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
sdot 119890minus12058751198901205871199105
minus
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
] sin(1205871199095
) minus
1205872
25
sdot
20 times 238 times 10minus6times 0675 times 10
11
261205872
23562
155485
cos (120587119910)
sdot sin(1205871199095
)
120590119909119910= [(minus(1 +
120587119910
5
)
sdot
2 times 238 times 10minus6times 0675 times 10
11times 120593 (minus5 + 120587)
261205872
119890minus12058751198901205871199105
minus (1 minus
120587119910
5
)
2 times 238 times 10minus6times 0675 times 10
11times 120593 (5 + 120587)
261205872
sdot 1198901205875119890minus1205871199105
)](minus
120587
5
) cos(1205871199095
) +
1205872
5
sdot
20 times 238 times 10minus6times 0675 times 10
11
261205872
sdot
119901119897(02)
120582119897
[119890minus238times10
minus6
119876119905int (119890
120572119876119905+ 1205720) 119889119905 minus int1205720 119889119905]
sdot sin (120587119910) cos(1205871199095
)
120593 =
23562
155485
119890minus238times10
minus6
times124912119905int (119890
238times10minus6
times124912119905+ 238
times10minus6
(minus09721 minus 12639119905)) 119889119905 minus int 238
times10minus6
((minus09721 minus 12639119905)) 119889119905
(20)
Indian Journal of Materials Science 5
minus35 minus30 minus25 minus20 minus15 minus10 minus5 00
05
1
15
2
25
3
Time(
t)
Temperature (T)
Figure 1 Temperature versus time
where 119909119868 = 15 119892119894119901= 1 119905 = 1 119905119868 = 15 120582
119897= 155485 119901
119897(02) =
23562 120593 = 02056 119876 = 124912 0 = minus45920 and
119879 = 293083 [119890minus29729119890minus04119905
4420611989044594119890minus04
] cos (120587119910)
sdot sin(1205871199095
)
119879 = 404676 [119890minus07369119905
] cos (120587119910) sin(1205871199095
)
120590119910119910= 119910[102767 times 02056119890
1205871199105+ 450219
times 02056119890minus1205871199105
] minus 59124 cos (120587119910) sin(1205871199095
)
120590119909119910= [(minus2318346 + 1456616119910) 119890
1205871199105+ (716544
minus 450205119910) 119890minus1205871199105
+ 555858 sin (120587119910)] cos(1205871199095
)
120590119909119909= [(minus4637527 + minus3690536119910) 119890
1205871199105
+ (1433047 minus 1140416119910) 119890minus1205871199105
+ 2779289]
sdot sin(1205871199095
)
(21)
6 Graphical Interpretation
See Figures 1ndash7
7 Discussion
In this article the three-dimensional nonhomogeneous heatconduction issue in a thin rectangular plate is studiedWe didnumerical computations for a thin rectangular plate made upof aluminumThe heat source 119892(119909 119910 119911 119905) is an instantaneouspoint heat source of strength 119892
119894 The thermoelastic behavior
is examined such as temperature and thermal stresses
0 05 1 15 2 25 30
05
1
15
2
25
3
35
4
45
120590yy
x-axis
Figure 2 120590119910119910
versus 119909
0 05 1 15 2 25 3minus150
minus100
minus50
0
50120590xy
x-axis
Figure 3 119909 versus 120590119909119910
0 05 1 15 2 25 3minus90
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
120590xx
x-axis
Figure 4 119909 versus 120590119909119909
6 Indian Journal of Materials Science
minus5 0 5 10 15 20 25 30 35 40 450
05
1
15
2
25
3
120590yy
y-axis
Figure 5 120590119910119910
versus 119910
minus100 0 100 200 300 400 500 600 700 8000
05
1
15
2
25
3
120590xy
y-axis
Figure 6 120590119909119910
versus 119910
minus9000 minus8000 minus7000 minus6000 minus5000 minus4000 minus3000 minus2000 minus1000 00
05
1
15
2
25
3
120590xx
y
Figure 7 120590119909119909
versus 119910
From Figure 1 it is found that at first when time iszero temperature is shrinking But as time increasestemperature develops as much as precise restrictionand it turns out to be regularFrom Figure 2 interatomic distance grew to beextensive up to precise value of119909 after specific value of119909 interatomic distance in a plate takes its function as itis When we provide temperature to aluminium plateinitially atoms in a plate get disturbed that is shortstress increases along 119909-axisThermal stress increasesinitially but it is observed that it remains constant andagain slightly decreasesFrom Figure 3 as temperature rises molecule beginsto vibrate more rapidly and push away from oneanother to increase separation between the atoms thatcause expansion in atoms In a plate position of atomsgets separated along 119909-axisThermal stress ofmaterialchanges from minimum of 119909 to maximum that isvariation observed along both 119909-axis and 119910-axisFrom Figure 4 initially when 119909 is zero stress is zerothat is in a plate interatomic constitution is constantbut when we change the worth of 119909 atomic distancegets compressed and at 119909 = 25 it turns into extracompression and again interatomic distance slowlyseparated Then it gets its customary positionFrom Figure 5 at first interatomic distance could bemuch closed as altering the worth of 119910 that distancegrew to be vast that is stress risesFrom Figure 6 stress alongside 119910-axis atomic struc-ture in a plate is rapidly changing its positions as wechange the value of 119910From Figure 7 it shows that originally at 119910 = 0 stressis incredibly minimum suggesting that interatomicdistance is compressed as value of 119910 changes it comesto its original interatomic distance so that it acquiresits original position
8 Conclusion
In this paper we carried out the nonhomogeneous thermoe-lastic problem solved using integral transform techniquesnumerically Results are obtained dependent on values of 119897119898 and 119899 which vary from 1 to infin Hence variation of heatby moving heat sources in a body changes infinitely Fromgraphical study when a body is provided with heat it affectsit in all directions Hence material shows expansion along 119909-axis 119910-axis and 119911-axis respectively We conclude that if timeincreases temperature will also increase Interatomic distancebecamewide along119910-axis As thin rectangular plate subjectedto point heat source which changes its place along 119909-axisinteratomic distance became narrow
The outcomes got here basically applicable in engineeringproblems especially for industrial machines subjected tothe heating such as the main shaft of a machine turbinesthe roll of rolling mill and practical applications in aircraftstructures
Indian Journal of Materials Science 7
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Nowinski Theory of Thermoelasticity with ApplicationSijthoff amp Noordhoff Alphen Aan Den Rijn The Netherlands1978
[2] W Nowacki ldquoThe state of stresses in a thick circular plate dueto temperature fieldrdquo Bulletin of the Polish Academy of SciencesTechnical Science vol 5 article 227 1957
[3] B A Boley and J H Weiner Theory of Thermal Stresses JohnWiley amp Sons New York NY USA 1960
[4] S K Roy Choudhary ldquoA note of quasi static stress in a thincircular plate due to transient temperature applied along the cir-cumference of a circle over the upper facerdquo Bulletin LrsquoAcademiePolonaise des Science Serie des Sciences Mathematiques vol 20pp 20ndash21 1972
[5] N W Khobragade and P C Wankhede ldquoAn inverse unsteadystate thermoelastic problem of a thin rectangular platerdquo TheJournal of Indian Academy of Mathematics vol 25 no 2 2003
[6] K R Gaikwad and K P Ghadle ldquoQuasi-static thermoelasticproblem of an infinitely long circular cylinderrdquo Journal of theKorean Society for Industrial and Applied Mathematics vol 14no 3 pp 141ndash149 2010
[7] V B Patil B R Ahirrao and N W Khobragade ldquoThermalstresses of semi infinite Rectangular slab with internal heatsourcerdquo IOSR Journals of Mathematics vol 8 no 6 pp 57ndash612013
[8] D T Solanke and M H Durge ldquoQuasi static thermal stressesin thin rectangular plate with internal moving line heat sourcerdquoScience Park Research Journal vol 1 no 44 pp 1ndash5 2014
[9] M S Thakare C S Sutar and N W Khobragade ldquoThermalstresses of a thin rectangular plate with internal moving heatsourcerdquo International Journal of Engineering and InnovativeTechnology vol 4 no 9 2015
[10] K R Gaikwad ldquoTwo-dimensional steady-state temperaturedistribution of a thin circular plate due to uniform internalenergy generationrdquo Cogent Mathematics vol 3 no 1 Article ID1135720 2016
[11] N M Ozisik Boundary Value Problem of Heat ConductionDover Mineola NY USA 1968
[12] N Noda R B Hetnarski and Y Tanigawa Thermal StressesLastran 2nd edition 2002
[13] I N SneddonTheUse of Integral Transform McGrawHill NewYork NY USA 1972
[14] E Marchi and A Fasulo ldquoHeat conduction in sectors of hollowcylinders with radiationrdquo Atti della Accademia delle Scienze diTorino no 1 pp 373ndash382 1967
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Indian Journal of Materials Science 5
minus35 minus30 minus25 minus20 minus15 minus10 minus5 00
05
1
15
2
25
3
Time(
t)
Temperature (T)
Figure 1 Temperature versus time
where 119909119868 = 15 119892119894119901= 1 119905 = 1 119905119868 = 15 120582
119897= 155485 119901
119897(02) =
23562 120593 = 02056 119876 = 124912 0 = minus45920 and
119879 = 293083 [119890minus29729119890minus04119905
4420611989044594119890minus04
] cos (120587119910)
sdot sin(1205871199095
)
119879 = 404676 [119890minus07369119905
] cos (120587119910) sin(1205871199095
)
120590119910119910= 119910[102767 times 02056119890
1205871199105+ 450219
times 02056119890minus1205871199105
] minus 59124 cos (120587119910) sin(1205871199095
)
120590119909119910= [(minus2318346 + 1456616119910) 119890
1205871199105+ (716544
minus 450205119910) 119890minus1205871199105
+ 555858 sin (120587119910)] cos(1205871199095
)
120590119909119909= [(minus4637527 + minus3690536119910) 119890
1205871199105
+ (1433047 minus 1140416119910) 119890minus1205871199105
+ 2779289]
sdot sin(1205871199095
)
(21)
6 Graphical Interpretation
See Figures 1ndash7
7 Discussion
In this article the three-dimensional nonhomogeneous heatconduction issue in a thin rectangular plate is studiedWe didnumerical computations for a thin rectangular plate made upof aluminumThe heat source 119892(119909 119910 119911 119905) is an instantaneouspoint heat source of strength 119892
119894 The thermoelastic behavior
is examined such as temperature and thermal stresses
0 05 1 15 2 25 30
05
1
15
2
25
3
35
4
45
120590yy
x-axis
Figure 2 120590119910119910
versus 119909
0 05 1 15 2 25 3minus150
minus100
minus50
0
50120590xy
x-axis
Figure 3 119909 versus 120590119909119910
0 05 1 15 2 25 3minus90
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
120590xx
x-axis
Figure 4 119909 versus 120590119909119909
6 Indian Journal of Materials Science
minus5 0 5 10 15 20 25 30 35 40 450
05
1
15
2
25
3
120590yy
y-axis
Figure 5 120590119910119910
versus 119910
minus100 0 100 200 300 400 500 600 700 8000
05
1
15
2
25
3
120590xy
y-axis
Figure 6 120590119909119910
versus 119910
minus9000 minus8000 minus7000 minus6000 minus5000 minus4000 minus3000 minus2000 minus1000 00
05
1
15
2
25
3
120590xx
y
Figure 7 120590119909119909
versus 119910
From Figure 1 it is found that at first when time iszero temperature is shrinking But as time increasestemperature develops as much as precise restrictionand it turns out to be regularFrom Figure 2 interatomic distance grew to beextensive up to precise value of119909 after specific value of119909 interatomic distance in a plate takes its function as itis When we provide temperature to aluminium plateinitially atoms in a plate get disturbed that is shortstress increases along 119909-axisThermal stress increasesinitially but it is observed that it remains constant andagain slightly decreasesFrom Figure 3 as temperature rises molecule beginsto vibrate more rapidly and push away from oneanother to increase separation between the atoms thatcause expansion in atoms In a plate position of atomsgets separated along 119909-axisThermal stress ofmaterialchanges from minimum of 119909 to maximum that isvariation observed along both 119909-axis and 119910-axisFrom Figure 4 initially when 119909 is zero stress is zerothat is in a plate interatomic constitution is constantbut when we change the worth of 119909 atomic distancegets compressed and at 119909 = 25 it turns into extracompression and again interatomic distance slowlyseparated Then it gets its customary positionFrom Figure 5 at first interatomic distance could bemuch closed as altering the worth of 119910 that distancegrew to be vast that is stress risesFrom Figure 6 stress alongside 119910-axis atomic struc-ture in a plate is rapidly changing its positions as wechange the value of 119910From Figure 7 it shows that originally at 119910 = 0 stressis incredibly minimum suggesting that interatomicdistance is compressed as value of 119910 changes it comesto its original interatomic distance so that it acquiresits original position
8 Conclusion
In this paper we carried out the nonhomogeneous thermoe-lastic problem solved using integral transform techniquesnumerically Results are obtained dependent on values of 119897119898 and 119899 which vary from 1 to infin Hence variation of heatby moving heat sources in a body changes infinitely Fromgraphical study when a body is provided with heat it affectsit in all directions Hence material shows expansion along 119909-axis 119910-axis and 119911-axis respectively We conclude that if timeincreases temperature will also increase Interatomic distancebecamewide along119910-axis As thin rectangular plate subjectedto point heat source which changes its place along 119909-axisinteratomic distance became narrow
The outcomes got here basically applicable in engineeringproblems especially for industrial machines subjected tothe heating such as the main shaft of a machine turbinesthe roll of rolling mill and practical applications in aircraftstructures
Indian Journal of Materials Science 7
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Nowinski Theory of Thermoelasticity with ApplicationSijthoff amp Noordhoff Alphen Aan Den Rijn The Netherlands1978
[2] W Nowacki ldquoThe state of stresses in a thick circular plate dueto temperature fieldrdquo Bulletin of the Polish Academy of SciencesTechnical Science vol 5 article 227 1957
[3] B A Boley and J H Weiner Theory of Thermal Stresses JohnWiley amp Sons New York NY USA 1960
[4] S K Roy Choudhary ldquoA note of quasi static stress in a thincircular plate due to transient temperature applied along the cir-cumference of a circle over the upper facerdquo Bulletin LrsquoAcademiePolonaise des Science Serie des Sciences Mathematiques vol 20pp 20ndash21 1972
[5] N W Khobragade and P C Wankhede ldquoAn inverse unsteadystate thermoelastic problem of a thin rectangular platerdquo TheJournal of Indian Academy of Mathematics vol 25 no 2 2003
[6] K R Gaikwad and K P Ghadle ldquoQuasi-static thermoelasticproblem of an infinitely long circular cylinderrdquo Journal of theKorean Society for Industrial and Applied Mathematics vol 14no 3 pp 141ndash149 2010
[7] V B Patil B R Ahirrao and N W Khobragade ldquoThermalstresses of semi infinite Rectangular slab with internal heatsourcerdquo IOSR Journals of Mathematics vol 8 no 6 pp 57ndash612013
[8] D T Solanke and M H Durge ldquoQuasi static thermal stressesin thin rectangular plate with internal moving line heat sourcerdquoScience Park Research Journal vol 1 no 44 pp 1ndash5 2014
[9] M S Thakare C S Sutar and N W Khobragade ldquoThermalstresses of a thin rectangular plate with internal moving heatsourcerdquo International Journal of Engineering and InnovativeTechnology vol 4 no 9 2015
[10] K R Gaikwad ldquoTwo-dimensional steady-state temperaturedistribution of a thin circular plate due to uniform internalenergy generationrdquo Cogent Mathematics vol 3 no 1 Article ID1135720 2016
[11] N M Ozisik Boundary Value Problem of Heat ConductionDover Mineola NY USA 1968
[12] N Noda R B Hetnarski and Y Tanigawa Thermal StressesLastran 2nd edition 2002
[13] I N SneddonTheUse of Integral Transform McGrawHill NewYork NY USA 1972
[14] E Marchi and A Fasulo ldquoHeat conduction in sectors of hollowcylinders with radiationrdquo Atti della Accademia delle Scienze diTorino no 1 pp 373ndash382 1967
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
6 Indian Journal of Materials Science
minus5 0 5 10 15 20 25 30 35 40 450
05
1
15
2
25
3
120590yy
y-axis
Figure 5 120590119910119910
versus 119910
minus100 0 100 200 300 400 500 600 700 8000
05
1
15
2
25
3
120590xy
y-axis
Figure 6 120590119909119910
versus 119910
minus9000 minus8000 minus7000 minus6000 minus5000 minus4000 minus3000 minus2000 minus1000 00
05
1
15
2
25
3
120590xx
y
Figure 7 120590119909119909
versus 119910
From Figure 1 it is found that at first when time iszero temperature is shrinking But as time increasestemperature develops as much as precise restrictionand it turns out to be regularFrom Figure 2 interatomic distance grew to beextensive up to precise value of119909 after specific value of119909 interatomic distance in a plate takes its function as itis When we provide temperature to aluminium plateinitially atoms in a plate get disturbed that is shortstress increases along 119909-axisThermal stress increasesinitially but it is observed that it remains constant andagain slightly decreasesFrom Figure 3 as temperature rises molecule beginsto vibrate more rapidly and push away from oneanother to increase separation between the atoms thatcause expansion in atoms In a plate position of atomsgets separated along 119909-axisThermal stress ofmaterialchanges from minimum of 119909 to maximum that isvariation observed along both 119909-axis and 119910-axisFrom Figure 4 initially when 119909 is zero stress is zerothat is in a plate interatomic constitution is constantbut when we change the worth of 119909 atomic distancegets compressed and at 119909 = 25 it turns into extracompression and again interatomic distance slowlyseparated Then it gets its customary positionFrom Figure 5 at first interatomic distance could bemuch closed as altering the worth of 119910 that distancegrew to be vast that is stress risesFrom Figure 6 stress alongside 119910-axis atomic struc-ture in a plate is rapidly changing its positions as wechange the value of 119910From Figure 7 it shows that originally at 119910 = 0 stressis incredibly minimum suggesting that interatomicdistance is compressed as value of 119910 changes it comesto its original interatomic distance so that it acquiresits original position
8 Conclusion
In this paper we carried out the nonhomogeneous thermoe-lastic problem solved using integral transform techniquesnumerically Results are obtained dependent on values of 119897119898 and 119899 which vary from 1 to infin Hence variation of heatby moving heat sources in a body changes infinitely Fromgraphical study when a body is provided with heat it affectsit in all directions Hence material shows expansion along 119909-axis 119910-axis and 119911-axis respectively We conclude that if timeincreases temperature will also increase Interatomic distancebecamewide along119910-axis As thin rectangular plate subjectedto point heat source which changes its place along 119909-axisinteratomic distance became narrow
The outcomes got here basically applicable in engineeringproblems especially for industrial machines subjected tothe heating such as the main shaft of a machine turbinesthe roll of rolling mill and practical applications in aircraftstructures
Indian Journal of Materials Science 7
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Nowinski Theory of Thermoelasticity with ApplicationSijthoff amp Noordhoff Alphen Aan Den Rijn The Netherlands1978
[2] W Nowacki ldquoThe state of stresses in a thick circular plate dueto temperature fieldrdquo Bulletin of the Polish Academy of SciencesTechnical Science vol 5 article 227 1957
[3] B A Boley and J H Weiner Theory of Thermal Stresses JohnWiley amp Sons New York NY USA 1960
[4] S K Roy Choudhary ldquoA note of quasi static stress in a thincircular plate due to transient temperature applied along the cir-cumference of a circle over the upper facerdquo Bulletin LrsquoAcademiePolonaise des Science Serie des Sciences Mathematiques vol 20pp 20ndash21 1972
[5] N W Khobragade and P C Wankhede ldquoAn inverse unsteadystate thermoelastic problem of a thin rectangular platerdquo TheJournal of Indian Academy of Mathematics vol 25 no 2 2003
[6] K R Gaikwad and K P Ghadle ldquoQuasi-static thermoelasticproblem of an infinitely long circular cylinderrdquo Journal of theKorean Society for Industrial and Applied Mathematics vol 14no 3 pp 141ndash149 2010
[7] V B Patil B R Ahirrao and N W Khobragade ldquoThermalstresses of semi infinite Rectangular slab with internal heatsourcerdquo IOSR Journals of Mathematics vol 8 no 6 pp 57ndash612013
[8] D T Solanke and M H Durge ldquoQuasi static thermal stressesin thin rectangular plate with internal moving line heat sourcerdquoScience Park Research Journal vol 1 no 44 pp 1ndash5 2014
[9] M S Thakare C S Sutar and N W Khobragade ldquoThermalstresses of a thin rectangular plate with internal moving heatsourcerdquo International Journal of Engineering and InnovativeTechnology vol 4 no 9 2015
[10] K R Gaikwad ldquoTwo-dimensional steady-state temperaturedistribution of a thin circular plate due to uniform internalenergy generationrdquo Cogent Mathematics vol 3 no 1 Article ID1135720 2016
[11] N M Ozisik Boundary Value Problem of Heat ConductionDover Mineola NY USA 1968
[12] N Noda R B Hetnarski and Y Tanigawa Thermal StressesLastran 2nd edition 2002
[13] I N SneddonTheUse of Integral Transform McGrawHill NewYork NY USA 1972
[14] E Marchi and A Fasulo ldquoHeat conduction in sectors of hollowcylinders with radiationrdquo Atti della Accademia delle Scienze diTorino no 1 pp 373ndash382 1967
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Indian Journal of Materials Science 7
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Nowinski Theory of Thermoelasticity with ApplicationSijthoff amp Noordhoff Alphen Aan Den Rijn The Netherlands1978
[2] W Nowacki ldquoThe state of stresses in a thick circular plate dueto temperature fieldrdquo Bulletin of the Polish Academy of SciencesTechnical Science vol 5 article 227 1957
[3] B A Boley and J H Weiner Theory of Thermal Stresses JohnWiley amp Sons New York NY USA 1960
[4] S K Roy Choudhary ldquoA note of quasi static stress in a thincircular plate due to transient temperature applied along the cir-cumference of a circle over the upper facerdquo Bulletin LrsquoAcademiePolonaise des Science Serie des Sciences Mathematiques vol 20pp 20ndash21 1972
[5] N W Khobragade and P C Wankhede ldquoAn inverse unsteadystate thermoelastic problem of a thin rectangular platerdquo TheJournal of Indian Academy of Mathematics vol 25 no 2 2003
[6] K R Gaikwad and K P Ghadle ldquoQuasi-static thermoelasticproblem of an infinitely long circular cylinderrdquo Journal of theKorean Society for Industrial and Applied Mathematics vol 14no 3 pp 141ndash149 2010
[7] V B Patil B R Ahirrao and N W Khobragade ldquoThermalstresses of semi infinite Rectangular slab with internal heatsourcerdquo IOSR Journals of Mathematics vol 8 no 6 pp 57ndash612013
[8] D T Solanke and M H Durge ldquoQuasi static thermal stressesin thin rectangular plate with internal moving line heat sourcerdquoScience Park Research Journal vol 1 no 44 pp 1ndash5 2014
[9] M S Thakare C S Sutar and N W Khobragade ldquoThermalstresses of a thin rectangular plate with internal moving heatsourcerdquo International Journal of Engineering and InnovativeTechnology vol 4 no 9 2015
[10] K R Gaikwad ldquoTwo-dimensional steady-state temperaturedistribution of a thin circular plate due to uniform internalenergy generationrdquo Cogent Mathematics vol 3 no 1 Article ID1135720 2016
[11] N M Ozisik Boundary Value Problem of Heat ConductionDover Mineola NY USA 1968
[12] N Noda R B Hetnarski and Y Tanigawa Thermal StressesLastran 2nd edition 2002
[13] I N SneddonTheUse of Integral Transform McGrawHill NewYork NY USA 1972
[14] E Marchi and A Fasulo ldquoHeat conduction in sectors of hollowcylinders with radiationrdquo Atti della Accademia delle Scienze diTorino no 1 pp 373ndash382 1967
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials