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

Submit your manuscripts athttpwwwhindawicom

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Biomaterials

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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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BioMed Research International

MaterialsJournal of

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Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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