Hindawi Publishing CorporationJournal of MaterialsVolume 2013 Article ID 809205 9 pageshttpdxdoiorg1011552013809205
Research ArticleFinite Difference Solution of Elastic-Plastic Thin RotatingAnnular Disk with Exponentially Variable Thickness andExponentially Variable Density
Sanjeev Sharma and Yadav Sanehlata
Department of Mathematics Jaypee Institute of Information Technology A-10 Sector 62 Noida-201307 Uttar Pradesh India
Correspondence should be addressed to Sanjeev Sharma sanjeevsharmajiitacin
Received 10 December 2012 Accepted 14 February 2013
Academic Editor Francois Peeters
Copyright copy 2013 S Sharma and Y Sanehlata This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Elastic-plastic stresses strains and displacements have been obtained for a thin rotating annular disk with exponentially variablethickness and exponentially variable density with nonlinear strain hardeningmaterial by finite differencemethod using Von-Misesrsquoyield criterion Results have been computed numerically and depicted graphically From the numerical results it can be concludedthat disk whose thickness decreases radially and density increases radially is on the safer side of design as compared to the diskwith exponentially varying thickness and exponentially varying density as well as to flat disk
1 Introduction
Due to wide applications of rotating disk circular diskspherical shells cylinders and shafts in engineering elastic-plastic analysis of rotating disk becoming more and moreactive topic in the field of solid mechanics The research onthem is always an important topic and their benefits havebeen included in some books [1ndash3] Sharma et al [4] uses theconcept of transition theory to evaluate the stresses for diskwith different parameters like variable thickness and variabledensities The elastic-plastic strain hardening problems ofannular diskswith constant thickness under external pressurewere firstly studied by Gamer [5] using linear stress-plasticstrain relation Later on the work was extended to annulardisk with some especially thickness functions by Guven[6] The elastic-plastic linear strain hardening problems ofrotating annular disks subjected to angular velocity are easierto deal with as compared to nonlinear strain hardeningproblems of annular disks Therefore main focus of researchnow these days is on rotating annular disk made of strainhardeningmaterial In contrast very few researchers are therewho investigated elastic-plastic deformation and stresses forrotating disks with nonlinear strain hardening
The obvious advantage using a linear strain hardeningstress-plastic strain is that a closed form solution can beobtained for annular disks with constant thickness and someespecially variable thickness functions However most ofthe materials exhibit nonlinear strain hardening behaviorthus this nonlinearity is obvious in the transition regionfrom elastic to plastic parts of stress-strain curve Due tothe previously reason a polynomial stress-strain relationof nonlinear strain hardening material is proposed in thepapers of You et al [7ndash9] Numerical method such as finitedifference method is an effective technique to the stressesand strains for these rotating disks However for scientificresearch and engineering analysis analytical methods andnumerical methods are still very active Therefore Sterner etal [10] proposed truncated Taylorrsquos series numerical methodto solve elastic problems of rotating disk with arbitraryvariable thickness You and Zhang [8] examine elastic-plasticstresses in rotating disks using Runge-Kuttarsquos method
In this paper we proposed a more straightforward andmost effective numerical method such as finite differencemethod which is most celebrated method to solve boundaryvalue problems The proposed method is used to analyze thestresses strains and displacements for annular disk having
2 Journal of Materials
FDMRKM119899 = 07 119898 = 07
119899 = 05 119898 = 05
119899 = 02 119898 = 01
0
05
1
15
2
25
3
35
4times10
8 Plot of stresses 119896 = 1 119905 = 05 ang vel = 500
01 015 02 025 03 035 04 045 05119903
Stre
sses
(119879119903119903
)(119879
119903119903)
(119879119903119903
)
(119879120579120579
)(119879
120579120579)
(119879120579120579
)
(a)
0
1
2
3
4
5
6
7
8
9Plot of stresses 119896 = 1 119905 = 05 ang vel = 780
Stre
sses
01 015 02 025 03 035 04 045 05119903
times108
FDMRKM119899 = 07 119898 = 07
119899 = 05 119898 = 05
119899 = 02 119898 = 01
(119879119903119903
)
(119879119903119903
)(119879
119903119903)
(119879120579120579
)
(119879120579120579
)
(119879120579120579
)
(b)
Figure 1 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with exponentially variable thickness and exponentially variable
density with 119899 = 07 02 05 and119898 = 07 01 05 under (a) 500 rads and (b) 780 rads for both approaches
exponential variable thickness and exponential variable den-sity with nonlinear strain hardening material behavior
2 Mathematical Formulation
Assuming that the stresses vary over the thickness of the diskthe theory of the disks of variable thickness can give goodresult as that of the disks of constant thickness as long asthey meet the assumption of plane stress For disk profile itis assumed that disk is symmetric with respect to the midplane This profile is defined by the thickness function ℎ(119903)
By the nonlinear strain-hardening material model pro-posed by You andZhang [8] the stress-strain relationship canbe written as
119890119890=
(120582 + 120583) 119879119890119890
120583 (3120582 + 2120583) 119890119890le 1198900
119890119901
119890= 11988611198793
119890119890+ 11988621198795
119890119890 119890119890gt 1198900
(11)
where 119890119890is the equivalent total strain and 119890
0the yield strain
and 1198861 1198862are material constants
Journal of Materials 5
0
05
1
15
2
25
FDM119898 = 01 119899 = 02
119898 = 05 119899 = 05
119898 = 07 119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
Plot of stresses angular velocity = 500
(a)
FDM119898 = 01 119899 = 02
119898 = 05 119899 = 05
119898 = 07 119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
1
2
3
4
5
6
Stre
sses
times108
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
Plot of stresses angular velocity = 780
(b)
Figure 4 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with 119899 = 02 05 07
and119898 = 01 05 07 under (a) 500 rads and (b) 780 rads for both approaches
Substitution of the second equation of (11) into (6) theplastic strains can be written as
119890119901
119903=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879119903119903minus1
2119879120579120579)
119890119901
120579=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879120579120579minus1
2119879119903119903)
(12)
The governing equation in the plastic region of therotating disks in terms of stresses and stress function can beobtained by substituting (12) into (10) as
times (2119879119903119903minus 119879120579120579) [1199031206011015840
minus (1 minus 119896119899(119903
119887)
119899
)120601]
+ (2119879120579120579minus 119879119903119903)
times (119903119896119899(119903
119887)
119899
1206011015840
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
)
+ (11988611198792
119890119890+ 11988621198794
119890119890)
times minus(1
2minus 119896119899(
119903
119887)
119899
) 1199031206011015840
+1
2(1 minus 119896119899(
119903
119887)
119899
)120601
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
+3
2119903ℎ0119890minus119896(119903119887)
119899
(119879120579120579minus 119879119903119903)
(13)
The values of the stress function 120601 at the elastic-plasticinterface radius are the same and therefore the stressfunction 120601 is continuous at the interface radius [9] It canbe seen from the continuity of the stress function and (3)
6 Journal of Materials
Plot of stresses 119898 = 01 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(a)
002040608
112141618
2times10
8
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119898 = 05 angular velocity = 500
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119898 = 07 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses (119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(c)
Figure 5 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119899 =
07 02 05 and119898 = 01 (b) 119899 = 07 02 05 and119898 = 05 and (c) 119899 = 07 02 05 and119898 = 07 under 500 rads for both approaches
(5)ndash(7) and (8) that the continuity conditions of the stressesand displacements at the elastic-plastic interface radius aresatisfied
The boundary conditions for the rotating annular disksare
119879119903119903= 0 at 119903 = 119886 119879
119903119903= 0 at 119903 = 119887 (14)
where 119886 and 119887 are the inner and outer radii of the rotatingdisks respectively
3 Finite Difference Algorithm
Todetermine the elastic-plastic stresses strains and displace-ment in thin rotating disks with a nonlinear strain hardeningmaterial we have to solve the second-order nonlinear differ-ential equation (13) under the given boundary condition (14)The general form of (13) can be written as
12060110158401015840
= 119891 (119903 120601 1206011015840
) (15)
31 Finite Difference Algorithm Steps The second-order dif-ferential equation (13) with the given boundary conditionscan be solved by using finite difference method
Journal of Materials 7
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119899 = 02 angular velocity = 500
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
(a)
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses
Plot of stresses 119899 = 05 angular velocity = 500
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119899 = 07 angular velocity = 500times107
0
2
4
6
8
10
12
14
16
18
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
(119879119903119903)(119879119903119903)
(119879119903119903)
Stre
sses
(119879120579120579)
(119879120579120579)
(119879120579120579)
(c)
Figure 6 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119898 =
01 05 07 and 119899 = 02 (b)119898 = 01 05 07 and 119899 = 05 and (c)119898 = 01 05 07 and 119899 = 07 under 500 rads for both approaches
(i) First partition the domain [01 05] into 119901 subinter-vals of length ℎ
(ii) To express the differential operators 119889120601119889119903 and1198892
1206011198891199032 in a discrete form we use the finite difference
approximations as
1198892
120601
1198891199032=120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
119889120601
119889119903=120601119894+1
minus 120601119894minus1
2ℎ (16)
(iii) With ℎ = 1119901 we have 119901 + 1 nodal points1206011 1206012 120601
119901+1 The values at the end points are
given by the boundary conditions that is we aregiven that 120593
1= 0 120593
119901+1= 0 Using the finite
difference approximation we get the following systemof equations
120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
= 119891(119903 120601119894120601119894+1
minus 120601119894minus1
2ℎ) 119894 = 2 3 119899
(17)
(iv) After simplifying and collecting coefficients of120601119894+1
120601119894
and120601119894minus1
in Step 3 the boundary value problem resultsin system of nonlinear equations
(v) The solution of the (119901 minus 1) system of nonlinearequations can be obtained using Newton-Raphsonrsquos
8 Journal of Materials
FDM119899 = 0 119898 = 0
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
Plot of stresses angular velocity = 500
0
05
1
15
2
25
(119879119903119903)
(119879120579120579)
(a)
01 015 02 025 03 035 04 045 050
1
2
3
4
5
6
119903
Stre
sses
times108
Plot of stresses angular velocity = 780
FDM119899 = 0 119898 = 0
RKM(b)
Figure 7 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with constant thickness and constant density under (a) 500 rads
and (b) 780 rads for both approaches
method where the unknowns are 1206012 1206013 120601
119901minus1 By
this process we will get the stress function and thenstresses strains and displacement
4 Numerical Illustration and Discussion
A two-dimensional plane stress analysis of rotating disk withnonuniform thickness and nonuniform density are carriedout using finite difference method The radius of the rotatingdisk is taken to be 119887 = 05 and its materials propertiesare material density 120588 = 8000Kgm3 Lamersquos constant are120582 = 7961GPa 120583 = 11942GPa and Poissonrsquos ratio ] = 03
41 Rotating Annular Disk with Exponentially Variable Thick-ness and Exponentially Variable Density It has been observedfrom Figures 1(a) and 1(b) that as thickness decreases anddensity increases radially circumferential stress is going ondecreasing Also circumferential stress is maximum at theinternal surface With the increase in angular velocity cir-cumferential stress is going on increasing Figures 2(a) 2(b)and 2(c) are sketched for uniform density and we observedthat circumferential stresses are maximum at internal surfaceand these stresses decreases with decrease in thickness radi-ally As with the increase in density radially circumferentialgoes on decreasingwhich gives appropriate design of the diskIt has been observed from Figures 3(a) 3(b) and 3(c) thatfor uniform thickness circumferential stress is maximumat internal surface Also circumferential stress for the diskwhose density decreases radially is maximum as comparedto the disk with high density Also decrease in thicknessradially yields decrease in circumferential stress which givesappropriate design of disk
From all previous analysis we can conclude that diskwhose thickness decreases radially and density increases
radially is on the safer side of the design as compared to otherthickness and density parameters because circumferentialstress is less for the disk whose thickness decreases radiallyand density increases radially as compared to other thicknessand density parameters
42 Rotating Annular Disk with Variable Thickness andVariable Density Using Power Law (ℎ = ℎ
0(119903119887)minus119899
120588 =
1205880(119903119887)119898
) It has been observed from Figures 4(a) and 4(b)that circumferential stress is maximum at internal surfaceCircumferential stress ismaximum for the diskwhose densityis less and thickness is high as compared to the disk whosethickness is less and density is high Also it has beenobserved from the figure that with the increase in angularspeed circumferential stress goes on increasing but againcircumferential stress is less as compared to other cases Ithas been observed from Figures 5(a) 5(b) and 5(c) thatfor uniform density disk circumferential stress is maximumat internal surface With the decrease in thickness radiallycircumferential stress goes on decreasing Also with theincrease in density circumferential stress further decreaseswhich provides an appropriate design of the disk It has beenobserved from Figures 6(a) 6(b) and 6(c) that for uniformthickness circumferential stresses are maximum at internalsurface Also it has been noted that circumferential stress isless for disk profile with high density as compared to diskprofiles with less density With the decrease in thicknessradially circumferential stress decreases which yields thatdisk whose thickness is less and density is high is appropriatefor the design as compared to other disk profiles
From the previous analysis we observed that disk withhigh density and less thickness is on the safer side of thedesign as compared to other parameters because circumfer-ential stress is less for previous case as compared to other
Journal of Materials 9
disk profilesThe results calculated for variable thickness andvariable density (ℎ = ℎ
0(119903119887)minus119899
120588 = 1205880(119903119887)119898
) using finitedifference method are in very good agreement with Runge-Kuttarsquos method studied by You et al [9] for all the rotatingdisks
43 Annular Disk with Constant Thickness and ConstantDensity It has been observed from Figures 7(a) and 7(b)that for constant thickness and constant density circumfer-ential stress is maximum at internal surface Also with theincrease in angular speed circumferential stress increasessignificantly
5 Conclusion
After analyzing all the three disk profiles it can be concludedthat circumferential stresses aremaximumat internal surfaceIt is also concluded that disk whose thickness decreasesradially and density increases radially is on the safer side ofdesign as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk This is because circumferential stresses for the diskswhose thickness decreases and density increases radially areless as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk
References
[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970
[2] S C Ugural and S K Fenster Advanced Strength and AppliedElasticity Elsevier New York NY USA 1987
[3] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998
[4] S Sharama M Sahni and R Kumar ldquoElastic-plastic analysisof a thin rotating disk of exponentially variable thickness withinclusionrdquo WSEAS Transactions on Mathematics vol 9 no 5pp 314ndash323 2010
[5] U Gamer ldquoElastic-plastic deformation of the rotating soliddiskrdquo Ingenieur-Archiv vol 54 no 5 pp 345ndash354 1984
[6] U Guven ldquoElastic-plastic stresses in a rotating annular disk ofvariable thickness and variable densityrdquo International Journal ofMechanical Sciences vol 34 no 2 pp 133ndash138 1992
[7] L H You S Y Long and J J Zhang ldquoPerturbation solution ofrotating solid disks with nonlinear strain-hardeningrdquoMechan-ics Research Communications vol 24 no 6 pp 649ndash658 1997
[8] L H You and J J Zhang ldquoElastic-plastic stresses in a rotatingsolid diskrdquo International Journal of Mechanical Sciences vol 41no 3 pp 269ndash282 1999
[9] L H You Y Y Tang J J Zhang and C Y Zheng ldquoNumericalanalysis of elastic-plastic rotating disks with arbitrary variablethickness and densityrdquo International Journal of Solids andStructures vol 37 no 52 pp 7809ndash7820 2000
[10] S C Sterner S Saigal W Kistler and D E Dietrich ldquoA unifiednumerical approach for the analysis of rotating disks includingturbine rotorsrdquo International Journal of Solids and Structuresvol 31 no 2 pp 269ndash277 1994
8 Plot of stresses 119896 = 1 119905 = 05 ang vel = 500
01 015 02 025 03 035 04 045 05119903
Stre
sses
(119879119903119903
)(119879
119903119903)
(119879119903119903
)
(119879120579120579
)(119879
120579120579)
(119879120579120579
)
(a)
0
1
2
3
4
5
6
7
8
9Plot of stresses 119896 = 1 119905 = 05 ang vel = 780
Stre
sses
01 015 02 025 03 035 04 045 05119903
times108
FDMRKM119899 = 07 119898 = 07
119899 = 05 119898 = 05
119899 = 02 119898 = 01
(119879119903119903
)
(119879119903119903
)(119879
119903119903)
(119879120579120579
)
(119879120579120579
)
(119879120579120579
)
(b)
Figure 1 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with exponentially variable thickness and exponentially variable
density with 119899 = 07 02 05 and119898 = 07 01 05 under (a) 500 rads and (b) 780 rads for both approaches
exponential variable thickness and exponential variable den-sity with nonlinear strain hardening material behavior
2 Mathematical Formulation
Assuming that the stresses vary over the thickness of the diskthe theory of the disks of variable thickness can give goodresult as that of the disks of constant thickness as long asthey meet the assumption of plane stress For disk profile itis assumed that disk is symmetric with respect to the midplane This profile is defined by the thickness function ℎ(119903)
By the nonlinear strain-hardening material model pro-posed by You andZhang [8] the stress-strain relationship canbe written as
119890119890=
(120582 + 120583) 119879119890119890
120583 (3120582 + 2120583) 119890119890le 1198900
119890119901
119890= 11988611198793
119890119890+ 11988621198795
119890119890 119890119890gt 1198900
(11)
where 119890119890is the equivalent total strain and 119890
0the yield strain
and 1198861 1198862are material constants
Journal of Materials 5
0
05
1
15
2
25
FDM119898 = 01 119899 = 02
119898 = 05 119899 = 05
119898 = 07 119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
Plot of stresses angular velocity = 500
(a)
FDM119898 = 01 119899 = 02
119898 = 05 119899 = 05
119898 = 07 119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
1
2
3
4
5
6
Stre
sses
times108
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
Plot of stresses angular velocity = 780
(b)
Figure 4 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with 119899 = 02 05 07
and119898 = 01 05 07 under (a) 500 rads and (b) 780 rads for both approaches
Substitution of the second equation of (11) into (6) theplastic strains can be written as
119890119901
119903=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879119903119903minus1
2119879120579120579)
119890119901
120579=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879120579120579minus1
2119879119903119903)
(12)
The governing equation in the plastic region of therotating disks in terms of stresses and stress function can beobtained by substituting (12) into (10) as
times (2119879119903119903minus 119879120579120579) [1199031206011015840
minus (1 minus 119896119899(119903
119887)
119899
)120601]
+ (2119879120579120579minus 119879119903119903)
times (119903119896119899(119903
119887)
119899
1206011015840
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
)
+ (11988611198792
119890119890+ 11988621198794
119890119890)
times minus(1
2minus 119896119899(
119903
119887)
119899
) 1199031206011015840
+1
2(1 minus 119896119899(
119903
119887)
119899
)120601
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
+3
2119903ℎ0119890minus119896(119903119887)
119899
(119879120579120579minus 119879119903119903)
(13)
The values of the stress function 120601 at the elastic-plasticinterface radius are the same and therefore the stressfunction 120601 is continuous at the interface radius [9] It canbe seen from the continuity of the stress function and (3)
6 Journal of Materials
Plot of stresses 119898 = 01 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(a)
002040608
112141618
2times10
8
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119898 = 05 angular velocity = 500
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119898 = 07 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses (119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(c)
Figure 5 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119899 =
07 02 05 and119898 = 01 (b) 119899 = 07 02 05 and119898 = 05 and (c) 119899 = 07 02 05 and119898 = 07 under 500 rads for both approaches
(5)ndash(7) and (8) that the continuity conditions of the stressesand displacements at the elastic-plastic interface radius aresatisfied
The boundary conditions for the rotating annular disksare
119879119903119903= 0 at 119903 = 119886 119879
119903119903= 0 at 119903 = 119887 (14)
where 119886 and 119887 are the inner and outer radii of the rotatingdisks respectively
3 Finite Difference Algorithm
Todetermine the elastic-plastic stresses strains and displace-ment in thin rotating disks with a nonlinear strain hardeningmaterial we have to solve the second-order nonlinear differ-ential equation (13) under the given boundary condition (14)The general form of (13) can be written as
12060110158401015840
= 119891 (119903 120601 1206011015840
) (15)
31 Finite Difference Algorithm Steps The second-order dif-ferential equation (13) with the given boundary conditionscan be solved by using finite difference method
Journal of Materials 7
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119899 = 02 angular velocity = 500
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
(a)
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses
Plot of stresses 119899 = 05 angular velocity = 500
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119899 = 07 angular velocity = 500times107
0
2
4
6
8
10
12
14
16
18
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
(119879119903119903)(119879119903119903)
(119879119903119903)
Stre
sses
(119879120579120579)
(119879120579120579)
(119879120579120579)
(c)
Figure 6 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119898 =
01 05 07 and 119899 = 02 (b)119898 = 01 05 07 and 119899 = 05 and (c)119898 = 01 05 07 and 119899 = 07 under 500 rads for both approaches
(i) First partition the domain [01 05] into 119901 subinter-vals of length ℎ
(ii) To express the differential operators 119889120601119889119903 and1198892
1206011198891199032 in a discrete form we use the finite difference
approximations as
1198892
120601
1198891199032=120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
119889120601
119889119903=120601119894+1
minus 120601119894minus1
2ℎ (16)
(iii) With ℎ = 1119901 we have 119901 + 1 nodal points1206011 1206012 120601
119901+1 The values at the end points are
given by the boundary conditions that is we aregiven that 120593
1= 0 120593
119901+1= 0 Using the finite
difference approximation we get the following systemof equations
120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
= 119891(119903 120601119894120601119894+1
minus 120601119894minus1
2ℎ) 119894 = 2 3 119899
(17)
(iv) After simplifying and collecting coefficients of120601119894+1
120601119894
and120601119894minus1
in Step 3 the boundary value problem resultsin system of nonlinear equations
(v) The solution of the (119901 minus 1) system of nonlinearequations can be obtained using Newton-Raphsonrsquos
8 Journal of Materials
FDM119899 = 0 119898 = 0
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
Plot of stresses angular velocity = 500
0
05
1
15
2
25
(119879119903119903)
(119879120579120579)
(a)
01 015 02 025 03 035 04 045 050
1
2
3
4
5
6
119903
Stre
sses
times108
Plot of stresses angular velocity = 780
FDM119899 = 0 119898 = 0
RKM(b)
Figure 7 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with constant thickness and constant density under (a) 500 rads
and (b) 780 rads for both approaches
method where the unknowns are 1206012 1206013 120601
119901minus1 By
this process we will get the stress function and thenstresses strains and displacement
4 Numerical Illustration and Discussion
A two-dimensional plane stress analysis of rotating disk withnonuniform thickness and nonuniform density are carriedout using finite difference method The radius of the rotatingdisk is taken to be 119887 = 05 and its materials propertiesare material density 120588 = 8000Kgm3 Lamersquos constant are120582 = 7961GPa 120583 = 11942GPa and Poissonrsquos ratio ] = 03
41 Rotating Annular Disk with Exponentially Variable Thick-ness and Exponentially Variable Density It has been observedfrom Figures 1(a) and 1(b) that as thickness decreases anddensity increases radially circumferential stress is going ondecreasing Also circumferential stress is maximum at theinternal surface With the increase in angular velocity cir-cumferential stress is going on increasing Figures 2(a) 2(b)and 2(c) are sketched for uniform density and we observedthat circumferential stresses are maximum at internal surfaceand these stresses decreases with decrease in thickness radi-ally As with the increase in density radially circumferentialgoes on decreasingwhich gives appropriate design of the diskIt has been observed from Figures 3(a) 3(b) and 3(c) thatfor uniform thickness circumferential stress is maximumat internal surface Also circumferential stress for the diskwhose density decreases radially is maximum as comparedto the disk with high density Also decrease in thicknessradially yields decrease in circumferential stress which givesappropriate design of disk
From all previous analysis we can conclude that diskwhose thickness decreases radially and density increases
radially is on the safer side of the design as compared to otherthickness and density parameters because circumferentialstress is less for the disk whose thickness decreases radiallyand density increases radially as compared to other thicknessand density parameters
42 Rotating Annular Disk with Variable Thickness andVariable Density Using Power Law (ℎ = ℎ
0(119903119887)minus119899
120588 =
1205880(119903119887)119898
) It has been observed from Figures 4(a) and 4(b)that circumferential stress is maximum at internal surfaceCircumferential stress ismaximum for the diskwhose densityis less and thickness is high as compared to the disk whosethickness is less and density is high Also it has beenobserved from the figure that with the increase in angularspeed circumferential stress goes on increasing but againcircumferential stress is less as compared to other cases Ithas been observed from Figures 5(a) 5(b) and 5(c) thatfor uniform density disk circumferential stress is maximumat internal surface With the decrease in thickness radiallycircumferential stress goes on decreasing Also with theincrease in density circumferential stress further decreaseswhich provides an appropriate design of the disk It has beenobserved from Figures 6(a) 6(b) and 6(c) that for uniformthickness circumferential stresses are maximum at internalsurface Also it has been noted that circumferential stress isless for disk profile with high density as compared to diskprofiles with less density With the decrease in thicknessradially circumferential stress decreases which yields thatdisk whose thickness is less and density is high is appropriatefor the design as compared to other disk profiles
From the previous analysis we observed that disk withhigh density and less thickness is on the safer side of thedesign as compared to other parameters because circumfer-ential stress is less for previous case as compared to other
Journal of Materials 9
disk profilesThe results calculated for variable thickness andvariable density (ℎ = ℎ
0(119903119887)minus119899
120588 = 1205880(119903119887)119898
) using finitedifference method are in very good agreement with Runge-Kuttarsquos method studied by You et al [9] for all the rotatingdisks
43 Annular Disk with Constant Thickness and ConstantDensity It has been observed from Figures 7(a) and 7(b)that for constant thickness and constant density circumfer-ential stress is maximum at internal surface Also with theincrease in angular speed circumferential stress increasessignificantly
5 Conclusion
After analyzing all the three disk profiles it can be concludedthat circumferential stresses aremaximumat internal surfaceIt is also concluded that disk whose thickness decreasesradially and density increases radially is on the safer side ofdesign as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk This is because circumferential stresses for the diskswhose thickness decreases and density increases radially areless as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk
References
[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970
[2] S C Ugural and S K Fenster Advanced Strength and AppliedElasticity Elsevier New York NY USA 1987
[3] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998
[4] S Sharama M Sahni and R Kumar ldquoElastic-plastic analysisof a thin rotating disk of exponentially variable thickness withinclusionrdquo WSEAS Transactions on Mathematics vol 9 no 5pp 314ndash323 2010
[5] U Gamer ldquoElastic-plastic deformation of the rotating soliddiskrdquo Ingenieur-Archiv vol 54 no 5 pp 345ndash354 1984
[6] U Guven ldquoElastic-plastic stresses in a rotating annular disk ofvariable thickness and variable densityrdquo International Journal ofMechanical Sciences vol 34 no 2 pp 133ndash138 1992
[7] L H You S Y Long and J J Zhang ldquoPerturbation solution ofrotating solid disks with nonlinear strain-hardeningrdquoMechan-ics Research Communications vol 24 no 6 pp 649ndash658 1997
[8] L H You and J J Zhang ldquoElastic-plastic stresses in a rotatingsolid diskrdquo International Journal of Mechanical Sciences vol 41no 3 pp 269ndash282 1999
[9] L H You Y Y Tang J J Zhang and C Y Zheng ldquoNumericalanalysis of elastic-plastic rotating disks with arbitrary variablethickness and densityrdquo International Journal of Solids andStructures vol 37 no 52 pp 7809ndash7820 2000
[10] S C Sterner S Saigal W Kistler and D E Dietrich ldquoA unifiednumerical approach for the analysis of rotating disks includingturbine rotorsrdquo International Journal of Solids and Structuresvol 31 no 2 pp 269ndash277 1994
By the nonlinear strain-hardening material model pro-posed by You andZhang [8] the stress-strain relationship canbe written as
119890119890=
(120582 + 120583) 119879119890119890
120583 (3120582 + 2120583) 119890119890le 1198900
119890119901
119890= 11988611198793
119890119890+ 11988621198795
119890119890 119890119890gt 1198900
(11)
where 119890119890is the equivalent total strain and 119890
0the yield strain
and 1198861 1198862are material constants
Journal of Materials 5
0
05
1
15
2
25
FDM119898 = 01 119899 = 02
119898 = 05 119899 = 05
119898 = 07 119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
Plot of stresses angular velocity = 500
(a)
FDM119898 = 01 119899 = 02
119898 = 05 119899 = 05
119898 = 07 119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
1
2
3
4
5
6
Stre
sses
times108
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
Plot of stresses angular velocity = 780
(b)
Figure 4 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with 119899 = 02 05 07
and119898 = 01 05 07 under (a) 500 rads and (b) 780 rads for both approaches
Substitution of the second equation of (11) into (6) theplastic strains can be written as
119890119901
119903=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879119903119903minus1
2119879120579120579)
119890119901
120579=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879120579120579minus1
2119879119903119903)
(12)
The governing equation in the plastic region of therotating disks in terms of stresses and stress function can beobtained by substituting (12) into (10) as
times (2119879119903119903minus 119879120579120579) [1199031206011015840
minus (1 minus 119896119899(119903
119887)
119899
)120601]
+ (2119879120579120579minus 119879119903119903)
times (119903119896119899(119903
119887)
119899
1206011015840
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
)
+ (11988611198792
119890119890+ 11988621198794
119890119890)
times minus(1
2minus 119896119899(
119903
119887)
119899
) 1199031206011015840
+1
2(1 minus 119896119899(
119903
119887)
119899
)120601
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
+3
2119903ℎ0119890minus119896(119903119887)
119899
(119879120579120579minus 119879119903119903)
(13)
The values of the stress function 120601 at the elastic-plasticinterface radius are the same and therefore the stressfunction 120601 is continuous at the interface radius [9] It canbe seen from the continuity of the stress function and (3)
6 Journal of Materials
Plot of stresses 119898 = 01 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(a)
002040608
112141618
2times10
8
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119898 = 05 angular velocity = 500
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119898 = 07 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses (119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(c)
Figure 5 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119899 =
07 02 05 and119898 = 01 (b) 119899 = 07 02 05 and119898 = 05 and (c) 119899 = 07 02 05 and119898 = 07 under 500 rads for both approaches
(5)ndash(7) and (8) that the continuity conditions of the stressesand displacements at the elastic-plastic interface radius aresatisfied
The boundary conditions for the rotating annular disksare
119879119903119903= 0 at 119903 = 119886 119879
119903119903= 0 at 119903 = 119887 (14)
where 119886 and 119887 are the inner and outer radii of the rotatingdisks respectively
3 Finite Difference Algorithm
Todetermine the elastic-plastic stresses strains and displace-ment in thin rotating disks with a nonlinear strain hardeningmaterial we have to solve the second-order nonlinear differ-ential equation (13) under the given boundary condition (14)The general form of (13) can be written as
12060110158401015840
= 119891 (119903 120601 1206011015840
) (15)
31 Finite Difference Algorithm Steps The second-order dif-ferential equation (13) with the given boundary conditionscan be solved by using finite difference method
Journal of Materials 7
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119899 = 02 angular velocity = 500
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
(a)
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses
Plot of stresses 119899 = 05 angular velocity = 500
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119899 = 07 angular velocity = 500times107
0
2
4
6
8
10
12
14
16
18
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
(119879119903119903)(119879119903119903)
(119879119903119903)
Stre
sses
(119879120579120579)
(119879120579120579)
(119879120579120579)
(c)
Figure 6 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119898 =
01 05 07 and 119899 = 02 (b)119898 = 01 05 07 and 119899 = 05 and (c)119898 = 01 05 07 and 119899 = 07 under 500 rads for both approaches
(i) First partition the domain [01 05] into 119901 subinter-vals of length ℎ
(ii) To express the differential operators 119889120601119889119903 and1198892
1206011198891199032 in a discrete form we use the finite difference
approximations as
1198892
120601
1198891199032=120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
119889120601
119889119903=120601119894+1
minus 120601119894minus1
2ℎ (16)
(iii) With ℎ = 1119901 we have 119901 + 1 nodal points1206011 1206012 120601
119901+1 The values at the end points are
given by the boundary conditions that is we aregiven that 120593
1= 0 120593
119901+1= 0 Using the finite
difference approximation we get the following systemof equations
120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
= 119891(119903 120601119894120601119894+1
minus 120601119894minus1
2ℎ) 119894 = 2 3 119899
(17)
(iv) After simplifying and collecting coefficients of120601119894+1
120601119894
and120601119894minus1
in Step 3 the boundary value problem resultsin system of nonlinear equations
(v) The solution of the (119901 minus 1) system of nonlinearequations can be obtained using Newton-Raphsonrsquos
8 Journal of Materials
FDM119899 = 0 119898 = 0
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
Plot of stresses angular velocity = 500
0
05
1
15
2
25
(119879119903119903)
(119879120579120579)
(a)
01 015 02 025 03 035 04 045 050
1
2
3
4
5
6
119903
Stre
sses
times108
Plot of stresses angular velocity = 780
FDM119899 = 0 119898 = 0
RKM(b)
Figure 7 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with constant thickness and constant density under (a) 500 rads
and (b) 780 rads for both approaches
method where the unknowns are 1206012 1206013 120601
119901minus1 By
this process we will get the stress function and thenstresses strains and displacement
4 Numerical Illustration and Discussion
A two-dimensional plane stress analysis of rotating disk withnonuniform thickness and nonuniform density are carriedout using finite difference method The radius of the rotatingdisk is taken to be 119887 = 05 and its materials propertiesare material density 120588 = 8000Kgm3 Lamersquos constant are120582 = 7961GPa 120583 = 11942GPa and Poissonrsquos ratio ] = 03
41 Rotating Annular Disk with Exponentially Variable Thick-ness and Exponentially Variable Density It has been observedfrom Figures 1(a) and 1(b) that as thickness decreases anddensity increases radially circumferential stress is going ondecreasing Also circumferential stress is maximum at theinternal surface With the increase in angular velocity cir-cumferential stress is going on increasing Figures 2(a) 2(b)and 2(c) are sketched for uniform density and we observedthat circumferential stresses are maximum at internal surfaceand these stresses decreases with decrease in thickness radi-ally As with the increase in density radially circumferentialgoes on decreasingwhich gives appropriate design of the diskIt has been observed from Figures 3(a) 3(b) and 3(c) thatfor uniform thickness circumferential stress is maximumat internal surface Also circumferential stress for the diskwhose density decreases radially is maximum as comparedto the disk with high density Also decrease in thicknessradially yields decrease in circumferential stress which givesappropriate design of disk
From all previous analysis we can conclude that diskwhose thickness decreases radially and density increases
radially is on the safer side of the design as compared to otherthickness and density parameters because circumferentialstress is less for the disk whose thickness decreases radiallyand density increases radially as compared to other thicknessand density parameters
42 Rotating Annular Disk with Variable Thickness andVariable Density Using Power Law (ℎ = ℎ
0(119903119887)minus119899
120588 =
1205880(119903119887)119898
) It has been observed from Figures 4(a) and 4(b)that circumferential stress is maximum at internal surfaceCircumferential stress ismaximum for the diskwhose densityis less and thickness is high as compared to the disk whosethickness is less and density is high Also it has beenobserved from the figure that with the increase in angularspeed circumferential stress goes on increasing but againcircumferential stress is less as compared to other cases Ithas been observed from Figures 5(a) 5(b) and 5(c) thatfor uniform density disk circumferential stress is maximumat internal surface With the decrease in thickness radiallycircumferential stress goes on decreasing Also with theincrease in density circumferential stress further decreaseswhich provides an appropriate design of the disk It has beenobserved from Figures 6(a) 6(b) and 6(c) that for uniformthickness circumferential stresses are maximum at internalsurface Also it has been noted that circumferential stress isless for disk profile with high density as compared to diskprofiles with less density With the decrease in thicknessradially circumferential stress decreases which yields thatdisk whose thickness is less and density is high is appropriatefor the design as compared to other disk profiles
From the previous analysis we observed that disk withhigh density and less thickness is on the safer side of thedesign as compared to other parameters because circumfer-ential stress is less for previous case as compared to other
Journal of Materials 9
disk profilesThe results calculated for variable thickness andvariable density (ℎ = ℎ
0(119903119887)minus119899
120588 = 1205880(119903119887)119898
) using finitedifference method are in very good agreement with Runge-Kuttarsquos method studied by You et al [9] for all the rotatingdisks
43 Annular Disk with Constant Thickness and ConstantDensity It has been observed from Figures 7(a) and 7(b)that for constant thickness and constant density circumfer-ential stress is maximum at internal surface Also with theincrease in angular speed circumferential stress increasessignificantly
5 Conclusion
After analyzing all the three disk profiles it can be concludedthat circumferential stresses aremaximumat internal surfaceIt is also concluded that disk whose thickness decreasesradially and density increases radially is on the safer side ofdesign as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk This is because circumferential stresses for the diskswhose thickness decreases and density increases radially areless as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk
References
[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970
[2] S C Ugural and S K Fenster Advanced Strength and AppliedElasticity Elsevier New York NY USA 1987
[3] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998
[4] S Sharama M Sahni and R Kumar ldquoElastic-plastic analysisof a thin rotating disk of exponentially variable thickness withinclusionrdquo WSEAS Transactions on Mathematics vol 9 no 5pp 314ndash323 2010
[5] U Gamer ldquoElastic-plastic deformation of the rotating soliddiskrdquo Ingenieur-Archiv vol 54 no 5 pp 345ndash354 1984
[6] U Guven ldquoElastic-plastic stresses in a rotating annular disk ofvariable thickness and variable densityrdquo International Journal ofMechanical Sciences vol 34 no 2 pp 133ndash138 1992
[7] L H You S Y Long and J J Zhang ldquoPerturbation solution ofrotating solid disks with nonlinear strain-hardeningrdquoMechan-ics Research Communications vol 24 no 6 pp 649ndash658 1997
[8] L H You and J J Zhang ldquoElastic-plastic stresses in a rotatingsolid diskrdquo International Journal of Mechanical Sciences vol 41no 3 pp 269ndash282 1999
[9] L H You Y Y Tang J J Zhang and C Y Zheng ldquoNumericalanalysis of elastic-plastic rotating disks with arbitrary variablethickness and densityrdquo International Journal of Solids andStructures vol 37 no 52 pp 7809ndash7820 2000
[10] S C Sterner S Saigal W Kistler and D E Dietrich ldquoA unifiednumerical approach for the analysis of rotating disks includingturbine rotorsrdquo International Journal of Solids and Structuresvol 31 no 2 pp 269ndash277 1994
By the nonlinear strain-hardening material model pro-posed by You andZhang [8] the stress-strain relationship canbe written as
119890119890=
(120582 + 120583) 119879119890119890
120583 (3120582 + 2120583) 119890119890le 1198900
119890119901
119890= 11988611198793
119890119890+ 11988621198795
119890119890 119890119890gt 1198900
(11)
where 119890119890is the equivalent total strain and 119890
0the yield strain
and 1198861 1198862are material constants
Journal of Materials 5
0
05
1
15
2
25
FDM119898 = 01 119899 = 02
119898 = 05 119899 = 05
119898 = 07 119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
Plot of stresses angular velocity = 500
(a)
FDM119898 = 01 119899 = 02
119898 = 05 119899 = 05
119898 = 07 119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
1
2
3
4
5
6
Stre
sses
times108
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
Plot of stresses angular velocity = 780
(b)
Figure 4 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with 119899 = 02 05 07
and119898 = 01 05 07 under (a) 500 rads and (b) 780 rads for both approaches
Substitution of the second equation of (11) into (6) theplastic strains can be written as
119890119901
119903=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879119903119903minus1
2119879120579120579)
119890119901
120579=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879120579120579minus1
2119879119903119903)
(12)
The governing equation in the plastic region of therotating disks in terms of stresses and stress function can beobtained by substituting (12) into (10) as
times (2119879119903119903minus 119879120579120579) [1199031206011015840
minus (1 minus 119896119899(119903
119887)
119899
)120601]
+ (2119879120579120579minus 119879119903119903)
times (119903119896119899(119903
119887)
119899
1206011015840
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
)
+ (11988611198792
119890119890+ 11988621198794
119890119890)
times minus(1
2minus 119896119899(
119903
119887)
119899
) 1199031206011015840
+1
2(1 minus 119896119899(
119903
119887)
119899
)120601
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
+3
2119903ℎ0119890minus119896(119903119887)
119899
(119879120579120579minus 119879119903119903)
(13)
The values of the stress function 120601 at the elastic-plasticinterface radius are the same and therefore the stressfunction 120601 is continuous at the interface radius [9] It canbe seen from the continuity of the stress function and (3)
6 Journal of Materials
Plot of stresses 119898 = 01 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(a)
002040608
112141618
2times10
8
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119898 = 05 angular velocity = 500
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119898 = 07 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses (119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(c)
Figure 5 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119899 =
07 02 05 and119898 = 01 (b) 119899 = 07 02 05 and119898 = 05 and (c) 119899 = 07 02 05 and119898 = 07 under 500 rads for both approaches
(5)ndash(7) and (8) that the continuity conditions of the stressesand displacements at the elastic-plastic interface radius aresatisfied
The boundary conditions for the rotating annular disksare
119879119903119903= 0 at 119903 = 119886 119879
119903119903= 0 at 119903 = 119887 (14)
where 119886 and 119887 are the inner and outer radii of the rotatingdisks respectively
3 Finite Difference Algorithm
Todetermine the elastic-plastic stresses strains and displace-ment in thin rotating disks with a nonlinear strain hardeningmaterial we have to solve the second-order nonlinear differ-ential equation (13) under the given boundary condition (14)The general form of (13) can be written as
12060110158401015840
= 119891 (119903 120601 1206011015840
) (15)
31 Finite Difference Algorithm Steps The second-order dif-ferential equation (13) with the given boundary conditionscan be solved by using finite difference method
Journal of Materials 7
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119899 = 02 angular velocity = 500
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
(a)
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses
Plot of stresses 119899 = 05 angular velocity = 500
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119899 = 07 angular velocity = 500times107
0
2
4
6
8
10
12
14
16
18
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
(119879119903119903)(119879119903119903)
(119879119903119903)
Stre
sses
(119879120579120579)
(119879120579120579)
(119879120579120579)
(c)
Figure 6 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119898 =
01 05 07 and 119899 = 02 (b)119898 = 01 05 07 and 119899 = 05 and (c)119898 = 01 05 07 and 119899 = 07 under 500 rads for both approaches
(i) First partition the domain [01 05] into 119901 subinter-vals of length ℎ
(ii) To express the differential operators 119889120601119889119903 and1198892
1206011198891199032 in a discrete form we use the finite difference
approximations as
1198892
120601
1198891199032=120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
119889120601
119889119903=120601119894+1
minus 120601119894minus1
2ℎ (16)
(iii) With ℎ = 1119901 we have 119901 + 1 nodal points1206011 1206012 120601
119901+1 The values at the end points are
given by the boundary conditions that is we aregiven that 120593
1= 0 120593
119901+1= 0 Using the finite
difference approximation we get the following systemof equations
120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
= 119891(119903 120601119894120601119894+1
minus 120601119894minus1
2ℎ) 119894 = 2 3 119899
(17)
(iv) After simplifying and collecting coefficients of120601119894+1
120601119894
and120601119894minus1
in Step 3 the boundary value problem resultsin system of nonlinear equations
(v) The solution of the (119901 minus 1) system of nonlinearequations can be obtained using Newton-Raphsonrsquos
8 Journal of Materials
FDM119899 = 0 119898 = 0
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
Plot of stresses angular velocity = 500
0
05
1
15
2
25
(119879119903119903)
(119879120579120579)
(a)
01 015 02 025 03 035 04 045 050
1
2
3
4
5
6
119903
Stre
sses
times108
Plot of stresses angular velocity = 780
FDM119899 = 0 119898 = 0
RKM(b)
Figure 7 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with constant thickness and constant density under (a) 500 rads
and (b) 780 rads for both approaches
method where the unknowns are 1206012 1206013 120601
119901minus1 By
this process we will get the stress function and thenstresses strains and displacement
4 Numerical Illustration and Discussion
A two-dimensional plane stress analysis of rotating disk withnonuniform thickness and nonuniform density are carriedout using finite difference method The radius of the rotatingdisk is taken to be 119887 = 05 and its materials propertiesare material density 120588 = 8000Kgm3 Lamersquos constant are120582 = 7961GPa 120583 = 11942GPa and Poissonrsquos ratio ] = 03
41 Rotating Annular Disk with Exponentially Variable Thick-ness and Exponentially Variable Density It has been observedfrom Figures 1(a) and 1(b) that as thickness decreases anddensity increases radially circumferential stress is going ondecreasing Also circumferential stress is maximum at theinternal surface With the increase in angular velocity cir-cumferential stress is going on increasing Figures 2(a) 2(b)and 2(c) are sketched for uniform density and we observedthat circumferential stresses are maximum at internal surfaceand these stresses decreases with decrease in thickness radi-ally As with the increase in density radially circumferentialgoes on decreasingwhich gives appropriate design of the diskIt has been observed from Figures 3(a) 3(b) and 3(c) thatfor uniform thickness circumferential stress is maximumat internal surface Also circumferential stress for the diskwhose density decreases radially is maximum as comparedto the disk with high density Also decrease in thicknessradially yields decrease in circumferential stress which givesappropriate design of disk
From all previous analysis we can conclude that diskwhose thickness decreases radially and density increases
radially is on the safer side of the design as compared to otherthickness and density parameters because circumferentialstress is less for the disk whose thickness decreases radiallyand density increases radially as compared to other thicknessand density parameters
42 Rotating Annular Disk with Variable Thickness andVariable Density Using Power Law (ℎ = ℎ
0(119903119887)minus119899
120588 =
1205880(119903119887)119898
) It has been observed from Figures 4(a) and 4(b)that circumferential stress is maximum at internal surfaceCircumferential stress ismaximum for the diskwhose densityis less and thickness is high as compared to the disk whosethickness is less and density is high Also it has beenobserved from the figure that with the increase in angularspeed circumferential stress goes on increasing but againcircumferential stress is less as compared to other cases Ithas been observed from Figures 5(a) 5(b) and 5(c) thatfor uniform density disk circumferential stress is maximumat internal surface With the decrease in thickness radiallycircumferential stress goes on decreasing Also with theincrease in density circumferential stress further decreaseswhich provides an appropriate design of the disk It has beenobserved from Figures 6(a) 6(b) and 6(c) that for uniformthickness circumferential stresses are maximum at internalsurface Also it has been noted that circumferential stress isless for disk profile with high density as compared to diskprofiles with less density With the decrease in thicknessradially circumferential stress decreases which yields thatdisk whose thickness is less and density is high is appropriatefor the design as compared to other disk profiles
From the previous analysis we observed that disk withhigh density and less thickness is on the safer side of thedesign as compared to other parameters because circumfer-ential stress is less for previous case as compared to other
Journal of Materials 9
disk profilesThe results calculated for variable thickness andvariable density (ℎ = ℎ
0(119903119887)minus119899
120588 = 1205880(119903119887)119898
) using finitedifference method are in very good agreement with Runge-Kuttarsquos method studied by You et al [9] for all the rotatingdisks
43 Annular Disk with Constant Thickness and ConstantDensity It has been observed from Figures 7(a) and 7(b)that for constant thickness and constant density circumfer-ential stress is maximum at internal surface Also with theincrease in angular speed circumferential stress increasessignificantly
5 Conclusion
After analyzing all the three disk profiles it can be concludedthat circumferential stresses aremaximumat internal surfaceIt is also concluded that disk whose thickness decreasesradially and density increases radially is on the safer side ofdesign as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk This is because circumferential stresses for the diskswhose thickness decreases and density increases radially areless as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk
References
[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970
[2] S C Ugural and S K Fenster Advanced Strength and AppliedElasticity Elsevier New York NY USA 1987
[3] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998
[4] S Sharama M Sahni and R Kumar ldquoElastic-plastic analysisof a thin rotating disk of exponentially variable thickness withinclusionrdquo WSEAS Transactions on Mathematics vol 9 no 5pp 314ndash323 2010
[5] U Gamer ldquoElastic-plastic deformation of the rotating soliddiskrdquo Ingenieur-Archiv vol 54 no 5 pp 345ndash354 1984
[6] U Guven ldquoElastic-plastic stresses in a rotating annular disk ofvariable thickness and variable densityrdquo International Journal ofMechanical Sciences vol 34 no 2 pp 133ndash138 1992
[7] L H You S Y Long and J J Zhang ldquoPerturbation solution ofrotating solid disks with nonlinear strain-hardeningrdquoMechan-ics Research Communications vol 24 no 6 pp 649ndash658 1997
[8] L H You and J J Zhang ldquoElastic-plastic stresses in a rotatingsolid diskrdquo International Journal of Mechanical Sciences vol 41no 3 pp 269ndash282 1999
[9] L H You Y Y Tang J J Zhang and C Y Zheng ldquoNumericalanalysis of elastic-plastic rotating disks with arbitrary variablethickness and densityrdquo International Journal of Solids andStructures vol 37 no 52 pp 7809ndash7820 2000
[10] S C Sterner S Saigal W Kistler and D E Dietrich ldquoA unifiednumerical approach for the analysis of rotating disks includingturbine rotorsrdquo International Journal of Solids and Structuresvol 31 no 2 pp 269ndash277 1994
Figure 4 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with 119899 = 02 05 07
and119898 = 01 05 07 under (a) 500 rads and (b) 780 rads for both approaches
Substitution of the second equation of (11) into (6) theplastic strains can be written as
119890119901
119903=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879119903119903minus1
2119879120579120579)
119890119901
120579=
(11988611198793
119890119890+ 11988621198795
119890119890)
119879119890119890
(119879120579120579minus1
2119879119903119903)
(12)
The governing equation in the plastic region of therotating disks in terms of stresses and stress function can beobtained by substituting (12) into (10) as
times (2119879119903119903minus 119879120579120579) [1199031206011015840
minus (1 minus 119896119899(119903
119887)
119899
)120601]
+ (2119879120579120579minus 119879119903119903)
times (119903119896119899(119903
119887)
119899
1206011015840
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
)
+ (11988611198792
119890119890+ 11988621198794
119890119890)
times minus(1
2minus 119896119899(
119903
119887)
119899
) 1199031206011015840
+1
2(1 minus 119896119899(
119903
119887)
119899
)120601
+ 2ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
1205962
1199033
+ ℎ01205880119890119905(119903119887)
119898
minus119896(119903119887)119899
119898119905(119903
119887)
119898
1205962
1199033
+3
2119903ℎ0119890minus119896(119903119887)
119899
(119879120579120579minus 119879119903119903)
(13)
The values of the stress function 120601 at the elastic-plasticinterface radius are the same and therefore the stressfunction 120601 is continuous at the interface radius [9] It canbe seen from the continuity of the stress function and (3)
6 Journal of Materials
Plot of stresses 119898 = 01 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(a)
002040608
112141618
2times10
8
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119898 = 05 angular velocity = 500
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119898 = 07 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses (119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(c)
Figure 5 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119899 =
07 02 05 and119898 = 01 (b) 119899 = 07 02 05 and119898 = 05 and (c) 119899 = 07 02 05 and119898 = 07 under 500 rads for both approaches
(5)ndash(7) and (8) that the continuity conditions of the stressesand displacements at the elastic-plastic interface radius aresatisfied
The boundary conditions for the rotating annular disksare
119879119903119903= 0 at 119903 = 119886 119879
119903119903= 0 at 119903 = 119887 (14)
where 119886 and 119887 are the inner and outer radii of the rotatingdisks respectively
3 Finite Difference Algorithm
Todetermine the elastic-plastic stresses strains and displace-ment in thin rotating disks with a nonlinear strain hardeningmaterial we have to solve the second-order nonlinear differ-ential equation (13) under the given boundary condition (14)The general form of (13) can be written as
12060110158401015840
= 119891 (119903 120601 1206011015840
) (15)
31 Finite Difference Algorithm Steps The second-order dif-ferential equation (13) with the given boundary conditionscan be solved by using finite difference method
Journal of Materials 7
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119899 = 02 angular velocity = 500
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
(a)
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses
Plot of stresses 119899 = 05 angular velocity = 500
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119899 = 07 angular velocity = 500times107
0
2
4
6
8
10
12
14
16
18
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
(119879119903119903)(119879119903119903)
(119879119903119903)
Stre
sses
(119879120579120579)
(119879120579120579)
(119879120579120579)
(c)
Figure 6 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119898 =
01 05 07 and 119899 = 02 (b)119898 = 01 05 07 and 119899 = 05 and (c)119898 = 01 05 07 and 119899 = 07 under 500 rads for both approaches
(i) First partition the domain [01 05] into 119901 subinter-vals of length ℎ
(ii) To express the differential operators 119889120601119889119903 and1198892
1206011198891199032 in a discrete form we use the finite difference
approximations as
1198892
120601
1198891199032=120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
119889120601
119889119903=120601119894+1
minus 120601119894minus1
2ℎ (16)
(iii) With ℎ = 1119901 we have 119901 + 1 nodal points1206011 1206012 120601
119901+1 The values at the end points are
given by the boundary conditions that is we aregiven that 120593
1= 0 120593
119901+1= 0 Using the finite
difference approximation we get the following systemof equations
120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
= 119891(119903 120601119894120601119894+1
minus 120601119894minus1
2ℎ) 119894 = 2 3 119899
(17)
(iv) After simplifying and collecting coefficients of120601119894+1
120601119894
and120601119894minus1
in Step 3 the boundary value problem resultsin system of nonlinear equations
(v) The solution of the (119901 minus 1) system of nonlinearequations can be obtained using Newton-Raphsonrsquos
8 Journal of Materials
FDM119899 = 0 119898 = 0
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
Plot of stresses angular velocity = 500
0
05
1
15
2
25
(119879119903119903)
(119879120579120579)
(a)
01 015 02 025 03 035 04 045 050
1
2
3
4
5
6
119903
Stre
sses
times108
Plot of stresses angular velocity = 780
FDM119899 = 0 119898 = 0
RKM(b)
Figure 7 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with constant thickness and constant density under (a) 500 rads
and (b) 780 rads for both approaches
method where the unknowns are 1206012 1206013 120601
119901minus1 By
this process we will get the stress function and thenstresses strains and displacement
4 Numerical Illustration and Discussion
A two-dimensional plane stress analysis of rotating disk withnonuniform thickness and nonuniform density are carriedout using finite difference method The radius of the rotatingdisk is taken to be 119887 = 05 and its materials propertiesare material density 120588 = 8000Kgm3 Lamersquos constant are120582 = 7961GPa 120583 = 11942GPa and Poissonrsquos ratio ] = 03
41 Rotating Annular Disk with Exponentially Variable Thick-ness and Exponentially Variable Density It has been observedfrom Figures 1(a) and 1(b) that as thickness decreases anddensity increases radially circumferential stress is going ondecreasing Also circumferential stress is maximum at theinternal surface With the increase in angular velocity cir-cumferential stress is going on increasing Figures 2(a) 2(b)and 2(c) are sketched for uniform density and we observedthat circumferential stresses are maximum at internal surfaceand these stresses decreases with decrease in thickness radi-ally As with the increase in density radially circumferentialgoes on decreasingwhich gives appropriate design of the diskIt has been observed from Figures 3(a) 3(b) and 3(c) thatfor uniform thickness circumferential stress is maximumat internal surface Also circumferential stress for the diskwhose density decreases radially is maximum as comparedto the disk with high density Also decrease in thicknessradially yields decrease in circumferential stress which givesappropriate design of disk
From all previous analysis we can conclude that diskwhose thickness decreases radially and density increases
radially is on the safer side of the design as compared to otherthickness and density parameters because circumferentialstress is less for the disk whose thickness decreases radiallyand density increases radially as compared to other thicknessand density parameters
42 Rotating Annular Disk with Variable Thickness andVariable Density Using Power Law (ℎ = ℎ
0(119903119887)minus119899
120588 =
1205880(119903119887)119898
) It has been observed from Figures 4(a) and 4(b)that circumferential stress is maximum at internal surfaceCircumferential stress ismaximum for the diskwhose densityis less and thickness is high as compared to the disk whosethickness is less and density is high Also it has beenobserved from the figure that with the increase in angularspeed circumferential stress goes on increasing but againcircumferential stress is less as compared to other cases Ithas been observed from Figures 5(a) 5(b) and 5(c) thatfor uniform density disk circumferential stress is maximumat internal surface With the decrease in thickness radiallycircumferential stress goes on decreasing Also with theincrease in density circumferential stress further decreaseswhich provides an appropriate design of the disk It has beenobserved from Figures 6(a) 6(b) and 6(c) that for uniformthickness circumferential stresses are maximum at internalsurface Also it has been noted that circumferential stress isless for disk profile with high density as compared to diskprofiles with less density With the decrease in thicknessradially circumferential stress decreases which yields thatdisk whose thickness is less and density is high is appropriatefor the design as compared to other disk profiles
From the previous analysis we observed that disk withhigh density and less thickness is on the safer side of thedesign as compared to other parameters because circumfer-ential stress is less for previous case as compared to other
Journal of Materials 9
disk profilesThe results calculated for variable thickness andvariable density (ℎ = ℎ
0(119903119887)minus119899
120588 = 1205880(119903119887)119898
) using finitedifference method are in very good agreement with Runge-Kuttarsquos method studied by You et al [9] for all the rotatingdisks
43 Annular Disk with Constant Thickness and ConstantDensity It has been observed from Figures 7(a) and 7(b)that for constant thickness and constant density circumfer-ential stress is maximum at internal surface Also with theincrease in angular speed circumferential stress increasessignificantly
5 Conclusion
After analyzing all the three disk profiles it can be concludedthat circumferential stresses aremaximumat internal surfaceIt is also concluded that disk whose thickness decreasesradially and density increases radially is on the safer side ofdesign as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk This is because circumferential stresses for the diskswhose thickness decreases and density increases radially areless as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk
References
[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970
[2] S C Ugural and S K Fenster Advanced Strength and AppliedElasticity Elsevier New York NY USA 1987
[3] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998
[4] S Sharama M Sahni and R Kumar ldquoElastic-plastic analysisof a thin rotating disk of exponentially variable thickness withinclusionrdquo WSEAS Transactions on Mathematics vol 9 no 5pp 314ndash323 2010
[5] U Gamer ldquoElastic-plastic deformation of the rotating soliddiskrdquo Ingenieur-Archiv vol 54 no 5 pp 345ndash354 1984
[6] U Guven ldquoElastic-plastic stresses in a rotating annular disk ofvariable thickness and variable densityrdquo International Journal ofMechanical Sciences vol 34 no 2 pp 133ndash138 1992
[7] L H You S Y Long and J J Zhang ldquoPerturbation solution ofrotating solid disks with nonlinear strain-hardeningrdquoMechan-ics Research Communications vol 24 no 6 pp 649ndash658 1997
[8] L H You and J J Zhang ldquoElastic-plastic stresses in a rotatingsolid diskrdquo International Journal of Mechanical Sciences vol 41no 3 pp 269ndash282 1999
[9] L H You Y Y Tang J J Zhang and C Y Zheng ldquoNumericalanalysis of elastic-plastic rotating disks with arbitrary variablethickness and densityrdquo International Journal of Solids andStructures vol 37 no 52 pp 7809ndash7820 2000
[10] S C Sterner S Saigal W Kistler and D E Dietrich ldquoA unifiednumerical approach for the analysis of rotating disks includingturbine rotorsrdquo International Journal of Solids and Structuresvol 31 no 2 pp 269ndash277 1994
Plot of stresses 119898 = 01 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(a)
002040608
112141618
2times10
8
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119898 = 05 angular velocity = 500
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119898 = 07 angular velocity = 500
FDM119899 = 02
119899 = 05
119899 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses (119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(c)
Figure 5 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119899 =
07 02 05 and119898 = 01 (b) 119899 = 07 02 05 and119898 = 05 and (c) 119899 = 07 02 05 and119898 = 07 under 500 rads for both approaches
(5)ndash(7) and (8) that the continuity conditions of the stressesand displacements at the elastic-plastic interface radius aresatisfied
The boundary conditions for the rotating annular disksare
119879119903119903= 0 at 119903 = 119886 119879
119903119903= 0 at 119903 = 119887 (14)
where 119886 and 119887 are the inner and outer radii of the rotatingdisks respectively
3 Finite Difference Algorithm
Todetermine the elastic-plastic stresses strains and displace-ment in thin rotating disks with a nonlinear strain hardeningmaterial we have to solve the second-order nonlinear differ-ential equation (13) under the given boundary condition (14)The general form of (13) can be written as
12060110158401015840
= 119891 (119903 120601 1206011015840
) (15)
31 Finite Difference Algorithm Steps The second-order dif-ferential equation (13) with the given boundary conditionscan be solved by using finite difference method
Journal of Materials 7
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
Plot of stresses 119899 = 02 angular velocity = 500
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
(a)
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses
Plot of stresses 119899 = 05 angular velocity = 500
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119899 = 07 angular velocity = 500times107
0
2
4
6
8
10
12
14
16
18
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
(119879119903119903)(119879119903119903)
(119879119903119903)
Stre
sses
(119879120579120579)
(119879120579120579)
(119879120579120579)
(c)
Figure 6 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119898 =
01 05 07 and 119899 = 02 (b)119898 = 01 05 07 and 119899 = 05 and (c)119898 = 01 05 07 and 119899 = 07 under 500 rads for both approaches
(i) First partition the domain [01 05] into 119901 subinter-vals of length ℎ
(ii) To express the differential operators 119889120601119889119903 and1198892
1206011198891199032 in a discrete form we use the finite difference
approximations as
1198892
120601
1198891199032=120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
119889120601
119889119903=120601119894+1
minus 120601119894minus1
2ℎ (16)
(iii) With ℎ = 1119901 we have 119901 + 1 nodal points1206011 1206012 120601
119901+1 The values at the end points are
given by the boundary conditions that is we aregiven that 120593
1= 0 120593
119901+1= 0 Using the finite
difference approximation we get the following systemof equations
120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
= 119891(119903 120601119894120601119894+1
minus 120601119894minus1
2ℎ) 119894 = 2 3 119899
(17)
(iv) After simplifying and collecting coefficients of120601119894+1
120601119894
and120601119894minus1
in Step 3 the boundary value problem resultsin system of nonlinear equations
(v) The solution of the (119901 minus 1) system of nonlinearequations can be obtained using Newton-Raphsonrsquos
8 Journal of Materials
FDM119899 = 0 119898 = 0
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
Plot of stresses angular velocity = 500
0
05
1
15
2
25
(119879119903119903)
(119879120579120579)
(a)
01 015 02 025 03 035 04 045 050
1
2
3
4
5
6
119903
Stre
sses
times108
Plot of stresses angular velocity = 780
FDM119899 = 0 119898 = 0
RKM(b)
Figure 7 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with constant thickness and constant density under (a) 500 rads
and (b) 780 rads for both approaches
method where the unknowns are 1206012 1206013 120601
119901minus1 By
this process we will get the stress function and thenstresses strains and displacement
4 Numerical Illustration and Discussion
A two-dimensional plane stress analysis of rotating disk withnonuniform thickness and nonuniform density are carriedout using finite difference method The radius of the rotatingdisk is taken to be 119887 = 05 and its materials propertiesare material density 120588 = 8000Kgm3 Lamersquos constant are120582 = 7961GPa 120583 = 11942GPa and Poissonrsquos ratio ] = 03
41 Rotating Annular Disk with Exponentially Variable Thick-ness and Exponentially Variable Density It has been observedfrom Figures 1(a) and 1(b) that as thickness decreases anddensity increases radially circumferential stress is going ondecreasing Also circumferential stress is maximum at theinternal surface With the increase in angular velocity cir-cumferential stress is going on increasing Figures 2(a) 2(b)and 2(c) are sketched for uniform density and we observedthat circumferential stresses are maximum at internal surfaceand these stresses decreases with decrease in thickness radi-ally As with the increase in density radially circumferentialgoes on decreasingwhich gives appropriate design of the diskIt has been observed from Figures 3(a) 3(b) and 3(c) thatfor uniform thickness circumferential stress is maximumat internal surface Also circumferential stress for the diskwhose density decreases radially is maximum as comparedto the disk with high density Also decrease in thicknessradially yields decrease in circumferential stress which givesappropriate design of disk
From all previous analysis we can conclude that diskwhose thickness decreases radially and density increases
radially is on the safer side of the design as compared to otherthickness and density parameters because circumferentialstress is less for the disk whose thickness decreases radiallyand density increases radially as compared to other thicknessand density parameters
42 Rotating Annular Disk with Variable Thickness andVariable Density Using Power Law (ℎ = ℎ
0(119903119887)minus119899
120588 =
1205880(119903119887)119898
) It has been observed from Figures 4(a) and 4(b)that circumferential stress is maximum at internal surfaceCircumferential stress ismaximum for the diskwhose densityis less and thickness is high as compared to the disk whosethickness is less and density is high Also it has beenobserved from the figure that with the increase in angularspeed circumferential stress goes on increasing but againcircumferential stress is less as compared to other cases Ithas been observed from Figures 5(a) 5(b) and 5(c) thatfor uniform density disk circumferential stress is maximumat internal surface With the decrease in thickness radiallycircumferential stress goes on decreasing Also with theincrease in density circumferential stress further decreaseswhich provides an appropriate design of the disk It has beenobserved from Figures 6(a) 6(b) and 6(c) that for uniformthickness circumferential stresses are maximum at internalsurface Also it has been noted that circumferential stress isless for disk profile with high density as compared to diskprofiles with less density With the decrease in thicknessradially circumferential stress decreases which yields thatdisk whose thickness is less and density is high is appropriatefor the design as compared to other disk profiles
From the previous analysis we observed that disk withhigh density and less thickness is on the safer side of thedesign as compared to other parameters because circumfer-ential stress is less for previous case as compared to other
Journal of Materials 9
disk profilesThe results calculated for variable thickness andvariable density (ℎ = ℎ
0(119903119887)minus119899
120588 = 1205880(119903119887)119898
) using finitedifference method are in very good agreement with Runge-Kuttarsquos method studied by You et al [9] for all the rotatingdisks
43 Annular Disk with Constant Thickness and ConstantDensity It has been observed from Figures 7(a) and 7(b)that for constant thickness and constant density circumfer-ential stress is maximum at internal surface Also with theincrease in angular speed circumferential stress increasessignificantly
5 Conclusion
After analyzing all the three disk profiles it can be concludedthat circumferential stresses aremaximumat internal surfaceIt is also concluded that disk whose thickness decreasesradially and density increases radially is on the safer side ofdesign as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk This is because circumferential stresses for the diskswhose thickness decreases and density increases radially areless as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk
References
[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970
[2] S C Ugural and S K Fenster Advanced Strength and AppliedElasticity Elsevier New York NY USA 1987
[3] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998
[4] S Sharama M Sahni and R Kumar ldquoElastic-plastic analysisof a thin rotating disk of exponentially variable thickness withinclusionrdquo WSEAS Transactions on Mathematics vol 9 no 5pp 314ndash323 2010
[5] U Gamer ldquoElastic-plastic deformation of the rotating soliddiskrdquo Ingenieur-Archiv vol 54 no 5 pp 345ndash354 1984
[6] U Guven ldquoElastic-plastic stresses in a rotating annular disk ofvariable thickness and variable densityrdquo International Journal ofMechanical Sciences vol 34 no 2 pp 133ndash138 1992
[7] L H You S Y Long and J J Zhang ldquoPerturbation solution ofrotating solid disks with nonlinear strain-hardeningrdquoMechan-ics Research Communications vol 24 no 6 pp 649ndash658 1997
[8] L H You and J J Zhang ldquoElastic-plastic stresses in a rotatingsolid diskrdquo International Journal of Mechanical Sciences vol 41no 3 pp 269ndash282 1999
[9] L H You Y Y Tang J J Zhang and C Y Zheng ldquoNumericalanalysis of elastic-plastic rotating disks with arbitrary variablethickness and densityrdquo International Journal of Solids andStructures vol 37 no 52 pp 7809ndash7820 2000
[10] S C Sterner S Saigal W Kistler and D E Dietrich ldquoA unifiednumerical approach for the analysis of rotating disks includingturbine rotorsrdquo International Journal of Solids and Structuresvol 31 no 2 pp 269ndash277 1994
Plot of stresses 119899 = 02 angular velocity = 500
0
05
1
15
2
25times10
8
Stre
sses
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)
(119879120579120579)
(119879120579120579)
(a)
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
times107
0
2
4
6
8
10
12
14
16
18
Stre
sses
Plot of stresses 119899 = 05 angular velocity = 500
(119879119903119903)(119879119903119903)
(119879119903119903)
(119879120579120579)(119879120579120579)
(119879120579120579)
(b)
Plot of stresses 119899 = 07 angular velocity = 500times107
0
2
4
6
8
10
12
14
16
18
FDM119898 = 01
119898 = 05
119898 = 07
RKM
01 015 02 025 03 035 04 045 05119903
(119879119903119903)(119879119903119903)
(119879119903119903)
Stre
sses
(119879120579120579)
(119879120579120579)
(119879120579120579)
(c)
Figure 6 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with variable thickness and variable density with (a) 119898 =
01 05 07 and 119899 = 02 (b)119898 = 01 05 07 and 119899 = 05 and (c)119898 = 01 05 07 and 119899 = 07 under 500 rads for both approaches
(i) First partition the domain [01 05] into 119901 subinter-vals of length ℎ
(ii) To express the differential operators 119889120601119889119903 and1198892
1206011198891199032 in a discrete form we use the finite difference
approximations as
1198892
120601
1198891199032=120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
119889120601
119889119903=120601119894+1
minus 120601119894minus1
2ℎ (16)
(iii) With ℎ = 1119901 we have 119901 + 1 nodal points1206011 1206012 120601
119901+1 The values at the end points are
given by the boundary conditions that is we aregiven that 120593
1= 0 120593
119901+1= 0 Using the finite
difference approximation we get the following systemof equations
120601119894+1
minus 2120601119894+ 120601119894minus1
ℎ2
= 119891(119903 120601119894120601119894+1
minus 120601119894minus1
2ℎ) 119894 = 2 3 119899
(17)
(iv) After simplifying and collecting coefficients of120601119894+1
120601119894
and120601119894minus1
in Step 3 the boundary value problem resultsin system of nonlinear equations
(v) The solution of the (119901 minus 1) system of nonlinearequations can be obtained using Newton-Raphsonrsquos
8 Journal of Materials
FDM119899 = 0 119898 = 0
RKM
01 015 02 025 03 035 04 045 05119903
Stre
sses
times108
Plot of stresses angular velocity = 500
0
05
1
15
2
25
(119879119903119903)
(119879120579120579)
(a)
01 015 02 025 03 035 04 045 050
1
2
3
4
5
6
119903
Stre
sses
times108
Plot of stresses angular velocity = 780
FDM119899 = 0 119898 = 0
RKM(b)
Figure 7 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with constant thickness and constant density under (a) 500 rads
and (b) 780 rads for both approaches
method where the unknowns are 1206012 1206013 120601
119901minus1 By
this process we will get the stress function and thenstresses strains and displacement
4 Numerical Illustration and Discussion
A two-dimensional plane stress analysis of rotating disk withnonuniform thickness and nonuniform density are carriedout using finite difference method The radius of the rotatingdisk is taken to be 119887 = 05 and its materials propertiesare material density 120588 = 8000Kgm3 Lamersquos constant are120582 = 7961GPa 120583 = 11942GPa and Poissonrsquos ratio ] = 03
41 Rotating Annular Disk with Exponentially Variable Thick-ness and Exponentially Variable Density It has been observedfrom Figures 1(a) and 1(b) that as thickness decreases anddensity increases radially circumferential stress is going ondecreasing Also circumferential stress is maximum at theinternal surface With the increase in angular velocity cir-cumferential stress is going on increasing Figures 2(a) 2(b)and 2(c) are sketched for uniform density and we observedthat circumferential stresses are maximum at internal surfaceand these stresses decreases with decrease in thickness radi-ally As with the increase in density radially circumferentialgoes on decreasingwhich gives appropriate design of the diskIt has been observed from Figures 3(a) 3(b) and 3(c) thatfor uniform thickness circumferential stress is maximumat internal surface Also circumferential stress for the diskwhose density decreases radially is maximum as comparedto the disk with high density Also decrease in thicknessradially yields decrease in circumferential stress which givesappropriate design of disk
From all previous analysis we can conclude that diskwhose thickness decreases radially and density increases
radially is on the safer side of the design as compared to otherthickness and density parameters because circumferentialstress is less for the disk whose thickness decreases radiallyand density increases radially as compared to other thicknessand density parameters
42 Rotating Annular Disk with Variable Thickness andVariable Density Using Power Law (ℎ = ℎ
0(119903119887)minus119899
120588 =
1205880(119903119887)119898
) It has been observed from Figures 4(a) and 4(b)that circumferential stress is maximum at internal surfaceCircumferential stress ismaximum for the diskwhose densityis less and thickness is high as compared to the disk whosethickness is less and density is high Also it has beenobserved from the figure that with the increase in angularspeed circumferential stress goes on increasing but againcircumferential stress is less as compared to other cases Ithas been observed from Figures 5(a) 5(b) and 5(c) thatfor uniform density disk circumferential stress is maximumat internal surface With the decrease in thickness radiallycircumferential stress goes on decreasing Also with theincrease in density circumferential stress further decreaseswhich provides an appropriate design of the disk It has beenobserved from Figures 6(a) 6(b) and 6(c) that for uniformthickness circumferential stresses are maximum at internalsurface Also it has been noted that circumferential stress isless for disk profile with high density as compared to diskprofiles with less density With the decrease in thicknessradially circumferential stress decreases which yields thatdisk whose thickness is less and density is high is appropriatefor the design as compared to other disk profiles
From the previous analysis we observed that disk withhigh density and less thickness is on the safer side of thedesign as compared to other parameters because circumfer-ential stress is less for previous case as compared to other
Journal of Materials 9
disk profilesThe results calculated for variable thickness andvariable density (ℎ = ℎ
0(119903119887)minus119899
120588 = 1205880(119903119887)119898
) using finitedifference method are in very good agreement with Runge-Kuttarsquos method studied by You et al [9] for all the rotatingdisks
43 Annular Disk with Constant Thickness and ConstantDensity It has been observed from Figures 7(a) and 7(b)that for constant thickness and constant density circumfer-ential stress is maximum at internal surface Also with theincrease in angular speed circumferential stress increasessignificantly
5 Conclusion
After analyzing all the three disk profiles it can be concludedthat circumferential stresses aremaximumat internal surfaceIt is also concluded that disk whose thickness decreasesradially and density increases radially is on the safer side ofdesign as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk This is because circumferential stresses for the diskswhose thickness decreases and density increases radially areless as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk
References
[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970
[2] S C Ugural and S K Fenster Advanced Strength and AppliedElasticity Elsevier New York NY USA 1987
[3] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998
[4] S Sharama M Sahni and R Kumar ldquoElastic-plastic analysisof a thin rotating disk of exponentially variable thickness withinclusionrdquo WSEAS Transactions on Mathematics vol 9 no 5pp 314ndash323 2010
[5] U Gamer ldquoElastic-plastic deformation of the rotating soliddiskrdquo Ingenieur-Archiv vol 54 no 5 pp 345ndash354 1984
[6] U Guven ldquoElastic-plastic stresses in a rotating annular disk ofvariable thickness and variable densityrdquo International Journal ofMechanical Sciences vol 34 no 2 pp 133ndash138 1992
[7] L H You S Y Long and J J Zhang ldquoPerturbation solution ofrotating solid disks with nonlinear strain-hardeningrdquoMechan-ics Research Communications vol 24 no 6 pp 649ndash658 1997
[8] L H You and J J Zhang ldquoElastic-plastic stresses in a rotatingsolid diskrdquo International Journal of Mechanical Sciences vol 41no 3 pp 269ndash282 1999
[9] L H You Y Y Tang J J Zhang and C Y Zheng ldquoNumericalanalysis of elastic-plastic rotating disks with arbitrary variablethickness and densityrdquo International Journal of Solids andStructures vol 37 no 52 pp 7809ndash7820 2000
[10] S C Sterner S Saigal W Kistler and D E Dietrich ldquoA unifiednumerical approach for the analysis of rotating disks includingturbine rotorsrdquo International Journal of Solids and Structuresvol 31 no 2 pp 269ndash277 1994
Figure 7 Radial stress (119879119903119903) and circumferential stress (119879
120579120579) for annular disk with constant thickness and constant density under (a) 500 rads
and (b) 780 rads for both approaches
method where the unknowns are 1206012 1206013 120601
119901minus1 By
this process we will get the stress function and thenstresses strains and displacement
4 Numerical Illustration and Discussion
A two-dimensional plane stress analysis of rotating disk withnonuniform thickness and nonuniform density are carriedout using finite difference method The radius of the rotatingdisk is taken to be 119887 = 05 and its materials propertiesare material density 120588 = 8000Kgm3 Lamersquos constant are120582 = 7961GPa 120583 = 11942GPa and Poissonrsquos ratio ] = 03
41 Rotating Annular Disk with Exponentially Variable Thick-ness and Exponentially Variable Density It has been observedfrom Figures 1(a) and 1(b) that as thickness decreases anddensity increases radially circumferential stress is going ondecreasing Also circumferential stress is maximum at theinternal surface With the increase in angular velocity cir-cumferential stress is going on increasing Figures 2(a) 2(b)and 2(c) are sketched for uniform density and we observedthat circumferential stresses are maximum at internal surfaceand these stresses decreases with decrease in thickness radi-ally As with the increase in density radially circumferentialgoes on decreasingwhich gives appropriate design of the diskIt has been observed from Figures 3(a) 3(b) and 3(c) thatfor uniform thickness circumferential stress is maximumat internal surface Also circumferential stress for the diskwhose density decreases radially is maximum as comparedto the disk with high density Also decrease in thicknessradially yields decrease in circumferential stress which givesappropriate design of disk
From all previous analysis we can conclude that diskwhose thickness decreases radially and density increases
radially is on the safer side of the design as compared to otherthickness and density parameters because circumferentialstress is less for the disk whose thickness decreases radiallyand density increases radially as compared to other thicknessand density parameters
42 Rotating Annular Disk with Variable Thickness andVariable Density Using Power Law (ℎ = ℎ
0(119903119887)minus119899
120588 =
1205880(119903119887)119898
) It has been observed from Figures 4(a) and 4(b)that circumferential stress is maximum at internal surfaceCircumferential stress ismaximum for the diskwhose densityis less and thickness is high as compared to the disk whosethickness is less and density is high Also it has beenobserved from the figure that with the increase in angularspeed circumferential stress goes on increasing but againcircumferential stress is less as compared to other cases Ithas been observed from Figures 5(a) 5(b) and 5(c) thatfor uniform density disk circumferential stress is maximumat internal surface With the decrease in thickness radiallycircumferential stress goes on decreasing Also with theincrease in density circumferential stress further decreaseswhich provides an appropriate design of the disk It has beenobserved from Figures 6(a) 6(b) and 6(c) that for uniformthickness circumferential stresses are maximum at internalsurface Also it has been noted that circumferential stress isless for disk profile with high density as compared to diskprofiles with less density With the decrease in thicknessradially circumferential stress decreases which yields thatdisk whose thickness is less and density is high is appropriatefor the design as compared to other disk profiles
From the previous analysis we observed that disk withhigh density and less thickness is on the safer side of thedesign as compared to other parameters because circumfer-ential stress is less for previous case as compared to other
Journal of Materials 9
disk profilesThe results calculated for variable thickness andvariable density (ℎ = ℎ
0(119903119887)minus119899
120588 = 1205880(119903119887)119898
) using finitedifference method are in very good agreement with Runge-Kuttarsquos method studied by You et al [9] for all the rotatingdisks
43 Annular Disk with Constant Thickness and ConstantDensity It has been observed from Figures 7(a) and 7(b)that for constant thickness and constant density circumfer-ential stress is maximum at internal surface Also with theincrease in angular speed circumferential stress increasessignificantly
5 Conclusion
After analyzing all the three disk profiles it can be concludedthat circumferential stresses aremaximumat internal surfaceIt is also concluded that disk whose thickness decreasesradially and density increases radially is on the safer side ofdesign as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk This is because circumferential stresses for the diskswhose thickness decreases and density increases radially areless as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk
References
[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970
[2] S C Ugural and S K Fenster Advanced Strength and AppliedElasticity Elsevier New York NY USA 1987
[3] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998
[4] S Sharama M Sahni and R Kumar ldquoElastic-plastic analysisof a thin rotating disk of exponentially variable thickness withinclusionrdquo WSEAS Transactions on Mathematics vol 9 no 5pp 314ndash323 2010
[5] U Gamer ldquoElastic-plastic deformation of the rotating soliddiskrdquo Ingenieur-Archiv vol 54 no 5 pp 345ndash354 1984
[6] U Guven ldquoElastic-plastic stresses in a rotating annular disk ofvariable thickness and variable densityrdquo International Journal ofMechanical Sciences vol 34 no 2 pp 133ndash138 1992
[7] L H You S Y Long and J J Zhang ldquoPerturbation solution ofrotating solid disks with nonlinear strain-hardeningrdquoMechan-ics Research Communications vol 24 no 6 pp 649ndash658 1997
[8] L H You and J J Zhang ldquoElastic-plastic stresses in a rotatingsolid diskrdquo International Journal of Mechanical Sciences vol 41no 3 pp 269ndash282 1999
[9] L H You Y Y Tang J J Zhang and C Y Zheng ldquoNumericalanalysis of elastic-plastic rotating disks with arbitrary variablethickness and densityrdquo International Journal of Solids andStructures vol 37 no 52 pp 7809ndash7820 2000
[10] S C Sterner S Saigal W Kistler and D E Dietrich ldquoA unifiednumerical approach for the analysis of rotating disks includingturbine rotorsrdquo International Journal of Solids and Structuresvol 31 no 2 pp 269ndash277 1994
disk profilesThe results calculated for variable thickness andvariable density (ℎ = ℎ
0(119903119887)minus119899
120588 = 1205880(119903119887)119898
) using finitedifference method are in very good agreement with Runge-Kuttarsquos method studied by You et al [9] for all the rotatingdisks
43 Annular Disk with Constant Thickness and ConstantDensity It has been observed from Figures 7(a) and 7(b)that for constant thickness and constant density circumfer-ential stress is maximum at internal surface Also with theincrease in angular speed circumferential stress increasessignificantly
5 Conclusion
After analyzing all the three disk profiles it can be concludedthat circumferential stresses aremaximumat internal surfaceIt is also concluded that disk whose thickness decreasesradially and density increases radially is on the safer side ofdesign as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk This is because circumferential stresses for the diskswhose thickness decreases and density increases radially areless as compared to the disk with exponentially varyingthickness and exponentially varying density as well as to flatdisk
References
[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970
[2] S C Ugural and S K Fenster Advanced Strength and AppliedElasticity Elsevier New York NY USA 1987
[3] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998
[4] S Sharama M Sahni and R Kumar ldquoElastic-plastic analysisof a thin rotating disk of exponentially variable thickness withinclusionrdquo WSEAS Transactions on Mathematics vol 9 no 5pp 314ndash323 2010
[5] U Gamer ldquoElastic-plastic deformation of the rotating soliddiskrdquo Ingenieur-Archiv vol 54 no 5 pp 345ndash354 1984
[6] U Guven ldquoElastic-plastic stresses in a rotating annular disk ofvariable thickness and variable densityrdquo International Journal ofMechanical Sciences vol 34 no 2 pp 133ndash138 1992
[7] L H You S Y Long and J J Zhang ldquoPerturbation solution ofrotating solid disks with nonlinear strain-hardeningrdquoMechan-ics Research Communications vol 24 no 6 pp 649ndash658 1997
[8] L H You and J J Zhang ldquoElastic-plastic stresses in a rotatingsolid diskrdquo International Journal of Mechanical Sciences vol 41no 3 pp 269ndash282 1999
[9] L H You Y Y Tang J J Zhang and C Y Zheng ldquoNumericalanalysis of elastic-plastic rotating disks with arbitrary variablethickness and densityrdquo International Journal of Solids andStructures vol 37 no 52 pp 7809ndash7820 2000
[10] S C Sterner S Saigal W Kistler and D E Dietrich ldquoA unifiednumerical approach for the analysis of rotating disks includingturbine rotorsrdquo International Journal of Solids and Structuresvol 31 no 2 pp 269ndash277 1994
