Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 676190 10 pageshttpdxdoiorg1011552013676190

Research ArticleSafety Analysis Using Lebesgue Strain Measure ofThick-Walled Cylinder for Functionally Graded Materialunder Internal and External Pressure

A K Aggarwal Richa Sharma and Sanjeev Sharma

Department of Mathematics Jaypee Institute of Information Technology A-10 Sector 62 Noida 201307 India

Correspondence should be addressed to Sanjeev Sharma sanjeevsharmajiitacin

Received 29 April 2013 Accepted 10 July 2013

Academic Editors Y-k Gao C-Q Hong and D-Y Ju

Copyright copy 2013 A K Aggarwal et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Safety analysis has been done for thick-walled circular cylinder under internal and external pressure using transition theory whichis based on the concept of generalized principal Lebesgue strain measure Results have been analyzed theoretically and discussednumerically From the analysis it can be concluded that circular cylinder made of functionally graded material is on the safer sideof the design as compared to homogeneous cylinder with internal and external pressure which leads to the idea of ldquostress savingrdquothat minimizes the possibility of fracture of cylinder

1 Introduction

The constantly increasing industrial demand for axisym-metrical cylindrical and spherical components or elementsof them has concentrated the attention of designers andscientists on this particular area of activity The researchon the prediction of stresses in thick-walled hollow circularcylinder has never ceased because of the importance of thesebasic structures in numerousmechanical civil electrical andcomputer engineering applications These days in nuclearindustry cylinders subjected to internal and external pressurehave become a point of interest due to their application toadvanced small and medium-sized light water reactors Forexample steam generator tubes in which primary coolantflows outside the tubes while secondary water flows insidethe tubes are typical examples of cylinders under internaland external pressure Another example is pipelines underseawater to transport gas oil and so forth Now for designand integrity evaluation of a cylinder under internal andexternal pressure one should carefully consider the failurecharacteristics of a cylinder under internal and externalpressure The failure mechanisms of such type of cylindermight be quite different from those of a typical one underinternal pressure Upon the estimation of load carryingcapacity of these thick-walled cylinders under internal and

external pressure and combined loading many numericaland experimental works have been also made to proposerelevant design criteria of thick-walled cylinders subjected tointernal and external pressure Plane strain and plane stressanalytical solutions of thick hollow cylinder problems in theelastic stress state have been available for many years in stan-dard and advanced textbooks [1ndash4] Thick-walled circularcylinder subjected to internal and external pressure is widelyused in various industries In general vessels under highpressure require a strict analysis for an optimum design forreliable and secure operational performance and thus effortswere continually made to increase reliability Solutions havebeen obtained either in analytical form or with numericalimplementations The literature includes solutions of Chen[5] who suggested an finite difference approach for theaxisymmetric plane strain problem beyond the elastic limitwhile Durban and Kubi [6] suggested an analytical methodfor pressurized elastic-plastic tubes in plane strain Jahedand Dubey [7] proposed a numerical method for solutionfor elastic-plastic tubes using total deformation theory ofplasticity Parker [8] implemented a numerical procedureto calculate pressure and associated residual stress fieldsfor open cylinder Dubey et al [9] obtained solutions foran elastic-plastic work hardening model using piecewiselinearization of constitutive law Olszak and Urbanowski

2 The Scientific World Journal

[10] calculate the stresses for nonhomogeneous thick-walledelastic-plastic cylinder subjected to internal pressure whileHodge and Balaban [11] calculated the stresses for rotatingcylinder Sharma [12] analyzed thick-walled cylinders underinternal pressure for isotropic nonhomogeneous elastic-plastic states using transition theory This paper is an exten-sion of Sharma [12] to include the effect of external pressurefor functionally graded material because these days cylindersmade of functionally graded material under internal andexternal pressure are an important design consideration innuclear industry

2 Generalized Lebesgue Strain Measure

The classical theory of elasticity and plasticity divides thespectrum of deformation of solids into two different statesone of which is elastic and other one is plastic In classicaltheory both field equations are solved separately and laterjoined together by yield conditionAs in the behavior ofmate-rials perfect elasticity and perfect plasticity are extremes butno one can draw a sharp line between these two states It istherefore natural to expect that there should be a transitionstate and at this transition a continuum approach necessarilymeans the introduction of nonlinearmeasure But in classicalmechanics the ordinary measure has been found sufficientand so no extension has been made This is because ofthe reason that in classical mechanics field equation forelastic and plastic regions is calculated separately and thenconnected by yield criterions which is an assumption Alsoif in a very small interval the number of fluctuation isvery large the ordinary measure based on Riemann integralfails and measures like that of Lebesgue have been usedThis generalized Lebesgue measure gives very satisfactoryresults in the problems like that of plasticity and creep Thegeneralized Lebesgue strain measure helps to bridge the gapbetween microscopic andmacroscopic descriptions of physi-cal system and eliminate semiempirical conditions like that ofTrescarsquos and von-mises creep strain laws that is Nortanrsquos lawwhich provides a coordination between the theoretical andexperimental results Seth [13] has defined the generalizedprincipal strainmeasure 120576

119894119894by taking the Lebesgue integral of

the weighted function 120576119894119894= int

119890119860

119894119894

0[(1 minus 2

119860

119890119894119894)]

(1198992)minus1

119889119890119860

119894119894= (1119899)

[1 minus (1 minus 2119890119860

119894119894)

(1198992)minus1

] where 119899 is the measure and 119890119860

119894119894is the

principal Almansi strain component

3 Objective

In this paper our aim is to calculate safety factor for thick-walled cylinder made of functionally graded material underinternal and external pressure using Sethrsquos transition theory[12 14ndash16] The stresses are calculated for transition as wellas for fully plastic state The constitutive equations for bothtransition and fully plastic states are also derived from theresults Non-homogeneity is taken as the compressibility ofmaterial in the cylinder as

119862 = 1198620119903minus119896 (1)

where 119886 le 119903 le 119887 1198620and 119896(le 0) are constants

Results obtained have been discussed numerically anddepicted graphically

4 Mathematical Formulation

Consider a nonhomogeneous thick-walled circular cylinderof internal and external radii 119886 and 119887 respectively subjectedto internal pressure 119901

1and external pressure 119901

2 The non-

homogeneity in the cylinder is due to variable compressibility119862 = 119862

0119903minus119896 The cylinder is taken so large that plane trans-

verse sections remain plane during the expansion and hencethe longitudinal strain is the same for all elements at eachstage of the expansion In cylindrical polar coordinates thedisplacements are given by [14ndash16]

119906 = 119903 (1 minus 120573) V = 0 119908 = 119889119911 (2)

where 120573 is a function of 119903 = radic1199092+ 1199102 and 119889 is a constant

The generalized strains are

119890119903119903

=

1

119899

[1 minus (1199031205731015840+ 120573)

119899

] 119890120579120579

=

1

119899

[1 minus 120573119899]

119890119911119911

=

1

119899

[1 minus (1 minus 119889)119899] 119890

119903120579= 119890120579119911

= 119890119911119903

= 0

(3)

where 119899 is the measure and 1205731015840= 119889120573119889119903

For isotropic material the stress-strain relation in elasticregion is given by

119879119894119895= 1205821205751198941198951198681+ 2120583119890

119894119895 (119894 119895 = 1 2 3) (4)

where 1198681= 119890119896119896 119879119894119895 119890119894119895are stress and strain tensors respec-

tively and 120575119894119895is Kroneckerrsquos delta

Equations of equilibrium are all satisfied except

119889

119889119903

(119879119903119903) +

(119879119903119903

minus 119879120579120579)

119903

= 0 (5)

A nonlinear differential equation in 120573 has been obtained bysubstituting (4) in (5) as

119899119875120573(119875 + 1)119899minus1 119889119875

119889120573

= [ 119903(

1205831015840

120583

minus

1198621015840

119862

)[(3 minus 2119862) minus (1 minus 119862) (1 minus 119889)119899

1

120573119899

minus (1 minus 119862) minus (119875 + 1)119899] + 119862

times [1 minus (119875 + 1)119899] + 119903119862

1015840[1 minus 2 minus (1 minus 119889)

119899

1

120573119899]

minus 119899119875 [(1 minus 119862) + (119875 + 1)119899] ]

(6)

where 1199031205731015840 = 120573119875 119862 = 2120583(120582 + 2120583)The transition point of 120573 in the previous equation is 119875 rarr

minus1 and 119875 rarr plusmninfin

The Scientific World Journal 3

The boundary conditions are given by

119879119903119903

= minus1199011

at 119903 = 119886

119879119903119903

= minus1199012

at 119903 = 119887

(7)

In the cylinder resultant axial force is given by

119871 = 2120587int

119887

119886

119903119879119911119911119889119903 (8)

5 Method of Solution

As elastic state can go to plastic state under external load-ing through a transition state and we consider only theprincipal stresses Therefore the transition can take placeeither through the principal stresses 119879

119903119903or 119879120579120579

becomingcritical or through the principal stress difference 119879

119903119903minus 119879120579120579

becoming critical It has been shown that the asymptoticsolution through the principal stress leads from elastic stateto plastic state at transition point 119875 rarr plusmninfin For finding theplastic stress at the transition point 119875 rarr plusmninfin we define thetransition function 119877 [14ndash16] in terms of 119879

119903119903as

119877 = 119879119903119903

minus

120582

119899

119896 equiv

2120583

119862119899

[119862 minus 120573119899(1 minus 119862) + (119875 + 1)

119899] (9)

Taking the logarithmic differentiation of (9) with respect to119903 we get

119889

119889119903

log119877 = [1199031198621015840(1 + 120573

119899) minus 119899119875120573

119899+1(119875 + 1)

119899minus1 119889119875

119889120573

minus 119899119875120573119899

times (1 minus 119862) + (119875 + 1)119899 + 119903(

1205831015840

120583

minus

1198621015840

119862

)

times 119862 minus 120573119899[(1 minus 119862) + (119875 + 1)

119899] ]

times (119903 [119862 minus 120573119899(1 minus 119862) + (119875 + 1)

119899])minus1

(10)

Take the asymptotic value 119875 rarr plusmninfin of (10) after substitut-ing 119889119875119889120573 from (6) This on integration gives

1198771= 119860 exp119891 (119903) (11)

where 119860 is a constant of integration and 119891(119903) = minusint(119862119903)119889119903By using (7) in (9) and (11) we get

119879119903119903

= 119860 [exp119891 (119903) minus exp119891 (119887)] minus 1199012 (12)

Using (12) in (5) we get

119879120579120579

= 119860 [(1 minus 119862) exp119891 (119903) minus exp119891 (119887)] minus 1199012 (13)

Equations (4) yields

119879119911119911

= (

1 minus 119862

2 minus 119862

) (119879119903119903

+ 119879120579120579) +

119862120582

(1 minus 119862)

(

3 minus 2119862

2 minus 119862

)

times

(1199012minus 1199011) 2120587 minus int

119887

119886(119903119888 (1 minus 119888) (2 minus 119888))(119879119903119903

+ 119879120579120579) 119889119903

120582 int

119887

119886(119903119888 (3 minus 2119888) (1 minus 119888) (2 minus 119888)) 119889119903

(14)

Taking the non-homogeneity in the cylinder due to variablecompressibility

119879119903119903

= 1198601[exp(

1198880119903minus119896

119896

) minus exp(

1198880119887minus119896

119896

)] minus 1199012

119879120579120579

= 1198601[(1 minus 119888

0119903minus119896) exp(

1198880119903minus119896

119896

) minus exp(

1198880119887minus119896

119896

)]

minus 1199012

119879119911119911

= (

1 minus 1198880119903minus119896

2 minus 1198880119903minus119896

) (119879119903119903

+ 119879120579120579)

+

1205821198880119903minus119896

(1 minus 1198880119903minus119896)

(

3 minus 21198880119903minus119896

2 minus 1198880119903minus119896

) 119890119911119911

(15)

Also 119879120579120579

minus 119879119903119903

=

(1199011minus 1199012) 1198880119903minus119896

exp (1198880119886minus119896119896) minus exp (119888

0119887minus119896119896)

times exp(

1198880119903minus119896

119896

)

(16)

It has been observed from (16) that |119879119903119903minus119879120579120579| is maximum at

119903 = (1198902119887119896)

1119896

= 1199031 that is yielding starts at 119903 = 119903

1 therefore

1003816100381610038161003816119879120579120579

minus 119879119903119903

1003816100381610038161003816119903=1199031

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1199011minus 1199012) 1198880119903minus119896

1exp (119888

01199031

minus119896119896)

exp (1198880119886minus119896119896) minus exp (119888

0119887minus119896119896)

10038161003816100381610038161003816100381610038161003816100381610038161003816

equiv 119884

(17)

Thus pressure required for initial yielding is given by

1003816100381610038161003816119875119894

1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890 [exp (minus1198902119877minus119896

0) minus exp (minus119890

2)]

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

119875119894=

1199011

119884

minus

1199012

119884

= 1198751198941minus 1198751198942

(18)

For full plasticity (1198620rarr 0) (16) becomes

1003816100381610038161003816119879120579120579

minus 119879119903119903

1003816100381610038161003816119903=119887

=

1003816100381610038161003816100381610038161003816100381610038161003816

(1199011minus 1199012) 119896119887minus119896

(119886minus119896

minus 119887minus119896)

1003816100381610038161003816100381610038161003816100381610038161003816

= 1198841

119875119891=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119887119896(119886minus119896

minus 119887minus119896)

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(19)

4 The Scientific World Journal

Now we introduced the following nondimensional quantitiesas

119877 = (

119903

119887

) 1198770= (

119886

119887

) 120590119903119903

= [

119879119903119903

119884

]

120590120579120579

= [

119879120579120579

119884

] 120590119911119911

= [

119879119911119911

119884

]

(20)

The necessary pressure required for initial yielding in nondi-mensional form is given by

1003816100381610038161003816119875119894

1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890 [exp (minus1198902119877minus119896

0) minus exp (minus119890

2)]

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(21)

The transitional stresses are obtained as

120590119903119903

= [

minus119875119894[exp (119888

0119887minus119896119896) (119877

minus119896

0minus 1) minus 1]

exp [(1198880119887minus119896119896) (119877

minus119896

0minus 1)] minus 1

] minus 1198751198942

120590120579120579

= [

minus119875119894

exp [(1198880119887minus119896119896) (119877

minus119896

0minus 1)] minus 1

]

times[(1 minus 1198880119887minus119896119877minus119896) exp

1198880119887minus119896

119896

(119877minus119896

0minus 1) minus 1] minus 119875

1198942

120590119911119911

= (

1 minus 1198880119887minus119896119877minus119896

2 minus 1198880119887minus119896119877minus119896

) (120590119903119903

+ 120590120579120579)

+

1205821198880119887minus119896119877minus119896

(3 minus 21198880(119887119877)minus119896)

(1 minus 1198880(119887119877)minus119896) (2 minus 119888

0(119887119877)minus119896)

119890119911119911

(22)

where

119890119911119911

= ([minus

(1198751198941minus 1198751198942)

2120587

] minus int

1

1198770

1198872119877(

1 minus 1198880(119887119877)minus119896

2 minus 1198880(119887119877)minus119896

)

times (120590119903119903

+ 120590120579120579) 119889119877)

times (120582int

1

1198770

1198880119887minus119896minus1

119877minus119896

(3 minus 21198880(119887119877)minus119896)

(1 minus 1198880(119887119877)minus119896) (2 minus 119888

0(119887119877)minus119896)

119889119877)

minus1

(23)

Also pressure required for fully plastic state is given by

10038161003816100381610038161003816119875119891

10038161003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

119877minus119896

0minus 1

119896

1003816100381610038161003816100381610038161003816100381610038161003816

119875119891=

1199011

1198841

minus

1199012

1198841

= 1198751198911

minus 1198751198912 (24)

and fully plastic stresses are obtained as

120590119903119903

= (minus119875119891)(

119877minus119896

minus 1

119877minus119896

0minus 1

) minus 1198751198912

120590120579120579

= 120590119903minus

119896119877minus119896

(minus119875119891)

(119877minus119896

0minus 1)

120590119911119911

=

119896120582119877minus119896

[minus1198751198912120587 minus (12) int

1

1198770

1198771198872(120590119903119903

+ 120590120579120579) 119889119877]

(119877minus119896

0minus 1)

(25)

Particular Case Nonhomogeneous Cylinder under InternalPressure OnlyThe stresses in fully plastic state are

120590119903119903

= (1198751198912)(

119877minus119896

minus 1

119877minus119896

0minus 1

) minus 1198751198912

120590120579120579

= 120590119903119903

minus

119896119877minus119896

(1198751198912)

(119877minus119896

0minus 1)

120590119911119911

=

119896120582119877minus119896

[11987511989122120587 minus (12) int

1

1198770

1198771198872(120590119903119903

+ 120590120579120579) 119889119877]

(119877minus119896

0minus 1)

(26)

These equations in nondimensional form are the same asthose obtained by Sharma [12]

6 Numerical Discussion

To observe the combined effect of pressure on a cylin-der made of homogeneous and nonhomogeneous materialgraphs have been drawn between pressure and radii ratios1198770= 01 (01) 05 For a homogeneous (119896 = 0) circular

cylinder yielding starts at internal surface whereas for acircular cylinder made of nonhomogeneous material (119896 lt 0non-homogeneity increases radially) yielding takes place atany radius 119903 where 119886 lt 119903 lt 119887 depending upon values of 119862

0

and 119896 Effective pressure is maximum at internal surface forcylinder made of nonhomogeneous as well as homogeneousmaterial It is seen from Figure 1 that for homogeneouscylinder high effective pressure is required for initial yieldingthan that of nonhomogeneous cylinder Also for cylindermade of homogeneous material effective pressure requiredfor initial yielding is less for highly compressible circularcylinder whereas for circular cylinder made of nonhomo-geneous materials high effective pressure is required forhighly compressible cylinder It is also seen from Figure 2that pressure (internalexternal) is maximum at externalsurface for cylinder made of nonhomogeneous as well ashomogeneous material It is also seen that homogeneouscylinder requires high pressure for initial yielding than that ofnonhomogeneous cylinder Also high pressure is required forinitial yielding for highly compressible homogeneous cylin-der whereas less pressure is required for highly compressiblenonhomogeneous cylinder

It has also been observed from Figure 3 that for homoge-neous and nonhomogeneous circular cylinder effective pres-sure required for fully plastic state is maximum at the internalsurface and for nonhomogeneous material less effectivepressure is required for fully plastic state for circular cylindermade of highly compressible material It is also observed thateffective pressure required for fully plastic state is more forcylinder made of homogeneous material than that of non-homogeneousmaterial For homogeneousnonhomogeneouscircular cylinder pressure (internalexternal) required forfully plastic state is maximum at the external surface Ithas been seen from Figure 4 that for nonhomogeneouscylinder made of highly compressible material high pressureis required for fully plastic state It is also observed fromFigure 4 that pressure required for fully plastic state is more

The Scientific World Journal 5

02 04 06 08 1

05

1

15

2

25

3

02 04 06 08 1

025

05

075

1

125

15

P

R0

P

R0

C = 050

C = 040

C = 035

C = 025

C = 015

k = minus125

k = minus100

k = minus075

k = minus050

k = minus025

Figure 1 Effective pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder for different compressibilityparameters

02 04 06 08 1

175

18

185

19

195

20

02 04 06 08 1

185

1875

1925

195

1975

20

P

R0

P

R0

Figure 2 External or internal pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

1

2

3

4

02 04 06 08 1

05

1

15

2

25

P

R0

P

R0

Figure 3 Effective pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder for differentcompressibility parameters

for cylinder made of nonhomogeneous material than that ofhomogeneous material

From Figures 5 and 6 it has been observed that for ho-mogeneous cylinder under external pressure only circum-ferential stresses are maximum at internal surface while for

nonhomogeneous cylinder stresses are maximum at externalsurfaceThese stresses increase significantly with the increasein external pressureWith internal pressure only as seen fromFigures 7 and 8 circumferential stresses are maximum atinternal surface for homogeneous cylinder while maximum

6 The Scientific World Journal

02 04 06 08 1

17

18

19

20

02 04 06 08 1175

185

19

195

20

P

R0

P

R0

Figure 4 External or internal pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

20

40

60

Stresses

02 04 06 08 1

50

100

150

200

Stresses

RR

120590rr for c = 035

120590rr for c = 05

120590rr for c = 01120590120579120579 for c = 05

120590120579120579 for c = 01

120590120579120579 for c = 035120590rr for k = minus5

120590rr for k = minus3

120590120579120579 for k = minus5

120590120579120579 for k = minus3

120590rr for k = minus1

120590120579120579 for k = minus1

Figure 5 Homogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

5

Stresses 02 04 06 08 1

10

20

Stresses

minus10

minus5

minus10

minus20

minus30

R R

Figure 6 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

at external surface for nonhomogeneous cylinder Also it hasbeen observed that the compressible circumferential stresseschange to tensile stresses It has also been observed fromFigures 9 and 10 that with the increase in pressure circum-ferential stresses increases significantly With the increasein external pressure (greater than that of internal pressure)circumferential stresses increases It has been observed fromFigures 11 and 12 (without internal pressure) that fully plasticstresses are maximum at external surface for cylinder made

of nonhomogeneous material and at internal surface forcylinder made of homogeneous material Also it has beenobserved that highly compressible cylinder is having lessstresswhereas less compressible cylinder is having high stressWith the increase in external pressure stresses increasessignificantly FromFigures 13 and 14 it has also been observedthat with the introduction of internal pressure (without exter-nal) compressible circumferential stresses are maximum atinternal surface for homogeneous while at external surface

The Scientific World Journal 7

02 04 06 08 1

Stresses

02 04 06 08 1

25

50

75

100

125

Stresses

minus20

minus40

minus60

minus25

R

R

Figure 7 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

5

Stresses

02 04 06 08 1

10

Stresses

minus5

minus10

minus10

minus20

minus30

R

R

Figure 8 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

02 04 06 08 1

02 04 06 08 1

20

40

60

minus25

minus20

minus50

minus75

minus100

minus125

minus150

Stresses

R

Stresses

R

Figure 9 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

1002 04 06 08 1

minus5

minus10

minus15

minus20

minus10

minus20

Stresses

R

Stresses

R

Figure 10 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

8 The Scientific World Journal

02 04 06 08 1

5

10

15

20

02 04 06 08 1

20

40

60

minus5

Stresses

R

Stresses

R

Figure 11 Homogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

25

5

02 04 06 08 1

10

minus125

minus10

minus75

minus5

minus25minus10

minus20

minus30

minus40

Stresses R

Stresses R

Figure 12 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

02 04 06 08 1

10

20

30

40

minus5

minus10

minus10

minus15

minus20

minus25

Stresses

R

Stresses

R

Figure 13 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

25

75

5

minus10

minus75

minus5

minus5

minus25Stresses

R

02 04 06 08 1

5

minus15

minus10

minus20

minus30

minus25

Stresses

R

Figure 14 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

The Scientific World Journal 9

02 04 06 08 1

minus10

minus20

minus30

minus40

minus50

Stresses

R

minus5

02 04 06 08 1

5

10

minus15

minus10

Stresses

R

Figure 15 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

minus5

02 04 06 08 1

5

10

15

minus15

minus10

minus20

R

Stresses

minus75

minus10

minus125

minus15

minus175

minus20

minus225

02 04 06 08 1R

Stresses

Figure 16 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

for nonhomogeneous cylinder As external pressure increaseand becomes more than that of internal one then stressesagain increase but are of compressive nature Also it has beenobserved that stresses increases significantlywith the increasein internal pressure as can be seen from Figures 15 and 16

7 Conclusions

From the above analysis we can conclude that nonhomo-geneous cylinder with internal and external pressure is onthe safer side of the design as compared to homogeneouscylinder because nonhomogeneous cylinder requires highpressure for initial yielding as compared to homogeneouscylinder It has also been concluded that highly compressiblenonhomogeneous cylinder is on the safer side of the designas compared to less compressible nonhomogeneous circularcylinder because highly compressible cylinder required highpressure for initial yielding as compared to less compressiblenonhomogeneous cylinder which leads to the idea of ldquostresssavingrdquo that minimizes the possibility of fracture of cylinder

References

[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw Hill 3rd edition 1970

[2] I S Sokolinokoff Mathematical Theory of Elasticity McGraw-Hill New York NY USA 1956

[3] J Chakrabarty Theory of Plasticity McGraw-Hill New YorkNY USA 1987

[4] A Mendelson Plasticity Theory and Application The Macmil-lan Company New York NY USA 1968

[5] P C T Chen ldquoA finite difference approach to axisymmetricplane strain problem beyond the elastic limitrdquo in Proceedingsof the 25th Transaction Conference Army Mathematicians pp661ndash674 1980

[6] D Durban andM Kubi ldquoA general solution for the pressurizedelastoplastic tuberdquo Journal of Applied Mechanics TransactionsASME vol 59 no 1 pp 20ndash26 1992

[7] H Jahed and R N Dubey ldquoAn axisymmetric method of elastic-plastic analysis capable of predicting residual stress fieldrdquoJournal of Pressure Vessel Technology Transactions of the ASMEvol 119 no 3 pp 264ndash273 1997

[8] A P Parker ldquoAutofrettage of open-end tubesmdashpressuresstresses strains and code comparisonsrdquo Journal of PressureVessel Technology Transactions of the ASME vol 123 no 3 pp271ndash281 2001

[9] R N Dubey R Seshadri and S Bedi ldquoAnalysis of thick elastic-plastic cylindersrdquo in Proceedings of the Plasticity Conference inWhistler British Columbia Canada 2000

[10] W Olszak and W Urbanowski ldquoNon-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressurerdquoArchiwumMechaniki Stosowanej vol 3 no 7 pp 315ndash336 1955

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

2 The Scientific World Journal

[10] calculate the stresses for nonhomogeneous thick-walledelastic-plastic cylinder subjected to internal pressure whileHodge and Balaban [11] calculated the stresses for rotatingcylinder Sharma [12] analyzed thick-walled cylinders underinternal pressure for isotropic nonhomogeneous elastic-plastic states using transition theory This paper is an exten-sion of Sharma [12] to include the effect of external pressurefor functionally graded material because these days cylindersmade of functionally graded material under internal andexternal pressure are an important design consideration innuclear industry

2 Generalized Lebesgue Strain Measure

The classical theory of elasticity and plasticity divides thespectrum of deformation of solids into two different statesone of which is elastic and other one is plastic In classicaltheory both field equations are solved separately and laterjoined together by yield conditionAs in the behavior ofmate-rials perfect elasticity and perfect plasticity are extremes butno one can draw a sharp line between these two states It istherefore natural to expect that there should be a transitionstate and at this transition a continuum approach necessarilymeans the introduction of nonlinearmeasure But in classicalmechanics the ordinary measure has been found sufficientand so no extension has been made This is because ofthe reason that in classical mechanics field equation forelastic and plastic regions is calculated separately and thenconnected by yield criterions which is an assumption Alsoif in a very small interval the number of fluctuation isvery large the ordinary measure based on Riemann integralfails and measures like that of Lebesgue have been usedThis generalized Lebesgue measure gives very satisfactoryresults in the problems like that of plasticity and creep Thegeneralized Lebesgue strain measure helps to bridge the gapbetween microscopic andmacroscopic descriptions of physi-cal system and eliminate semiempirical conditions like that ofTrescarsquos and von-mises creep strain laws that is Nortanrsquos lawwhich provides a coordination between the theoretical andexperimental results Seth [13] has defined the generalizedprincipal strainmeasure 120576

119894119894by taking the Lebesgue integral of

the weighted function 120576119894119894= int

119890119860

119894119894

0[(1 minus 2

119860

119890119894119894)]

(1198992)minus1

119889119890119860

119894119894= (1119899)

[1 minus (1 minus 2119890119860

119894119894)

(1198992)minus1

] where 119899 is the measure and 119890119860

119894119894is the

principal Almansi strain component

3 Objective

In this paper our aim is to calculate safety factor for thick-walled cylinder made of functionally graded material underinternal and external pressure using Sethrsquos transition theory[12 14ndash16] The stresses are calculated for transition as wellas for fully plastic state The constitutive equations for bothtransition and fully plastic states are also derived from theresults Non-homogeneity is taken as the compressibility ofmaterial in the cylinder as

119862 = 1198620119903minus119896 (1)

where 119886 le 119903 le 119887 1198620and 119896(le 0) are constants

Results obtained have been discussed numerically anddepicted graphically

4 Mathematical Formulation

Consider a nonhomogeneous thick-walled circular cylinderof internal and external radii 119886 and 119887 respectively subjectedto internal pressure 119901

1and external pressure 119901

2 The non-

homogeneity in the cylinder is due to variable compressibility119862 = 119862

0119903minus119896 The cylinder is taken so large that plane trans-

verse sections remain plane during the expansion and hencethe longitudinal strain is the same for all elements at eachstage of the expansion In cylindrical polar coordinates thedisplacements are given by [14ndash16]

119906 = 119903 (1 minus 120573) V = 0 119908 = 119889119911 (2)

where 120573 is a function of 119903 = radic1199092+ 1199102 and 119889 is a constant

The generalized strains are

119890119903119903

=

1

119899

[1 minus (1199031205731015840+ 120573)

119899

] 119890120579120579

=

1

119899

[1 minus 120573119899]

119890119911119911

=

1

119899

[1 minus (1 minus 119889)119899] 119890

119903120579= 119890120579119911

= 119890119911119903

= 0

(3)

where 119899 is the measure and 1205731015840= 119889120573119889119903

For isotropic material the stress-strain relation in elasticregion is given by

119879119894119895= 1205821205751198941198951198681+ 2120583119890

119894119895 (119894 119895 = 1 2 3) (4)

where 1198681= 119890119896119896 119879119894119895 119890119894119895are stress and strain tensors respec-

tively and 120575119894119895is Kroneckerrsquos delta

Equations of equilibrium are all satisfied except

119889

119889119903

(119879119903119903) +

(119879119903119903

minus 119879120579120579)

119903

= 0 (5)

A nonlinear differential equation in 120573 has been obtained bysubstituting (4) in (5) as

119899119875120573(119875 + 1)119899minus1 119889119875

119889120573

= [ 119903(

1205831015840

120583

minus

1198621015840

119862

)[(3 minus 2119862) minus (1 minus 119862) (1 minus 119889)119899

1

120573119899

minus (1 minus 119862) minus (119875 + 1)119899] + 119862

times [1 minus (119875 + 1)119899] + 119903119862

1015840[1 minus 2 minus (1 minus 119889)

119899

1

120573119899]

minus 119899119875 [(1 minus 119862) + (119875 + 1)119899] ]

(6)

where 1199031205731015840 = 120573119875 119862 = 2120583(120582 + 2120583)The transition point of 120573 in the previous equation is 119875 rarr

minus1 and 119875 rarr plusmninfin

The Scientific World Journal 3

The boundary conditions are given by

119879119903119903

= minus1199011

at 119903 = 119886

119879119903119903

= minus1199012

at 119903 = 119887

(7)

In the cylinder resultant axial force is given by

119871 = 2120587int

119887

119886

119903119879119911119911119889119903 (8)

5 Method of Solution

As elastic state can go to plastic state under external load-ing through a transition state and we consider only theprincipal stresses Therefore the transition can take placeeither through the principal stresses 119879

119903119903or 119879120579120579

becomingcritical or through the principal stress difference 119879

119903119903minus 119879120579120579

becoming critical It has been shown that the asymptoticsolution through the principal stress leads from elastic stateto plastic state at transition point 119875 rarr plusmninfin For finding theplastic stress at the transition point 119875 rarr plusmninfin we define thetransition function 119877 [14ndash16] in terms of 119879

119903119903as

119877 = 119879119903119903

minus

120582

119899

119896 equiv

2120583

119862119899

[119862 minus 120573119899(1 minus 119862) + (119875 + 1)

119899] (9)

Taking the logarithmic differentiation of (9) with respect to119903 we get

119889

119889119903

log119877 = [1199031198621015840(1 + 120573

119899) minus 119899119875120573

119899+1(119875 + 1)

119899minus1 119889119875

119889120573

minus 119899119875120573119899

times (1 minus 119862) + (119875 + 1)119899 + 119903(

1205831015840

120583

minus

1198621015840

119862

)

times 119862 minus 120573119899[(1 minus 119862) + (119875 + 1)

119899] ]

times (119903 [119862 minus 120573119899(1 minus 119862) + (119875 + 1)

119899])minus1

(10)

Take the asymptotic value 119875 rarr plusmninfin of (10) after substitut-ing 119889119875119889120573 from (6) This on integration gives

1198771= 119860 exp119891 (119903) (11)

where 119860 is a constant of integration and 119891(119903) = minusint(119862119903)119889119903By using (7) in (9) and (11) we get

119879119903119903

= 119860 [exp119891 (119903) minus exp119891 (119887)] minus 1199012 (12)

Using (12) in (5) we get

119879120579120579

= 119860 [(1 minus 119862) exp119891 (119903) minus exp119891 (119887)] minus 1199012 (13)

Equations (4) yields

119879119911119911

= (

1 minus 119862

2 minus 119862

) (119879119903119903

+ 119879120579120579) +

119862120582

(1 minus 119862)

(

3 minus 2119862

2 minus 119862

)

times

(1199012minus 1199011) 2120587 minus int

119887

119886(119903119888 (1 minus 119888) (2 minus 119888))(119879119903119903

+ 119879120579120579) 119889119903

120582 int

119887

119886(119903119888 (3 minus 2119888) (1 minus 119888) (2 minus 119888)) 119889119903

(14)

Taking the non-homogeneity in the cylinder due to variablecompressibility

119879119903119903

= 1198601[exp(

1198880119903minus119896

119896

) minus exp(

1198880119887minus119896

119896

)] minus 1199012

119879120579120579

= 1198601[(1 minus 119888

0119903minus119896) exp(

1198880119903minus119896

119896

) minus exp(

1198880119887minus119896

119896

)]

minus 1199012

119879119911119911

= (

1 minus 1198880119903minus119896

2 minus 1198880119903minus119896

) (119879119903119903

+ 119879120579120579)

+

1205821198880119903minus119896

(1 minus 1198880119903minus119896)

(

3 minus 21198880119903minus119896

2 minus 1198880119903minus119896

) 119890119911119911

(15)

Also 119879120579120579

minus 119879119903119903

=

(1199011minus 1199012) 1198880119903minus119896

exp (1198880119886minus119896119896) minus exp (119888

0119887minus119896119896)

times exp(

1198880119903minus119896

119896

)

(16)

It has been observed from (16) that |119879119903119903minus119879120579120579| is maximum at

119903 = (1198902119887119896)

1119896

= 1199031 that is yielding starts at 119903 = 119903

1 therefore

1003816100381610038161003816119879120579120579

minus 119879119903119903

1003816100381610038161003816119903=1199031

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1199011minus 1199012) 1198880119903minus119896

1exp (119888

01199031

minus119896119896)

exp (1198880119886minus119896119896) minus exp (119888

0119887minus119896119896)

10038161003816100381610038161003816100381610038161003816100381610038161003816

equiv 119884

(17)

Thus pressure required for initial yielding is given by

1003816100381610038161003816119875119894

1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890 [exp (minus1198902119877minus119896

0) minus exp (minus119890

2)]

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

119875119894=

1199011

119884

minus

1199012

119884

= 1198751198941minus 1198751198942

(18)

For full plasticity (1198620rarr 0) (16) becomes

1003816100381610038161003816119879120579120579

minus 119879119903119903

1003816100381610038161003816119903=119887

=

1003816100381610038161003816100381610038161003816100381610038161003816

(1199011minus 1199012) 119896119887minus119896

(119886minus119896

minus 119887minus119896)

1003816100381610038161003816100381610038161003816100381610038161003816

= 1198841

119875119891=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119887119896(119886minus119896

minus 119887minus119896)

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(19)

4 The Scientific World Journal

Now we introduced the following nondimensional quantitiesas

119877 = (

119903

119887

) 1198770= (

119886

119887

) 120590119903119903

= [

119879119903119903

119884

]

120590120579120579

= [

119879120579120579

119884

] 120590119911119911

= [

119879119911119911

119884

]

(20)

The necessary pressure required for initial yielding in nondi-mensional form is given by

1003816100381610038161003816119875119894

1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890 [exp (minus1198902119877minus119896

0) minus exp (minus119890

2)]

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(21)

The transitional stresses are obtained as

120590119903119903

= [

minus119875119894[exp (119888

0119887minus119896119896) (119877

minus119896

0minus 1) minus 1]

exp [(1198880119887minus119896119896) (119877

minus119896

0minus 1)] minus 1

] minus 1198751198942

120590120579120579

= [

minus119875119894

exp [(1198880119887minus119896119896) (119877

minus119896

0minus 1)] minus 1

]

times[(1 minus 1198880119887minus119896119877minus119896) exp

1198880119887minus119896

119896

(119877minus119896

0minus 1) minus 1] minus 119875

1198942

120590119911119911

= (

1 minus 1198880119887minus119896119877minus119896

2 minus 1198880119887minus119896119877minus119896

) (120590119903119903

+ 120590120579120579)

+

1205821198880119887minus119896119877minus119896

(3 minus 21198880(119887119877)minus119896)

(1 minus 1198880(119887119877)minus119896) (2 minus 119888

0(119887119877)minus119896)

119890119911119911

(22)

where

119890119911119911

= ([minus

(1198751198941minus 1198751198942)

2120587

] minus int

1

1198770

1198872119877(

1 minus 1198880(119887119877)minus119896

2 minus 1198880(119887119877)minus119896

)

times (120590119903119903

+ 120590120579120579) 119889119877)

times (120582int

1

1198770

1198880119887minus119896minus1

119877minus119896

(3 minus 21198880(119887119877)minus119896)

(1 minus 1198880(119887119877)minus119896) (2 minus 119888

0(119887119877)minus119896)

119889119877)

minus1

(23)

Also pressure required for fully plastic state is given by

10038161003816100381610038161003816119875119891

10038161003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

119877minus119896

0minus 1

119896

1003816100381610038161003816100381610038161003816100381610038161003816

119875119891=

1199011

1198841

minus

1199012

1198841

= 1198751198911

minus 1198751198912 (24)

and fully plastic stresses are obtained as

120590119903119903

= (minus119875119891)(

119877minus119896

minus 1

119877minus119896

0minus 1

) minus 1198751198912

120590120579120579

= 120590119903minus

119896119877minus119896

(minus119875119891)

(119877minus119896

0minus 1)

120590119911119911

=

119896120582119877minus119896

[minus1198751198912120587 minus (12) int

1

1198770

1198771198872(120590119903119903

+ 120590120579120579) 119889119877]

(119877minus119896

0minus 1)

(25)

Particular Case Nonhomogeneous Cylinder under InternalPressure OnlyThe stresses in fully plastic state are

120590119903119903

= (1198751198912)(

119877minus119896

minus 1

119877minus119896

0minus 1

) minus 1198751198912

120590120579120579

= 120590119903119903

minus

119896119877minus119896

(1198751198912)

(119877minus119896

0minus 1)

120590119911119911

=

119896120582119877minus119896

[11987511989122120587 minus (12) int

1

1198770

1198771198872(120590119903119903

+ 120590120579120579) 119889119877]

(119877minus119896

0minus 1)

(26)

These equations in nondimensional form are the same asthose obtained by Sharma [12]

6 Numerical Discussion

To observe the combined effect of pressure on a cylin-der made of homogeneous and nonhomogeneous materialgraphs have been drawn between pressure and radii ratios1198770= 01 (01) 05 For a homogeneous (119896 = 0) circular

cylinder yielding starts at internal surface whereas for acircular cylinder made of nonhomogeneous material (119896 lt 0non-homogeneity increases radially) yielding takes place atany radius 119903 where 119886 lt 119903 lt 119887 depending upon values of 119862

0

and 119896 Effective pressure is maximum at internal surface forcylinder made of nonhomogeneous as well as homogeneousmaterial It is seen from Figure 1 that for homogeneouscylinder high effective pressure is required for initial yieldingthan that of nonhomogeneous cylinder Also for cylindermade of homogeneous material effective pressure requiredfor initial yielding is less for highly compressible circularcylinder whereas for circular cylinder made of nonhomo-geneous materials high effective pressure is required forhighly compressible cylinder It is also seen from Figure 2that pressure (internalexternal) is maximum at externalsurface for cylinder made of nonhomogeneous as well ashomogeneous material It is also seen that homogeneouscylinder requires high pressure for initial yielding than that ofnonhomogeneous cylinder Also high pressure is required forinitial yielding for highly compressible homogeneous cylin-der whereas less pressure is required for highly compressiblenonhomogeneous cylinder

It has also been observed from Figure 3 that for homoge-neous and nonhomogeneous circular cylinder effective pres-sure required for fully plastic state is maximum at the internalsurface and for nonhomogeneous material less effectivepressure is required for fully plastic state for circular cylindermade of highly compressible material It is also observed thateffective pressure required for fully plastic state is more forcylinder made of homogeneous material than that of non-homogeneousmaterial For homogeneousnonhomogeneouscircular cylinder pressure (internalexternal) required forfully plastic state is maximum at the external surface Ithas been seen from Figure 4 that for nonhomogeneouscylinder made of highly compressible material high pressureis required for fully plastic state It is also observed fromFigure 4 that pressure required for fully plastic state is more

The Scientific World Journal 5

02 04 06 08 1

05

1

15

2

25

3

02 04 06 08 1

025

05

075

1

125

15

P

R0

P

R0

C = 050

C = 040

C = 035

C = 025

C = 015

k = minus125

k = minus100

k = minus075

k = minus050

k = minus025

Figure 1 Effective pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder for different compressibilityparameters

02 04 06 08 1

175

18

185

19

195

20

02 04 06 08 1

185

1875

1925

195

1975

20

P

R0

P

R0

Figure 2 External or internal pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

1

2

3

4

02 04 06 08 1

05

1

15

2

25

P

R0

P

R0

Figure 3 Effective pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder for differentcompressibility parameters

for cylinder made of nonhomogeneous material than that ofhomogeneous material

From Figures 5 and 6 it has been observed that for ho-mogeneous cylinder under external pressure only circum-ferential stresses are maximum at internal surface while for

nonhomogeneous cylinder stresses are maximum at externalsurfaceThese stresses increase significantly with the increasein external pressureWith internal pressure only as seen fromFigures 7 and 8 circumferential stresses are maximum atinternal surface for homogeneous cylinder while maximum

6 The Scientific World Journal

02 04 06 08 1

17

18

19

20

02 04 06 08 1175

185

19

195

20

P

R0

P

R0

Figure 4 External or internal pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

20

40

60

Stresses

02 04 06 08 1

50

100

150

200

Stresses

RR

120590rr for c = 035

120590rr for c = 05

120590rr for c = 01120590120579120579 for c = 05

120590120579120579 for c = 01

120590120579120579 for c = 035120590rr for k = minus5

120590rr for k = minus3

120590120579120579 for k = minus5

120590120579120579 for k = minus3

120590rr for k = minus1

120590120579120579 for k = minus1

Figure 5 Homogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

5

Stresses 02 04 06 08 1

10

20

Stresses

minus10

minus5

minus10

minus20

minus30

R R

Figure 6 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

at external surface for nonhomogeneous cylinder Also it hasbeen observed that the compressible circumferential stresseschange to tensile stresses It has also been observed fromFigures 9 and 10 that with the increase in pressure circum-ferential stresses increases significantly With the increasein external pressure (greater than that of internal pressure)circumferential stresses increases It has been observed fromFigures 11 and 12 (without internal pressure) that fully plasticstresses are maximum at external surface for cylinder made

of nonhomogeneous material and at internal surface forcylinder made of homogeneous material Also it has beenobserved that highly compressible cylinder is having lessstresswhereas less compressible cylinder is having high stressWith the increase in external pressure stresses increasessignificantly FromFigures 13 and 14 it has also been observedthat with the introduction of internal pressure (without exter-nal) compressible circumferential stresses are maximum atinternal surface for homogeneous while at external surface

The Scientific World Journal 7

02 04 06 08 1

Stresses

02 04 06 08 1

25

50

75

100

125

Stresses

minus20

minus40

minus60

minus25

R

R

Figure 7 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

5

Stresses

02 04 06 08 1

10

Stresses

minus5

minus10

minus10

minus20

minus30

R

R

Figure 8 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

02 04 06 08 1

02 04 06 08 1

20

40

60

minus25

minus20

minus50

minus75

minus100

minus125

minus150

Stresses

R

Stresses

R

Figure 9 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

1002 04 06 08 1

minus5

minus10

minus15

minus20

minus10

minus20

Stresses

R

Stresses

R

Figure 10 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

8 The Scientific World Journal

02 04 06 08 1

5

10

15

20

02 04 06 08 1

20

40

60

minus5

Stresses

R

Stresses

R

Figure 11 Homogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

25

5

02 04 06 08 1

10

minus125

minus10

minus75

minus5

minus25minus10

minus20

minus30

minus40

Stresses R

Stresses R

Figure 12 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

02 04 06 08 1

10

20

30

40

minus5

minus10

minus10

minus15

minus20

minus25

Stresses

R

Stresses

R

Figure 13 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

25

75

5

minus10

minus75

minus5

minus5

minus25Stresses

R

02 04 06 08 1

5

minus15

minus10

minus20

minus30

minus25

Stresses

R

Figure 14 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

The Scientific World Journal 9

02 04 06 08 1

minus10

minus20

minus30

minus40

minus50

Stresses

R

minus5

02 04 06 08 1

5

10

minus15

minus10

Stresses

R

Figure 15 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

minus5

02 04 06 08 1

5

10

15

minus15

minus10

minus20

R

Stresses

minus75

minus10

minus125

minus15

minus175

minus20

minus225

02 04 06 08 1R

Stresses

Figure 16 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

for nonhomogeneous cylinder As external pressure increaseand becomes more than that of internal one then stressesagain increase but are of compressive nature Also it has beenobserved that stresses increases significantlywith the increasein internal pressure as can be seen from Figures 15 and 16

7 Conclusions

From the above analysis we can conclude that nonhomo-geneous cylinder with internal and external pressure is onthe safer side of the design as compared to homogeneouscylinder because nonhomogeneous cylinder requires highpressure for initial yielding as compared to homogeneouscylinder It has also been concluded that highly compressiblenonhomogeneous cylinder is on the safer side of the designas compared to less compressible nonhomogeneous circularcylinder because highly compressible cylinder required highpressure for initial yielding as compared to less compressiblenonhomogeneous cylinder which leads to the idea of ldquostresssavingrdquo that minimizes the possibility of fracture of cylinder

References

[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw Hill 3rd edition 1970

[2] I S Sokolinokoff Mathematical Theory of Elasticity McGraw-Hill New York NY USA 1956

[3] J Chakrabarty Theory of Plasticity McGraw-Hill New YorkNY USA 1987

[4] A Mendelson Plasticity Theory and Application The Macmil-lan Company New York NY USA 1968

[5] P C T Chen ldquoA finite difference approach to axisymmetricplane strain problem beyond the elastic limitrdquo in Proceedingsof the 25th Transaction Conference Army Mathematicians pp661ndash674 1980

[6] D Durban andM Kubi ldquoA general solution for the pressurizedelastoplastic tuberdquo Journal of Applied Mechanics TransactionsASME vol 59 no 1 pp 20ndash26 1992

[7] H Jahed and R N Dubey ldquoAn axisymmetric method of elastic-plastic analysis capable of predicting residual stress fieldrdquoJournal of Pressure Vessel Technology Transactions of the ASMEvol 119 no 3 pp 264ndash273 1997

[8] A P Parker ldquoAutofrettage of open-end tubesmdashpressuresstresses strains and code comparisonsrdquo Journal of PressureVessel Technology Transactions of the ASME vol 123 no 3 pp271ndash281 2001

[9] R N Dubey R Seshadri and S Bedi ldquoAnalysis of thick elastic-plastic cylindersrdquo in Proceedings of the Plasticity Conference inWhistler British Columbia Canada 2000

[10] W Olszak and W Urbanowski ldquoNon-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressurerdquoArchiwumMechaniki Stosowanej vol 3 no 7 pp 315ndash336 1955

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

The Scientific World Journal 3

The boundary conditions are given by

119879119903119903

= minus1199011

at 119903 = 119886

119879119903119903

= minus1199012

at 119903 = 119887

(7)

In the cylinder resultant axial force is given by

119871 = 2120587int

119887

119886

119903119879119911119911119889119903 (8)

5 Method of Solution

As elastic state can go to plastic state under external load-ing through a transition state and we consider only theprincipal stresses Therefore the transition can take placeeither through the principal stresses 119879

119903119903or 119879120579120579

becomingcritical or through the principal stress difference 119879

119903119903minus 119879120579120579

becoming critical It has been shown that the asymptoticsolution through the principal stress leads from elastic stateto plastic state at transition point 119875 rarr plusmninfin For finding theplastic stress at the transition point 119875 rarr plusmninfin we define thetransition function 119877 [14ndash16] in terms of 119879

119903119903as

119877 = 119879119903119903

minus

120582

119899

119896 equiv

2120583

119862119899

[119862 minus 120573119899(1 minus 119862) + (119875 + 1)

119899] (9)

Taking the logarithmic differentiation of (9) with respect to119903 we get

119889

119889119903

log119877 = [1199031198621015840(1 + 120573

119899) minus 119899119875120573

119899+1(119875 + 1)

119899minus1 119889119875

119889120573

minus 119899119875120573119899

times (1 minus 119862) + (119875 + 1)119899 + 119903(

1205831015840

120583

minus

1198621015840

119862

)

times 119862 minus 120573119899[(1 minus 119862) + (119875 + 1)

119899] ]

times (119903 [119862 minus 120573119899(1 minus 119862) + (119875 + 1)

119899])minus1

(10)

Take the asymptotic value 119875 rarr plusmninfin of (10) after substitut-ing 119889119875119889120573 from (6) This on integration gives

1198771= 119860 exp119891 (119903) (11)

where 119860 is a constant of integration and 119891(119903) = minusint(119862119903)119889119903By using (7) in (9) and (11) we get

119879119903119903

= 119860 [exp119891 (119903) minus exp119891 (119887)] minus 1199012 (12)

Using (12) in (5) we get

119879120579120579

= 119860 [(1 minus 119862) exp119891 (119903) minus exp119891 (119887)] minus 1199012 (13)

Equations (4) yields

119879119911119911

= (

1 minus 119862

2 minus 119862

) (119879119903119903

+ 119879120579120579) +

119862120582

(1 minus 119862)

(

3 minus 2119862

2 minus 119862

)

times

(1199012minus 1199011) 2120587 minus int

119887

119886(119903119888 (1 minus 119888) (2 minus 119888))(119879119903119903

+ 119879120579120579) 119889119903

120582 int

119887

119886(119903119888 (3 minus 2119888) (1 minus 119888) (2 minus 119888)) 119889119903

(14)

Taking the non-homogeneity in the cylinder due to variablecompressibility

119879119903119903

= 1198601[exp(

1198880119903minus119896

119896

) minus exp(

1198880119887minus119896

119896

)] minus 1199012

119879120579120579

= 1198601[(1 minus 119888

0119903minus119896) exp(

1198880119903minus119896

119896

) minus exp(

1198880119887minus119896

119896

)]

minus 1199012

119879119911119911

= (

1 minus 1198880119903minus119896

2 minus 1198880119903minus119896

) (119879119903119903

+ 119879120579120579)

+

1205821198880119903minus119896

(1 minus 1198880119903minus119896)

(

3 minus 21198880119903minus119896

2 minus 1198880119903minus119896

) 119890119911119911

(15)

Also 119879120579120579

minus 119879119903119903

=

(1199011minus 1199012) 1198880119903minus119896

exp (1198880119886minus119896119896) minus exp (119888

0119887minus119896119896)

times exp(

1198880119903minus119896

119896

)

(16)

It has been observed from (16) that |119879119903119903minus119879120579120579| is maximum at

119903 = (1198902119887119896)

1119896

= 1199031 that is yielding starts at 119903 = 119903

1 therefore

1003816100381610038161003816119879120579120579

minus 119879119903119903

1003816100381610038161003816119903=1199031

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1199011minus 1199012) 1198880119903minus119896

1exp (119888

01199031

minus119896119896)

exp (1198880119886minus119896119896) minus exp (119888

0119887minus119896119896)

10038161003816100381610038161003816100381610038161003816100381610038161003816

equiv 119884

(17)

Thus pressure required for initial yielding is given by

1003816100381610038161003816119875119894

1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890 [exp (minus1198902119877minus119896

0) minus exp (minus119890

2)]

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

119875119894=

1199011

119884

minus

1199012

119884

= 1198751198941minus 1198751198942

(18)

For full plasticity (1198620rarr 0) (16) becomes

1003816100381610038161003816119879120579120579

minus 119879119903119903

1003816100381610038161003816119903=119887

=

1003816100381610038161003816100381610038161003816100381610038161003816

(1199011minus 1199012) 119896119887minus119896

(119886minus119896

minus 119887minus119896)

1003816100381610038161003816100381610038161003816100381610038161003816

= 1198841

119875119891=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119887119896(119886minus119896

minus 119887minus119896)

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(19)

4 The Scientific World Journal

Now we introduced the following nondimensional quantitiesas

119877 = (

119903

119887

) 1198770= (

119886

119887

) 120590119903119903

= [

119879119903119903

119884

]

120590120579120579

= [

119879120579120579

119884

] 120590119911119911

= [

119879119911119911

119884

]

(20)

The necessary pressure required for initial yielding in nondi-mensional form is given by

1003816100381610038161003816119875119894

1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890 [exp (minus1198902119877minus119896

0) minus exp (minus119890

2)]

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(21)

The transitional stresses are obtained as

120590119903119903

= [

minus119875119894[exp (119888

0119887minus119896119896) (119877

minus119896

0minus 1) minus 1]

exp [(1198880119887minus119896119896) (119877

minus119896

0minus 1)] minus 1

] minus 1198751198942

120590120579120579

= [

minus119875119894

exp [(1198880119887minus119896119896) (119877

minus119896

0minus 1)] minus 1

]

times[(1 minus 1198880119887minus119896119877minus119896) exp

1198880119887minus119896

119896

(119877minus119896

0minus 1) minus 1] minus 119875

1198942

120590119911119911

= (

1 minus 1198880119887minus119896119877minus119896

2 minus 1198880119887minus119896119877minus119896

) (120590119903119903

+ 120590120579120579)

+

1205821198880119887minus119896119877minus119896

(3 minus 21198880(119887119877)minus119896)

(1 minus 1198880(119887119877)minus119896) (2 minus 119888

0(119887119877)minus119896)

119890119911119911

(22)

where

119890119911119911

= ([minus

(1198751198941minus 1198751198942)

2120587

] minus int

1

1198770

1198872119877(

1 minus 1198880(119887119877)minus119896

2 minus 1198880(119887119877)minus119896

)

times (120590119903119903

+ 120590120579120579) 119889119877)

times (120582int

1

1198770

1198880119887minus119896minus1

119877minus119896

(3 minus 21198880(119887119877)minus119896)

(1 minus 1198880(119887119877)minus119896) (2 minus 119888

0(119887119877)minus119896)

119889119877)

minus1

(23)

Also pressure required for fully plastic state is given by

10038161003816100381610038161003816119875119891

10038161003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

119877minus119896

0minus 1

119896

1003816100381610038161003816100381610038161003816100381610038161003816

119875119891=

1199011

1198841

minus

1199012

1198841

= 1198751198911

minus 1198751198912 (24)

and fully plastic stresses are obtained as

120590119903119903

= (minus119875119891)(

119877minus119896

minus 1

119877minus119896

0minus 1

) minus 1198751198912

120590120579120579

= 120590119903minus

119896119877minus119896

(minus119875119891)

(119877minus119896

0minus 1)

120590119911119911

=

119896120582119877minus119896

[minus1198751198912120587 minus (12) int

1

1198770

1198771198872(120590119903119903

+ 120590120579120579) 119889119877]

(119877minus119896

0minus 1)

(25)

Particular Case Nonhomogeneous Cylinder under InternalPressure OnlyThe stresses in fully plastic state are

120590119903119903

= (1198751198912)(

119877minus119896

minus 1

119877minus119896

0minus 1

) minus 1198751198912

120590120579120579

= 120590119903119903

minus

119896119877minus119896

(1198751198912)

(119877minus119896

0minus 1)

120590119911119911

=

119896120582119877minus119896

[11987511989122120587 minus (12) int

1

1198770

1198771198872(120590119903119903

+ 120590120579120579) 119889119877]

(119877minus119896

0minus 1)

(26)

These equations in nondimensional form are the same asthose obtained by Sharma [12]

6 Numerical Discussion

To observe the combined effect of pressure on a cylin-der made of homogeneous and nonhomogeneous materialgraphs have been drawn between pressure and radii ratios1198770= 01 (01) 05 For a homogeneous (119896 = 0) circular

cylinder yielding starts at internal surface whereas for acircular cylinder made of nonhomogeneous material (119896 lt 0non-homogeneity increases radially) yielding takes place atany radius 119903 where 119886 lt 119903 lt 119887 depending upon values of 119862

0

and 119896 Effective pressure is maximum at internal surface forcylinder made of nonhomogeneous as well as homogeneousmaterial It is seen from Figure 1 that for homogeneouscylinder high effective pressure is required for initial yieldingthan that of nonhomogeneous cylinder Also for cylindermade of homogeneous material effective pressure requiredfor initial yielding is less for highly compressible circularcylinder whereas for circular cylinder made of nonhomo-geneous materials high effective pressure is required forhighly compressible cylinder It is also seen from Figure 2that pressure (internalexternal) is maximum at externalsurface for cylinder made of nonhomogeneous as well ashomogeneous material It is also seen that homogeneouscylinder requires high pressure for initial yielding than that ofnonhomogeneous cylinder Also high pressure is required forinitial yielding for highly compressible homogeneous cylin-der whereas less pressure is required for highly compressiblenonhomogeneous cylinder

It has also been observed from Figure 3 that for homoge-neous and nonhomogeneous circular cylinder effective pres-sure required for fully plastic state is maximum at the internalsurface and for nonhomogeneous material less effectivepressure is required for fully plastic state for circular cylindermade of highly compressible material It is also observed thateffective pressure required for fully plastic state is more forcylinder made of homogeneous material than that of non-homogeneousmaterial For homogeneousnonhomogeneouscircular cylinder pressure (internalexternal) required forfully plastic state is maximum at the external surface Ithas been seen from Figure 4 that for nonhomogeneouscylinder made of highly compressible material high pressureis required for fully plastic state It is also observed fromFigure 4 that pressure required for fully plastic state is more

The Scientific World Journal 5

02 04 06 08 1

05

1

15

2

25

3

02 04 06 08 1

025

05

075

1

125

15

P

R0

P

R0

C = 050

C = 040

C = 035

C = 025

C = 015

k = minus125

k = minus100

k = minus075

k = minus050

k = minus025

Figure 1 Effective pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder for different compressibilityparameters

02 04 06 08 1

175

18

185

19

195

20

02 04 06 08 1

185

1875

1925

195

1975

20

P

R0

P

R0

Figure 2 External or internal pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

1

2

3

4

02 04 06 08 1

05

1

15

2

25

P

R0

P

R0

Figure 3 Effective pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder for differentcompressibility parameters

for cylinder made of nonhomogeneous material than that ofhomogeneous material

From Figures 5 and 6 it has been observed that for ho-mogeneous cylinder under external pressure only circum-ferential stresses are maximum at internal surface while for

nonhomogeneous cylinder stresses are maximum at externalsurfaceThese stresses increase significantly with the increasein external pressureWith internal pressure only as seen fromFigures 7 and 8 circumferential stresses are maximum atinternal surface for homogeneous cylinder while maximum

6 The Scientific World Journal

02 04 06 08 1

17

18

19

20

02 04 06 08 1175

185

19

195

20

P

R0

P

R0

Figure 4 External or internal pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

20

40

60

Stresses

02 04 06 08 1

50

100

150

200

Stresses

RR

120590rr for c = 035

120590rr for c = 05

120590rr for c = 01120590120579120579 for c = 05

120590120579120579 for c = 01

120590120579120579 for c = 035120590rr for k = minus5

120590rr for k = minus3

120590120579120579 for k = minus5

120590120579120579 for k = minus3

120590rr for k = minus1

120590120579120579 for k = minus1

Figure 5 Homogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

5

Stresses 02 04 06 08 1

10

20

Stresses

minus10

minus5

minus10

minus20

minus30

R R

Figure 6 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

at external surface for nonhomogeneous cylinder Also it hasbeen observed that the compressible circumferential stresseschange to tensile stresses It has also been observed fromFigures 9 and 10 that with the increase in pressure circum-ferential stresses increases significantly With the increasein external pressure (greater than that of internal pressure)circumferential stresses increases It has been observed fromFigures 11 and 12 (without internal pressure) that fully plasticstresses are maximum at external surface for cylinder made

of nonhomogeneous material and at internal surface forcylinder made of homogeneous material Also it has beenobserved that highly compressible cylinder is having lessstresswhereas less compressible cylinder is having high stressWith the increase in external pressure stresses increasessignificantly FromFigures 13 and 14 it has also been observedthat with the introduction of internal pressure (without exter-nal) compressible circumferential stresses are maximum atinternal surface for homogeneous while at external surface

The Scientific World Journal 7

02 04 06 08 1

Stresses

02 04 06 08 1

25

50

75

100

125

Stresses

minus20

minus40

minus60

minus25

R

R

Figure 7 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

5

Stresses

02 04 06 08 1

10

Stresses

minus5

minus10

minus10

minus20

minus30

R

R

Figure 8 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

02 04 06 08 1

02 04 06 08 1

20

40

60

minus25

minus20

minus50

minus75

minus100

minus125

minus150

Stresses

R

Stresses

R

Figure 9 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

1002 04 06 08 1

minus5

minus10

minus15

minus20

minus10

minus20

Stresses

R

Stresses

R

Figure 10 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

8 The Scientific World Journal

02 04 06 08 1

5

10

15

20

02 04 06 08 1

20

40

60

minus5

Stresses

R

Stresses

R

Figure 11 Homogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

25

5

02 04 06 08 1

10

minus125

minus10

minus75

minus5

minus25minus10

minus20

minus30

minus40

Stresses R

Stresses R

Figure 12 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

02 04 06 08 1

10

20

30

40

minus5

minus10

minus10

minus15

minus20

minus25

Stresses

R

Stresses

R

Figure 13 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

25

75

5

minus10

minus75

minus5

minus5

minus25Stresses

R

02 04 06 08 1

5

minus15

minus10

minus20

minus30

minus25

Stresses

R

Figure 14 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

The Scientific World Journal 9

02 04 06 08 1

minus10

minus20

minus30

minus40

minus50

Stresses

R

minus5

02 04 06 08 1

5

10

minus15

minus10

Stresses

R

Figure 15 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

minus5

02 04 06 08 1

5

10

15

minus15

minus10

minus20

R

Stresses

minus75

minus10

minus125

minus15

minus175

minus20

minus225

02 04 06 08 1R

Stresses

Figure 16 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

for nonhomogeneous cylinder As external pressure increaseand becomes more than that of internal one then stressesagain increase but are of compressive nature Also it has beenobserved that stresses increases significantlywith the increasein internal pressure as can be seen from Figures 15 and 16

7 Conclusions

From the above analysis we can conclude that nonhomo-geneous cylinder with internal and external pressure is onthe safer side of the design as compared to homogeneouscylinder because nonhomogeneous cylinder requires highpressure for initial yielding as compared to homogeneouscylinder It has also been concluded that highly compressiblenonhomogeneous cylinder is on the safer side of the designas compared to less compressible nonhomogeneous circularcylinder because highly compressible cylinder required highpressure for initial yielding as compared to less compressiblenonhomogeneous cylinder which leads to the idea of ldquostresssavingrdquo that minimizes the possibility of fracture of cylinder

References

[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw Hill 3rd edition 1970

[2] I S Sokolinokoff Mathematical Theory of Elasticity McGraw-Hill New York NY USA 1956

[3] J Chakrabarty Theory of Plasticity McGraw-Hill New YorkNY USA 1987

[4] A Mendelson Plasticity Theory and Application The Macmil-lan Company New York NY USA 1968

[5] P C T Chen ldquoA finite difference approach to axisymmetricplane strain problem beyond the elastic limitrdquo in Proceedingsof the 25th Transaction Conference Army Mathematicians pp661ndash674 1980

[6] D Durban andM Kubi ldquoA general solution for the pressurizedelastoplastic tuberdquo Journal of Applied Mechanics TransactionsASME vol 59 no 1 pp 20ndash26 1992

[7] H Jahed and R N Dubey ldquoAn axisymmetric method of elastic-plastic analysis capable of predicting residual stress fieldrdquoJournal of Pressure Vessel Technology Transactions of the ASMEvol 119 no 3 pp 264ndash273 1997

[8] A P Parker ldquoAutofrettage of open-end tubesmdashpressuresstresses strains and code comparisonsrdquo Journal of PressureVessel Technology Transactions of the ASME vol 123 no 3 pp271ndash281 2001

[9] R N Dubey R Seshadri and S Bedi ldquoAnalysis of thick elastic-plastic cylindersrdquo in Proceedings of the Plasticity Conference inWhistler British Columbia Canada 2000

[10] W Olszak and W Urbanowski ldquoNon-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressurerdquoArchiwumMechaniki Stosowanej vol 3 no 7 pp 315ndash336 1955

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

4 The Scientific World Journal

Now we introduced the following nondimensional quantitiesas

119877 = (

119903

119887

) 1198770= (

119886

119887

) 120590119903119903

= [

119879119903119903

119884

]

120590120579120579

= [

119879120579120579

119884

] 120590119911119911

= [

119879119911119911

119884

]

(20)

The necessary pressure required for initial yielding in nondi-mensional form is given by

1003816100381610038161003816119875119894

1003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890 [exp (minus1198902119877minus119896

0) minus exp (minus119890

2)]

119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(21)

The transitional stresses are obtained as

120590119903119903

= [

minus119875119894[exp (119888

0119887minus119896119896) (119877

minus119896

0minus 1) minus 1]

exp [(1198880119887minus119896119896) (119877

minus119896

0minus 1)] minus 1

] minus 1198751198942

120590120579120579

= [

minus119875119894

exp [(1198880119887minus119896119896) (119877

minus119896

0minus 1)] minus 1

]

times[(1 minus 1198880119887minus119896119877minus119896) exp

1198880119887minus119896

119896

(119877minus119896

0minus 1) minus 1] minus 119875

1198942

120590119911119911

= (

1 minus 1198880119887minus119896119877minus119896

2 minus 1198880119887minus119896119877minus119896

) (120590119903119903

+ 120590120579120579)

+

1205821198880119887minus119896119877minus119896

(3 minus 21198880(119887119877)minus119896)

(1 minus 1198880(119887119877)minus119896) (2 minus 119888

0(119887119877)minus119896)

119890119911119911

(22)

where

119890119911119911

= ([minus

(1198751198941minus 1198751198942)

2120587

] minus int

1

1198770

1198872119877(

1 minus 1198880(119887119877)minus119896

2 minus 1198880(119887119877)minus119896

)

times (120590119903119903

+ 120590120579120579) 119889119877)

times (120582int

1

1198770

1198880119887minus119896minus1

119877minus119896

(3 minus 21198880(119887119877)minus119896)

(1 minus 1198880(119887119877)minus119896) (2 minus 119888

0(119887119877)minus119896)

119889119877)

minus1

(23)

Also pressure required for fully plastic state is given by

10038161003816100381610038161003816119875119891

10038161003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

119877minus119896

0minus 1

119896

1003816100381610038161003816100381610038161003816100381610038161003816

119875119891=

1199011

1198841

minus

1199012

1198841

= 1198751198911

minus 1198751198912 (24)

and fully plastic stresses are obtained as

120590119903119903

= (minus119875119891)(

119877minus119896

minus 1

119877minus119896

0minus 1

) minus 1198751198912

120590120579120579

= 120590119903minus

119896119877minus119896

(minus119875119891)

(119877minus119896

0minus 1)

120590119911119911

=

119896120582119877minus119896

[minus1198751198912120587 minus (12) int

1

1198770

1198771198872(120590119903119903

+ 120590120579120579) 119889119877]

(119877minus119896

0minus 1)

(25)

Particular Case Nonhomogeneous Cylinder under InternalPressure OnlyThe stresses in fully plastic state are

120590119903119903

= (1198751198912)(

119877minus119896

minus 1

119877minus119896

0minus 1

) minus 1198751198912

120590120579120579

= 120590119903119903

minus

119896119877minus119896

(1198751198912)

(119877minus119896

0minus 1)

120590119911119911

=

119896120582119877minus119896

[11987511989122120587 minus (12) int

1

1198770

1198771198872(120590119903119903

+ 120590120579120579) 119889119877]

(119877minus119896

0minus 1)

(26)

These equations in nondimensional form are the same asthose obtained by Sharma [12]

6 Numerical Discussion

To observe the combined effect of pressure on a cylin-der made of homogeneous and nonhomogeneous materialgraphs have been drawn between pressure and radii ratios1198770= 01 (01) 05 For a homogeneous (119896 = 0) circular

cylinder yielding starts at internal surface whereas for acircular cylinder made of nonhomogeneous material (119896 lt 0non-homogeneity increases radially) yielding takes place atany radius 119903 where 119886 lt 119903 lt 119887 depending upon values of 119862

0

and 119896 Effective pressure is maximum at internal surface forcylinder made of nonhomogeneous as well as homogeneousmaterial It is seen from Figure 1 that for homogeneouscylinder high effective pressure is required for initial yieldingthan that of nonhomogeneous cylinder Also for cylindermade of homogeneous material effective pressure requiredfor initial yielding is less for highly compressible circularcylinder whereas for circular cylinder made of nonhomo-geneous materials high effective pressure is required forhighly compressible cylinder It is also seen from Figure 2that pressure (internalexternal) is maximum at externalsurface for cylinder made of nonhomogeneous as well ashomogeneous material It is also seen that homogeneouscylinder requires high pressure for initial yielding than that ofnonhomogeneous cylinder Also high pressure is required forinitial yielding for highly compressible homogeneous cylin-der whereas less pressure is required for highly compressiblenonhomogeneous cylinder

It has also been observed from Figure 3 that for homoge-neous and nonhomogeneous circular cylinder effective pres-sure required for fully plastic state is maximum at the internalsurface and for nonhomogeneous material less effectivepressure is required for fully plastic state for circular cylindermade of highly compressible material It is also observed thateffective pressure required for fully plastic state is more forcylinder made of homogeneous material than that of non-homogeneousmaterial For homogeneousnonhomogeneouscircular cylinder pressure (internalexternal) required forfully plastic state is maximum at the external surface Ithas been seen from Figure 4 that for nonhomogeneouscylinder made of highly compressible material high pressureis required for fully plastic state It is also observed fromFigure 4 that pressure required for fully plastic state is more

The Scientific World Journal 5

02 04 06 08 1

05

1

15

2

25

3

02 04 06 08 1

025

05

075

1

125

15

P

R0

P

R0

C = 050

C = 040

C = 035

C = 025

C = 015

k = minus125

k = minus100

k = minus075

k = minus050

k = minus025

Figure 1 Effective pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder for different compressibilityparameters

02 04 06 08 1

175

18

185

19

195

20

02 04 06 08 1

185

1875

1925

195

1975

20

P

R0

P

R0

Figure 2 External or internal pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

1

2

3

4

02 04 06 08 1

05

1

15

2

25

P

R0

P

R0

Figure 3 Effective pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder for differentcompressibility parameters

for cylinder made of nonhomogeneous material than that ofhomogeneous material

From Figures 5 and 6 it has been observed that for ho-mogeneous cylinder under external pressure only circum-ferential stresses are maximum at internal surface while for

nonhomogeneous cylinder stresses are maximum at externalsurfaceThese stresses increase significantly with the increasein external pressureWith internal pressure only as seen fromFigures 7 and 8 circumferential stresses are maximum atinternal surface for homogeneous cylinder while maximum

6 The Scientific World Journal

02 04 06 08 1

17

18

19

20

02 04 06 08 1175

185

19

195

20

P

R0

P

R0

Figure 4 External or internal pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

20

40

60

Stresses

02 04 06 08 1

50

100

150

200

Stresses

RR

120590rr for c = 035

120590rr for c = 05

120590rr for c = 01120590120579120579 for c = 05

120590120579120579 for c = 01

120590120579120579 for c = 035120590rr for k = minus5

120590rr for k = minus3

120590120579120579 for k = minus5

120590120579120579 for k = minus3

120590rr for k = minus1

120590120579120579 for k = minus1

Figure 5 Homogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

5

Stresses 02 04 06 08 1

10

20

Stresses

minus10

minus5

minus10

minus20

minus30

R R

Figure 6 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

at external surface for nonhomogeneous cylinder Also it hasbeen observed that the compressible circumferential stresseschange to tensile stresses It has also been observed fromFigures 9 and 10 that with the increase in pressure circum-ferential stresses increases significantly With the increasein external pressure (greater than that of internal pressure)circumferential stresses increases It has been observed fromFigures 11 and 12 (without internal pressure) that fully plasticstresses are maximum at external surface for cylinder made

of nonhomogeneous material and at internal surface forcylinder made of homogeneous material Also it has beenobserved that highly compressible cylinder is having lessstresswhereas less compressible cylinder is having high stressWith the increase in external pressure stresses increasessignificantly FromFigures 13 and 14 it has also been observedthat with the introduction of internal pressure (without exter-nal) compressible circumferential stresses are maximum atinternal surface for homogeneous while at external surface

The Scientific World Journal 7

02 04 06 08 1

Stresses

02 04 06 08 1

25

50

75

100

125

Stresses

minus20

minus40

minus60

minus25

R

R

Figure 7 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

5

Stresses

02 04 06 08 1

10

Stresses

minus5

minus10

minus10

minus20

minus30

R

R

Figure 8 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

02 04 06 08 1

02 04 06 08 1

20

40

60

minus25

minus20

minus50

minus75

minus100

minus125

minus150

Stresses

R

Stresses

R

Figure 9 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

1002 04 06 08 1

minus5

minus10

minus15

minus20

minus10

minus20

Stresses

R

Stresses

R

Figure 10 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

8 The Scientific World Journal

02 04 06 08 1

5

10

15

20

02 04 06 08 1

20

40

60

minus5

Stresses

R

Stresses

R

Figure 11 Homogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

25

5

02 04 06 08 1

10

minus125

minus10

minus75

minus5

minus25minus10

minus20

minus30

minus40

Stresses R

Stresses R

Figure 12 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

02 04 06 08 1

10

20

30

40

minus5

minus10

minus10

minus15

minus20

minus25

Stresses

R

Stresses

R

Figure 13 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

25

75

5

minus10

minus75

minus5

minus5

minus25Stresses

R

02 04 06 08 1

5

minus15

minus10

minus20

minus30

minus25

Stresses

R

Figure 14 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

The Scientific World Journal 9

02 04 06 08 1

minus10

minus20

minus30

minus40

minus50

Stresses

R

minus5

02 04 06 08 1

5

10

minus15

minus10

Stresses

R

Figure 15 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

minus5

02 04 06 08 1

5

10

15

minus15

minus10

minus20

R

Stresses

minus75

minus10

minus125

minus15

minus175

minus20

minus225

02 04 06 08 1R

Stresses

Figure 16 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

for nonhomogeneous cylinder As external pressure increaseand becomes more than that of internal one then stressesagain increase but are of compressive nature Also it has beenobserved that stresses increases significantlywith the increasein internal pressure as can be seen from Figures 15 and 16

7 Conclusions

From the above analysis we can conclude that nonhomo-geneous cylinder with internal and external pressure is onthe safer side of the design as compared to homogeneouscylinder because nonhomogeneous cylinder requires highpressure for initial yielding as compared to homogeneouscylinder It has also been concluded that highly compressiblenonhomogeneous cylinder is on the safer side of the designas compared to less compressible nonhomogeneous circularcylinder because highly compressible cylinder required highpressure for initial yielding as compared to less compressiblenonhomogeneous cylinder which leads to the idea of ldquostresssavingrdquo that minimizes the possibility of fracture of cylinder

References

[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw Hill 3rd edition 1970

[2] I S Sokolinokoff Mathematical Theory of Elasticity McGraw-Hill New York NY USA 1956

[3] J Chakrabarty Theory of Plasticity McGraw-Hill New YorkNY USA 1987

[4] A Mendelson Plasticity Theory and Application The Macmil-lan Company New York NY USA 1968

[5] P C T Chen ldquoA finite difference approach to axisymmetricplane strain problem beyond the elastic limitrdquo in Proceedingsof the 25th Transaction Conference Army Mathematicians pp661ndash674 1980

[6] D Durban andM Kubi ldquoA general solution for the pressurizedelastoplastic tuberdquo Journal of Applied Mechanics TransactionsASME vol 59 no 1 pp 20ndash26 1992

[7] H Jahed and R N Dubey ldquoAn axisymmetric method of elastic-plastic analysis capable of predicting residual stress fieldrdquoJournal of Pressure Vessel Technology Transactions of the ASMEvol 119 no 3 pp 264ndash273 1997

[8] A P Parker ldquoAutofrettage of open-end tubesmdashpressuresstresses strains and code comparisonsrdquo Journal of PressureVessel Technology Transactions of the ASME vol 123 no 3 pp271ndash281 2001

[9] R N Dubey R Seshadri and S Bedi ldquoAnalysis of thick elastic-plastic cylindersrdquo in Proceedings of the Plasticity Conference inWhistler British Columbia Canada 2000

[10] W Olszak and W Urbanowski ldquoNon-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressurerdquoArchiwumMechaniki Stosowanej vol 3 no 7 pp 315ndash336 1955

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

The Scientific World Journal 5

02 04 06 08 1

05

1

15

2

25

3

02 04 06 08 1

025

05

075

1

125

15

P

R0

P

R0

C = 050

C = 040

C = 035

C = 025

C = 015

k = minus125

k = minus100

k = minus075

k = minus050

k = minus025

Figure 1 Effective pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder for different compressibilityparameters

02 04 06 08 1

175

18

185

19

195

20

02 04 06 08 1

185

1875

1925

195

1975

20

P

R0

P

R0

Figure 2 External or internal pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

1

2

3

4

02 04 06 08 1

05

1

15

2

25

P

R0

P

R0

Figure 3 Effective pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder for differentcompressibility parameters

for cylinder made of nonhomogeneous material than that ofhomogeneous material

From Figures 5 and 6 it has been observed that for ho-mogeneous cylinder under external pressure only circum-ferential stresses are maximum at internal surface while for

nonhomogeneous cylinder stresses are maximum at externalsurfaceThese stresses increase significantly with the increasein external pressureWith internal pressure only as seen fromFigures 7 and 8 circumferential stresses are maximum atinternal surface for homogeneous cylinder while maximum

6 The Scientific World Journal

02 04 06 08 1

17

18

19

20

02 04 06 08 1175

185

19

195

20

P

R0

P

R0

Figure 4 External or internal pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

20

40

60

Stresses

02 04 06 08 1

50

100

150

200

Stresses

RR

120590rr for c = 035

120590rr for c = 05

120590rr for c = 01120590120579120579 for c = 05

120590120579120579 for c = 01

120590120579120579 for c = 035120590rr for k = minus5

120590rr for k = minus3

120590120579120579 for k = minus5

120590120579120579 for k = minus3

120590rr for k = minus1

120590120579120579 for k = minus1

Figure 5 Homogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

5

Stresses 02 04 06 08 1

10

20

Stresses

minus10

minus5

minus10

minus20

minus30

R R

Figure 6 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

at external surface for nonhomogeneous cylinder Also it hasbeen observed that the compressible circumferential stresseschange to tensile stresses It has also been observed fromFigures 9 and 10 that with the increase in pressure circum-ferential stresses increases significantly With the increasein external pressure (greater than that of internal pressure)circumferential stresses increases It has been observed fromFigures 11 and 12 (without internal pressure) that fully plasticstresses are maximum at external surface for cylinder made

of nonhomogeneous material and at internal surface forcylinder made of homogeneous material Also it has beenobserved that highly compressible cylinder is having lessstresswhereas less compressible cylinder is having high stressWith the increase in external pressure stresses increasessignificantly FromFigures 13 and 14 it has also been observedthat with the introduction of internal pressure (without exter-nal) compressible circumferential stresses are maximum atinternal surface for homogeneous while at external surface

The Scientific World Journal 7

02 04 06 08 1

Stresses

02 04 06 08 1

25

50

75

100

125

Stresses

minus20

minus40

minus60

minus25

R

R

Figure 7 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

5

Stresses

02 04 06 08 1

10

Stresses

minus5

minus10

minus10

minus20

minus30

R

R

Figure 8 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

02 04 06 08 1

02 04 06 08 1

20

40

60

minus25

minus20

minus50

minus75

minus100

minus125

minus150

Stresses

R

Stresses

R

Figure 9 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

1002 04 06 08 1

minus5

minus10

minus15

minus20

minus10

minus20

Stresses

R

Stresses

R

Figure 10 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

8 The Scientific World Journal

02 04 06 08 1

5

10

15

20

02 04 06 08 1

20

40

60

minus5

Stresses

R

Stresses

R

Figure 11 Homogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

25

5

02 04 06 08 1

10

minus125

minus10

minus75

minus5

minus25minus10

minus20

minus30

minus40

Stresses R

Stresses R

Figure 12 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

02 04 06 08 1

10

20

30

40

minus5

minus10

minus10

minus15

minus20

minus25

Stresses

R

Stresses

R

Figure 13 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

25

75

5

minus10

minus75

minus5

minus5

minus25Stresses

R

02 04 06 08 1

5

minus15

minus10

minus20

minus30

minus25

Stresses

R

Figure 14 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

The Scientific World Journal 9

02 04 06 08 1

minus10

minus20

minus30

minus40

minus50

Stresses

R

minus5

02 04 06 08 1

5

10

minus15

minus10

Stresses

R

Figure 15 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

minus5

02 04 06 08 1

5

10

15

minus15

minus10

minus20

R

Stresses

minus75

minus10

minus125

minus15

minus175

minus20

minus225

02 04 06 08 1R

Stresses

Figure 16 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

for nonhomogeneous cylinder As external pressure increaseand becomes more than that of internal one then stressesagain increase but are of compressive nature Also it has beenobserved that stresses increases significantlywith the increasein internal pressure as can be seen from Figures 15 and 16

7 Conclusions

From the above analysis we can conclude that nonhomo-geneous cylinder with internal and external pressure is onthe safer side of the design as compared to homogeneouscylinder because nonhomogeneous cylinder requires highpressure for initial yielding as compared to homogeneouscylinder It has also been concluded that highly compressiblenonhomogeneous cylinder is on the safer side of the designas compared to less compressible nonhomogeneous circularcylinder because highly compressible cylinder required highpressure for initial yielding as compared to less compressiblenonhomogeneous cylinder which leads to the idea of ldquostresssavingrdquo that minimizes the possibility of fracture of cylinder

References

[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw Hill 3rd edition 1970

[2] I S Sokolinokoff Mathematical Theory of Elasticity McGraw-Hill New York NY USA 1956

[3] J Chakrabarty Theory of Plasticity McGraw-Hill New YorkNY USA 1987

[4] A Mendelson Plasticity Theory and Application The Macmil-lan Company New York NY USA 1968

[5] P C T Chen ldquoA finite difference approach to axisymmetricplane strain problem beyond the elastic limitrdquo in Proceedingsof the 25th Transaction Conference Army Mathematicians pp661ndash674 1980

[6] D Durban andM Kubi ldquoA general solution for the pressurizedelastoplastic tuberdquo Journal of Applied Mechanics TransactionsASME vol 59 no 1 pp 20ndash26 1992

[7] H Jahed and R N Dubey ldquoAn axisymmetric method of elastic-plastic analysis capable of predicting residual stress fieldrdquoJournal of Pressure Vessel Technology Transactions of the ASMEvol 119 no 3 pp 264ndash273 1997

[8] A P Parker ldquoAutofrettage of open-end tubesmdashpressuresstresses strains and code comparisonsrdquo Journal of PressureVessel Technology Transactions of the ASME vol 123 no 3 pp271ndash281 2001

[9] R N Dubey R Seshadri and S Bedi ldquoAnalysis of thick elastic-plastic cylindersrdquo in Proceedings of the Plasticity Conference inWhistler British Columbia Canada 2000

[10] W Olszak and W Urbanowski ldquoNon-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressurerdquoArchiwumMechaniki Stosowanej vol 3 no 7 pp 315ndash336 1955

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 The Scientific World Journal

02 04 06 08 1

17

18

19

20

02 04 06 08 1175

185

19

195

20

P

R0

P

R0

Figure 4 External or internal pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder (internal orexternal = 20) for different compressibility parameters

02 04 06 08 1

20

40

60

Stresses

02 04 06 08 1

50

100

150

200

Stresses

RR

120590rr for c = 035

120590rr for c = 05

120590rr for c = 01120590120579120579 for c = 05

120590120579120579 for c = 01

120590120579120579 for c = 035120590rr for k = minus5

120590rr for k = minus3

120590120579120579 for k = minus5

120590120579120579 for k = minus3

120590rr for k = minus1

120590120579120579 for k = minus1

Figure 5 Homogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

5

Stresses 02 04 06 08 1

10

20

Stresses

minus10

minus5

minus10

minus20

minus30

R R

Figure 6 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

at external surface for nonhomogeneous cylinder Also it hasbeen observed that the compressible circumferential stresseschange to tensile stresses It has also been observed fromFigures 9 and 10 that with the increase in pressure circum-ferential stresses increases significantly With the increasein external pressure (greater than that of internal pressure)circumferential stresses increases It has been observed fromFigures 11 and 12 (without internal pressure) that fully plasticstresses are maximum at external surface for cylinder made

of nonhomogeneous material and at internal surface forcylinder made of homogeneous material Also it has beenobserved that highly compressible cylinder is having lessstresswhereas less compressible cylinder is having high stressWith the increase in external pressure stresses increasessignificantly FromFigures 13 and 14 it has also been observedthat with the introduction of internal pressure (without exter-nal) compressible circumferential stresses are maximum atinternal surface for homogeneous while at external surface

The Scientific World Journal 7

02 04 06 08 1

Stresses

02 04 06 08 1

25

50

75

100

125

Stresses

minus20

minus40

minus60

minus25

R

R

Figure 7 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

5

Stresses

02 04 06 08 1

10

Stresses

minus5

minus10

minus10

minus20

minus30

R

R

Figure 8 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

02 04 06 08 1

02 04 06 08 1

20

40

60

minus25

minus20

minus50

minus75

minus100

minus125

minus150

Stresses

R

Stresses

R

Figure 9 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

1002 04 06 08 1

minus5

minus10

minus15

minus20

minus10

minus20

Stresses

R

Stresses

R

Figure 10 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

8 The Scientific World Journal

02 04 06 08 1

5

10

15

20

02 04 06 08 1

20

40

60

minus5

Stresses

R

Stresses

R

Figure 11 Homogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

25

5

02 04 06 08 1

10

minus125

minus10

minus75

minus5

minus25minus10

minus20

minus30

minus40

Stresses R

Stresses R

Figure 12 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

02 04 06 08 1

10

20

30

40

minus5

minus10

minus10

minus15

minus20

minus25

Stresses

R

Stresses

R

Figure 13 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

25

75

5

minus10

minus75

minus5

minus5

minus25Stresses

R

02 04 06 08 1

5

minus15

minus10

minus20

minus30

minus25

Stresses

R

Figure 14 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

The Scientific World Journal 9

02 04 06 08 1

minus10

minus20

minus30

minus40

minus50

Stresses

R

minus5

02 04 06 08 1

5

10

minus15

minus10

Stresses

R

Figure 15 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

minus5

02 04 06 08 1

5

10

15

minus15

minus10

minus20

R

Stresses

minus75

minus10

minus125

minus15

minus175

minus20

minus225

02 04 06 08 1R

Stresses

Figure 16 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

for nonhomogeneous cylinder As external pressure increaseand becomes more than that of internal one then stressesagain increase but are of compressive nature Also it has beenobserved that stresses increases significantlywith the increasein internal pressure as can be seen from Figures 15 and 16

7 Conclusions

From the above analysis we can conclude that nonhomo-geneous cylinder with internal and external pressure is onthe safer side of the design as compared to homogeneouscylinder because nonhomogeneous cylinder requires highpressure for initial yielding as compared to homogeneouscylinder It has also been concluded that highly compressiblenonhomogeneous cylinder is on the safer side of the designas compared to less compressible nonhomogeneous circularcylinder because highly compressible cylinder required highpressure for initial yielding as compared to less compressiblenonhomogeneous cylinder which leads to the idea of ldquostresssavingrdquo that minimizes the possibility of fracture of cylinder

References

[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw Hill 3rd edition 1970

[2] I S Sokolinokoff Mathematical Theory of Elasticity McGraw-Hill New York NY USA 1956

[3] J Chakrabarty Theory of Plasticity McGraw-Hill New YorkNY USA 1987

[4] A Mendelson Plasticity Theory and Application The Macmil-lan Company New York NY USA 1968

[5] P C T Chen ldquoA finite difference approach to axisymmetricplane strain problem beyond the elastic limitrdquo in Proceedingsof the 25th Transaction Conference Army Mathematicians pp661ndash674 1980

[6] D Durban andM Kubi ldquoA general solution for the pressurizedelastoplastic tuberdquo Journal of Applied Mechanics TransactionsASME vol 59 no 1 pp 20ndash26 1992

[7] H Jahed and R N Dubey ldquoAn axisymmetric method of elastic-plastic analysis capable of predicting residual stress fieldrdquoJournal of Pressure Vessel Technology Transactions of the ASMEvol 119 no 3 pp 264ndash273 1997

[8] A P Parker ldquoAutofrettage of open-end tubesmdashpressuresstresses strains and code comparisonsrdquo Journal of PressureVessel Technology Transactions of the ASME vol 123 no 3 pp271ndash281 2001

[9] R N Dubey R Seshadri and S Bedi ldquoAnalysis of thick elastic-plastic cylindersrdquo in Proceedings of the Plasticity Conference inWhistler British Columbia Canada 2000

[10] W Olszak and W Urbanowski ldquoNon-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressurerdquoArchiwumMechaniki Stosowanej vol 3 no 7 pp 315ndash336 1955

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

The Scientific World Journal 7

02 04 06 08 1

Stresses

02 04 06 08 1

25

50

75

100

125

Stresses

minus20

minus40

minus60

minus25

R

R

Figure 7 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

5

Stresses

02 04 06 08 1

10

Stresses

minus5

minus10

minus10

minus20

minus30

R

R

Figure 8 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

02 04 06 08 1

02 04 06 08 1

20

40

60

minus25

minus20

minus50

minus75

minus100

minus125

minus150

Stresses

R

Stresses

R

Figure 9 Homogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

1002 04 06 08 1

minus5

minus10

minus15

minus20

minus10

minus20

Stresses

R

Stresses

R

Figure 10 Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

8 The Scientific World Journal

02 04 06 08 1

5

10

15

20

02 04 06 08 1

20

40

60

minus5

Stresses

R

Stresses

R

Figure 11 Homogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

25

5

02 04 06 08 1

10

minus125

minus10

minus75

minus5

minus25minus10

minus20

minus30

minus40

Stresses R

Stresses R

Figure 12 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

02 04 06 08 1

10

20

30

40

minus5

minus10

minus10

minus15

minus20

minus25

Stresses

R

Stresses

R

Figure 13 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

25

75

5

minus10

minus75

minus5

minus5

minus25Stresses

R

02 04 06 08 1

5

minus15

minus10

minus20

minus30

minus25

Stresses

R

Figure 14 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

The Scientific World Journal 9

02 04 06 08 1

minus10

minus20

minus30

minus40

minus50

Stresses

R

minus5

02 04 06 08 1

5

10

minus15

minus10

Stresses

R

Figure 15 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

minus5

02 04 06 08 1

5

10

15

minus15

minus10

minus20

R

Stresses

minus75

minus10

minus125

minus15

minus175

minus20

minus225

02 04 06 08 1R

Stresses

Figure 16 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

for nonhomogeneous cylinder As external pressure increaseand becomes more than that of internal one then stressesagain increase but are of compressive nature Also it has beenobserved that stresses increases significantlywith the increasein internal pressure as can be seen from Figures 15 and 16

7 Conclusions

From the above analysis we can conclude that nonhomo-geneous cylinder with internal and external pressure is onthe safer side of the design as compared to homogeneouscylinder because nonhomogeneous cylinder requires highpressure for initial yielding as compared to homogeneouscylinder It has also been concluded that highly compressiblenonhomogeneous cylinder is on the safer side of the designas compared to less compressible nonhomogeneous circularcylinder because highly compressible cylinder required highpressure for initial yielding as compared to less compressiblenonhomogeneous cylinder which leads to the idea of ldquostresssavingrdquo that minimizes the possibility of fracture of cylinder

References

[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw Hill 3rd edition 1970

[2] I S Sokolinokoff Mathematical Theory of Elasticity McGraw-Hill New York NY USA 1956

[3] J Chakrabarty Theory of Plasticity McGraw-Hill New YorkNY USA 1987

[4] A Mendelson Plasticity Theory and Application The Macmil-lan Company New York NY USA 1968

[5] P C T Chen ldquoA finite difference approach to axisymmetricplane strain problem beyond the elastic limitrdquo in Proceedingsof the 25th Transaction Conference Army Mathematicians pp661ndash674 1980

[6] D Durban andM Kubi ldquoA general solution for the pressurizedelastoplastic tuberdquo Journal of Applied Mechanics TransactionsASME vol 59 no 1 pp 20ndash26 1992

[7] H Jahed and R N Dubey ldquoAn axisymmetric method of elastic-plastic analysis capable of predicting residual stress fieldrdquoJournal of Pressure Vessel Technology Transactions of the ASMEvol 119 no 3 pp 264ndash273 1997

[8] A P Parker ldquoAutofrettage of open-end tubesmdashpressuresstresses strains and code comparisonsrdquo Journal of PressureVessel Technology Transactions of the ASME vol 123 no 3 pp271ndash281 2001

[9] R N Dubey R Seshadri and S Bedi ldquoAnalysis of thick elastic-plastic cylindersrdquo in Proceedings of the Plasticity Conference inWhistler British Columbia Canada 2000

[10] W Olszak and W Urbanowski ldquoNon-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressurerdquoArchiwumMechaniki Stosowanej vol 3 no 7 pp 315ndash336 1955

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 The Scientific World Journal

02 04 06 08 1

5

10

15

20

02 04 06 08 1

20

40

60

minus5

Stresses

R

Stresses

R

Figure 11 Homogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

25

5

02 04 06 08 1

10

minus125

minus10

minus75

minus5

minus25minus10

minus20

minus30

minus40

Stresses R

Stresses R

Figure 12 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (1198752= 5 and 15)

02 04 06 08 1

02 04 06 08 1

10

20

30

40

minus5

minus10

minus10

minus15

minus20

minus25

Stresses

R

Stresses

R

Figure 13 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0 and

15)

02 04 06 08 1

25

75

5

minus10

minus75

minus5

minus5

minus25Stresses

R

02 04 06 08 1

5

minus15

minus10

minus20

minus30

minus25

Stresses

R

Figure 14 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 5) and external pressure (119875

2= 0

and 15)

The Scientific World Journal 9

02 04 06 08 1

minus10

minus20

minus30

minus40

minus50

Stresses

R

minus5

02 04 06 08 1

5

10

minus15

minus10

Stresses

R

Figure 15 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

minus5

02 04 06 08 1

5

10

15

minus15

minus10

minus20

R

Stresses

minus75

minus10

minus125

minus15

minus175

minus20

minus225

02 04 06 08 1R

Stresses

Figure 16 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

for nonhomogeneous cylinder As external pressure increaseand becomes more than that of internal one then stressesagain increase but are of compressive nature Also it has beenobserved that stresses increases significantlywith the increasein internal pressure as can be seen from Figures 15 and 16

7 Conclusions

From the above analysis we can conclude that nonhomo-geneous cylinder with internal and external pressure is onthe safer side of the design as compared to homogeneouscylinder because nonhomogeneous cylinder requires highpressure for initial yielding as compared to homogeneouscylinder It has also been concluded that highly compressiblenonhomogeneous cylinder is on the safer side of the designas compared to less compressible nonhomogeneous circularcylinder because highly compressible cylinder required highpressure for initial yielding as compared to less compressiblenonhomogeneous cylinder which leads to the idea of ldquostresssavingrdquo that minimizes the possibility of fracture of cylinder

References

[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw Hill 3rd edition 1970

[2] I S Sokolinokoff Mathematical Theory of Elasticity McGraw-Hill New York NY USA 1956

[3] J Chakrabarty Theory of Plasticity McGraw-Hill New YorkNY USA 1987

[4] A Mendelson Plasticity Theory and Application The Macmil-lan Company New York NY USA 1968

[5] P C T Chen ldquoA finite difference approach to axisymmetricplane strain problem beyond the elastic limitrdquo in Proceedingsof the 25th Transaction Conference Army Mathematicians pp661ndash674 1980

[6] D Durban andM Kubi ldquoA general solution for the pressurizedelastoplastic tuberdquo Journal of Applied Mechanics TransactionsASME vol 59 no 1 pp 20ndash26 1992

[7] H Jahed and R N Dubey ldquoAn axisymmetric method of elastic-plastic analysis capable of predicting residual stress fieldrdquoJournal of Pressure Vessel Technology Transactions of the ASMEvol 119 no 3 pp 264ndash273 1997

[8] A P Parker ldquoAutofrettage of open-end tubesmdashpressuresstresses strains and code comparisonsrdquo Journal of PressureVessel Technology Transactions of the ASME vol 123 no 3 pp271ndash281 2001

[9] R N Dubey R Seshadri and S Bedi ldquoAnalysis of thick elastic-plastic cylindersrdquo in Proceedings of the Plasticity Conference inWhistler British Columbia Canada 2000

[10] W Olszak and W Urbanowski ldquoNon-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressurerdquoArchiwumMechaniki Stosowanej vol 3 no 7 pp 315ndash336 1955

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

The Scientific World Journal 9

02 04 06 08 1

minus10

minus20

minus30

minus40

minus50

Stresses

R

minus5

02 04 06 08 1

5

10

minus15

minus10

Stresses

R

Figure 15 Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0 and

15)

minus5

02 04 06 08 1

5

10

15

minus15

minus10

minus20

R

Stresses

minus75

minus10

minus125

minus15

minus175

minus20

minus225

02 04 06 08 1R

Stresses

Figure 16 Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (1198751= 10) and external pressure (119875

2= 0

and 15)

for nonhomogeneous cylinder As external pressure increaseand becomes more than that of internal one then stressesagain increase but are of compressive nature Also it has beenobserved that stresses increases significantlywith the increasein internal pressure as can be seen from Figures 15 and 16

7 Conclusions

From the above analysis we can conclude that nonhomo-geneous cylinder with internal and external pressure is onthe safer side of the design as compared to homogeneouscylinder because nonhomogeneous cylinder requires highpressure for initial yielding as compared to homogeneouscylinder It has also been concluded that highly compressiblenonhomogeneous cylinder is on the safer side of the designas compared to less compressible nonhomogeneous circularcylinder because highly compressible cylinder required highpressure for initial yielding as compared to less compressiblenonhomogeneous cylinder which leads to the idea of ldquostresssavingrdquo that minimizes the possibility of fracture of cylinder

References

[1] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw Hill 3rd edition 1970

[2] I S Sokolinokoff Mathematical Theory of Elasticity McGraw-Hill New York NY USA 1956

[3] J Chakrabarty Theory of Plasticity McGraw-Hill New YorkNY USA 1987

[4] A Mendelson Plasticity Theory and Application The Macmil-lan Company New York NY USA 1968

[5] P C T Chen ldquoA finite difference approach to axisymmetricplane strain problem beyond the elastic limitrdquo in Proceedingsof the 25th Transaction Conference Army Mathematicians pp661ndash674 1980

[6] D Durban andM Kubi ldquoA general solution for the pressurizedelastoplastic tuberdquo Journal of Applied Mechanics TransactionsASME vol 59 no 1 pp 20ndash26 1992

[7] H Jahed and R N Dubey ldquoAn axisymmetric method of elastic-plastic analysis capable of predicting residual stress fieldrdquoJournal of Pressure Vessel Technology Transactions of the ASMEvol 119 no 3 pp 264ndash273 1997

[8] A P Parker ldquoAutofrettage of open-end tubesmdashpressuresstresses strains and code comparisonsrdquo Journal of PressureVessel Technology Transactions of the ASME vol 123 no 3 pp271ndash281 2001

[9] R N Dubey R Seshadri and S Bedi ldquoAnalysis of thick elastic-plastic cylindersrdquo in Proceedings of the Plasticity Conference inWhistler British Columbia Canada 2000

[10] W Olszak and W Urbanowski ldquoNon-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressurerdquoArchiwumMechaniki Stosowanej vol 3 no 7 pp 315ndash336 1955

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

10 The Scientific World Journal

[11] P G Hodge and M Balaban ldquoElasticmdashplastic analysis of arotating cylinderrdquo International Journal of Mechanical Sciencesvol 4 no 6 pp 465ndash476 1962

[12] S Sharma ldquoElastic-plastic transition of a non-homogeneousthick-walled circular cylinder under internal pressurerdquoDefenceScience Journal vol 54 no 2 pp 135ndash141 2004

[13] B R Seth ldquoMeasure-concept in mechanicsrdquo InternationalJournal of Non-Linear Mechanics vol 1 no 1 pp 35ndash40 1966

[14] B R Seth ldquoTransition theory of elastic-plastic deformationcreep and relaxationrdquo Nature vol 195 no 4844 pp 896ndash8971962

[15] S B Seth ldquoTransition analysis of collapse of thick cylindersrdquo ZAngew Math Mech vol 50 no 10 pp 617ndash621 1970

[16] B N Borah ldquoThermo elastic-plastic transitionrdquo ContemporaryMathematics vol 379 pp 93ndash111 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials