Advances in Materials Science and Engineering

Advances in Materials Science and Engineering / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 810508 | https://doi.org/10.1155/2013/810508

Sanjeev Sharma, Sanehlata Yadav, "Thermo Elastic-Plastic Analysis of Rotating Functionally Graded Stainless Steel Composite Cylinder under Internal and External Pressure Using Finite Difference Method", Advances in Materials Science and Engineering, vol. 2013, Article ID 810508, 11 pages, 2013. https://doi.org/10.1155/2013/810508

Thermo Elastic-Plastic Analysis of Rotating Functionally Graded Stainless Steel Composite Cylinder under Internal and External Pressure Using Finite Difference Method

Academic Editor: Mohd Sapuan Salit
Received22 Jul 2013
Accepted26 Oct 2013
Published02 Dec 2013

Abstract

The thermal elastic-plastic stresses have been investigated for a rotating functionally graded stainless steel composite cylinder under internal and external pressure with general nonlinear strain hardening law and von Mises’ yield criterion using finite difference method. The modulus of elasticity in the rotating cylinder varies radially according to power law and the temperature distribution satisfies Laplace heat equation in radial direction. From the analysis, we can conclude that cylinder made of functionally graded stainless steel composite material with variable thickness and variable density under thermal loading for Swift’s strain hardening measure is better choice of the design as compared to homogeneous cylinder. This is because of the reason that circumferential stress is less for functionally graded stainless steel composite cylinder as compared to homogeneous cylinder for Swift’s strain hardening measure under internal and external pressure. This leads to the idea of “stress saving” which minimizes the possibility of fracture of cylinder.

1. Introduction

Thermo elastic-plastic analysis of functionally graded materials under internal and external pressure is an active topic for engineering mechanics. The study of thick-walled cylinder is an interesting area of research and highly used due to vast utilization in the pressure vessels and pipes and so forth. Functionally graded materials are nothing but nonhomogeneous composite materials which are highly heat resistant and very promising in high-tech engineering fields [1, 2]. The demands of functionally graded materials are increasing under high rotation and temperature. These materials find their applications in many areas, that is, flywheels, aerospace, nuclear reactors, compressors, and so forth. The problems of rotating thick-walled cylinder and plates can be found in many text books of elasticity [3, 4]. Obata et al. [5] calculated stresses for a hollow circular cylinder and hollow sphere under thermal loading using perturbation method. For these materials they found that the perturbation solutions had good convergence. Perry and Aboudi [6] calculated the residual stresses in a homogeneous cylinder using finite difference method and concluded that it is very effective in solving autofrettage problem numerically. Gao [7] investigated elastic-plastic stresses, strains, and displacements in a cylinder under internal pressure using elastic strain hardening in the plane strain conditions. Tutuncu and Ozturk [8] studied the closed form solution for stresses and displacement in cylindrical and spherical vessels made of functionally graded material subjected to internal pressure using infinitesimal theory of elasticity. They compared stress distributions in functionally graded material with homogeneous material. Singh and Gupta [9] developed a mathematical model to describe steady-state creep in an isotropic functionally graded composite subjected to internal pressure which contains linearly varying silicon carbide particles in a matrix of pure aluminum. They observed that the radial stress in the cylinder decreases throughout with the increase in reinforcement gradient, whereas the tangential, axial, and effective stresses increase significantly near the inner radius but show significant decrease towards the outer radius. Aggarwal et al. [10] investigated safety factors in terms of elastic-plastic stresses for functionally graded thick-walled circular cylinder under internal and external pressure and concluded that functionally graded thick-walled cylinder minimizes the possibility of fracture of the cylinder. Aggarwal et al. [11] calculated thermal creep stresses for nonhomogeneous thick-walled cylinder under internal and external pressure using Lebesgue strain measure and concluded that nonhomogeneous cylinder is better choice of design as compared to homogeneous cylinder. Parvizi et al. [12] studied a mathematical model to predict the yielding in a functionally graded Al A359/SiCp cylinder and find closed form solution for plastic stresses using Tresca’s yield criterion subjected to internal pressure with thermal loading. They observed that there is a point in cylinder where the hoop stress changes from compressive to tensile and the position of this point is independent of the temperature gradient and depends on material properties and geometry of the functionally graded cylinder. Eraslan and Akgül [13] calculated the numerical solution for elastic-plastic stresses in a rotating disk with von Mises’ yield criterion using general nonlinear strain hardening rule.

In this paper, thermal elastic-plastic stresses have been calculated for cylinder made up of functionally graded stainless steel composite material under internal and external pressure using finite difference method. In this problem, a general nonlinear strain hardening law with von Mises’ yield criterion has been considered. Results have been discussed numerically with the help of graphs and tables.

2. Objective

For a rotating cylinder with varying material properties, circumferential stress at the hub does not exceed the allowable value which tells the designers little more than that the design of the cylinder is safe at the given pressure. Thus our prime objective is to calculate allowable thermal elastic-plastic stresses in an open ended functionally graded stainless steel composite rotating cylinder under pressure for varying expansion to incorporate a “safety factor” that prevents the cylinder from bursting under pressure and thermal loading.

3. Mathematical Formulation

3.1. Distribution of Material Properties

Consider a long open ended axisymmetric cylinder made up of functionally graded stainless steel composite material with inner and outer radii and , respectively, and the cylinder is subjected to internal pressure and external pressure as shown in Figure 1. The cylindrical polar coordinates under plane stress condition have been considered in this problem.

In this study, Poisson’s ratio and thermal expansion coefficient () are assumed to be constants. The other properties, that is, Young’s modulus which is defined by power law, temperature distribution that follows Laplace heat equation in radial direction, thickness, and density, are varying radially and are expressed as where , is the radius of the cylinder, , , , and are material constants, and , , and are the geometric parameters.

3.2. Basic Equations

The equilibrium equation for the cylinder in the absence of body forces is where and are radial and circumferential stresses, respectively.

Using infinitesimal theory of elasticity, the relations between strains and radial displacements are where and are radial and circumferential strains, respectively, and is the radial displacement.

The equation of compatibility can be derived from (3) as

From infinitesimal theory of elasticity, the stress-strain relations are where , , and are the elastic radial, circumferential, and axial strains, respectively.

Due to geometric symmetry of the cylinder, circumferential displacement, shear stresses, and strains are assumed to be zero.

Using deformation theory of plasticity, the relation between the stresses and plastic strains can be determined as where is the equivalent stress, is the equivalent plastic strain, and , , and are the plastic radial, circumferential, and axial strains, respectively.

von Mises’ yield criterion is given by

The total radial, circumferential, and axial strains in thick-walled rotating cylinder are

The temperature field satisfying Laplace heat equation is with at and at , where is a constant, given by .

We define the stress function for thick-walled rotating cylinder which is related to radial and hoop stresses as

Since it has been assumed that the cylinder is long and open ended and there is plane stress condition, therefore axial stress is zero; that is, .

Substituting (10) and (5) into (8), we have

Substituting of (11) into compatibility (4), we have where , , and   .

The relation between the yield stress and the equivalent plastic strain for Swift’s hardening law can be expressed as where , , , , and are hardening parameter, material parameter, yield limit, equivalent total strain, and yield strain, respectively.

Substituting from (13) into (6) results in

Substituting (14) into (12), we have

Equation (15) is the differential equation of the functionally graded stainless steel composite rotating cylinder with nonlinear strain hardening subjected to thermal loading in the plastic region in terms of stresses and stress function.

Equation (15) can be described in the general form in terms of stress function as

Equation (16) is a nonlinear two point boundary value problem and can be solved numerically, subjected to the boundary conditions where and are the inner and outer radii of the cylinder and and are internal and external pressures, respectively.

Using finite difference method with central difference in (16), we get the following system of equations:

Equation (18) consists of algebraic system of equations with the boundary conditions and . After solving (18) with boundary conditions we get a stress function . Then, the radial and circumferential stresses can be obtained from (10) after substituting the value of stress function .

4. Numerical Discussion

The properties of a functionally graded stainless steel composite thick-walled rotating cylinder under internal and external pressure and  MPa, respectively, subjected to thermal loading () are defined as follows: the radii of the cylinder are taken as  m and  m, Poisson’s ratio , Young’s modulus  GPa, and thermal expansion coefficient . The geometric parameters of the cylinder are taken as in Young’s modulus function and is nonlinear strain hardening measure.

To show the effect of internal and external pressure on a functionally graded stainless steel composite rotating cylinder with strain hardening measure having constant thickness and constant density, Tables 1 and 2 show the circumferential stresses with different parameters of Young’s modulus .


MPa
,
0.10.20.30.40.5

,
,
08.34545301413.40649158470.0677187483−3.9596784280−8.9602457103
13.37996417852.71715349610.8969909345−2.6654005915−8.5548978826
21.15030245051.81183571651.2081310260−1.5371690373−7.8899736228
,
,
08.81886551763.64715906770.2649872301−3.7776104911−8.7852151029
13.60496429962.94215356891.1219909709−2.4404004448−8.3298983560
21.18697075791.97747643901.4492915544−1.2424185539−7.5536537230
,
,
08.34545291343.40649144740.0677186071−3.9596781512−8.9602463330
13.37996411632.71715351460.8969910927−2.6654006381−8.5548985793
21.15030245071.81183571521.2081310262−1.5371690375−7.8899736082
,
,
08.81886612753.64715912560.2649873453−3.7776106334−8.7852145930
13.60496445282.94215360401.1219910130−2.4404005339−8.3298984893
21.18697075781.97747643891.4492915543−1.2424185538−7.5536537227
,
,
08.34545327133.40649161330.0677188786−3.9596783083−8.9602460739
10.37996444172.71715370780.8969909208−2.6654005810−8.5548984885
21.15030245101.81183571651.2081310255−1.5371690384−7.8899735899
,
,
08.81886585253.64715971970.2649871677−3.7776107003−8.7852160047
13.60496398602.94215351651.1219910039−2.4404004285−8.3298983452
21.18697075771.97747643871.4492915543−1.2424185537−7.5536537224

MPa
,
0.10.20.30.40.5

,
,
01.11996436810.45482878970.0269123600−0.4863252783−1.2191626393
10.62728130800.50957688730.1910926579−0.4339086008−1.5351994013
20.32639054980.50380394360.3586769803−0.3274307737−1.9402396397
,
,
01.15129037440.47610737450.0479630332−0.4638575699−1.1948611058
10.64975060920.53206564630.2136358300−0.4115308637−1.5126545536
20.33544525680.52693444330.3869603568−0.3022694774−1.9278605585
,
,
01.11996447870.45482873860.0269122948−0.4863252189−1.2191625155
10.62728129780.50957691020.1910926887−0.4339085404−1.5351993416
20.32639057550.50380383000.3586769873−0.3274306978−1.9402449592
,
,
01.15129047880.47610732180.0479628568−0.4638573696−1.1948613670
10.64975060750.53206563320.2136358153−0.4115307488−1.5126545365
20.33544526790.52693446960.3869603369−0.3022695403−1.9278600529
,
,
01.11996427150.45482880450.0269123491−0.4863251255−1.2191624769
11.11996447870.45482873860.0269122948−0.4863252189−1.2191625155
20.32639050070.50380381900.3586771087−0.3274306558−1.9402425453
,
,
01.15129037650.47610731250.0479629909−0.4638574341−1.1948612858
10.64975059610.53206560170.2136358897−0.4115308258−1.5126545465
20.33544524770.52693442950.3869603736−0.3022695113−1.9278604854


MPa
,
0.10.20.30.40.5

,
,
08.34538272893.40649008990.0677286065−3.9596681544−8.9602414309
13.37994565802.71714353510.8969901053−2.6653896254−8.5548804228
21.15029694951.81182791751.2081251826−1.5371621151−7.8899506975
,
,
08.81879362963.64715913170.2649975386−3.7776011860−8.7852121667
13.60494455862.94214317661.1219899984−2.4403890591−8.3298798957
21.18696498551.97746830141.4492853591−1.2424112423−7.5536293888
,
,
08.34538332023.40649032760.0677284767−3.9596684114−8.9602417913
13.37994545092.71714341890.8969901711−2.6653895895−8.5548805034
21.15029695011.81182791781.2081251821−1.5371621158−7.8899507229
,
,
08.81879220563.64715848950.2649978782−3.7776012451−8.7852123681
13.60494444702.94214302331.1219900447−2.4403890386−8.3298802292
21.18696498541.97746830131.4492853591−1.2424112422−7.5536293885
,
,
08.34538200243.40648963390.0677286501−3.9596681965−8.9602415728
13.37994595332.71714372100.8969899701−2.6653898309−8.5548807795
21.15029694901.81182791761.2081251818−1.5371621155−7.8899507012
,
,
08.81879291203.64715862480.2649979578−3.7776012681−8.7852105705
13.60494462232.94214297981.1219900968−2.4403889573−8.3298789707
21.18696498531.97746830111.4492853590−1.2424112421−7.5536293882

MPa
,
0.10.20.30.40.5

,
,
01.11995831520.45482586950.0269146492−0.4863265526−1.2191654300
10.62727991790.50957706380.1910931884−0.4339078950−1.5352039429
20.32639037390.50380387500.3586772209−0.3274306376−1.9402344736
,
,
01.15125672360.47610581460.0480074980−0.4638915755−1.1948352560
10.64966080080.53203256490.2137603620−0.4118738110−1.5125508783
20.33544237600.52693118560.3869645881−0.3022635649−1.9278571554
,
,
01.11995796310.45482583180.0269147731−0.4863261541−1.2191655802
10.62727992980.50957707430.1910931873−0.4339077941−1.5352040493
20.32639046490.50380391940.3586771853−0.3274308913−1.9402307798
,
,
01.15125694020.47610574470.0480074391−0.4638917626−1.1948352393
10.64966070270.53203251830.2137605230−0.4118740006−1.5125508832
20.33544236760.52693123060.3869645451−0.3022636353−1.9278567951
,
,
01.11995791020.45482588490.0269148341−0.4863259137−1.2191656124
10.62727992920.50957705570.1910931905−0.4339078242−1.5352038916
20.32639032690.50380373030.3586773178−0.3274301493−1.9402403280
,
,
01.15125638660.47610574690.0480078293−0.4638922712−1.1948350571
10.64966070650.53203250640.2137604900−0.4118740845−1.5125507865
20.33544237340.52693123340.3869645540−0.3022636247−1.9278571251

It has been observed from Table 1 that, when external pressure is greater than the internal pressure, circumferential stresses approaches tensile to compressible. Also, these stresses are maximum at external surface for homogeneous as well as functionally graded stainless steel composite cylinder. These stresses are less for functionally graded stainless steel composite cylinder as compared to homogeneous cylinder. As nonhomogeneity changes from to , circumferential stresses decrease significantly. With the introduction of thermal effects circumferential stresses increase for homogeneous as well as for functionally graded stainless steel composite cylinder () but decrease for functionally graded stainless steel composite cylinder with . It has also been noticed from Table 1 that with the increase in thermal effects these stresses decrease significantly for homogeneous as well as for functionally graded stainless steel composite cylinder and are less for the cylinder with nonhomogeneity parameter . With the increase in angular speed, circumferential stresses increase significantly. When external pressure is less than the internal pressure, circumferential stresses are maximum at internal surface for homogeneous cylinder while maximum at external surface for functionally graded stainless steel composite cylinder and these stresses decreased with the change in nonhomogeneity measure from to . With the introduction of thermal effects circumferential stresses increase for homogenous as well as for functionally graded stainless steel composite measure while decrease for functionally graded stainless steel composite cylinder with . With the increase in thermal effects these circumferential stresses decrease significantly while increase with the increase in angular speed. It has been observed from Table 2 that the behavior of homogeneous and functionally graded stainless steel composite cylinder same as discussed in Table 1 but it has been observed that, with increase in strain hardening measure from to , these stresses decrease significantly for functionally graded stainless steel composite cylinder.

Tables 3 and 4 have been made for circumferential stresses in rotating cylinders with variable thickness and variable density with different parameters of Young’s modulus and strain hardening measure .


MPa
, , ,
0.10.20.30.40.5

,
,
04.0786101211.692083635−0.362927882−3.599668076−8.388122376
11.8440238461.5363475660.332616455−2.596999032−8.186025126
20.6985885561.1667186270.723851443−1.632455674−7.703313292
,
,
04.4684986591.869943306−0.216047724−3.455691317−8.239259816
12.0399465351.7218583200.520942472−2.403128817−7.985940189
20.7254521051.3141265610.957769537−1.332711442−7.351503187
,
,
04.0786109261.692083796−0.362928200−3.599667939−8.388124156
11.8440237591.5363473600.332616655−2.596998868−8.186026405
20.6985884851.1667186530.723851463−1.632455697−7.703312785
,
,
04.4684983291.869943517−0.216047300−3.455691093−8.239251730
12.0399466051.7218582380.520942423−2.403128796−7.985940383
20.7254521281.3141265660.957769488−1.332711458−7.351503426
,
,
04.0786102781.692083456−0.362927548−3.599667594−8.388125027
11.8440237661.5363475650.332616556−2.596998868−8.186026291
20.6985884731.1667186540.723851445−1.632455676−7.703312834
,
,
04.4684982961.869943321−0.216047655−3.455691489−8.239237723
12.0399462001.7218582880.520942491−2.403128819−7.985940064
20.7254520821.3141265210.957769539−1.332711402−7.351503219

MPa
, , ,
0.10.20.30.40.5

,
,
00.5023440540.209118575−0.032085608−0.411840852−1.075755914
10.3016906850.2539188280.067872877−0.387759274−1.337560991
20.1703383870.2726225550.177093634−0.328795174−1.672186161
, ,
00.5206025720.223194821−0.013327208−0.385714307−1.041663604
10.3148583380.2691512500.087139893−0.362225697−1.303362831
20.1755331960.2895171040.201213547−0.301959686−1.648231058
,
,
00.5023440760.209118532−0.032085631−0.411840853−1.075756093
10.3016905460.2539187310.067873081−0.387757210−1.337562051
20.1703383980.2726225410.177093558−0.328795106−1.672185933
,
,
00.5206026190.223194813−0.013327161−0.385714149−1.041664533
10.3148583260.2691511990.087139890−0.362225687−1.303362773
20.1755331910.2895170930.201213591−0.301959717−1.648231090
,
,
00.5023440410.209118525−0.032085593−0.411841019−1.075755756
10.3016905480.2539187490.067873075−0.387757974−1.337560151
20.1703384130.2726225280.177093598−0.328795120−1.672185805
,
,
00.5206025320.223194802−0.013327191−0.385714162−1.041664483
10.3148583700.2691512150.087139921−0.362225740−1.303362723
20.1755331680.2895171590.201213488−0.301959669−1.648230944


MPa
, , ,
0.10.20.30.40.5

,
,
04.0785507871.692092827−0.362917634−3.599662505−8.388124303
11.8440088541.5363440370.332619755−2.596989139−8.186011860
20.6985839181.1667134980.723849363−1.632447308−7.703291360
,
,
04.4684388681.869957168−0.216034432−3.455686049−8.239492345
12.0399301941.7218543130.520946730−2.403118071−7.985927195
20.7254470531.3141207760.957767141−1.332701862−7.351479020
,
,
04.0785507581.692093003−0.362917541−3.599662684−8.388124758
11.8440090341.5363441700.332619845−2.596989055−8.186013323
20.6985839891.1667135070.723849336−1.632447294−7.703291352
,
,
04.4684408411.869957567−0.216033626−3.455685275−8.239588040
12.0399302221.7218544400.520946513−2.403117933−7.985926001
20.7254471071.3141208820.957767050−1.332701999−7.351479230
,
,
04.0785510891.692092847−0.362917694−3.599662435−8.388126479
11.8440086981.5363439570.332619859−2.596988916−8.186012239
20.6985839891.1667135240.723849328−1.632447317−7.703291542
,
,
04.4684381201.869956228−0.216035341−3.455687051−8.239368632
12.0399300391.7218545690.520946538−2.403117966−7.985925731
20.7254470491.3141207680.957767137−1.332701833−7.351479029

MPa
, , ,
0.10.20.30.40.5

,
,
00.5023391900.209119023−0.032080434−0.411843003−1.075751575
10.3016894410.2539191370.067872897−0.387758158−1.337567443
20.1703380210.2726224330.177094183−0.328793821−1.672191050
,
,
00.5206002600.223196743−0.013328691−0.385729358−1.041635950
10.3148578280.2691519550.087140006−0.362234515−1.303350674
20.1755328790.2895170020.201213980−0.301960040−1.648233514
,
,
00.5023391730.209119009−0.032080335−0.411843048−1.075752013
10.3016890290.2539189980.067873493−0.387754850−1.337566825
20.1703380610.2726224050.177094107−0.328793799−1.672190560
,
,
00.5206002680.223196732−0.013328653−0.385729330−1.041636334
10.3148578440.2691519360.087139999−0.362234465−1.303350722
20.1755328560.2895169800.201214076−0.301960031−1.648233768
,
,
00.5023391020.209118939−0.032080279−0.411842835−1.075752159
10.3016892520.2539190920.067873202−0.387759066−1.337565663
20.1703380110.2726224320.177094145−0.328793808−1.672190610
,
,
00.5206003450.223196689−0.013328705−0.385729338−1.041635876
10.3148578460.2691519490.087139963−0.362234522−1.303350759
20.1755328430.2895169950.201214043−0.301960031−1.648233661

It has been observed from Table 3 that for cylinder with varying thickness and density, whose external pressure is greater than the internal pressure, circumferential stresses approach from tensile to compressible and are maximum at external surface for homogeneous as well as functionally graded stainless steel composite cylinder. These stresses are less for functionally graded stainless steel composite cylinder as compared to homogeneous cylinder with varying thickness and density as well as cylinder with constant thickness and density. With increase in strain hardening measure from to these circumferential stresses decrease significantly for functionally graded stainless steel composite cylinder with varying thickness and density as can be seen from Table 4.

Figures 24 have been drawn to discuss the effect of internal and external pressure on stresses in rotating cylinder made of functionally graded stainless steel composite material with constant thickness and constant density with nonlinear strain hardening measure.

It has been observed from Figure 2 that circumferential stress approaches towards compressive from tensile. It has also been observed that when external pressure is greater than the internal pressure, these stresses are maximum at external surface for homogeneous as well as functionally graded stainless steel composite cylinder. Also, it has been observed that circumferential stress is maximum at internal surface for homogeneous cylinder while maximum at external surface for functionally graded stainless steel composite cylinder when external pressure is less than the internal pressure. Also, with the increase in angular speed, circumferential stresses increase significantly. From Figure 3 it can be seen that, as circumferential stresses increase for homogeneous cylinder as well as for nonhomogenous cylinder but with the change in nonhomogeneity from to , circumferential stresses decrease when external pressure is greater than the internal pressure with thermal effects. Also, these stresses increase for homogenous cylinder as well as for functionally graded stainless steel composite cylinder () while decrease for functionally graded stainless steel composite cylinder with . With the increase in temperature these stresses decrease significantly for homogeneous as well as for functionally graded stainless steel composite cylinder as can be seen from Figure 4.

Figures 57 have been drawn to discuss the effect of internal and external pressure on stresses in rotating cylinder made of functionally graded stainless steel composite material with variable thickness and variable density with nonlinear strain hardening measure.

It has been observed from Figure 5 that circumferential stresses are maximum at external surface for homogeneous as well as functionally graded stainless steel composite cylinder. It has also been observed that with the increase in angular speed circumferential stresses increase significantly. With the introduction of thermal effects, circumferential stresses increase for homogeneous cylinder as well as for functionally graded stainless steel composite cylinder (), but with the change in nonhomogeneity from to , circumferential stresses decrease when external pressure is greater than the internal pressure as can be seen from Figure 6. Circumferential stresses decrease for homogeneous cylinder while increase for functionally graded stainless steel composite cylinder when external pressure is less than the internal pressure. It has also been observed from Figure 7 that with the increase in temperature these stresses increase significantly for homogeneous cylinder as well for functionally graded stainless steel composite cylinder with while decrease for functionally graded stainless steel composite cylinder with when external pressure is greater than the internal pressure, while these stresses decrease significantly for homogeneous cylinder as well for functionally graded stainless steel composite cylinder when external pressure is less than the internal pressure.

5. Conclusion

From the analysis, we can conclude that rotating cylinder made of functionally graded stainless steel composite material having variable thickness and variable density with Swift’s strain hardening measure and thermal loading is better choice for designers as compared to rotating cylinder with constant thickness and constant density. This is because of the reason that circumferential stress is less for functionally graded stainless steel composite cylinder with variable thickness and variable density as compared to other cases, which leads to the idea of stress saving that minimizes the possibility of fracture of cylinder.

Conflict of Interests

The authors declare that they have no conflict of interest.

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Copyright © 2013 Sanjeev Sharma and Sanehlata Yadav. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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