numerical solution of thermal elastic-plastic …the study of stress distribution at high angular...

Sharma, S., et al.: Numerical Solution of Thermal Elastic-Plastic Functionally Graded ... THERMAL SCIENCE: Year 2019, Vol. 23, No. 1, pp. 125-136 125

NUMERICAL SOLUTION OF THERMAL ELASTIC-PLASTIC FUNCTIONALLY GRADED THIN ROTATING DISK WITH

EXPONENTIALLY VARIABLE THICKNESS AND VARIABLE DENSITY

by

Sanjeev SHARMA** and Sanehlata YADAV Department of Mathematics, Jaypee Institute of Information Technology, Noida, India

Original scientific paper https://doi.org/10.2298/TSCI131001136S

Thermal elastic-plastic stresses and strains have been obtained for rotating an-nular disk by using finite difference method with Von-Mises’ yield criterion and non-linear strain hardening measure. The compressibility of the disk is assumed to be varying in the radial direction. From the numerical results, we can conclude that thermal rotating disk made of functionally graded material whose thickness decreases exponentially and density increases exponentially with non-linear strain hardening measure (m = 0.2) is on the safe side of the design as compared to disk made of homogenous material. This is because of the reason that circumferential stress is less for functionally graded disk as compared to homogenous disk. Also, plastic strains are high for functionally graded disk as compared to homogenous disk. It means that disk made of functionally graded material reduces the possibil-ity of fracture at the bore as compared to the disk made of homogeneous material which leads to the idea of stress saving.Key words: disk, rotation, thin, thickness, density, thermal, functionally graded

Introduction

The study of stress distribution at high angular velocity and temperature in rotating disk made of non-homogenous material is an active topic due to large number of industrial and mechanical applications. Some analytical solutions to rotating disk in elastic state and plastic state are available in [1, 2] and research on them is always an important topic. Eraslan [3] investigated inelastic stresses and displacements in rotating solid disks of exponentially varying thickness using Tresca’s and Von-Mises’ yield criterion with linear strain hardening. Plastic limit angular velocities have been calculated for different disk profiles. Further, Eraslan and Akgul [4] extended his work to find the numerical solution for elastic-plastic stresses in a rotating disk with Von-Mis-es’ yield criterion using general non-linear strain hardening rule. The previous work mainly con-centrates on the material whose mechanical and physical properties are constant. In contrast, non-homogenous materials have different spatial distribution of material properties which can be designed according to different engineering applications. Gupta and Shukla [5] studied the effect of non-homogeneity on elastic-plastic stresses in a rotating disk using transition theory and con-cluded that non-homogeneous thin rotating disk require high angular speed for initial yielding as compared to homogenous disk. Further, Gupta et al. [6] studied rotating disk with variable thick-ness and variable density to analyze creep stresses and concluded that a rotating disk with variable

* Corresponding author, e-mail: [email protected], [email protected]

Sharma, S., et al.: Numerical Solution of Thermal Elastic-Plastic Functionally Graded ... 126 THERMAL SCIENCE: Year 2019, Vol. 23, No. 1, pp. 125-136

thickness and density is on the safer side of design in comparison to flat disk with variable density. Yuriy et al. [7] discussed the control problem of thermal stresses in axissymmetrical infinite cylin-der using technique of integral transform. Sharma and Sahni [8] investigated elastic-plastic stress-es for rotating disk made of transversely isotropic material and concluded that rotating disk made of transversely isotropic material is on the safer side of design as compared to rotating disc made of isotropic material. Reza et al. [9] discussed finite element simulation of residual stresses during the quenching process. A 3-D non-linear stress analysis model is used to estimate stress fields of UIC60. You et al. [10] calculated the elastic-plastic stresses with polynomial non-linear strain hardening model for rotating solid disk using perturbation technique. Further, with this polynomi-al non-linear strain hardening model You et al. [11] investigated elastic-plastic stresses for rotating disk having arbitrary variable thickness and density using Runge-Kutta method. Zhanling et al. [12] used finite element analysis of thermal-structure coupling to investigate stress and tempera-ture when material properties are temperature dependent in a drag disk break application. Sharma and Yadav [13] investigated elastic-plastic stresses for rotating disk made up of isotropic material having exponentially variable thickness and exponentially variable density with non-linear strain hardening using finite difference method. They observed that disk whose thickness decreases ra-dially and density increases radially is on the safer side of the design as compared to disk whose thickness and density varying exponentially as well as flat disk. Deepak et al. [14] studied creep stresses in a rotating disk made up of composite of silicon carbide particles in a matrix of pure alu-minums and their study indicates that with the increase in particle gradient in the disk, the radial stress increases throughout the disk, whereas the tangential and effective stresses increase near the inner radius but decrease near the outer radius.

In this paper, we investigated thermal elastic-plastic stresses and strains for rotating disk made up of functionally graded material using finite difference method. The thickness and density are assumed to vary exponentially along the radius. Results have been discussed numer-ically and propounded graphically.

Mathematical formulation

Distribution of material properties and thickness profile with basic equations

A thermal annular axi-symmetrical disk has been considered with inner radius, a , and external radius, b , rotating with angular velocity, ω . The disk is made up of functionally grad-ed material having exponentially varying thickness and density with plane stress condition i. e.

0zzT = . The coefficient of thermal expansion, compressibility, density, temperature distribution

of the material and thickness profile of the rotating annular disks with radius, r , are expressed:

( ) ( ) ( )

( ) ( )

1

0 0 0

00

, , e ,

log ,log

exp

d

n

rkb

rb

r rr C r C rb b

rr h r ha bb

α

α α ρ ρ

θθ

− −

−

= = =

= =

(1)

where 0α , 0C , 0ρ , 0θ , and 0h are material constants and k , d , 1α , and n are the geometric pa-rameters.

The equation of equilibrium for the rotating disk is:


2 2d ( ) – 0d rrhrT hT h r

r θθ ρω+ = (2)

The relation between strains and radial displacements are:

d , ,dr zu ue e e tr rθ= = = (3)

where u is the radial displacement and t is a constant.The equation of compatibility can be derived from eq. (3):

d 0 d rer e erθ

θ+ − = (4)

The total radial and circumferential strains in rotating annular disks are:

e pr r re e e αθ= + + , e pe e eθ θ θ αθ= + + (5)

The relationship between stresses and strains can be represented by Hooke’s Law in elasticity:

( ) ( ) ( ) ( )1 12 1 , 2 13 3

e er rr rre C T C T e C T C Tθθ θ θθ= − − − = − − − (6)

where C is the compressibility, rrT and Tθθ are radial and circumferential stresses, respectively.The temperature field satisfying Laplace heat equation is with:

2 0θ∇ = with 0θ θ= at r a= , 0θ = at r b= , ( ) 0 loglog

rr a bb

θθ = , 0θ is a constant (7)

Let us define radial and circumferential stress in terms of stress function:

2 21 d, drrT T r

hr h rθθφ φ ρω= = + (8)

Substituting eq. (8) into eq. (6) and expressing strain components of eq. (5) in terms of stress function:

( )

( ) 2 2

2 111 2

1 d

d3

pr r

p

C Ce

e

ehr

rh

C Cr

eθθ

φαθ

φ αθρω

+ + +

− − − = −

−

−

(9)

Substituting eq. (9) into eq. (4), we have a governing differential equation for thermal elastic-plastic strain hardening rotating disks:

( ) ( )

2 2

2 4 2 3

1 31 12 2 2 2

7 4 32 2 2

p p pr

h C C h Cr r r r r r hrh C C h C C

C Ce h r h r hr e eC C Cθ θ

φ φ φ

θα αθ ω ρ ρ ρω′

′ ′ ′ ′ − ′′ ′+ − − − − − + ⋅ − − − − ′ − ′ ′ ′⋅ + + = − − + − − − −

(10)


where

2

2ddrφφ′′ = , d

drφφ′ = , d

dCCr

′ = , dd

pp ee

rθ

θ′ = , d

drθθ ′ = , d

drαα′ = , d

dhhr

′ = , ddrρρ′ =

In the elastic region ( 0)p pre eθ= = , eq. (10) reduces to:

( )

2

2 2 4 2 3

11 12 2 2

3 7 42 2 2

h C C h Cr r r r r rh C C h C

C Chr h r h rC C C

φ φ φ

θα αθ ω ρ ρ ρω

′ ′ ′ ′ − ′′ ′+ − − − − − + − − − ′ − ′ ′ ′+ + = − − − − −

(11)

For the plastic deformation, the relations between stresses and plastic strains can be determined according to the deformation theory in plasticity [2]:

1 1,2 2

p pp pe er rr rr

ee ee

e ee T T e T TT Tθθ θθθ

= − = −

(12)

where pre and peθ are the plastic radial and circumferential strains, p

ee is the equivalent plastic strain which depends on the material model used and eeT is the equivalent stress.

The von Mises’ equivalent stress is given by the expression:

( ) ( ) ( )2 2 212ee rr zz zz rrT T T T T T Tθθ θθ= − + − + − (13)

The stress-strain relationship for Swift’s hardening law can be written:

00

01, 1 , ,

mpee ee

e e e eT Te e e e e eE Tη

= ≤ = − >

(14)

where 0T is the yield limit, ee – the equivalent total strain, 0e – the yield strain and eeT – the equivalent stress.

Substituting pee from eq. (14) into eq. (12) results in:

0 0

1 11 1

2

1 1,2

m m

e

p p

ee ee

e er

er rr re

TT T e T T

T TT T

T θθ θθθ

η η = − = −

− −

(15)

The boundary conditions for the rotating annular disks are:

0rrT = at r a= , 0rrT = at r b= (16)

Substituting eq. (15) into eq. (10), we have a non-linear differential equation in terms of φ as:

( )222

0 0

3 1 11 2 1 2 1 ( 1)2 22

m mee ee

ee rreeee

T Tr T T T mC T T TT θθφ

η

′′ + − + − + − − −


( )

( )

( )

2 4 2 3

0

2 2 4

22 3

0

1 1 2 1 2 4

3 1 22 2 1 1 ( 1) 2 2 2

21

mee

ee

rrm

eeee

rr rree

T h hT r r r h r h rT h h

hT T r h rhC T Tm T T h r T T

T Thr rh

θθ

θθ θθ

φ φ ρ ω ρω

φ ρ ωη

ρω

φ φ

′ ′ ′ ′− + − + − − + ′ ′ ′− − + +− − + + − − + + − ⋅

′ ′⋅ − +

( ) ( )

[ ]

0

2

2 4

9 1 12 2 2

1 312 2 2

2

mee

rree

Thr h CT T r r rT C T h C

C h Cr r hrC h C C

Ch rC

θθ φη

φ θα αθ

ω ρ ρ

−

′ ′ ′− − − + − − − − −

′ ′ − ′ ′− − − + + = − − − ′ ′= − −

2 3 7 42

Ch rC

ρω − − −

Equation (17) is a non-linear differential equation in the plastic region for rotating disk made up of functionally graded material with Swift’s non-linear strain hardening mea-sure having non-uniform thickness and whose material properties subjected to thermal loading. When compressibility is assumed to be constants without thermal effects, eq. (17) becomes same as that obtained by Sharma and Yadav [13].

Finite difference algorithm

To determine thermal elastic-plastic non-homogeneous stresses and strains in thin rotating disks with non-linear strain hardening material, we have to solve the second order non-linear differential eq. (17) under the boundary conditions (16). The general form of eq. (17) can be written:

( ) , ,f rφ φ φ′′ ′= (18)

– First partitioned the disk domain [ ]r a b= into p subintervals of length r∆ and express the differential operator φ′ and φ′′ in finite difference form:

( )

21 1

2 2 2 d

di i i

r rφφ φ φ+ −− +

=∆

and 1 1 d d 2

i i

r rφφφ + −−

=∆

– With h = 1/p, we have p + 1 nodal points 11 2 , , pφφ φ +… . The values at the end points are given by eq. (16), i. e. 1 10, 0 pφ φ += = . Using the finite difference approximation, we get the following system of equations:

( )

1 1 1 12

2 , , , 2,3,...2

i i i i iif r i p

rrφ

φφφ φ φ+ − + −− + − = = ∆ ∆

Stress function, stresses and strains can be obtained from the aforementioned system of ( )1p − non-linear equations using Newton-Raphson method.

(17)


Numerical discussion

The material properties of the annular disks made of functionally graded material with angular velocity ( 300,700 rad/s)ω = and thermal effects 0( 0,400,700 C)θ = ° are defined as: material density 0 7850ρ = kg/m3, compressibility coefficient 0 0.5C = , thermal expansion co-efficient 6

0 17.8 10α −= ⋅ oC–1, Poisson’s ratio 0.3ν = . The inner and outer radii of the disk are taken as 0.1a = m and 0.5b = m, respectively. The geometric parameters of the disk are taken as 0.5, 1n = in thickness function, 0.7, 1d = in density function, 1 0α = in coefficient of thermal expansion and 0, 0.5, 1, 1.5k = − − − in compressibility function.

In order to explain the effect of rotation and compressibility on stresses in a disk made up of homogenous and functionally graded materials, tab. 1 and curve have been drawn in figs*. 1-3 between stresses and radii r = 0.1:0.1:0.5.

Table 1 has made for circumferential (hoop) stresses with different parameters of com-pressibility and angular speed for non-linear strain hardening measure 0.2m = . From tab. 1, it is observed that without thermal effects circumferential stresses are maximum at the internal sur-face and increases significantly with the increase in angular speed. For functionally graded disk, circumferential stresses increases with the increase in compressibility radially. It is also noticed that circumferential stress is high for homogenous disk (k = 0) as compared to functionally grad-ed disk ( 0.5, 1, 1.5)k = − − − . Also, circumferential stresses are high for highly functionally grad-ed disk (k = –5) as compared to homogenous disk and these stresses increases with the increase in compressibility ( 5, 6, 7, .)k etc= − − − . With the introduction of thermal effects, circumferen-tial stresses decrease and these stresses further decrease with the increase in temperature. With the increase in Swift’s strain hardening measure, circumferential stress increases whereas these stresses decrease with the exponential decrease in thickness and increase in density.

Radial stress and strain (ω = 300 rad/s, d = 0.7, n = 0.5)Hoop stress and strain (ω = 300 rad/s, d = 0.7, n = 0.5)Radial stress and strain (ω = 300 rad/s, d = 0.7, n = 1)Hoop stress and strain (ω = 300 rad/s, d = 0.7, n = 1)

Radial stress and strain (ω = 700 rad/s, d = 0.7, n = 0.5)Hoop stress and strain (ω = 700 rad/s, d = 0.7, n = 0.5)Radial stress and strain (ω = 700 rad/s, d = 0.7, n = 1)Hoop stress and strain (ω = 700 rad/s, d = 0.7, n = 1)

Plot of stresses (k = 0, θ0 = 0 °C)

r [m]

Stre

sses

[MPa

]

r [m]

Plot of stresses (k = –0.5, θ0 = 0 °C)

r [m] r [m]

Stre

sses

[MPa

]

Plot of stresses (k = –1, θ0 = 0 °C) Plot of stresses (k = –1.5, θ0 = 0 °C)

Stre

sses

[MPa

]

Stre

sses

[MPa

]

Figure 1. Elastic-plastic stresses in a rotating disk with parameters = − − −( 0, 0.5, 1, 1.5)k* Legend is valid for all figures (figs. 1-6)


Plot of stresses (k = 0, θ0 = 400 °C) Plot of stresses (k = –0.5, θ0 = 400 °C)


r [m] r [m]

r [m] r [m]

Stre

sses

[MPa

]

Stre

sses

[MPa

]

Stre

sses

[MPa

]

Stre

sses

[MPa

]

Figure 2. Thermal elastic-plastic stresses in a rotating disk with thermal effects ( = °0 400 Cθ ) with parameters = − − −( 0, 0.5, 1, 1.5)k

From fig. 1, it is observed that circumferential stresses are maximum at internal sur-face for the disks made up of both homogenous and functionally graded material. Also, circum-ferential stresses are maximum for homogenous disk as compared to functionally graded disk. As compressibility measure changes from linear to non-linear, circumferential stresses decreas-es. With the increase in non-linearity, stresses are again increases i. e. 257.843MPaθσ = at

0.5k = − , 258.195MPaθσ = at 1k = − , and 260.190MPaθσ = at 1.5k = − . From fig. 2, it has also been observed that with the increase in angular velocity from 300 rad/sω = to

700 rad/sω = , circumferential stresses increase significantly and maximum at internal surface of the disk. With the increase in thickness measure from 0.5n = to 1n = and density measure from 0.7d = to 1d = , circumferential stress decreases. It can be observed from fig. 2 that circumferential stresses with thermal effects are maximum at internal surface for disk made of homogenous and functionally graded material. These circumferential stresses are maximum for homogenous disk as compared to functionally graded disk. With the increase in angular veloc-ity circumferential stress increases significantly while with the increase in temperature, thick-ness and density, circumferential stress decreases as can be seen from fig. 3.

In order to explain the effect of rotation and compressibility on strains in a disk made up of homogenous and functionally graded materials, tab. 2 and graphs have been drawn in figs. 4-6 between strains and radii r = 0.1:0.1:0.5.

It has been noticed from tab. 2 that without thermal effects, circumferential strains are maximum at external surface of the disk. Also, these circumferential strains are compres-sive in nature. With the increase in angular speed, circumferential strains decrease signifi-cantly. Also, these circumferential strains increase when the thickness of disk decreases while density of the disk increases exponentially. These strains decreases with the increase in tem-perature while increases with the increase in angular speed. It has also been noted that cir-


θ 0 [°

C]

ω =

300

rad/

s, d

= 0.

7, n

= 0

.5ω

= 3

00 ra

d/s,

d =

1, n

= 1

ω =

700

rad/

s, d

= 1,

n =

1ω

= 7

00 ra

d/s,

d =

0.7,

n =

0.5

p e θ

Tabl

e 2.

Hoo

p st

rain

in r

otat

ing

annu

lar

disk

hav

ing

diffe

rent

geo

met

ric

para

met

ers o

f com

pres

sibi

lity

=−

−−

(0,

0.5,

1,1.

5)k

w

ith n

on-li

near

stra

in h

arde

ning

mea

sure

=

0.2

m, a

ngul

ar v

eloc

ity ω

[rad

s–1],

and

tem

pera

ture

0θ [°

C]

θ 0 [°

C]

T θθ [

MPa

]ω

= 3

00 ra

d/s,

d =

0.7,

n =

0.5

ω =

300

rad/

s, d

= 1,

n =

1ω

= 7

00 ra

d/s,

d =

1, n

= 1

ω =

700

rad/

s, d

= 0.

7, n

= 0

.5

Tabl

e 1.

Hoo

p st

ress

in r

otat

ing

annu

lar

disk

hav

ing

diffe

rent

geo

met

ric

para

met

ers o

f com

pres

sibili

ty=

−−

−(

0,0.

5,1,

1.5)

k

with

non

-line

ar st

rain

har

deni

ng m

easu

re m

= 0

.2, a

ngul

ar v

eloc

ity ω

[rad

s–1],

and

tem

pera

ture

θ0 [

°C]

cumferential strains are less for homogenous disk (k = 0) as compared to disk made of func-tionally graded material and these strains decreases with the increase in compressibility measure ( 0.5, 1, 1.5, 2)k = − − − − , etc.

From fig. 4, it is observed that plastic strains are maximum at external surface for homogenous as well as functionally graded disk. Plastic strains decreases significantly for functionally graded disk when compressibility measure changes from 1k = − to 1.5k = − as


Figure 3. Thermal elastic-plastic stresses in a rotating disk with thermal effects ( = °0 700 Cθ ) with parameters = − − −( 0, 0.5, 1, 1.5)k

Plot of stresses (k = 0, θ0 = 700 °C) Plot of stresses (k = –0.5, θ0 = 700 °C)


r [m] r [m]

Stre

sses

[MPa

]

Stre

sses

[MPa

]

r [m] r [m]

Stre

sses

[MPa

]

Stre

sses

[MPa

]

(k = 0, θ0 = 0 °C) (k = –0.5, θ0 = 0 °C)

(k = –1, θ0 = 0 °C) (k = –1.5, θ0 = 0 °C)

r [m] r [m]

r [m] r [m]

Plas

tic st

rain

s

Plas

tic st

rain

s

Plas

tic st

rain

s

Plas

tic st

rain

s

Figure 4. Plastic strains in a rotating disk with parameters = − − −( 0, 0.5, 1, 1.5)k


can be seen from fig. 4. Plastic strains decreases significantly when angular velocity chang-es from ω = 300 rad/s to ω = 700 rad/s, but as thickness measure changes from 0.5h = to

1h = and density measure from 0.7d = to 1d = , these strains increases significantly. With the increase in temperature, plastic strains increases remarkably. From fig. 5, it has been observed that circumferential strains increases with the introduction of temperature. These strains further increases with the increase in temperature as can be seen from fig. 6. Also circumferential strains decrease significantly with the increase in angular velocity while increases with the increase in thickness and density.

(k = 0, θ0 = 400 °C) (k = –0.5, θ0 = 400 °C)

(k = –1, θ0 = 400 °C) (k = –1.5, θ0 = 400 °C)

r [m] r [m]

r [m] r [m]

Plas

tic st

rain

s

Plas

tic st

rain

s

Plas

tic st

rain

s

Plas

tic st

rain

s

Figure 5. Thermal plastic strains in a rotating disk with thermal effects ( = °0 400 Cθ ) with parameters ( = − − −0, 0.5, 1, 1.5k )

Conclusion

From the previous analysis, it can be conclude that disk made of functionally grad-ed material having non-linear strain hardening m = 0.2 with thermal effects whose thickness decreases exponentially and density increases exponentially is on the safer side of the de-sign as compared to disk made of homogenous material. This is because of the reason that circumferential stress is less for functionally graded disk as compared to homogenous disk. Also, plastic strains are high for functionally graded disk as compared to homogenous disk. It means that disk made of functionally graded material reduces the possibility of fracture at the bore as compared to the disk made of homogeneous material which leads to the idea of stress saving.

Nomenclaturea – inner radii of the diskb – outer radii of the diskC – compressibility of the disk

C0, h0 – material constantsd, k, n – geometric parametersE – Young’s modulus


Figure 6. Thermo plastic strains in a rotating disk with thermal effects ( = °0 700 Cθ ) with parameters = − − −( 0, 0.5, 1, 1.5)k

r [m] r [m]

r [m] r [m]

(k = 0, θ0 = 700 °C) (k = –0.5, θ0 = 700 °C)

(k = –1, θ0 = 700 °C) (k = –1.5, θ0 = 700 °C)

Plas

tic st

rain

s

Plas

tic st

rain

s

Plas

tic st

rain

s

Plas

tic st

rain

s

e0 – yield strainee – equivalent strain

ere – radial elasticpre – plastic strains

er – total radial strainsez – circumferential strains

eeθ – circumferential elasticpeθ – plastic strains

eθ – axial strainsh – thickness of the diskr – radius of the diskTee – equivalent stress

Trr, Tθθ, Tzz – radial, circumferential, and axial stresses

u – radial displacement

Greek symbols

α – thermal expansionα0, θ0, ρ0 – material constantsα1 – geometric parameterθ – temperatureρ – density of the diskϕ – stress functionω – angular velocity of the diskν – Possion’s ratio

References[1] Timoshenko, S. P., Goodier, J. N., Theory of Elasticity, McGraw-Hill, 3rd ed., New York, USA, 1970[2] Hill, R., The Mathematical Theory of Plasticity, Oxford University Press, Oxford, UK, 1998[3] Eraslan, A. N., Inelastic Deformations of Rotating Variable Thickness Solid Disks by Tresca and

von-Mises’ Criteria, Int. J. Comp. Eng. Sci., 3 (2002), 1, pp. 89-101[4] Eraslan, A. N., Akgul, F., Yielding and Elasto-Plastic Deformation of Annular Disks of a Parabolic

Section Subject to External Compression, Turkish J. Eng. Env. Sci., 29 (2005), 1, pp. 51-60[5] Gupta, S. K, Shukla, R. K., Effect of Non-Homogeneity on Elastic-Plastic Transition in a Thin Rotating

Disk, Indian J. Pure Appl. Math., 25 (1994), 10, pp. 1089-1097[6] Gupta, S. K, et al., Creep Transition in a Thin Rotating Disk Having Variable Thickness and Variable

Density, Indian J. Pure Appl. Math., 31 (2000), 10, pp. 1235-1248[7] Yuriy, P., et al., Control of Thermal Stresses in Axissymmetric Problems of Fractional Thermoelasticity

for an Infinite Cylindrical Domain, Thermal Science, 21 (2017), 1A, pp. 19-28


[8] Sharma, S., Sahni, M., Elastic-Plastic Transition of Transversely Isotropic Thin Rotating Disk, Contem-porary Eng. Sci., 2 (2009), 9, pp. 433-440

[9] Reza, M. N., et al., Three-Dimensional Finite Element Simulation of Residual Stresses in Uic60 Rails during the Quenching Process, Thermal Science, 21 (2017), 3, pp. 1301-1307

[10] You, L. H., et al., Perturbation Solution of Rotating Solid Disks with Nonlinear Strain-Hardening, Me-chanics Research Communications, 24 (1997), 6, pp. 649-658

[11] You, L. H., et al., Numerical Analysis of Elastic-Plastic Rotating Disks with Arbitrary Variable Thick-ness and Density, International Journal of Solids and Structures, 37 (2000), 52, pp. 7809-7820

[12] Zhanling, J., et al., Elastoplastic Finite Element Analysis for Wet Multidisc Brake during Lasting Brak-ing, Thermal Science, 19 (2015), 6, pp. 2205-2217

[13] Sharma, S., Yadav. S., Finite Difference Solution of Elastic-Plastic Thin Rotating Annular Disk with Exponentially Variable Thickness and Exponentially Variable Density, Journal of Materials, (2013), pp. 1-9

[14] Deepak, D., et al., Creep Modeling In Functionally Graded Rotating Disc of Variable Thickness, Jour-nal of Mechanical Science And Technology, 24 (2010), 11, pp. 2221-2232

Paper submitted: October 1, 2013Paper revised: April 9, 2018Paper accepted: April 29, 2018

© 2019 Society of Thermal Engineers of SerbiaPublished by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia.

This is an open access article distributed under the CC BY-NC-ND 4.0 terms and conditions

numerical solution of thermal elastic-plastic …the study of stress distribution at high angular...

Documents