Archive of Applied Mechanics 71 (2001) 221±232 Ó Springer-Verlag 2001

Temperature and stress analysis of an elastic circular cylinder in contact with heated rigid stamps F. Ashida

Summary The present paper discusses a plane strain problem of transient thermoelasticity in a circular cylinder which is in partial contact with two heated rigid stamps, in the case where the coef®cient of relative heat transfer on the contact surface of the cylinder is different from that on the traction-free surface. A ®nite difference method with respect to the time variable and Airy's thermal stress function is employed to analyze the temperature and thermoelastic ®elds. The problem is formulated in terms of two dual-series equations derived not only from the thermal boundary conditions but also from the mechanical boundary conditions. Since the radial, hoop and axial stresses have singularities at the end of the contact surface of the cylinder, the stress singularity coef®cients are de®ned and then the relationship among these three coef®cients is also obtained. Finally, numerical results are illustrated graphically. Key words stress singularity, asymmetric contact, circular cylinder, thermoelasticity, plane strain

1 Introduction When a structural body in contact with another body is subjected to a thermal load, stresses produced in the body concentrate at the end of the contact surface. The stress concentration may lead to a crack at the end of the contact surface so that the structural body may fracture. Evaluation of the stress concentration is very important for the safety of the structural body in service. Therefore, many papers have dealt with various thermoelastic contact problems, [1, 2]. Among the recent investigations in thermoelastic contact, [3] discussed uniqueness and continuous dependence of solutions to a contact problem of thermoelasticity. Steady thermoelastic contact problems in an elastic half-space and a rigid body were solved in [4], [5] and [6]. Papers [7] and [8] investigated the stress singularity in bonded dissimilar materials, while [9] analyzed a steady thermoelastic contact problem in two plates with bolted joints by means of a ®nite element method. In papers [10], [11] and [12] axisymmetric contact problems of steady thermoelasticity with frictional heating were studied on the assumption that the shearing stress on the contact surface was negligible. However, the models analyzed in these papers are semi-in®nite bodies, semi-in®nite quarter bodies and plates with ¯at surfaces. In contrast, thermoelastic contact problems in circular cylinders have attracted little attention. Axisymmetric contact problems in isotropic circular cylinders inserted into rigid rings were solved in [13±15], while similar problems in transversely isotropic circular cylinders inserted into rigid rings were analyzed in [16±18]. Although there are many structural systems whose components are in asymmetric contact with other bodies, an asymmetric contact problem of thermoelasticity in a circular cylinder has not been analyzed. The present paper deals with a plane strain problem of transient thermoelasticity in a circular cylinder which is in partial contact with two heated rigid stamps. It is assumed that the coef®cient of relative heat transfer on the contact surface of the cylinder is different from that on the traction-free surface. This problem is analyzed by means of a ®nite difference formulation with respect to a time variable as well as of Airy's thermal stress function. Stress singularity coef®cients are derived from the radial, hoop and axial stresses. Numerical results for Received 3 March 2000; accepted for publication 12 July 2000 F. Ashida Department of Electronic and Control Systems Engineering, Shimane University, Matsue 690-8504, Japan e-mail: [email protected]

221

the temperature, radial displacement, radial stress and stress singularity coef®cient are illustrated graphically.

222

2 Analysis Let us analyze a transient thermoelastic problem in a circular cylinder which is in partial contact with two heated rigid stamps without friction, as shown in Fig. 1. In order to simplify the analysis of this problem, it is presumed that the thermal and mechanical conditions of the cylinder are symmetric with respect to x and y axes. Therefore, we only have to analyze the thermal and elastic ®elds in the ®rst quadrant. Moreover, assuming that the cylinder and rigid stamps are long and the heating temperature distributes uniformly in the axial direction, this problem can be reduced to a plane strain problem. 2.1 Temperature field It is assumed that the cylinder, initially at uniform zero temperature, is suddenly subjected to heating temperature on the contact surface. Then, the initial and boundary conditions for the temperature ®eld are given by T ˆ 0 at t ˆ 0 ;

…1†

T;r ‡ h1 T ˆ h1 T0 q…h†; T;r ‡ h2 T ˆ 0;

0  h < h0 ; h0 < h  p2 ;

 on r ˆ a ;

…2†

where T is temperature, t is time, h1 and h2 are coef®cients of relative heat transfer on the contact and traction-free surfaces, T0 denotes constant temperature, q…h† is a prescribed function assumed to be

q…h† ˆ q… h† ˆ q…p

h† ;

and the comma denotes partial differentiation with respect to variable. The temperature is governed by Fourier's heat conduction equation

r2 T ˆ j 1 T;t ;

…3†

where j is thermal diffusivity and

r2 ˆ

o2 ‡r or2

1

o ‡r or

2

o : oh2

Since it is dif®cult to solve the governing equation (3) under the initial and boundary conditions (1) and (2) by means of the Laplace transform technique, a ®nite difference formulation with respect to the time variable, [17], is introduced. The time variable is given by

Fig. 1. Geometry of a circular cylinder in asymmetric contact with two rigid stamps

j X

tj ˆ

Dtk ;

…4†

kˆ1

where Dtk are time differences. The temperature is expressed as

Tj ˆ

j X

wjk DTk ;

…5†

kˆ1

where DTk are temperature differences and

wjk ˆ djk ‡ …1

223

w…j 1†k Dtk djk † ; Dtk Dtj

…6†

in which djk is Kronecker's delta. Considering the initial condition (1), Eqs. (3) and (2) become

r2 …DTj †

DTj ˆ0 ; jDtj

…7†

…DTj †;r ‡ h1 DTj ˆ Dj q…h†; …DTj †;r ‡ h2 DTj ˆ 0;

0  h < h0 ; h0 < h  p2 ;

 on r ˆ a ;

…8†

where

Dj ˆ h1 T0

…1

d1j †

j 1 X

wjk Dk :

…9†

kˆ1

The solution for Eq. (7) is taken to be

DTj ˆ

1 X nˆ0

A2nj I2n …sj r† cos…2nh† ;

…10†

where A2nj are unknown coef®cients, In …r† are the modi®ed Bessel function of the ®rst kind and

1 sj ˆ p : jDtj

…11†

Substituting Eq. (10) into Eqs. (8), a dual-series equation is obtained 1 X nˆ0 1 X nˆ0

U2nj E2nj cos…2nh† ˆ Dj q…h†; E2nj cos…2nh† ˆ 0;

0  h < h0 ;

p h0 < h  ; 2

…12†

where E2nj are unknown coef®cients related to the unknown coef®cients A2nj and U2nj are known coef®cients as follows:

E2nj

   2n ‡ h2 I2n …sj a† ‡ sj I2n‡1 …sj a† ; ˆ A2nj a

U2nj ˆ

…2n ‡ h1 a†I2n …sj a† ‡ sj aI2n‡1 …sj a† : …2n ‡ h2 a†I2n …sj a† ‡ sj aI2n‡1 …sj a†

…13† …14†

In order to solve Eqs. (12), we introduce the following series, which identically satis®es the second of Eqs. (12)

1 h0 X … 1†l xlj fJ2l‡2 …2nh0 † ‡ J2l …2nh0 †g ; 1 ‡ dn0 lˆ0

E2nj ˆ

…15†

where xlj are new unknown coef®cients and Jl …h† are the Bessel functions of the ®rst kind. Substituting Eq. (15) into the ®rst of Eqs. (12) and applying the series

q…h† ˆ

1 X nˆ0

224

q2n cos…2nh†;

4 q2n ˆ …1 ‡ dn0 †p

Zh0 q…h† cos…2nh†dh ;

…16†

0

and

cos…2nh† ˆ

1 X mˆ0

em cos…2mb†J2m …2nh0 †;

  h ; em ˆ 2 b ˆ sin h0 1

dm0 ;

…17†

the following in®nite system of simultaneous algebraic equations is obtained:

h0

1 X

… 1†l xlj

lˆ0

ˆ Dj

1 X nˆ0

1 X U2nj fJ2l‡2 …2nh0 † ‡ J2l …2nh0 †g nˆ0

q2n Z2m …2nh0 †;

1 ‡ dn0

Z2m …2nh0 †

…m ˆ 0; 1; 2; . . .† ;

…18†

where

Z2m …c† ˆ …m ‡ 3†fJ2m …c†

J2m‡4 …c†g

…m ‡ 1†fJ2m‡4 …c†

J2m‡8 …c†g :

…19†

Therefore, the unknown coef®cients xlj can be determined by solving the linear equations (18) numerically. Finally, substituting Eqs. (10) and (15) into Eqs. (5) and (13), respectively, the temperature is given by

Tj ˆ

j X kˆ1

wjk

1 X nˆ0

A2nk I2n …sk r† cos…2nh† ;

…20†

where

A2nj

P l h0 a 1 lˆ0 … 1† xlj ‰J2l‡2 …2nh0 † ‡ J2l …2nh0 †Š : ˆ …1 ‡ dn0 †f…2n ‡ h2 a†I2n …sj a† ‡ sj aI2n‡1 …sj a†g

…21†

2.2 Thermoelastic field The plane strain problem of transient thermoelasticity in a circular cylinder which is in partial contact with two rigid stamps, as shown in Fig. 1, can be analyzed by means of Airy's thermal stress function. Applying the ®nite difference formulation, Airy's thermal stress function vj at t ˆ tj is expressed as vj ˆ

j X kˆ1

wjk Dvk ;

where Dvk are governed by

…22†

r2 r2 …Dvj † ˆ

aE 1

m

r2 …DTj † ;

…23†

in which a is the coef®cient of linear thermal expansion, E is the Young's modulus and m is the Poisson's ratio. The stress components are expressed in terms of Airy's thermal stress function as follows:

rrrj ˆ r 2 vj;hh ‡ r 1 vj;r ; rhhj ˆ vj;rr ; 2

rrhj ˆ r vj;h

…24† 1

r vj;hr :

225

The admissible solution for Eq. (23) is taken to be

Dvj ˆ

1 X

(

cos…2nh†

nˆ0

aE 1

I2n …sj r† r2n‡2 r2n A2nj ‡ F ‡ G 2nj 2nj 2n 2 a2n a s2j m

) ;

…25†

where F2nj and G2nj are unknown coef®cients. Substituting Eq. (25) into Eqs. (22) and (24), the stress components are obtained as follows:

rrrj

) I2n …sk r† I2n‡1 …sk r† A2nk 2n…2n 1† ˆ wjk cos…2nh† 1 m sk r …sk r†2 nˆ0 kˆ1 #  r 2n  r 2n 2 2…n 1†…2n ‡ 1†F2nk ; 2n…2n 1†G2nk a a

rhhj ˆ

j X

"

1 X

(

! 2n…2n 1† A2nk I2n …sk r† wjk cos…2nh† 1‡ 1 m …sk r†2 nˆ0 kˆ1 #  r 2n  r 2n 2 ; 2n…2n 1†G2nk 2…n ‡ 1†…2n ‡ 1†F2nk a a j X

"

1 X

rzzj ˆ m…rrrj ‡ rhhj † rrhj ˆ

(

aE

aE

) I2n‡1 …sk r† sk r

aETj ; "

1 X

…27† …28†

) 2n 1 I2n‡1 …sk r† 2 wjk n sin…2nh† I2n …sk r† ‡ A2nk 1 m sk r …sk r†2 nˆ0 kˆ1 #  r 2n  r 2n 2 …2n ‡ 1†F2nk : …2n 1†G2nk a a j X

…26†

aE

(

…29†

The radial displacement is derived from the radial strain as

( ) " j 1 X 1‡m X aE 2nI2n …sk r† I2n‡1 …sk r† urj ˆ wjk cos…2nh† ‡ r A2nk E 1 m sk r …sk r†2 nˆ0 kˆ1 #  r 2n  r 2n 2 2…n 1 ‡ 2m†F2nk 2nG2nk : a a

…30†

It is assumed that the cylinder is in smooth contact with two rigid stamps without friction and the rigid stamps are ®xed. In this case, the mechanical boundary conditions are taken to be

rrhj ˆ 0 on r ˆ a ; urj ˆ 0; 0  h < h0 rrrj ˆ 0; h0 < h  p2

…31†  on r ˆ a :

…32†

Substituting Eq. (29) into Eq. (31), we have

"

1

G2nj ˆ

2n

1

(

aE

…2n ‡ 1†F2nj

1

m

A2nj

I2n‡1 …sj a† 2n 1 2 I2n …sj a† ‡ sj a …sj a†

)# :

…33†

Substitution of Eqs. (30) and (26) into Eqs. (32) leads to a dual-series equation

226

1 X …X2n L2nj ‡ W2nj † cos…2nh† ˆ 0 …0  h < h0 †; nˆ0

1 X nˆ0

L2nj

…34†

p cos…2nh† ˆ 0 …h0 < h  † ; 2

where L2nj are unknown coef®cients related to the unknown coef®cients F2nj , and X2n and W2nj are known coef®cients as follows:

 L2nj ˆ …2n ‡ 1† 2F2nj

aE 1

m

A2nj

 I2n‡1 …sj a† ; sj a

…35†

…1 ‡ m†f1 2m 4n…1 m†ga ; …2n ‡ 1†…2n 1†E I2n‡1 …sj a† ˆ 2…1 ‡ m†aA2nj : sj

X2n ˆ W2nj

…36†

In order to solve Eqs. (34), we introduce the following series, which identically satis®es the second of Eqs. (34):

L2nj ˆ

1 2h0 X ylj J2l …2nh0 † ; 1 ‡ dn0 lˆ0

…37†

where ylj are new unknown coef®cients. Substituting Eq. (37) into the ®rst of Eqs. (34) and applying the series given in Eq. (17), the following in®nite system of simultaneous algebraic equations is obtained:

2h0

1 X lˆ0

ylj

1 X X2n J2l …2nh0 † nˆ0

1 ‡ dn0

Z2m …2nh0 † ˆ

1 X nˆ0

W2nj Z2m …2nh0 †;

…m ˆ 0; 1; 2; . . .† :

…38†

Therefore, the unknown coef®cients ylj can be determined by solving the linear equations (38) numerically. Substituting Eqs. (33), (35), (37) and (20) into Eqs. (26)±(28), the radial, hoop and axial stresses on the surface of the cylinder are expressed by

‰rrrj Šrˆa

  j 1 X H…h0 h† X h l ; ˆ q wjk … 1† ylk T2l h0 lˆ0 1 h2 =h20 kˆ1

‰rhhj Šrˆa ˆ

j X kˆ1

" wjk

  1 H…h0 h† X h l q … 1† ylk T2l h0 1 h2 =h20 lˆ0  1 aE X A2nk cos…2nh† I2n …sk a† 1 m nˆ0

…39†

# I2n‡1 …sk a† ; 2…2n ‡ 1† sk a

…40†

‰rzzj Šrˆa ˆ

"

  1 2mH…h0 h† X h wjk q … 1†l ylk T2l h 2 2 0 kˆ1 1 h =h0 lˆ0  1 aE X A2nk cos…2nh† I2n …sk a† 1 m nˆ0

j X

I2n‡1 …sk a† 2…2n ‡ 1†m sk a

# ; …41†

where H…h† is the Heaviside's unit step function and Tl …h† are Tchebycheff's polynomials of the ®rst kind. Expressions (39)±(41) indicate that the radial, hoop and axial stresses have singularities at the end of the contact surface of the cylinder. In order to evaluate the strength of these singularities, the following stress singularity coef®cients are introduced:

s   h ‰rrrj ; rhhj ; rzzj Šrˆa : ‰Krj ; Khj ; Kzj Š ˆ lim 2 1 h!h0 h0

…42†

Substituting Eqs. (39)±(41) into Eq. (42), we have j

Krj ˆ Khj ˆ

1 X Kzj X ˆ wjk … 1†l ylk : 2m lˆ0 kˆ1

…43†

2.3 Exact solution When a coef®cient of relative heat transfer on the contact surface of the cylinder is identical with that on the traction-free surface, namely h1 ˆ h2 ˆ h, the exact solution is obtained by employing the Laplace transform technique as follows: ( ) 1 1  r 2n X X Tˆ cos…2nh† T2n ‡ T2nm J2n …bm r† ; …44† a nˆ0 mˆ1 1 Kz X ˆ … 1†l yl ; Kr ˆ Kh ˆ 2m lˆ0

…45†

where

T2n

haT0 q2n ; ˆ 2n ‡ ha

2

T2nm ˆ

…2n ‡ ha†J2n …bm a†

…4n2

2haT0 q2n e jbm t ; b2m a2 h2 a2 †J2n …bm a†

bm a J2n‡1 …bm a† ˆ 0 ;

and yl are the solutions for the linear equations when W2nj in Eqs. (38) are replaced by the following coef®cients:

(

W2n

1 X T2n J2n …bm a† ‡ 2…2n ‡ ha† ˆ …1 ‡ m†aa T2nm 2n ‡ 1 b2m a2 mˆ1

) :

3 Numerical results The following dimensionless quantities are introduced for convenience in the presentation of numerical results: jtj Tj ; Bi ˆ ahi ; Tj ˆ ; 2 a T0 …1 m†uij …1 m†rikj …1 m†Kij ; rikj ˆ ; Kij ˆ : uij ˆ aa…1 ‡ m†T0 aET0 aET0 r r ˆ ; a

tj ˆ

227

For an illustrative example, the heating temperature distribution and the Poisson's ratio are assumed to be

q…h† ˆ H…h0

h†;

m ˆ 0:3 ;

and the time variable tj is given by j 1

4

tj ˆ 10 6

j 23 24

tj ˆ 10 228

2

j 75 5

tj ˆ 10

j 7

…j ˆ 1±6†;

3

…j ˆ 7±22†;

j 47 28

1

tj ˆ 10 16

…j ˆ 23±46†;

tj ˆ 10

…j ˆ 47±74†;

…j  75† :

It is necessary to examine the convergences of the in®nite series P involving P1the formulations P1 such as Eqs. (18) and (38). The upper limits of in®nite series of 1 nˆo ; n0 ˆ0 and lˆ0 are P P 1 1 0 represented by N; N and L, where nˆ0 and n0 ˆ0 denote the in®nite series of the left- and right-hand sides of Eqs.(18) and (38) respectively. The temperature at the center of the contact surface of the cylinder as well as the radial stress singularity coef®cient was calculated for various upper limit numbers of the in®nite series for B1 ˆ B2 ˆ 1 and 2h0 ˆ 30 . The ®nite difference results for the temperature at the centre of contact surface of the cylinder as well as the radial stress singularity coef®cient are shown in Tables 1 and 2 and compared to the exact solutions. These tables show that the in®nite series converge for N ˆ 45000; N 0 ˆ 7500 and L ˆ 80 in the case of temperature and for N ˆ 45000, N 0 ˆ 7500 and L ˆ 95 in the case of radial stress singularity coef®cient at a reasonable rate, and then the ®nite difference results agree quite well with the exact solutions. Figures 2±4 illustrate the distributions of the temperature, radial displacement and radial stress on the surface of the cylinder at selected Fourier's numbers for B1 ˆ 10, B2 ˆ 1 and 2h0 ˆ 30 . Figures 5 and 6 illustrate how the range of the contact region in¯uences the temperature distribution on the surface of the cylinder as well as the radial stress singularity Table 1. Effects of upper limit numbers of in®nite series on ®nite difference results for temperature at the center of contact surface of the cylinder in the case of 2h0 ˆ 30 and B1 ˆ B2 ˆ 1 Fourier's number tj

Upper limits of series N

N0

L

10

45000 50000 40000 45000 45000 45000 45000

7500 7500 7500 8000 7000 7500 7500

80 80 80 80 80 85 75

0.03461 0.03461 0.03461 0.03476 0.03436 0.03436 0.03360

0.10610 0.10610 0.10610 0.10630 0.10576 0.10576 0.10473

0.03520

0.10681

Exact solution

3

10

2

100

101

102

0.23211 0.23212 0.23212 0.23237 0.23171 0.23170 0.23044

0.34123 0.33976 0.34031 0.34205 0.33941 0.34040 0.33871

0.36879 0.36879 0.36878 0.36908 0.36831 0.36831 0.36684

0.36884 0.36884 0.36884 0.36913 0.36837 0.36836 0.36690

0.23298

0.34144

0.36816

0.36816

10

1

Table 2. Effects of upper limit numbers of in®nite series on ®nite difference results for radial stress singularity coef®cient in the case of 2h0 ˆ 30 and B1 ˆ B2 ˆ 1 Fourier's number tj

Upper limits of series N

N0

L

10

45000 50000 40000 45000 45000 45000 45000

7500 7500 7500 8000 7000 7500 7500

95 95 95 95 95 100 90

0.00283 0.00284 0.00283 0.00283 0.00284 0.00285 0.00280

0.00334 0.00334 0.00334 0.00285 0.00336 0.00328 0.00326

0.00292

0.00348

Exact solution

3

10

2

100

101

102

)0.06056 )0.06059 )0.06052 )0.06053 )0.06060 )0.06063 )0.06003

)0.28503 )0.28513 )0.28479 )0.28480 )0.28528 )0.28487 )0.28207

)0.35629 )0.35644 )0.35603 )0.35599 )0.35662 )0.35610 )0.35268

)0.35644 )0.35658 )0.35617 )0.35614 )0.35676 )0.35625 )0.35283

)0.06126

)0.28831

)0.35633

)0.35633

10

1

229

Fig. 2. Temperature distributions on the surface of the cylinder

Fig. 3. Radial displacement distributions on the surface of the cylinder

Fig. 4. Radial stress distributions on the surface of the cylinder

230 Fig. 5. Effect of the range of the contact region on temperature distributions on the surface of the cylinder

Fig. 6. Effect of the range of the contact region on radial stress singularity coef®cients

coef®cient for B1 ˆ 10 and B2 ˆ 1. In these ®gures, the dotted and broken lines represent the numerical results for B1 ˆ B2 ˆ 1 and B1 ˆ B2 ˆ 10, respectively. Figures 2±4 and 6 show that the temperature, radial displacement, radial stress and radial stress singularity coef®cient for B1 ˆ 10 and B2 ˆ 1 are much larger than those in both cases of B1 ˆ B2 ˆ 1 and B1 ˆ B2 ˆ 10. It is seen from Fig. 3 that the radial displacement is negative for small Fourier's number, and then turns to positive for large Fourier's number. Conversely, as shown in Fig. 4, the radial stress in the neighbourhood of the end of contact surface is positive when Fourier's number is small, and then turns to negative as time proceeds. Figures 5 and 6 show that the temperature and radial stress singularity coef®cient increase according to the range of the contact region. Now, it is seen from Fig. 2 that when 2h0 ˆ 30 and tj ˆ 1, the temperature on the tractionfree surface of the cylinder for B1 ˆ B2 ˆ 1 is higher than that in the case of B1 ˆ B2 ˆ 10, because the quantity of the heat transfer by convection on the traction- free surface increases according to Biot's number on its surface. A similar trend applies to the radial displacement and radial stress singularity coef®cient, as it evident from Figs. 3 and 6. Figures 7 and 8 illustrate how Biot's numbers on the contact and traction-free surfaces have in¯uence on the radial stress singularity coef®cient for the case of 2h0 ˆ 30 . The radial stress singularity coef®cient increases with increasing Biot's number on the contact surface, but decreases with increasing Biot's number on the traction-free surface, because the quantity of the heat transfer by convection increases according to the each Biot's number.

231 Fig. 7. Effect of Biot's number at the contact surface of the cylinder on radial stress singularity coef®cient

Fig. 8. Effect of Biot's number at the tractionfree surface of the cylinder on radial stress singularity coef®cient

In the present problem, the cylinder is assumed to be in complete contact with the rigid stamp without friction, and so Eqs. (2), (31) and (32) are given. As shown in Figs. 3 and 4, these boundary conditions bring about the curious result that the radial displacement is negative and the radial stress is positive near the end of contact surface of the cylinder, when Fourier's number is very small. If separation between the cylinder and rigid stamp is acceptable to the boundary conditions, we can deduce that separation occurs at the end of rigid stamp and then the stress singularity does not appear, while the radial displacement near the end of the contact surface is negative. In this case, the stress singularity is considered to occur after the radial displacement near the end of the contact surface has turned from negative to positive. Therefore, positive and negative of the stress singularity coef®cient shown in Fig. 6 indicate the behavior of elastic deformation near the end of the contact surface.

References

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