UDC 539.3 AXIS-SYMMETRIC THERMOELASTICITY TASK OF THE ELASTIC CYLINDER PRESSURE ON THE ELASTIC LAYER TAKING INTO A CCOUNT NON-IDEAL HEAT CONTACT

https://doi.org/10.33108/visnyk_tntu

Scientific Journal of the Ternopil National Technical University

2019, № 1 (93) https://doi.org/10.33108/visnyk_tntu2019.01 ISSN 2522-4433. Web: visnyk.tntu.edu.ua

UDC 539.3

AXIS-SYMMETRIC THERMOELASTICITY TASK OF THE ELASTIC

CYLINDER PRESSURE ON THE ELASTIC LAYER TAKING INTO A

CCOUNT NON-IDEAL HEAT CONTACT

Bogdan Okrepkyi

1

; Boris Shelestovs’kyi

2

1

_{Ternopil National Economic University, Ternopil, Ukraine}

2

_{Termopil Ivan Puluj National Technical University, Ternopil, Ukraine}

Summary. Solution of the axis-symmetric contact thermoelasticity task of the elastic round isotropic cylinder pressure on the elastic isotropic layer of the finite thickness, taking into account non-ideal heat contact between the cylinder and the layer, has been built. Using the Henkel’s method of integral transformation the equation of heat-conductivity and heat-elasticity for the layer was solved, and using the Fourier’s method – for the cylinder. The temperature field, displacement and stress in the cylinder are presented by the coefficients, which satisfy the non-finite system of algebraic equations.

Key words: elastic cylinder, layer, temperature, non-ideal heat contact, normal contact stresses.

https://doi.org/10.33108/visnyk_tntu2019.01.019 Received 12.03.2019

Staement of the problem. Determination of contact strains and stresses taking into

account temperature factors is an important task for the investigation of the machine parts and structural elements strength in the areas of their interaction while calculating structures on the elastic basis for effective structure use and base carrying capacity.

Analysis of the available investigations. The influence of temperature factors on the

nature of bodies interaction is investigated in papers [1−4]. Particularly, in [2−3] axisymmetric contact problems of thermoelasticity concerning hot circular die pressure on the isotropic half-space and the layer and in [4] concerning the elastic circular cylinder on the elastic half-half-space taking into account the nonideal thermal contact are solved. However, the influence of the conditions of nonideal thermal contact between isotropic elastic cylinders and the layer on the magnitude and nature of the normal stresses distribution in the contact area is not sufficiently investigated.

The objective of the paper is to construct the solution of the axisymmetric contact

thermoelasticity problem concerning the pressure of elastic isotropic circular cylinder with the flat base on elastic isotropic layer of finite thickness under nonideal thermal contact and to derive formulas for temperature and normal contact stresses determination; to investigate the influence of contact conductivity and Young's modulus of the cylinder and the layer on the distribution of the normal stresses temperature component.

Statement of the problem. Let us suppose the elastic cylinder with length L and a radius

R with the flat base is pressed by force P into the elastic layer of finite thickness H. The bodies

materials are assumed to be isotropic. All points of the cylinder end under the action of external loading are shifted to the same value  . The surfaces of the cylinder outside the contact area are free from external forces. On the contact area the tangential tensions are _rz 0.

with the external environment according to Newton's law takes place on the free layer surfaces. For given assumptions it is necessary to determine the temperature fields and contact normal stresses.

The cylindrical coordinate system r, , z with the center located on the layer surface, and axis 0z is directed along the cylinder axis. All quantities (stresses, displacement, temperature, elastic constants, coefficients of heat conduction and heat exchange, coefficients of linear thermal expansion), related to the layer are marked with the index «1», and those related to the cylinder are without indexes.

Thus, the proposed problem is solved under the following boundary conditions:





0, 0 , . T T  r R zL (1)





0, ; 0 . T r R z L r  _{   }  (2)









1 1 1 1 1 0 , , 0 , 0 . z z z T T T h T T r R z z z z               (3)





1 1 1 2 0, , 0 . T H T R r z z  _{     }  (4)





1 1 1 1 0, 0 , . T H T r z H z          (5)





, 0 , 0 , . z rz U      r R zL (6)





0, 0, , 0 . r rz r R z L        (7)





0, 0 , 0 . rz r R z      (8)





1 0, 0 , 0 . r z r z       (9)





(1) (1) , , 0 , 0 . z z z z U U    r R z (10)





1 0, , 0 . z R r z       (11)





1 1 0, 0, 0 , . z rz U      r z H (12)

The solution of the boundary value problems for thermal conductivity and thermal elasticity equations. It is known [5], that in the axisymmetric case, the thermoelastic potential

and the temperature field for the isotropic body are determined from the equations

2 1 2 , 0, 1 T T T           (13)

and the temperature stresses and displacements are calculated by the formulas:

2 2 ( ) ( ) ( ) 2 1 , 2 , 2 T T T z rz z U z r z r r r  _{ }  _{ }           _  _    _   _, 2 2 1 2 , T r z r r      _   _     (14)

where _T is the coefficient of linear temperature expansion;  , is shear modulus and Poisson ratio.

To determine the temperature field in the layer, let us introduce Hankel transformation of zero order functions T r z1( , )





   

1 1 0 0 , , , T  z r T r z J r dr  

_

(15)

due to which, according to the second equation (13), we find T1



 ,



through the arbitrary functions  1

 

and  2

 

:





 

 

 

1 1 2 0 0 , T     e   e  J  d     

_

_  _ , ₍₁₆₎

where J0

 

 is Bessel function of the first kind from the actual argument;

, / , .

r

z R R

R

    

The temperature field in the cylinder is found by Fourier method. The general solution is as follows:

 



2 2









0 0 0 0 1 , _{k k k k k} k T r z A z B D r rz J  r A sh z B ch z       



 

  



0 1 sin cos , k k k k k k I  r C  z D  z   



 (17)

where A B C_k, _k, _k, D_k are arbitrary constants; I0

 

kr is Bessel function of the first kind of the

imaginary argument;  k, k are eigen values determined from the boundary conditions.

a) for the layer:

 

1 ₁

 

₂

 

₀

 

0 1 1 1 , ; 2 1 T r z  e e J d               _{ }  _  _ 



(18)

b) for the cylinder:

 

3 2 4 4 0 0 0 0 1 1 1 1 , 6 4 16 6 r z A z B r C r D z       (19)

The components of temperature stresses and displacements are calculated by the formulas (14). With formulas for temperature stresses and displacements, it is possible to solve the problem under mechanical boundary conditions. In order to do this, it is necessary to add components of stresses and displacements from the biharmonic potential to the values calculated according to formulas (14) [1].

To satisfy the boundary condition (2) in formula (17) it is necessary to put

0 0, k 0, k 0 ( 1, )

D  D  C  k  ;_k _k /R, where _k are the roots of the equation

 

1 k 0 J   ; . k k R    (20)

The boundary condition (1) taking into account the orthogonality of Bessel functions results in such relations between constants B B₀, _n and A₀, A_n



n 1,



.

0 0 0 , n n n, / .

B T A R B  th A L R (21)

Having satisfied the boundary conditions (3−5) taking into account (21) we get the system of integral relations connecting the functions  1

 

and  2

 

with the coefficients



1



 



1



 

 





2 1 2 2 0 0 0, 1 . K    K    J  d           



(24)



1



 



1



 

 





1 1 1 2 0 0 0, 0 , h h K e  K e J d                      



(25) where hH R/ ; K₁1 H R K₁1 , ₂1H R h₂1 , ₀1h R₀ /_z1 .

Applying the formula of the inverse transformation of the integral Hankel transformation to equation (26) and introducing the notation

 



1



 



1



 

2 1 2 2

K K

          we get the system of equations respectively to functions  ₁

 

and  ₂

 

having the following solution:

 



1



   

1 1 1 / 2 h K e Q        , ₂

 

1



₁1



   

/ , 2 h K e  Q   _ _    

 



2 1 1





1 1



1 2 1 2 . Q    K K sh h  K K ch h (26)

Substituting the functions  ₁

 

and  ₂

 

(26) into equation (24), we get:

   

0





0 0 , 1 . J d        



(27)

The boundary conditions (9), (12) for the stress  1_z



, 0



and displacement U1_z



, 0



on the layer surface give:







    

1

_{   }

   

 

1 2 1 1 0 0 1 0 0 1 1 1 , 0 1 2 . z _T Q b U R h J d R J d b Q Q                     

_

_  _ 

_

₍₂₈₎





   

1 1 3 0 0 , 0 2 , z b J d    

_

    (29)

 

*

 

1 *

 

*

 

1 *

 

1 1 2 3 1 4 , R F b F F b F R R            

 

 

1 2 2 1 2 2 , 2 ( 1 ) , Q  sh h h Q    sh h ch h  K shh h sh h





_{ }

2 1 1 2 2 , h h e G h Q        





1 1 0 1 1 1 1 1 b       .

, k. k k k L R       (30)

Besides the same conditions result in the following relations between arbitrary constants:





* * * 1 3 * * * 2 3 2 2 2 1 1 0, / , 2 1 k k n n T k k k k k k k A b D N F b R E F b cth b b  _{    }             (31)





3 1 1 3 2 1 1 1 2 1 T _k k k k k k b R b B cth A b            _   _  _{ }, * * 1 0 1 2 / , 1, . k k b k k k A B I I k b     _{   }   _  _{   }        

Demanding the fulfillment of the first boundary condition (10) and the boundary condition (11) taking into account (31) we derive the system of integral equations respectively to functions  

 

and  

 

:

   

* * * 2





0





0 0 2 0 1 1 1 0 1 1 1 1 1 2 2 1 k k k k k T k k k k J J d A F LN J R A R R b               _{ }          _  _  _  _    











    

1

_{   }

   

 





2 0 0 0 1 0 0 1 2 , 0 1 . T Q G h J J d Q Q                  

_



_

  (32)

   

0





0 0, 1 . J d        



(33)

Introducing the function f t( ) by relationship

 

11 1

 

1 1 0 cos , 1 b f t tdt b     



(34)

equation (33) is satisfied identically, and equation (32) is reduced to Abel equation

 

_{ }

2 2 0 , f t dt g t     



(35)

which solution according to [6] is determined by the formula

where

 









* 0 * * 2 0 0 3 1 1 1 2 1 1 1 1 2 2 1 k k k k k T k k k k J A g F N J A R R b                    _  _     







    

   

   

 

1 2 0 0 0 1 0 0 1 2 _T Q . G h J d J d Q Q                  







(37)

Substituting expression (37) to formula (36), taking into account (37), we get Fredholm integral equation of the second kind relatively to the function f t

 

:

 

* 2 * * 0 3 1 1 4 2 1 cos 2 k k k k k f t A F N R R   b          _  _  



(38)

 





1 1 0 0 cos 1 1 2 2 cos cos 1 k T k k k t A f x dx G h x td  _{   }            







   

   





1 2 0 1 0 2 1 cos , 0 1 . T Q td t Q Q             

_

 

Having satisfied the first condition (6) taking into account the conditions of Bessel functions orthogonality, we obtain

* 1 * * 2 2 0 4 0 0 0 1 1 2 3 , 2 A b RB RC 4 R 1 TA  _          * * 3 2 1 3 1 1 . 2 1 k k k k T k A F b th M   R b       (39)

Taking into account equations (31), (39), we satisfy the boundary condition (7) for

normal stress



_r. Multiplying the obtained expression by cos



n



,



n 0,



and integrating it with  in the range from 0 to , taking into account the trigonometric functions orthogonality, we obtain the relation between constants B0*





2 1 2 1 2 k k b b I k         _{ }  

 

2 2 0 1 , , 2 2 1 1 , 1 , . n k n k n k k k k k I I U t ch n     _         _{  } _{ }                

The contact stresses  1_z



, 0



taking into account (29) and (34) are determined by the formula:





 

1

 





1 1 0 _{2 2 2} 1 , 0 , 1 , 1 z f f t dt t                    



 (41) where 1 1 1 1 3 0 1 1 2 1 b b b    .

Using the cylinder equilibrium condition

 

1

2 1

0

2R   _z d

  

_

and formula (41) taking into account (31), (39), the integral equation (38) is reduced to the form:

 

1

 





0 0 2 sin 2 cos cos f t f x dx G h x t  d    _{ }  _  _   





(42)

   

   

1 2 0 1 0 2 1 sin cos T Q t d Q     _{ }  _        _{ }  _  _   







2 * 1 3 2 1 1 1 0 sin 1 2 cos , 0 1 . 2 k k k k k k k b th t t R b R                _  _      



By satisfying the second boundary condition (10) and integrating the resulting expression on

 multiplied by , and J0

 

n in the range from 0 to 1 taking into account Bessel

functions orthogonality conditions, we get the relations between the constants





* * *

0, 0, k, 0, k 1,

B C M A A k  and the function f t

 

:

 

 

2 2 2 , 1 2 2 0 0 2 1 2 1 1 n k k n n k k k t I J A R n k              _{  }  _ _   _  



(44)

 

 

2 3 2 2 , 1 2 , 0 2 1 1 1 2 1 k n k m k m m m k k m m k t k I t J A ch                 _  _       _   _   









2

 

0 1 1 2 . 1 Tth n J n An  _{   }     

In order to determine the function  

 

we extend the equation (27) to the entire interval



0  



   

0



   



0 1 , 0 , J d              



(45)

here

 

x is Heaviside function; 

 

 is unknown function taking in the form

 

0 0 0



 



1 , 0 1 , N k k k T a a J          _  _   



 (46)

where a_k



k0,N



are unknown coefficients to be defined; value N is chosen from the condition of satisfying the desired accuracy of the problem solution.

Applying to both parts of equation (45) the formula of the inversion of integral Hankel transformation, we derive the function 

 

through unknown coefficients a_k:

 

 

2

 

0

 

0 0 1 1 2 2 1 . N k k k k a J T a J J             _  _  



 (47)

Multiplying both part of the equations (48), (49) by and J₀



 _n



and integrating them by , in the range from 0 to 1 taking into account Bessel functions orthogonality, we get the system of linear algebraic equations respectively to unknown





(1) (2) (1) (2)

, , , , 0,

k k k k k

a X X y y k  N . To calculate the temperature fields in the cylinder, we have the following formulas:

a) cylindrical area



0  1, 0 











1 ( 2) 1



_{ }



( 2), 0 0, 2 0 0 1 0 , 1 2 2 , N N m m k z z k k k m k k m z z m m sh T T a a ch J                         _    _   



 

 (50) b) sphere



  h  0, 0  







11





1*



_{ }

,



1

  

_

0

_{  }



1 0 0 1 1 1 1 1 0 1 1 2 1 2 1 , 2 m m m m m m m P y K y I y K h T T a K y I y K K h K K Q iy                    _{ }  _ _         



(51)









   

_



_

_{ }



  

_

_{  }



2 * 1 , 0 1 , 1 0 0 2 2 1 1 0 1 2 0 k k N m m m m k k k k m m k m m m P J y P y K y I y a Q J K y I y y Q iy                         _{ } _{   }    _{  }       





,









1





1 , 1 P x  xchx h K shx h , P1*



ym,



 ymcosym



h



K11sinym



h



.

At 0  1 the upper expression in round parentheses is taken as multiplier, at1 − the lower one.

In order to determine the contact normal stresses 1





, 0

z

  we get the following expression:









 1





1 , 0 , 0 T , 0 z z z        ; (52)  

_

_

1

_{  }

_

_{ }

(1) 1 0 2 1 2 ₂ 1 0, 5 1 1 , 0 1 2 1 1 N k P z k k k P X k T X R     _        _    _  



;





1

  



 

1 1 1 ( ) 0 2 ( 2) 0 ₂ 0 2 1 1 1 , 0 1 2 1 1 N k T z _T k k k T  X k X T            _    _  



, where

 

1 1 0 ₂ 1 ; 2 1 E     _     

 

2k 1

T _  is Chebyshev function;  _z



, 0



is power stress component, ( )





, 0

T z

  is temperature stress component.

possible ratios of thermophysical and elastic characteristics of the bodies. Taking this into account, the solution is found by the method of reduction from truncated systems. For numerical calculations the systems of 30 linear algebraic equations with 30 unknowns were solved.

The distribution of the dimensionless temperature 1 0 T T

  along the dimensionless coordinate  at 1 1 2 0.1; 2; 2, , 0.5 z z h k k 

       is shown in Fig. 1. Fig. 2 and 3 represent

the distribution of dimensionless normal stress

( ) 2 0 T z TT E     at 1 1 1 1; 0.1; 0.3 T z z T   _{ }       .

Figure 1. Temperature distribution for different values

of contact conductivity, curves: 1 0 1h 0.1 1 0 2h 1; 1 0 3h 5; 1 0 4 h  

Figure 2. Temperature component distribution of the contact normal stresses for different values of the young’s modulus relation 1

/ ;

E E

  curves:

1  0.5; 2  1; 3  2

Figure 3. Temperature component distribution of the contact normal stresses for different values of

the contact conductivity, curves:

Conclusions. While pressing the elastic cylinder into the elastic layer, the nonideal

thermal contact between the cylinder and the layer significantly affects the nature of the distribution of the temperature components of normal contact stresses, in the same way as it is in the corresponding problem [7].

References

1. Grilickij D. V., Kizyma YA. Osesimmetrichnye kontaktnye zadachi teorii uprugosti i termouprugosti. L'vov:Vishcha shkola.-Izd. pri L'vovskom universitete, 1981. 135 p. [Іn Russian].

2. Okrepkyi B. S., Shelestovska M. Tysk tsylindrychnoho kruhovoho shtampa na pruzhnyi pivprostir z urakhuvanniam neidealnoho teplovoho kontaktu. Visnyk TNTU. 2006. No 3, P. 26−33. [Іn Ukrainian]. 3. Okrepkyi B. S. Shelestovska M. Tysk tsylindrychnoho kruhovoho shtampa na pruzhnyi shar z

urakhuvanniam neidealnoho teplovoho kontaktu. Visnyk TNTU. 2011. Vol. 16. No. 2. P. 42−52. [Іn Ukrainian].

4. Okrepkyi B. S., Shelestovska M. Osesymetrychna kontaktna zadacha termopruzhnosti pro tysk pruzhnoho tsylindra na pruzhnyi pivprostir z urakhuvanniam neidealnoho teplovoho kontaktu. Visnyk Ternopilskoho derzhavnoho tekhnichnoho universytetu. 2014. No 2. P. 65−76. [Іn Ukrainian].

5. Kovalenko A. D. Osnovy termouprugosti. Kiev: Nauk. dumka, 1970. 304 p. [Іn Russian]. 6. Uitteker E. T., Vatson D. Kurs sovremennogo analiza. M.: Fizmat, 1963. 343 p. [Іn Russian].

7. Kizyma YA. N. Davlenie uprugogo cilindra na uprugij sloj konechnoj tolshchiny. Izd. AN SSSR, MTT, 1972. No. 3 [Іn Russian].