Copyright © 2019 N.N. Dikhtyaruk, O.V. Kurinienko, E.A. Poplavskaya 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. 

 

 

 
Problems of Tribologyy, 92 (2) (2019)6-12 

 

Problems of Tribology 
Website: http://tribology.khnu.km.ua/index.php/ProbTrib 

E-mail: tribosenator@gmail.com 
 

DOI: 10.31891/2079-1372-2019-92-2-6-12 
 

Contact problem of elastic strips with initial stresses with periodically 
placed resilient protective strap 

 
N.N. Dikhtyaruk1, O.V. Kurinienko1*, E.A. Poplavskaya1 

 
1Khmelnitsky National University,  Ukraine 

*E-mail: o.v.kurinyenko@gmail.com 

 
 
Abstract 
 
The present article considers the formulation and solution of the problem on contact interaction of a pre-stressed band 

with regularly placed  elastic cover plates. The study has been carried out within the framework of the linearized elasticity 
theory in general form for the theory of large (finite) initial deformations and two variants of the theory of small initial de-
formations with an arbitrary structure of the elastic potential [1]. The solution of the problem comes down to a singular inte-
gro-differential equation for unknown contact stresses with a kernel expressed in the form of a sum of the kernel and some 
regular kernel under certain boundary conditions. The solution sought is given in the form of a series of Jacobi polynomials. 
A quasiregular infinite system of linear equations is obtained in order to determine the unknown coefficients of the series [2]. 

 
Key words: linearized theory of elasticity, initial (residual) stress, contact problems, Fourier integral transforma-

tions. 
 
Introduction 
 
Contact interaction of initially stressed bodies is one of topical issues in the field of mechanics of deform-

able solids. The solution to such problems is related to a wide range of questions that arise in engineering, con-
struction and other industries. Strict conditions have been formulated recently: high (guaranteed) durability of 
structures and machine parts, reliability on the one hand and low material consumption of the latter made with 
the use of new structural materials. This problem is particularly relevant nowadays and it is of great interest to 
researchers who solve fundamental and applied problems in the sphere of contact interaction of bodies with ini-
tial (residual) stresses. Problems arising with the transfer of stress from the elastic band to regularly placed elas-
tic cover plates – the classical theory of elasticity – are becoming relevant again with the appearance of initial 
(residual) stress in the band. 

 
Literature review 
 
In spite of the fact that researches of influence of initial tension began to be conducted actively in our 

country and abroad only at the end of the 20th century, it is possible to list many names, researches and publica-
tions of this subject [1, 2]. In stringent formulation of contact problems for elastic bodies with initial stresses, 
there is a need to involve the apparatus of nonlinear elasticity theory, which greatly complicates the construction 
of analytical solutions. But in the case of large (finite) stresses (deformations) it is possible to be limited to con-
sideration the linearized theory of elasticity [1]. There are two directions of research of contact problems within 
the framework of the linearized theory of elasticity. The first is connected with studies of contact interaction of 
bodies with a concrete form of elastic potential [3]. In the second one - the problem is put in the general form for 
compressible (noncomputable) bodies with the potential of an arbitrary structure based on the linearized theory 
of elasticity [1, 2, 6 - 11]. 

In this article, using the relations of the linearized theory of elasticity [2], the solving of the contact prob-
lem for the contact interaction of an infinite and finite stringer with a prestressed strip is presented. The study 
was carried in a general form for compressible and incompressible bodies for the theory of large (finite) initial 
strains and two variants of the theory of small initial strains at an arbitrary structure of elastic potential [6, 9]. 

 
Aim and Problem Statement 



Problems of Tribology 7 
 

 
The aim of this work is to study the effect of initial (residual) stresses on the law of distributing stresses 

and displacement under regularly placed elastic cover plates on the contact line with a pre-stressed band. Let us 
consider an elastic initially stressed infinite band whose thickness is t and which is supported by elastic cover 
plates h thick regularly placed on its surface plane 1 0y   at finite intervals 
 2 , 2 , ( , 0 1, 2...)a lk a lk l a k        with the period of l2  while surface plane 2 0y   is 
tightly fixed (Fig. 1). 

 

 
 

Fig. 1. Band reinforcement 
 
It is necessary to determine the influence of the initial (residual) stresses in the elastic band with initial 

(residual) stresses on the law of distribution of normal and tangential contact stresses in the contact area of elas-
tic cover plates with an elastic band, when the latter is under effect of vertical and horizontal loads with the in-
tensity  0 1p y   and  0 1q y   respectively. 

In view of the frequency of the considered problem the influence of initial (residual) stresses on kL  at  

2 0y   under elastic thin cover plates will be equal [5], i.e. contact stresses arising on kL  – intervals from the 
side of the elastic cover plates will be periodic functions with the period 2l . 

       
21 0 1 1 1

2 2xy yy y y l y l         ; 

     22 1 22 1 22 12 2Q y Q y l Q y l    ; 
22Q      2 1 1 2 1 1 2 1 12 2Q y Q y l Q y l    . 

(1) 

So, we can confine ourselves to examining one of them, for example, the one for which 0k  . We will 
also consider that there are some valid assumptions for the elastic band with initial (residual) stresses and elastic 
cover plates, namely: the elastic band with initial (residual) stresses is in the condition of planar deformation, and 
for the elastic strap loaded both by vertical and horizontal forces the common model of beam bending in combi-
nation with the model of a uni-axial stress and strain state of the elastic strap is true. This means that the elastic 
strap bends like a regular beam in the vertical direction 2Oy , it contracts or stretches in the horizontal direction 
of the 1Oy  axis like a normal shank in a uni-axial stress and strain state with finite rigidity. From this perspec-
tive the discussed problem will have the following equation [4]: 

         1 1 1 1 2 2 1 1; ; 2 2 ;u y u y v y v y y a kl y a kl         
 0, 1, 2,...k    . 

(2) 

Here ,u v  – are respectively horizontal and vertical movements of the elastic strap in the designations 
of the classic theory of elasticity;  1 1u y ,  2 2u y  – are respectively horizontal and vertical movements of the 
limiting points of the elastic band with initial (residual) stress;  1p y ,  1q y  – intensity of normal and hori-
zontal contact stress. 

 
Presentation of the main material 
 
1. Deriving singular and integro-differential equations 
When deriving the basic system of differential equations that enable us to solve the given problem we 

should note that based on the principle of superposition the function of influence for vertical and horizontal 
movement of the limiting points of the elastic band plane 2 0y    and taking into account the regularity of the 
latter [5] can be written as: 



8 Problems of Tribology 
 

           
2

22 1 11 1 11 1
2

a a l

a a l

h y p d h y p d h y p d
 

   

                

           
2 4 4

12 1 11 1 11 1
2 4 4

...
a l a l a l

a l a l a l

h y p d h y p d h y p d
  

     

                
 (3) 

For receiving the value of the integral (3) the regularity of normal and tangential contact stresses  p    
and  q   is used. 

           
2

22 1 11 1 11 1
2

a a l

a a l

h y p d h y p d h y p d
 

   

                
 

       

   

2 4

1 2 1 1 1 1
2 4

4

1 1 1
4

...;

a l a l

a l a l
a l

a l

h y p d h y p d

            h y p d

 

   



 

          

     

 


 

     2 2 . .. ;p p k l p k l         
     2 2 ...q q kl q kl  .         

(4) 

Substituting variables  2 , 1, 1, 2,...kl k       taking into account the value of displace-
ments under the action of the single normal and tangential force [3] after a series of transformations leads to the 
following form of the integral (5): 

         11 1 11 1
1

1
cos 1 2 cos 2 .

2

a

ka

h y p d H y s kls ds p d
  

  

   
              

           (5) 
This integral can be some what simplified if you use the value [4]: 

 1
1

2
1 2 cos 2 ;

k k
ky y k

l

 

 


     

 

   1 1
1

.y y   


 

(6) 

Here  1y  – Dirac unit function [5]. 
After a series of transformations we obtain: 

         011 1 11 11 1
1

1
0 cos .

2

a

ka

P k k
h y p d H H y p d

l l l l

 

 

    
              

   (7) 
Applying the same procedure to other integrals we have: 

         
1

0
12 1 12 12 1

11

1
0 sin .

2 k
Q k k

h y q d H H y q d
l l l l

 

 

    
               

   

         021 1 21 21 1
1

1
0 sin .

2

a

ka

P k k
h y p d H H y p d

l l l l

 

 

    
                

   

         022 1 22 22 1
1

1
0 cos .

2

a

ka

Q k k
h y q d H H y q d

l l l l

 

 

    
               

   

(8) 

Let us write down asymptotic expressions  ijH i , 1, 2j  , at 0   і    . 
     11 120 0lim ; lim ;H A H O         

     122 1 110lim ; lim ;H A H B O


 
        (9) 

lim        1 122 12 1lim ; lim .H B O H B O 
 

           



Problems of Tribology 9 
 

Here 1 2, , ,A A B B  are values characterizing the initial (residual) stress and strain state in the elastic 
band and are determined for compressible and noncompressible bodies in the case of a specific structure of elas-
tic potentials. 

If we consider the value of the sum given below [4]: 

1

cos
ln 2 sin ,

2k
kx x

k





   (10) 
then integrals (5) and (8) can be expressed in the following form 

      111 1 0 ln 2 sin2 2
a

a

yA B
h y p d P p dt

l l



 

  
          

 

   11 1
1

1
cos ;

a

ka

k Bl k
H y p d

l l k l





     
          

  

     1 122 1 0 ln 2 sin2 2
a

a

A yB
h y q d Q q d

l l



 

 
            

 

   22 1
1

1
cos .

a

ka

k Bl k
H y q d

l l k l





     
          

  

(11) 

 11 1
1

sin
sin    2 2 ,

2k

yky
qy y

k






       (12) 

integrals (11) can be written as the formulas below: 

       112 1 12 1
1

1
sin sin

a

ka

B lk k
h y q d H y q d

l l k l

 

 

       
                   

   

   1 1 01 0 1 1sin ;    ,2 2
a

a

B B
q q y d Q y M y a

l l
          

       121 1 21 1
1

1
sin

a

ka

B lk k
h y p d H y p d

l l k l

 

 

       
                  

 
 

     1 11 0 1 0 1sin ;    .2 2
a

a

B B
p q y d P y M y a

l l
       

   (13) 

where 

   0 0, .
a a

a a

M x q x dx M x p x d
 

     (14) 
Taking into account (8), (11), (13) we find the displacement of the limiting points in the contact area 

 aay ,1   on the surface plane 2 0y   of the elastic band with initial (residual) stress. 

      11 1 1ln 2 sin 2 2
a

a

BB
u y q y d

l

 
        

 

         012 1 12 1 1 0 012
a a

a a

L y t p d L y t q d B Q y M AP
l 
            , 

1 .y a  

         12 1 1 1ln 2 sin 2 2
a a

a a

BB
u y q y d p siqn y d

l 

 
           

 

         22 1 12 1 1 0 0 1 0
1

.
2

a a

a a

L y q d N y p d B P y M A Q
l 

           
 

(15)



10 Problems of Tribology 
 

 
We introduce a designation here: 

   11 1 11 1
1

1
cos ;

k

k Bl k
L y H y

l l k l





    
        

  

   112 1 12 1
1

1
sin ;

k

B lk k
L y H y

l l k l





    
       

  

   22 1 22 1
1

1
cos .

k

k Bl k
L y H y

l l k l





    
        

  

(16) 

In order to obtain initial functional equations, from which unknown contact stresses are determined 
(taking into account (15)) we have to make the following system of singular integro-differential equations. 

     
4

1 1
14

1

ln 2 sin
2 2

a a

a a

y Bd B
D p d q siqn y d

dy l 

  
         

 

             011 1 12 1 1 0 0 1 0 11 .2
a a

a a

L y p d L y q d B Q y M AP p y p y
l 

             

     1 1 1
1

ln 2 sin
2 2

a a

a a

y Bd B
q d d p siqn y d

dy l 

    
            

   

         22 1 12 1 0 0 0 1 0
1
2

a a

a a

L y q d L y p d B P y M A P
l 

            
 

    0
1

1
,

a

a

q q d
E h 

       1 .y a                                         (17) 
This system is solved with such boundary conditions: 
where 1y a   

  0 ;
a

a

p d M


        0 ;
a

a

q d Q


       0 .
a

a

p d P


                                      (18) 
2. Bringing a singular integro-differential equation to an infinite system of linear algebraic equations 
The system of singular integral equations can be simplified by introducing new functions. 
 

    ( );X p i q        

   
0 0

1 1
; ;

l l
p p q q

P Q
    

          
   (19) 

   0 0 0
0 0

1 1
; .

l l
p p q q

P Q
    

          
    

After complex cumbersome transformations and replacement of variables  

1 ,yl l
 

      (20) 

we obtain an infinite system of linear algebraic equations: 
           1 2 1 2 3 4, , 0 0

1
, 1, 2,...m n m n n m n n m m m m

n
l X P X P X P X P X P R P m





                  (21) 

For using numerical methods to the system (21) it is necessary to determine the coefficient 0X  and 
constant R , which are in the right part. 

1
0

0
0

cos
2 1 .

2

c h Q
X i

l P


   

  
 

 (22) 

Constant R   can be found from a system of algebraic equations (21) which is completely quasiregular.  
 



Problems of Tribology 11 
 

Conclusions 
 
For obtaining a numerical result we have studied a case where all regularly placed cover plates that rein-

force the elastic band with initial (residual) stresses are loaded with tangential force  0Q Q  . 
Fig. 2, 3, 4 illustrate the influence of initial (residual) stresses in the elastic band on the law of distribut-

ing contact stresses under regularly placed cover plates from the action of tangential force  0 1 0q y Q   for 
dimentionless values (19). The graphs show the place of curves for the parameter value 

1 7,0;8,0;9,0;1;1,1;2,1;3,1  which characterizes the initial (residual) stress and deformed state. Value 

1  = 1 corresponds to an elastic band without initial (residual) stress and on the graphs it is depicted as a dotted 
line, i.e., the value of contact stresses for a similar problem solved within the classical theory of elasticity. 

 
 

Fig. 2. Distribution of loads under the finite stringers  
(Harmonic potential) 

Fig. 3. Distribution of loads under the finite stringers  
(Potential of Bartenev-Khazanovich) 

 

 
 

Fig. 4 .  Distribution of loads under the finite stringers (Potential of Treloar) 
 
It follows from the graphs that the regularity of the influence of initial stresses in the elastic band on the 

distribution of contact tangential stresses essentially depends on the structure of the elastic potential in the latter. 
Initial stresses have a more significant quantitative effect in highly elastic materials compared with rigid ones, 
the qualitative effects subside. Moreover, at compression  1 1   contact stresses are reduced, and in the case 
of stretching  1 1  they increase. This result can be efficiently used to regulate contact forces  when calculat-
ing durability for designing constructions. 

 
 
 



12 Problems of Tribology 
 

References 
 
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Діхтярук М.М., Курінєнко О.В., Поплавська О.А. Контактна задача про взаємодію смуги з 

початковими напруженнями з періодично розміщеними пружними накладками 
 
В даній статті розглядається постановка і розв’язок задачі про контактну взаємодію попередньо 

напруженої смуги з періодично розміщеними пружними накладками. Всі дослідження проводилося в 
рамках лінеаризованої теорії пружності у загальному вигляді для теорії великих (скінчених) початкових 
деформацій та двох варіантів теорії малих початкових деформацій при довільній структурі пружного 
потенціалу [1]. Розв’язок задачі зводиться до розв’язування сингулярного інтегро-диференційного 
рівняння для невідомих контактних напружень з ядром, записаним у вигляді суми ядра і деякого 
регулярного ядра, при певних граничних умовах. Шуканий розв’язок подано у вигляді ряду за 
поліномами Якобі і для визначення невідомих коефіцієнтів ряду отримана цілком квазірегулярна 
нескінчена система лінійних рівнянь [2]. 
 

Ключові слова: лінеаризована теорія пружності, початкові (залишкові) напруження, контактні 
задачі, інтегральні перетворення Фур’є.