paramath

* B6


SOLVING THE VISCOUS COMPOSITE CYLINDER PROBLEM BY SOKOLOV’S METHOD  
                                             

                                                 By             CO . H . TRAN , PHONG . T. NGO  

 
 

                                 Faculty of Mathematics & Informatics , University of Natural Sciences – VNU-HCM 

 
 
 
 

Abstract : The paper presents some thoughts about the plane strain problem of the viscous orthotropic composite materials cylinder under internal and external pressure with 

 

respect to using the average approximating method . To compute the interior stress , from the elastic solution we use the Volterra’s principle and Sokolov’s method in the 

 

corresponding integral equation to find the viscous solution .   

 
 
 
 

I. The axial symmetric plane strain problem of cylinder : 

 

We examine an orthotropic viscoelastic composite material cylinder which has the horizontal section within limit of 2 circles : r = a , r = b ( a < b ) .          

Choosing the cylindrical coordinates r ,       , z ( the axial z is along with the cylinder ) . The components of stress and deformation         

 
are functions of r , t   respectively .   The two components of deformation-tensor :
 

                                                                              (1.1)

 

and the differential equation of equilibrium is :

 

                                                                                          (1.2)

 

when t = 0 , boundary conditions :                                (1.3)

 
 

II . The Volterra Integral equation of the second kind :

 

The displacement -differential equation of the cylinder in the case of viscoelastic plane-deformation :  

 
 

                                                               (2.1)

The general solution is :                                           (2.2)         

 

   ,     are the arbitrary constants .    

 
 

The elastic constants in (2.1) will be substituted by operators respectively , from the Volterra’s principle we have : ([1])  

 
 

                                                                                  (2.3) 

 
The equation (2.3) is rewritten as : 
 
 

                                                  (2.4)

                                                                                                                              

 
 

Assume that        (in [2] ) we obtain the solution of (2.4) from : 

 

                                                                                    (2.5)

 
 

The kernel expression of :                                        (2.6)

 
 

 And the formula of f(x,t) is :                              (2.7)

 
 

In this case :                                                                                  (2.8)

 
 

                                                                                                         (2.9)

 
 
 

From the experimental test , if we have     :                              (2.10)

 
 

Then   :                                                                                                             (2.11)

 
 

From the results of (2.5) we obtain the analytical solution of      ,      and   .            

 
        

III. The approximate solution of the Volterra Integral equation of the second kind

 

               3.1 The Sokolov’s average approximate method : 

 
 
 

The convergence speed of calculation process can be increased by Sokolov’s method . The basic contents of this method is described as following :

 

From the integral equation :                                                (3.1)    

                      
 

The n-order expression of  u(t)  is :                       (3.2)      

 

This relation has connection with the adjustment quantity  : 

 

                                                                                                                                       (3.3)  

          

 The difference between two results       :                                         (3.4)     

 

The recurrence relation     :                                  (3.5)  

 

Note that the convergence -condition of Sokolov’s method is                 

                                      

 Here     is the project operator from the Banach ‘s space   B into its subspace Bo (   u ∈ B )   ([2])

 
 
  
3.2 The flow chart of Sokolov’s method :
 
 
 
 
 

    Find  ,   define 

            T  

          Compute D =

 

            3.3 . Problem definition – parameters and the analytical solution :

 
 

In the following , modules    and    are given of the exponential expressions :

a = 1 , b = 2 ,       ;      ; 

 

1. Parameters Information :  :>m:=(1/x^2)*(100.+(t/To)^(1/10))*(t/To)^(2/5)/(100.+(t/To)^(1/2));To:=1;

We should choose   ,   , 

 

2. Activate the procedure :   > xapxi ( m/x, -1 , 1 , 2 , 4 , K   ) ;

 

   ;  

 

 

 

 

 

 

 Comment :  In this procedure note that we should choose the number n   of recurrence   from 4 up to 6 , so we get the result which is coincide to the solution of Schapery in 

 

[4] . In the comparison with the result in [3] the analytical solution of   and  obtained by Sokolov’s method has a better smoothness . Figure 1 and 2 describe 

 

the convergence of  and  it is a waste of time to treat the problem in detail ,   moreover the accuracy of  

 

solution will be influenced by accumulation of error in programme .

 

Using Maple version 6.0  the convergence of method can be found correctly when n = 5 .

 

                                                                                                                         
 

                                                                                                                         Fig 1 . graph of strain   

 
 

 

                                                                                                                  Fig 2 . convergence of strain     

 
 

Figure 3. and   4.   show the manners of the graphs of stress coincide to the work of R.A.Schapery . To obtain these result   , the most important work stage is the 

 

transformation differential equation to the correspondent integral equation and setting up the suitable boundary conditions.  

                             

 

                                                                                                                     Fig 3 . graph of stress    ,   a = 1

 
 

 

                                                                                                                         Fig 4 . graph of stress    ,   b = 1

In Figure 5 we notice the schematic representation of the displacement at  x = 1 and x = 2 ; the manners of the graphs of strain    and convergence process of  

 

 according to the order   { n -2 , n-1 ,n }   can be shown in figure 6  .

                                                                                 

                                                          Fig 5 . graph of displacement                                   Fig 6 graph of  .

 

                              

                      

                                                    

                                                                                 Fig 7 . graph of      .                            Fig 8 . graph of     .

 
                                 
 
 

IV . Conclusion :  

 

The plane strain problem of the viscous orthotropic composite materials cylinder under internal and external pressure can be solved by other approximate methods : direct , collocation , quasi-elastic , Laplace transform inversion 

 

… ([3]) .   Almost the algorithm regularly depends on the displacement differential equation and the establishment of procedure are also carried out from it . The Average Approximating Method on Functional Adjustment 

 

Quantity ( Sokolov's method ) and the Volterra’s principle corresponding to the integral equation may be used to find the viscous solution . Moreover   this application makes increasing for the convergence speed of the 

 

solution   . From the first approximation of the solution u , we find the adjustment quantity for the next and so on . Note that we should choose the number n   of recurrence and the expressions of   

 

appropriately to get the analytical result which is satisfied the given boundary conditions of the problem .

 
  
 
                                            REFERENCE 
 
 

[1] Yu. N. Rabotnov , Elements of hereditary solid mechanics , MIR Publishers ,Moscow 1980 .

 

[2] Phan Van Hap , Cac phuong phap giai gan dung , Nxb Đai hoc va trung hoc chuyen nghiep , Ha noi , 1981 . ( in Vietnamese )

 

[3] Ngo Thanh Phong , Nguyen Thoi Trung , Nguyen Dình Hien , Ap dung phuong phap gan dung bien doi Laplace nguoc de giai bai toan bien dang phang trong vat lieu 

 

composite dan nhot truc huong , Tap chí phat trien KHCN , tap 7 , so 4 & 5 / 2004 . ( in Vietnamese ) 

 

[4] R.A. Schapery , Stress Analysis of Viscoelastic Composite Materials , Volume 2 , Edited by G.P.Sendeckyj ,Academic Press , Newyork –London , 1971 . 

 
 
 
 

B6.COHONGTRAN-VISCOCYSOKOLOV
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