* B6
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
(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)
(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])
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 .
[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 .
