1 SOLVING THE VISCOUS COMPOSITE CYLINDER PROBLEM BY SOKOLOV S METHOD NGO By CO. H. TRAN, PHONG. T. Faculty of Mathematics & Infomatics, Univesity of Natual Sciences VNU-HCM Abstact : The pape pesents some thoughts about the plane stain poblem of the viscous othotopic composite mateials cylinde unde intenal and extenal pessue with espect to using the aveage appoximating method. To compute the inteio stess, fom the elastic solution we use the Voltea s pinciple and Sokolov s method in the coesponding integal equation to find the viscous solution. I. The axial symmetic plane stain poblem of cylinde : We examine an othotopic viscoelastic composite mateial cylinde which has the hoizontal section within limit of cicles : = a, = b ( a < b.
Choosing the cylindical coodinates,, z ( the axial z is along with the cylinde. The components of stess and defomation ae functions of, t espectively. The two components of defomation-tenso : (1.1 u(, ε (, = ; ε (, = u(, and the diffeential equation of equilibium is : (1. σ (, + σ (, - σ (, = 0 when t = 0, bounday conditions : (1.3 σ ( a,0 = P ; σ ( b,0 = Q II. The Voltea Integal equation of the second kind : The displacement -diffeential equation of the cylinde in the case of viscoelastic plane-defomation : (.1 The geneal solution is : (. u (, + = E u (, t u (, 0 E E E u (, t := C 1 + C E E C 1 C, ae the abitay constants. The elastic constants in (.1 will be substituted by opeatos espectively, fom the Voltea s pinciple we have : ([1] (.3 u(, u(, + Eˆ Eˆ u(, = 0 The equation (.3 is ewitten as :
3 (.4 u(, u(, + a 1( + a 1 a1(, =, a ( Eˆ Eˆ u(, = 0 m( (, = = Assume that fom : u ( a, = C = C 0, u '( a, 1 (in [] we obtain the solution of (.4 (.5 x φ( x, + K[( x, y; t] φ( y, dy = f ( x, a The kenel expession of : (.6 ( x y K[( x, y; t] = a1( x, + a( x, 1! [ ] And the fomula of f(x, is : f ( x, = C0a1( x, C1x + C0 a( x, (.7 In this case : (.8 1 m( x y K[( x, y; t] = x x (.9 m := ( 100. + t ( / 100. + t 110 t ( 5 / Fom the expeimental test, if we have : (.10 σ = 0, ε (,0 = Const Then : (.11 σ ( =. E ( ε
4 Fom the esults of (.5 we obtain the analytical solution of u ( x,, σ ( a, and σ ( b,. III. The appoximate solution of the Voltea Integal equation of the second kind : 3.1 The Sokolov s aveage appoximate method : The convegence speed of calculation pocess can be inceased by Sokolov s method. The basic contents of this method is descibed as following : Fom the integal equation : (3.1 b u( = v( + λ K( t, τ u( τ dτ a The n-ode expession of u( is : u ( = v( + λ K( t, τ { u ( τ α } dτ (3. b n n + a This elation has connection with the adjustment quantity : n (3.3 α n b 1 = b a Δ a n ( dt The diffeence between two esults : (3.4 Δ ( t = u ( t u 1( t n n n The ecuence elation : (3.5 b b λ αn = K( t, τ n 1 n 1 R( λ a a { Δ ( τ α } dτdt Note that the convegence -condition of Sokolov s method is L λ = λ( I λkπ 1 K ( I Π < 1 Hee Π is the poject opeato fom the Banach s space B into its subspace Bo ( u B ([]
5 3. The flow chat of Sokolov s method : Begin. Input the paametes. The initial solution u ( x 0 Constuction the integal equation b u( = v( + λ K( t, τ u( τ dτ Find α 1, define T 1( x = u1( x Compute D = bb R( λ = b a λ K( t, τ dτdt define u 0( x, T ( x 0 Y N i N esult i = i + 1 Compute Ti 1( x, Ti ( x and α i + 1, α i + estimate eo saiso[i] End
6 3.3. Poblem definition paametes and the analytical solution : In the following, modules E (, and E (, ae given of the exponential expessions : a = 1, b =, C 0 = 0, C1 = 1 ; E e = 0.5, T = 0 1 ; 1. Paametes Infomation : :>m:=(1/x^*(100.+(t/to^(1/10*(t/to^(/5/(100.+(t/to^(1/;to:=1; We should choose n := 4, n := 5, n := 6. Activate the pocedue : > xapxi ( m/x, -1, 1,, 4, K ; m ( x y 1 K := + x x ; To := 1 "--------------------------------KETQUA-------------------------------- " E (, := 100 + 1 t 0.1 Ee σ ( x, ln( x x ( 100. + t ( / := 100. + t 110 t ( 5 / 0.1013313763 10 30 ( 100. + t ( / + 1 ( 100. + t 110 1 1. ( 100. + t ( / 100. + t t ( 4/ 5 110 t ( 5 / x 100 + 1 Ee/ x t 0.1 0.00500000000 Comment : In this pocedue note that we should choose the numbe n of ecuence fom 4 up to 6, so we get the esult which is coincide to the solution of Schapey in [4]. In the compaison with the esult in [3] the analytical solution of σ ( a, and σ ( b, obtained by Sokolov s method has a bette smoothness. Figue 1 and descibe
7 the convegence of ε (, and ε (,. Especially, if n > 6 and using Maple vesion 9.5 it is a waste of time to teat the poblem in detail, moeove the accuacy of solution will be influenced by accumulation of eo in pogamme. Using Maple vesion 6.0 the convegence of method can be found coectly when n = 5. gaph of stain ε (, Fig 1.
8 convegence of stain ε (, t Fig. Figue 3. and 4. show the mannes of the gaphs of stess coincide to the wok of R.A.Schapey. To obtain these esult, the most impotant wok stage is the tansfomation diffeential equation to the coespondent integal equation and setting up the suitable bounday conditions.
9 gaph of stess σ ( a,, a = 1 Fig 3. Fig 4. gaph of stess σ ( b, t, b = 1 In Figue 5 we notice the schematic epesentation of the displacement at x = 1 and x = ; the mannes of the gaphs of stain ε (, and convegence pocess of ε (, accoding to the ode { n -, n-1,n } can be shown in figue 6.
10 Fig 6 gaph of ε (,. Fig 5. gaph of displacement u(,
11 Fig 8. gaph of σ (, t. Fig 7. gaph of ε (, t.
1 IV. Conclusion : The plane stain poblem of the viscous othotopic composite mateials cylinde unde intenal and extenal pessue can be solved by othe appoximate methods : diect, collocation, quasi-elastic, Laplace tansfom invesion ([3]. Almost the algoithm egulaly depends on the displacement diffeential equation and the establishment of pocedue ae also caied out fom it. The Aveage Appoximating Method on Functional Adjustment Quantity ( Sokolov's method and the Voltea s pinciple coesponding to the integal equation may be used to find the viscous solution. Moeove this application makes inceasing fo the convegence speed of the solution u n (. Fom the fist appoximation of the solution u, we find the adjustment quantity fo the next and so on. Note that we should choose the numbe n of ecuence and the expessions of a x,, a ( x, 1( t appopiately to get the analytical esult which is satisfied the given bounday conditions of the poblem. REFERENCE [1] Yu. N. Rabotnov, Elements of heeditay solid mechanics, MIR Publishes,Moscow 1980. [] Phan Van Hap, Cac phuong phap giai gan dung, Nxb Đai hoc va tung hoc chuyen nghiep, Ha noi, 1981. ( in Vietnamese [3] Ngo Thanh Phong, Nguyen Thoi Tung, Nguyen Dình Hien, Ap dung phuong phap gan dung bien doi Laplace nguoc de giai bai toan bien dang phang tong vat lieu composite dan nhot tuc huong, Tap chí phat tien KHCN, tap 7, so 4 & 5 / 004. ( in Vietnamese [4] R.A. Schapey, Stess Analysis of Viscoelastic Composite Mateials, Volume, Edited by G.P.Sendeckyj,Academic Pess, Newyok London, 1971.