THE SOLUTION OF A VARIABLE BOUNDARY PROBLEM

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THE SOLUTION OF A VARIABLE BOUNDARY PROBLEM

Co. H. Tran - Phong . T . Ngo

Faculty of Mathematics, University of Natural Sciences - VNU-HCM

coth123@math.com & cohtran@math.com ntphong_6@yahoo.com

Copyright 2007

August 16 2007

NOTE:

This worksheet demonstrates Maple's capabilities in researching the numerical and graphical solution of the variable boundary problem of a thick-walled cylinder of material enclosed in a thin metallic shell .

All rights reserved. Copying or transmitting of this material without the permission of the authors is not allowed .

Abstract :

The relations between stress and strain in linear viscoelastic theory are discussed from the viewpoint of application to problems of stress analysis. This consideration includes some important differences from the estimation of linear viscoelastic laws for the representation of material properties , and the integral operators expressing the creep function or relaxation function can be applied . By using of the differential operator for the relation between stress and strain it is usually most convenient to solve some problems which have the variable boundary .

REFERENCES

[1] ю.н. РАБОТНОВ , Пoлзyчесть элементов конструкций , издательство “наука” , MOCKBA 1966

[2] Lee E .H ., Radok J.R.M., Woodward W.B., Stress Analysis for Linear Viscoelastic Materials , Trans. Soc. Rheol., 3, 41-59 (1959)

Disclaimer: While every effort has been made to validate the solutions in this worksheet, the author is not responsible for any errors contained and are not liable for any damages resulting from the use of this material.

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.

The Solution Of A Variable Boundary Problem

Publish at Calameo or read more publications
Author : Co Hong Tran MMPC-VN.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.

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 Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
 Pure mathematics is, in its way, the poetry of logical ideas. 

 Albert Einstein .

THE RELAXATION FUNCTION PROBLEM .

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.

The Relaxation function problem of an orthotropic cylinder .

Co. H. Tran. Faculty of Mathematics, University of Natural Sciences - VNU-HCMcoth123@math.com & cohtran@math.com
Copyright 2007
June 06 2007

NOTE:This worksheet demonstrates Maple's capabilities in researching the numerical and graphical solution of the relaxation function problem of an orthotropic cylinder .
All rights reserved. Copying or transmitting of this material without the permission of the authors is not allowed .

Use Maple 10


Abstract
The worksheet 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 direct method . To compute the interior stress , from the elastic solution we use the correspondence principle and the inverse Laplace transform .

1. Analysis of the composite orthotropic cylinder :


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

2. Direct method : The direct method is an approximate inversion technic based on the direct relation between the time dependence and the transformed solution . If the plot of the viscoelastic solution has small curvature when plotted with variables logt then : (1) where C is Euler's constant .. (1) is exact if , is proportional to logt . (1) can be rewritten : (2) Note that (2) is used when , has small curvature with respect to logt . From the correspondence principle we obtain the viscoelastic solution . (3) (4) (5) (6) The operator moduli : (7) We consider the relaxation test , in which , is a constant at t = 0 (8) . We have , , (9) By the similar way , we find out : (10) Assume that the relaxation moduli have power form : (11) where are constants . By applying the Laplace transfom for (11) , we obtain the operator moduli : (12) with the values of Gamma function : ; (13)

3. Parameters - The Numerical and Graphical Solution : >
restart;cycrstrecom:=proc(T,Gamma1,c1,P1,Q1,M1,d1) global P,Q,sigmaat1,sigmaat2,sigmabt2,sigmabt1,sigmaatisotropic,sigmabtisotropic ; local To,E,E1,M,d,j,Gamma,Gamma_form,gamma;with(inttrans):with(plottools):with(plots):print(" PARAMETERS DEFINITION : ");print( T=To,gamma=Gamma1,c=c1);;;print(" REPRESENTATION OF STRESS : ");;;;;print(" LAPLACE TRANSFORM OF MODULI : ");;;;;Gamma_form:=sqrt(E1[theta]/E1[r]);print(" EXPRESSION OF : ",gamma=Gamma_form;;print(" SUBSTITUTE ",c=c1 ,p =1/(2*t),gamma=Gamma) ;;;;print(" CHANGE THE PRESENTATION OF TIME INTO LOG(t/To) ");;;print(" OUTPUT DATA ");;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;M:=M1;;;d:=d1;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;printf(" s=log(t/To) sigma[Theta](a)(s)/P \n\n");
>
for j from 0 to M do printf("%10.1f %10.4f \n", -d*(10-j), subs(s=-d*(10-j),sigmaat2)) ; end do;
>
;;;;;;;;;;;;;;;;;;;;;;;;;;;;for j from 1 to M do printf("%10.1f %10.4f \n", d*j, subs(s=d*j,sigmaat2)) ; end do;
>
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;print(" NUMERICAL AND GRAPHICAL SOLUTION ");;printf("\n%s"," KET THUC BAI TOAN ONG TRU COMPOSITE DAN NHOT TRUC HUONG BANG PHUONG PHAP TRUC TIEP "); ;plot([sigmaat2,sigmaat2,sigmaatisotropic],s=-10..30,y=0.85..5.2,color=[grey,black,black],style=[line,point,point],thickness=1,symbol=[cross,diamond,cross],linestyle=1,axes=boxed,labels=["logt/To","sigma(a,t)/P"],legend=[`sigma(a,t)/P`,`sigma(a,t)/P`,`Isotropic solution`],title="Numerical solution");;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
>
end:>
cycrstrecom(1, .83, 1/2, 1, 0, 30, 1);


s=log(t/To) sigma[Theta](a)(s)/P -10.0 1.4286 -9.0 1.4286 -8.0 1.4287 -7.0 1.4289 -6.0 1.4294 -5.0 1.4305 -4.0 1.4335 -3.0 1.4409 -2.0 1.4595 -1.0 1.5056 0.0 1.6182 1.0 1.8804 2.0 2.4264 3.0 3.3300 4.0 4.3240 5.0 4.8725 6.0 4.8419 7.0 4.5127 8.0 4.1110 9.0 3.7260 10.0 3.3828 11.0 3.0857 12.0 2.8323 13.0 2.6184 14.0 2.4396 15.0 2.2913 16.0 2.1691 17.0 2.0691 18.0 1.9877 19.0 1.9217 20.0 1.8684 1.0 1.8804 2.0 2.4264 3.0 3.3300 4.0 4.3240 5.0 4.8725 6.0 4.8419 7.0 4.5127 8.0 4.1110 9.0 3.7260 10.0 3.3828 11.0 3.0857 12.0 2.8323 13.0 2.6184 14.0 2.4396 15.0 2.2913 16.0 2.1691 17.0 2.0691 18.0 1.9877 19.0 1.9217 20.0 1.8684 21.0 1.8255 22.0 1.7910 23.0 1.7634 24.0 1.7413 25.0 1.7237 26.0 1.7096 27.0 1.6984 28.0 1.6894 29.0 1.6823 30.0 1.6766

KET THUC BAI TOAN ONG TRU COMPOSITE DAN NHOT TRUC HUONG BANG PHUONG PHAP TRUC TIEP

>

REFERENCES

[1] Ngo Thanh Phong , Nguyen Thoi Trung , Nguyen Dình Hien , Ap dung
phap gan dung bien doi Laplace nguoc de giai bai toan bien dang phang trong
lieu composite dan nhot truc huong , Tap chí phat trien KHCN , tap 7 , so 4 &
in Vietnamese ) , 2002 .

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


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The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.

View details online at the link : http://www.maplesoft.com/applications/app_center_view.aspx?AID=2136

The Relaxation function problem of an orthotropic cylinder .

Publish at Calameo or read more publications
Author : Co Hong Tran MMPC-VN.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License. -------------------------------------------------------------------------------------------
Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
Pure mathematics is, in its way, the poetry of logical ideas. 

Albert Einstein . 

BÀI TOÁN CHÙNG ỨNG SUẤT

BÀI TOÁN  CHÙNG ỨNG SUẤT





This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.

s=log(t/To) sigma[Theta](a)(s)/P

-10.0 1.4286
-9.0 1.4286
-8.0 1.4287
-7.0 1.4289
-6.0 1.4294
-5.0 1.4305
-4.0 1.4335
-3.0 1.4409
-2.0 1.4595
-1.0 1.5056 7.0 4.5127 8.0 4.1110 9.0 3.7260 10.0 3.3828 11.0 3.0857 12.0 2.8323 13.0 2.6184 14.0 2.4396 15.0 2.2913 16.0 2.1691 17.0 2.0691 18.0 1.9877 19.0 1.9217 20.0 1.8684 1.0 1.8804 2.0 2.4264 3.0 3.3300 4.0 4.3240 5.0 4.8725
0.0 1.6182 6.0 4.8419 7.0 4.5127 8.0 4.1110 9.0 3.7260 10.0 3.3828 11.0 3.0857 12.0 2.8323 13.02.6184 14.0 2.4396 15.0 2.2913 16.0 2.1691 17.0 2.0691 18.0 1.9877 19.0 1.9217 20.0 1.8684 21.0 1.8255 22.0 1.7910 23.0 1.7634 24.0 1.7413 25.0 1.7237 26.0 1.7096 27।0 1.6984 28.0 1.6894 29.0 1.6823 30.0 1.6766
1.0 1.8804 2.0 2.4264 3.0 3.3300 4.0 4.3240 5.0 4.8725 6।0 4.8419

KET THUC BAI TOAN ONG TRU COMPOSITE DAN NHOT TRUC HUONG BANG PHUONG PHAP TRUC TIEP
Moi thac mac xin lien he :TRAN-HONG-CO .^^./ E-mail : cohtran@mail.com ^._.^ phone : ( 0 8 ) 54 2 5 0 8 7 4

The Relaxation function problem of an orthotropic cylinder .

Publish at Calameo or read more publications
Author : Co Hong Tran MMPC-VN.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.


-------------------------------------------------------------------------------------------
 Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
Pure mathematics is, in its way, the poetry of logical ideas. 

 Albert Einstein .

NHẬT KÝ KHOA HỌC

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Sunday, May 20, 2007

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(comments 1-4)paramath: tac gia Tran Hong Co ða ðang toan bo luan van tren http://www.freewebs.com/paramath/tailieu gom 2 phan :
*p1 /. Luan van ða bao ve thanh cong tai Hoi ðong
*p2/. Luan van ( lan 1) hoan thanh thang 11 nam 2005 , sau ðo ðuoc bo sung vao p1/. Muc ðich cua cong viec nay la chong sao chep , tham khao neu khong co su cho phep cua chinh tac gia . June 10 , 2007 HCMC VIETNAM
June 10, 2007, 07:22 PM (UTC 8)paramath: ngày 10 tháng 6 , 2007 Frac-Mec ( Morozov)
June 10, 2007, 01:16 AM (UTC 8)

Solving the Viscous Composite Cylinder Problem by Sokolov's Method

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Solving the Viscous Composite Cylinder Problem by Sokolov's Method
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Author:
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Application Type:
White Paper
Publish date:
**NEW** July, 2006
Related Products:
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View as PDF (.pdf, 1,199.5kb)
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


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Solving The Viscous Composite Cylinder Problem By Sokolov’s Method

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Author : Co Hong Tran MMPC-VN.

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KHÔNG tri KHÔNG lý tâm THANH TỊNH

ĐẠO nơi ẨN NGHĨA ấy TỰ NHIÊN .


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Ý quy VÔ Ý TÂM THIỀN ĐỊNH

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Bản gốc :  09/01/2006




This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.

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Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
 Pure mathematics is, in its way, the poetry of logical ideas. 

Albert Einstein .