paramath - * B8

* B8

The Relaxation function problem of an orthotropic cylinder

Co. H. Tran.
Faculty of Mathematics, University of Natural Sciences - VNU-HCM

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 .

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 < 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 :

and the differential equation of equilibrium :

The boundary conditions :

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 .

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 : ");
sigma[theta](at):=(gamma*P*(1+c^(2*gamma))-gamma*Q*2*c^(gamma-1))/(1-c^(2*gamma));
print(sigma[theta](a)=sigma[theta](at));
sigma[theta](bt):=(gamma*P*2*c^(gamma+1)-Q*(1+c^(2*gamma)))/(1-c^(2*gamma));
print(sigma[theta](b)=sigma[theta](bt));
P:=P1;Q:=Q1;
To:=T;E[rt]:=(100*(t/To)^(-0.5)+1)*E[e];E[thetat]:=(100*(t/To)^(-0.1)+1)*E[e];
print(E[r]=E[rt]);
print(E[theta]=E[thetat]);
print(" LAPLACE TRANSFORM OF MODULI : ");
E1[rp]:=p*evalf(laplace(E[rt],t,p),3);
print(E1[r]=E1[rp]);
E1[thetap]:=p*evalf(laplace(E[thetat],t,p),3);
print(E1[theta]=E1[thetap]);
Gamma_form:=sqrt(E1[theta]/E1[r]);
print(" EXPRESSION OF : ",gamma=Gamma_form);
Gamma:=evalf(sqrt(E1[thetap]/E1[rp]),5):
Gamma:=simplify(Gamma);
print(gamma=Gamma);
sigma[theta](a):=(Gamma*P1*(1+c1^(2*Gamma))-Gamma*Q1*2*c1^(Gamma-1))/(1-c1^(2*Gamma));
sigma[theta](b):=(Gamma*P1*2*c1^(Gamma+1)-Q1*(1+c1^(2*Gamma)))/(1-c1^(2*Gamma));
print(sigma[Theta](a)=sigma[theta](a));;print(sigma[Theta](b)=sigma[theta](b));
sigma[theta](at):=(gamma*P*(1+c^(2*gamma))-gamma*Q*2*c^(gamma-1))/(1-c^(2*gamma));
print(" SUBSTITUTE ",c=c1 ,p =1/(2*t),gamma=Gamma);
sigmaat1:=evalf(subs(c=(1/2),p=(1/(2*t)),gamma=Gamma,P=P1,Q=Q1,sigma[theta](a)),3);
sigmaat1:=evalf(simplify(sigmaat1),2);
print(sigma[Theta](a)=sigmaat1);
sigmaat2:=subs(t=10^(s)*To,sigmaat1):
sigmaat2:=evalf(simplify(sigmaat2),2)/P1;
sigmabt1:=evalf(subs(c=(1/2),p=(1/(2*t)),gamma=Gamma,P=P1,Q=Q1,sigma[theta](b)),3);
sigmabt1:=evalf(simplify(sigmabt1),2);
print(sigma[Theta](b)=sigmabt1);
sigmabt2:=subs(t=10^(s)*To,sigmabt1):
sigmabt2:=evalf(simplify(sigmabt2),2);
print(" CHANGE THE PRESENTATION OF TIME INTO LOG(t/To) ");
print(sigma[Theta](a)=sigmaat2);
print(sigma[Theta](b)=sigmabt2);
sigmaatisotropic:=subs(s=0,sigmaat2);
sigmaatisotropic:=evalf(simplify(sigmaatisotropic),2);
print(sigmaa_isotropic=sigmaatisotropic);
sigmabtisotropic:=subs(s=0,sigmabt2):
sigmabtisotropic:=evalf(simplify(sigmabtisotropic),2);
print("   OUTPUT   DATA ");
M:=M1;
d:=d1;
printf("    s=log(t/To)         sigma[Theta](a)(s)/P       nn");
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: