Stability of a thermoelastic rod

Toby Driscoll, 8th November 2011

(Chebfun example ode-eig/ThermoelasticRod.m)

Suppose a thermoelastic rod is fixed to a wall at one end and may expand to make contact with a wall at the other end. J. R. Barber [1] proposed a boundary condition that models a physically realistic transition between thermal insulation, when far from contact, and perfect thermal contact.

Linear stability analysis suggests a change from stable to unstable behavior as the temperature difference between the walls increases. The eigenvalue problem governing the stability of the perturbation phi(x) is nondimensionally

   phi''(x) = lambda*phi(x),   0 < x < 1
                                            / 1
                                           |
   phi(0) = 0,  phi'(1) + phi(1) = 4 delta |    phi(x) dx
                                           |
                                         0/

where delta is a function of the thermal gradient. The transition from stable to unstable happens at delta=1. The presence of the integral of phi in the boundary condition makes the problem unusual from a classical standpoint, but from the Chebfun point of view it's just another linear boundary condition.

LW = 'linewidth';
format long,

First, we solve the eigenvalue problem in a stable case.

N = chebop( @(u) diff(u,2), [0 1] );    % operator on 0<x<1
N.lbc = 0;     % fixed end
delta = 0.96;  % stable choice
N.bc = @(x,u) feval(diff(u),1) + u(1) - 4*delta*sum(u);  % Barber condition
[Vs,Ls] = eigs(N,4,0);  % eigenmodes closest to zero

The eigenvalues are all negative, indicating stability:

diag(Ls)

ans =
   1.0e+02 *
  -0.001601435706615
  -0.251462532662429
  -0.626486098335208
  -1.234915472724216

Here is what happens in a slightly unstable case:

delta = 1.02;  % unstable choice
N.bc = @(x,u) feval(diff(u),1) + u(1) - 4*delta*sum(u);  % Barber condition
[Vu,Lu] = eigs(N,4,0);
diag(Lu)

ans =
   1.0e+02 *
   0.000799646107482
  -0.252000055361213
  -0.625884455972660
  -1.235278901225324

Here we see the perturbation which is least stable in the first case, or unstable in the second case.

subplot(1,2,1)
plot(Vs(:,1),LW,1.6)
title(sprintf('Stable, \\lambda = %.3f',Ls(1,1)))
subplot(1,2,2)
plot(Vu(:,1),LW,1.6)
title(sprintf('Unstable, \\lambda = %.3f',Lu(1,1)))

The solutions above look linear, but they do have significant Chebyshev coefficients out to degree 8.

Without knowing the transition value delta=1 in advance, we could locate it through a simple Chebfun rootfinding search. First, we parameterize the boundary conditions and the maximum real eigenvalue.

BC = @(delta) @(x,u) [u(0), feval(diff(u),1) + u(1) - 4*delta*sum(u)];
maxlam = @(delta) eigs( chebop([0,1],@(u)diff(u,2),BC(delta)), 1, 0 );

Then, we construct a chebfun for the maximum lambda. A polynomial of degree 10 captures the behavior of the maximum eigenvalue to about 11 digits.

stability = chebfun(maxlam,[0.5,2],'eps',1e-11,'vectorize')

stability = 
   chebfun column (1 smooth piece)
       interval       length   endpoint values   
[     0.5,       2]       11       -2      3.9   
vertical scale = 3.9

Finally, the transition in stability occurs when the eigenvalue passes through zero.

dstar = find(stability==0)
clf, plot(stability,LW,1.6), hold on, plot(dstar,0,'r*')
xlabel('\delta'), ylabel('max \lambda'), grid on

dstar =
   0.999999999997509

References:

[1] J. R. Barber, "Contact problems involving a cooled punch," J. Elast. 8 (1978), 409-423.

[2] J. A. Pelesko, "Nonlinear stability, thermoelastic contact, and the Barber condition", J. Appl. Mech. 68 (2001), 28-33.