Thermal Runaway

Spontaneous Thermal Runaway as an Ultimate Failure Mechanism of Materials

S. Braeck and Y.Y. Podladchikov

PRL 98, 095504 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e095504


Very interesting article! I find that the PRL editor's recommendations almost always prove to be interesting reads.

The authors investigated a feedback process that can occur in viscoelastic systems (which, I just learned, means systems that show features of both viscous and elastic materials). Two observations allowed them to obtain simplified equations of the strain profile:
⁃ Conservation of momentum implies the strain does no depend explicitly on position.
⁃ Vanishing velocity at the boundaries allows one to write the time derivative of the strain field in terms of an integral over the temperature profile.

The temperature profile is the solution of a diffusion equation with a forcing term that describes viscous dissipation. This set of coupled differential equations is the starting point for the authors' investigation.

After deriving the coupled set of partial differential equations, the authors first performed a linear stability analysis (LSA). I've seen this in a couple other papers. It seems like the basic approach is to approximate the equations of motion by linear equations, then solve these. If the solutions grow exponentially, the system is unstable. If they decay exponentially, the system is stable.

Applying LSA to the system under study, the authors find that the crossover from stable to unstable solutions depend on two dimensionless ratios:
⁃ \sigma_0/\sigma_c, the ratio of the strain at the boundaries to a critical strain that depends only on the physical properties of the material in questions
⁃ \tau_r/\tau_d, the ratio of the stress relaxation time of the system to the thermal diffusion time.

When the stress relaxation time is much smaller than the diffusion time, then the condition for stability is very simple: the applied strain must be smaller than the critical strain: \sigma_0 < \sigma_c.

Next the authors investigate the maximum temperature rise \Delta T during the evolution of the system as a function of all the control parameters. The authors say there are 13 dimensional parameters in the equations. Depending on how I count, I can get this number, or a few more: L, h, x, \sigma_0, \sigma, T, T_{bg}, T_0, G, E, A, n, \kappa, R, and C. The authors simplify this down to 6 dimensionless combinations using dimensional analysis.

Investigating the dependence on control parameters numerically, the authors find that the scaled maximum temperature is a function of only two dimensionless parameters --- the same two that emerged from their LSA! The phase diagram of \Delta T is divided into two regions: a stable deformation region and an adiabatic thermal runaway region.

The boundary between these is a critical region where the temperature distribution is highly localized at the center of the slab. In the adiabatic region, the temperature profile is roughly constant across the slab, but in the critical region, it is sharply peaked at the center, with the maximum value being several orders of magnitude larger than the rest of the slab. (This is beautifully illustrated in Figure 2.)

The nonlinear equations give rise to a feedback between temperature and strain profiles that result in a self-localizing runaway process. This results in a shear band that is localized to a region much smaller than the width of the perturbed region. I did not see any expression for the width of this region. It would be interesting to know how it scales with the other control parameters of the system, or whether it is infinitely localized.

The motivation for the work was to explain why real crystals seem to fail at stresses lower than a limit established by Frenkel in 1926: \sigma = G/10. The authors use material parameters from mantle rocks and metallic glass and find thermal runaway occurs at values lower than Frenkel's limit. The range of values given by the authors includes the values given by experimental data on both systems. They conclude by noting that the critical stress is of order 1 GPa because the kinetic terms do not appear in the expression for \sigma_c.

I thought this was a rather well-written paper. The authors did a good job of motivating their research and gave a very clear explanation of exactly what they did. The simplification gained through linear stability analysis and dimensional analysis gives a lot of insight into the problem. It is nice to see the numerics justify the conclusions drawn from the simplified version of the system, which I would not always expect from a nonlinear system.