The Coupled and Quasi-static Approximation

Abstract

This chapter is given over to an investigation of the effect of thermomechanical coupling in isolation from the effect of inertia. To this end we make the coupled and quasi-static approximation which entails that the coupling constant a be positive while the inertial constant b is set equal to zero. Thus, we retain the first of the thermoelastic equations (1.1.1) but replace (1.1.2) by the approximate equation

$$ \frac{{{\partial^2}u}}{{\partial {x^2}}} = \sqrt a \cdot \frac{{\partial \theta }}{{\partial x}}. $$

(2.1.1)

. In the light of the boundary conditions (1.1.3) the strain ∂u/∂x must satisfy the condition

$$ \int_0^1 {\frac{{\partial u}}{{\partial x}}dx = 0} $$

and if we integrate (2.1.1) with respect to x and appeal to this condition we can express the strain in terms of the temperature by way of the equation

$$ \frac{{\partial u}}{{\partial x}}(x,t) = \sqrt a \cdot \theta (x,t) = \sqrt a \cdot \int_0^1 {\theta (y,t)dy.} $$

(2.1.2)

. When we differentiate with respect to t and substitute for the strain rate in (1.1.1) we conclude that the temperature is a solution of the equation.

$$ \frac{{{\partial^2}\theta }}{{\partial {x^2}}}(x,t) = (1 + a)\frac{{\partial \theta }}{{\partial t}}(x,t) - a\frac{d}{{dt}}\int_0^1 {\theta (y,t)dy.} $$

(2.1.3)