The Coupled and Quasi-static Approximation

• William Alan Day
Chapter
Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 30)

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)

Keywords

Maximum Principle Heat Equation Theta Function Monotone Property Nonlocal Boundary
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References

1. †.
The most suitable reference is to the notes [23] taken by J. W. Green of the lectures of H. Lewy. In order to establish what we require it suffices to make only minor modifications to the arguments of Sections 5–10 of Chapter II of those notes.Google Scholar
2. †.
Protter and Weinberger [31] attribute the Maximum Principle to E. E. Levi [27, 28].Google Scholar
3. †.
See, for example, Widder [32, Theorem 5.1].Google Scholar
4. †.
Kac [25, 26] has made the corresponding observation for the heat equation in higher dimensions the basis from which to deduce certain very interesting results concerning the eigenvalues of the Laplace operator. In Kac’s vivid language, the solution does not feel the boundary initially.Google Scholar
5. †.
Hardy, Littlewood, and Pólya [24, Theorem 257].Google Scholar
6. †.
See [12, 13]. My results have been extensively generalized by Friedman [22] to embrace parabolic equations in ℝn which satisfy nonlocal boundary conditions.Google Scholar