Thermoelasticity

Abstract

Most materials tend to expand if their temperature rises and, to a first approximation, the expansion is proportional to the temperature change. If the expansion is unrestrained, all dimensions will expand equally — i.e. there will be a uniform dilatation described by

$$e_{xx} = e_{yy} = e_{zz} = \alpha T$$

(14.1)

$$e_{xy} = e_{yz} = e_{zx} = 0$$

(14.2)

where α is the coefficient of linear thermal expansion. Notice that no shear strains are induced in unrestrained thermal expansion, so that a body which is heated to a uniformly higher temperature will get larger, but will retain the same shape.

Thermal strains are additive to the elastic strains due to local stresses, so that Hooke’s law is modified to the form

$$e_{xx} = \frac{{\sigma _{xx} }}{E} - \frac{{v\sigma _{yy} }}{E} - \frac{{v\sigma _{zz} }}{E} + \alpha T$$

(14.3)

$$e_{xy} = \frac{{\sigma _{xy} (1 + v)}}{E}.$$

(14.4)