Elasticity pp 219-225 | Cite as

# Thermoelasticity

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## 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 bywhere 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.

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

(14.1)

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

(14.2)

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)

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© Springer Science+Business Media B.V. 2010