3.8. Thermal Stresses

Consider the consequences of increasing or decreasing the uniform temperature of an entirely unconstrained elastic body. The resultant expansion or contraction occurs in such a way as to cause a cubic element of the solid to remain cubic, while experiencing changes of length on each of its sides. Normal strains occur in each direction unaccompanied by normal stresses. In addition, there are neither shear strains nor shear stresses. If the body is heated in such a way as to produce a nonuniform temperature field, or if the thermal expansions are prohibited from taking place freely because of restrictions placed on the boundary even if the temperature is uniform, or if the material exhibits anisotropy in a uniform temperature field, thermal stresses will occur. The effects of such stresses can be severe, especially since the most adverse thermal environments are often associated with design requirements involving unusually stringent constraints as to weight and volume. This is especially true in aerospace applications but is of considerable importance, too, in many everyday machine design applications.

Equations of Thermoelasticity

Solution of thermal stress problems requires reformulation of the stress–strain relationships accomplished by superposition of the strain attributable to stress and that due to temperature. For a change in temperature T(x, y), the change of length, δL, of a small linear element of length L in an unconstrained body is δL = αLT. Here α, usually a positive number, is termed the coefficient of linear thermal expansion. The thermal strain ε_t associated with the free expansion at a point is then