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When the temperature of an unrestrained body is uniformly increased, the body expands, and the normal strain is
epsilon_x = epsilon_y = epsilon_z
= alpha * delta_T [2-67]
where alpha is the coefficient of thermal expansion and
delta_T is the temperature change, in degrees.
In this action the body experiences a simple volume
increase with the components of shear strain all zero.
If a straight bar is restrained at the ends so as to prevent lengthwise expansion and then is subjected to a uniform increase in temperature, a compressive stress will develop because of the axial constraint. The stress is
sigma = epsilon*E = alpha*delta_T*E [2-68]
In a similar manner, if a uniform flat plate is restrained at the edges and also subjected to a uniform temperature rise, the compressive stress developed is given by the equation
sigma = (alpha*delta_T*E)/(1-mu) [2-69]
The stresses represented by equations [2-68] and [2-69], though due to temperature, are not thermal stresses insomuch as they result from the fact that the edges were restrained. A thermal stress is one which arises because of the existance of a temperature gradient in a body.
Figure [2-30] shows the internal stresses within a slab of infinite dimensions during heating and cooling. During cooling, the maximum stress is the surface tension. At the same time, force equilibrium requires a compressive stress at the center of the slab. During heating, the external surfaces are hot and tend to expand but are restrained by the cooler center. This causes compression in the surface and tension in the center as shown.
![Figure [2-30]](fig0230.GIF)
Table [2-4] lists approximate values of the coefficient alpha for various engineering materials.
TABLE [2-4] COEFFICIENTS OF THERMAL EXPANSION (LINEAR MEAN COEFFICIENTS FOR THE TEMPERATURE RANGE 0 - 100 DEGREES CENTEGRADE)