handbook of modern sensors

3.11 Temperature and Thermal Properties of Materials 97
(A)
(B)
Fig. 3.38. (A) Warping of a laminated plate where two materials have different coefficients of
thermal expansion; (B) a bimetal coil used as a temperature transducer.
where T = T 2 − T 1 . Table A.16 gives values of α for different materials 16 . Strictly
speaking, α depends on the actual temperature. However, for most engineering purposes,
small variations in α may be neglected. For the so-called isotropic materials,
α is the same for any direction. The fractional change in area of an object and its
volume with a high degree of accuracy can be represented, respectively, by
A = 2αAT , (3.111)
V = 3αV T . (3.112)
Thermal expansion is a useful phenomenon that can be employed in many sensors
where thermal energy is either measured or used as an excitation signal. Consider two
laminated plates, X and Y, that are fused together (Fig. 3.38A). The plates have the
same thickness and surface area and identical moduli of elasticity. Their coefficients
of thermal expansion, α 1 and α 2 , however, are different. The fused plates are anchored
at the left-hand side to the reference wall. Now, if we apply heat to the structure, (i.e.,
if we increase its temperature from T 1 to T 2 ), plate X will expand more than plate
Y (for α 1 >α 2 ). The lamination area will restrain plate X from a uniform expansion
while forcing plate Y to expand more than its coefficient of expansion would require.
This results in the formation of the internal stress and the structure will warp downward.
Contrary, if we cool the structure, it will warp upward. The radius of warping
can be estimated from [36]
2j
r ≈
3(α X − α Y )(T 2 − T 1 ) . (3.113)
The warping results in deflection of the tip that is strongest at the end of the structure
anchored at the other end. This deflection can be measured as a representative of the
16 More precisely, thermal expansion can be modeled by higher-order polynomials: l 2 = l 1 [1 +
α 1 (T 2 − T 1 ) + α 2 (T 2 − T 1 ) 2 + α 3 (T 2 − T 1 ) 3 +···]; however, for the majority of practical
purposes, a linear approximation is usually sufficient.