section epub:type=”chapter”> When solid materials are heated, most expand (although some very flexible rubbers actually contract). This expansion is defined by the linear coefficient of thermal expansion, α. If a rod-shaped specimen (having initial and final lengths L0 and L) is heated through a temperature interval ΔT, then This equation can be rewritten as. where ɛ is the (linear) thermal strain. For an isotropic material, the volume strain is given by We have already come across thermal expansion in several places: Section 17.4 (thermal cracking of PUR foam insulation panels); Chapter 18, Worked Example 1 (thermal fatigue of copper water-cooling plates); Example 18.7 (thermal fatigue in a boiler); Section 19.5 (residual stresses in welds); Example 29.11 (shrink fitting of compressor wheels). α is an important material property, with a huge range of applications—and consequences. In Section 17.4, we said that the thermal stress σ in the PUR foam was where E is Young’s modulus and υ is Poisson’s ratio. But how do we prove this? We saw in Example 7.11 that the elastic stress-strain relations for an isotropic material are We can easily modify these relations to include the effect of thermal expansion by simply adding the thermal strain to each strain component, giving Now let’s look at the free surface of the foam (see Figure 17.7). We locate principal axes 1p and 2p in the plane of the surface, and 3p perpendicular to the surface. σ3 = 0 (a free surface cannot support a stress) and ɛ1 = ɛ2 = 0 (the surface cannot contract sideways, because it is fixed to the steel plate). So This gives σ1 = σ2 = σ (equibiaxial), (Note that ΔT is negative, because the free surface of the foam has been cooled. So σ is a tensile stress.)■ How do we measure α? As a rough approximation for many metals and ceramics, a temperature rise of 100°C produces a thermal strain of one in a thousand (equivalent to 1 thou per inch). Measuring α for these materials therefore requires very accurate measurement of length and precise control of temperature. This presents considerable challenges, especially when measurements have to be done at high temperatures. If traditional mechanical methods (e.g., micrometers, displacement transducers) are used, great care must be taken that the measuring equipment itself does not expand (or if it does, this is calibrated out). Most of these problems have now been solved using noncontact methods, such as laser interferometry (which is also extremely accurate). Data for α are shown in Table 31.1. Within each materials class the values vary a lot, although the range of values for metals and ceramics is much the same. What is notable, however, are the large values for polymers (and some polymer-based composites)—typically 5 or 10 times those for metals and ceramics. α itself is a weak function of temperature—the values shown in the table are average values over a finite temperature range (often 25–125°C). For exacting applications, data must be obtained over the relevant temperature range, which may be hundreds of degrees higher.
Thermal Expansion
31.1 Introduction
Worked Example 1
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31.2 Coefficients of Thermal Expansion
