1.3 Principles of thermal deformation

Thermal expansion is the tendency of matter to increase in volume when heated. For solids and liquids the amount of expansion will vary depending on the material's coefficient of thermal expansion. While for gases the change in volume or pressure is related to the container in which the gas is contained. When solids expand tensile forces are created and when solids contract compressive forces are created.

The coefficient of thermal expansion describes how the size of an object changes with a change in temperature. For solid materials with a significant length, an estimate of the amount of thermal expansion can be described by the ratio  ΔL / Linitial or linear strain:

where ΔL / Linitial [-] is the ratio of thermal strain, Linitial [m] is the initial length before the change of temperature and Lfinal [m] is the final length recorded after the change of temperature. The total strain is a combination of mechanical and thermal strain.

For most solids, thermal expansion scales linear with temperature difference:

where ΔT is the temperature difference [K]. Thus, the change in strain can be estimated by:

where α [K-1] is the coefficient of thermal expansion (CTE) in inverse Kelvin, Tfinal the final temperature [°C] or [K] and Tinitial [°C] or [K] the initial temperature and ΔT [°C] or [K] is the difference of the temperature between the two recorded strains. 

Three principles can be distinguished. These principles are described for a plate in the xy-plane (for example a machine part) with length Lx in x-direction, which is shown in Figure 1.

  • Longitudinal translation ∆Lx: in case of a uniform temperature change and/or gradient in x-direction (Gx), the beam will expand in axial (x) direction. The translation ∆Lx (x) is then the integrated sum of expansions of infinite small lengths dx:

  • Rotation β: a temperature gradient in y-direction (Gy) introduces a local rotation dβ(x) around the z-axis caused by different expansions dLx at different heights y1 and y2. The rotation dβ(x) is obtained by integrating all local rotations dβ(x) over x:

  • Transversal translation caused by rotation f : a temperature gradient in y-direction (Gy) also causes a translation ∆y in the (negative) y-direction. This transversal translation can be calculated by integration rotation β over length Lx:

The shortening in y-direction introduced by the rotation β is a second order effect. For rotation, shortening ∆y=Linital (1-cos⁡(α) ), for α is 0.1 mrad, ∆y=5∙10-9Linitial.

It must be noted that material properties for most materials change as a function of temperature, which must be taken into account for e.g. cryogenic applications.


Figure 1: Deformations caused by expansions (or contractions)