# 1.3 Principles of thermal deformation

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...

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 Δ*L* / *L*_{initial} or linear strain:

where Δ*L* / *L*_{initial} [-] is the ratio of thermal strain, *L*_{initial} [m] is the initial length before the change of temperature and *L*_{final} [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 Δ*T* 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, *T*_{final} the final temperature [°C] or [K] and *T*_{initial} [°C] or [K] the initial temperature and Δ*T* [°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 *L*_{x} in *x*-direction, which is shown in Figure 1.

- Longitudinal translation ∆
*L*_{x}: in case of a uniform temperature change and/or gradient in*x*-direction (*G*_{x}), the beam will expand in axial (*x*) direction. The translation ∆*L*_{x}(*x*) is then the integrated sum of expansions of infinite small lengths d*x*:

- Rotation
*β*: a temperature gradient in*y*-direction (*G*_{y}) introduces a local rotation d*β*(*x*) around the*z*-axis caused by different expansions d*L*_{x}at different heights*y*_{1}and*y*_{2}. The rotation d*β*(x) is obtained by integrating all local rotations d*β*(*x*) over*x*:

- Transversal translation caused by rotation
*f*: a temperature gradient in*y*-direction (*G*_{y}) also causes a translation ∆*y*in the (negative)*y*-direction. This transversal translation can be calculated by integration rotation*β*over length*L*_{x}:

The shortening in *y*-direction introduced by the rotation *β* is a second order effect. For rotation, shortening ∆*y*=*L*_{inital} (1-cos(*α*) ), for α is 0.1 mrad, ∆*y*=5∙10^{-9} ∙ *L*_{initial}.

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)

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