Thermal Expansion: CTE Definitions and Thermal Strain

This entry discusses different definitions of CTE, their relation to thermal strain, how to convert between the different forms, and how to use them in a model. The forms discussed below include instantaneous coefficient of thermal expansion (CTE), secant coefficient of thermal expansion, and direct use of a thermal strain function.

nonlinear thermal strain
Figure 1

Hooke’s law for a linear, isotropic elastic material may be written as

    \[ \epsilon_x=\frac{1}{E} \left( \sigma_x - \nu \left( \sigma_y + \sigma_z \right) \right) + \epsilon_T \]

    \[ \epsilon_y=\frac{1}{E} \left( \sigma_y - \nu \left( \sigma_z + \sigma_x \right) \right) + \epsilon_T \]

    \[ \epsilon_z=\frac{1}{E} \left( \sigma_z - \nu \left( \sigma_x + \sigma_y \right) \right) + \epsilon_T \]

    \[ \gamma_{xy} = \frac{\tau_{xy}}{G}, \quad \gamma_{yz} = \frac{\tau_{yz}}{G},  \quad \gamma_{zx} = \frac{\tau_{zx}}{G} \]

where \epsilon_T is the thermal strain. The thermal strain appears only in the normal strains but not the shear strains. Thermal strain is a volumetric expansion or contraction and one can show via symmetry arguments that thermal strain should not appear in the shear strains for an isotropic material.

linear thermal strain
Figure 2

The most common model of thermal strain is with a linear function of temperature and constant CTE (coefficient of thermal expansion) (see Figure~2)

(1)   \begin{equation*} \epsilon_T= \alpha \left( T - T_0 \right) \end{equation*}

where \alpha is the coefficient of thermal expansion, or CTE, T is the temperature of the element of interest, and T_0 is the reference temperature at which there is zero thermal strain.

However if the thermal strain is nonlinear, then a different expression must be used to calculate the thermal strain at a given temperature (see Figure~1). First, one can use the thermal strain function itself. Within Ansys one can specify the thermal strain material property using THSX at the temperatures specified with MPTEMP. Secondly, one can specify the instantaneous coefficient of thermal expansion. The instantaneous CTE is the derivative of the thermal strain function at the temperature of interest.

    \[ \alpha_{inst} \left( T \right) = \frac{d \epsilon_T}{dT} \]

To calcuate the thermal strain from the instantaneous coefficient of thermal expansion, one must use integration between and thus know the thermal strain at one temperature to obtain the integration constant.

    \[ \epsilon_T \left( T \right) = \int_{T_0}^T \alpha_{inst} \left( T \right) dT \]

The instantaneous thermal strain function may be input into ansys using the material property CTEX at the temperatures specified with MPTEMP.

Another way of specifying the thermal expansion is with an average slope of the thermal strain curve between the temperature of interest and a reference temperature T_0 as shown in Figure~1. This is called the secant coefficient of thermal expansion.

    \[ \alpha_{secant} \left( T \right) = \frac{\epsilon_T \left( T \right) - \epsilon_T \left( T_0 \right) }{T-T_0} \]

One may express the thermal strain as a function of the secant CTE

    \[ \epsilon_T \left( T \right) = \epsilon_T \left( T_0 \right) + \alpha_{secant} \left( T \right) \left( T-T_0 \right) \]

The secant thermal strain function may be input into ansys using the material property ALPX at the temperatures specified with MPTEMP.

One can also convert between instantaneous and secant CTE using

    \[ \alpha_{secant} \left( T \right) \left( T - T_0 \right) = \int_{T_0}^T \alpha_{inst} \left( T \right) dT \]

where T is the temperature of interest and T_0 is the reference temperature. This relation is derived by the fact that the area under the instantaneous CTE curve and area under the averaged or secant CTE line are both equal to the thermal strain.

Specification of coefficients of thermal expansion

Information on the CTE for speciic materials can often be found in manufacturers’ catalogs, textbooks, or websites like MatWeb, but sometimes it is difficult to know what exactly is being specified.

For a material with a varying CTE, the CTE is typically specified over a temperature range. Let us take as an example the low expansion glass ceramic Zerodur (see for example Schott TIE-37: Thermal expansion of Zerodur http://fp.optics.arizona.edu/optomech/references/glass/Schott/tie-37_thermal_expansion_of_zerodur_v2_us.pdf). They specify the CTE over the range 0~deg~C to 50~deg~C. They are specifying the secant CTE. Within this temperature range, the instantaneous CTE may deviate from the secant CTE. As shown above, if the thermal strain curve is linear in a given temperature range, then the secant CTE is the same as the instantaneous CTE. This document on Zerodur also includes what is called TCL (total change of length) curves, which are identical to thermal strain \delta \ell / \ell.

Often in precision engineering applications, one finds that the catalog values do not contain enough detail of the thermal strain or CTE for calculations. For low CTE materials such as ULE, Zerodur, Super Invar, Cordierite, the manufacturer can typically supply more precise values of thermal strain or instantaneous CTE than is listed in the catalogs. Additionally, the CTE of these materials can be tailored by the manufacturer to meet specific requirements.

 

The Ansys Mechanical APDL Theory Reference manual serves as as a good resource for this topic.