# Determine the Thermal Expansion Coefficient

In this article you can read how the coefficient of thermal coefficient of expansion of aluminium can be determined using "mismatched" foil strain gauges.

When there is a change in temperature, each strain gauge quarter bridge registers a measurement signal, the "apparent strain". The apparent strain of a strain gauge measuring point exposed to a temperature difference Δϑ, can be described as follows:

The following applies here:

εs                     Apparent strain of the strain gauge
αr                     Temperature coefficient of the electrical resistance
αb                    Thermal expansion coefficient of the measurement object
αm                   Thermal expansion coefficient of the measuring grid material
k                       K factor of the strain gauge
Δϑ                    Temperature difference that triggers the apparent strain

On all of their strain gage packs, HBM shows the apparent strain as a function of temperature in a chart and also as a polynomial. Of course, these data only ever give useful results if the thermal coefficient of linear expansion of the material to be tested matches the data on the strain gage pack.

The following then applies:

## Determining the Thermal Coefficient of Linear Expansion α

But the apparent strain can also be used perfectly well for measurement purposes, if the coefficient of thermal expansion αm is to be determined. In this situation, the following formula can be used.

Transposed, this produces:

εa                     Strain indicated at the amplifier
εb                     The strain triggered by the mechanical load
αDMS                Thermal coefficient of linear expansion as per the strain gauge pack

In a practical test, four HBM strain gauges of the LG11-6/350 type, adapted to steel (α=10.8 10-6/K), were installed on an aluminum workpiece.
A four-wire circuit was used to eliminate cable influences. According to the data supplied by the manufacturer for the material, α=23.00 *10-6/K for T= 0 … 100°C.

 ϑ (°C) εa(*10-6) εs(*10-6) εa-εs(*10-6) αb(*10-6)/K -10 -396.9 -38.017 -358.883 0 -254.35 -16.9 -237.45 22.9433 10 -122.5 -5.003 -117.497 22.7953 20 0 -1.084 1.084 22.6581 30 118.75 -3.901 122.651 22.9567 40 232.4 -12.212 244.612 22.9961 50 344.32 -24.775 369.095 23.2483 60 453.27 -40.348 493.618 23.2523 70 562.12 -57.689 619.809 23.4191 80 671.6 -75.556 747.156 23.5347 90 781.82 -92.707 874.527 23.5371 100 894.07 -107.9 1001.97 23.5443 110 1010.45 -119.893 1130.343 23.6373 120 1132.32 -127.444 1259.764 23.7421

Tab. 1 Measurement results for a strain gauge adapted for ferrit. steel, installed on aluminum

If you calculate αm for the specified interval, you obtain 23.19 *10-6/K, which corresponds to a deviation from the theoretical value of 0.19 *10-6/K (0.84%).
To run the experiment, it is first necessary to install several strain gauges on the object under investigation (to attain experimental reliability). The sample must be flat in the direction of the measuring grid.
In the next step, the strains are determined subject to the temperature. Care must be taken to ensure that thermal equilibrium is established.
First εa-εs is calculated. To determine the thermal coefficient of linear expansion, you subtract the two calculated values (εa-εs ) from each other and divide this by the corresponding temperature interval. The coefficient of thermal expansion αDMS as per the pack data must then be added to this.
Example:
In the interval from 20 to 40 degrees, the coefficient of thermal expansion is calculated as follows (using calculation shown in Fig. 4)

During this measurement, the strain gauge creep is an undesirable effect. So in the interest of maximum accuracy, it is advisable to use HBM series K strain gauges, which have three different creep adjustments as standard and of these, use the strain gauge with the greatest end loop length.
Also, when the measuring temperatures are over 60 °C, it is advisable to use hot curing adhesives for installation.