Determination of the Coefficient of Thermal Expansion of a Material Using Strain Gauges

Determine the Thermal Expansion Coefficient

In this article you will learn how the coefficient of thermal 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:

ε_s Apparent strain of the strain gauge
α_r Temperature coefficient of the electrical resistance
α_b Thermal expansion coefficient of the measurement object
α_m 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:

ε_a Strain indicated at the amplifier
ε_m The strain triggered by the mechanical load
α_DMS 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	-38.0	-358.9
0	-254.4	-16.9	-237.5	22.9
10	-122.5	-5.0	-117.5	22.8
20	0	-1.1	1.1	22.7
30	118.8	-3.9	122.7	23.0
40	232.4	-12.2	244.6	23.0
50	344.3	-24.8	369.1	23.2
60	453.3	-40.3	493.6	23.3
70	562.1	-57.7	619.8	23.4
80	671.6	-75.6	747.2	23.5
90	781.8	-92.7	874.5	23.5
100	894.1	-107.9	1002.0	23.5
110	1010.5	-119.9	1130.3	23.6
120	1132.3	-127.4	1259.8	23.7

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 α_DMS 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 Formula 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.

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Note: Subject to modifications. All product descriptions are for general information only. They are not to be understood as a guarantee of quality or durability.