 # IN OUR ELEMENT: HOW DO YOU CALCULATE A MATERIAL’S THERMAL EXPANSION?

By: Nyomi Martinez, Customer Technical Services Engineer, Performance Alloys and Composites

The coefficient of thermal expansion (α), often abbreviated as CTE, provides a measure of how much a material expands or shrinks when heated or cooled. Its units are expressed in strain per time, which becomes reciprocal temperature (F-1or C-1). The equation for calculating the change in length (ΔL) of a component of length (L) due to a change in temperature is: ΔL = L · α · ΔT. Because the change in length divided by the original length is strain, thermal strain can be calculated as: Sometimes you will see this written as the coefficient of linear thermal expansion because the coefficient of thermal expansion described above deals with expansion in only one direction. The coefficient of area expansion and the coefficient of volume expansion describe change in area per degree temperature rise and the change in volume per degree temperature rise, respectively. For isotropic materials, these values are respectively twice and three times the linear coefficient of thermal expansion: Note that this is for isotropic (non-directional) solid materials only, which means that their properties are the same in all directions. The value reported on the solid material’s data sheet will almost always be the linear expansion coefficient. More than one value may be included, particularly for composite materials where the rate of expansion might be different in different directions.

Thermal strain expands in all three directions, while mechanical strain expands in one direction and contracts in the other two. A thermal strain of 20% would make a part 20% longer, thicker, and wider. A 20% mechanical strain would make a part 20% longer, but only 6% narrower and thinner (assuming Poisson’s ratio of 0.3, which is typical for most metals).

The thermal expansion coefficient is not a constant, but instead is a function of temperature, as shown in Figure 1. To accurately determine the thermal expansion and thermal strain, you will need to know not only the change in temperature, but also the starting and ending temperatures so that you can use the proper value for CTE. Figure 1. Two Different Methods of Presenting Thermal Expansion Data

Figure 1 shows a curve of instantaneous CTE vs. temperature. This is a plot of the instantaneous slope of the thermal strain vs. temperature curve. Another way of presenting thermal expansion data would be as a table of linear expansion rates to be used in a particular range of temperatures, as shown in the inset on Figure 1.

The CTE does have practical implications. For example, you can use known expansion vs. temperature behavior to construct a simple temperature gauge using two materials that expand at different rates. Bimetallic sensors and switches make use of two materials with different thermal expansion rates that are bonded together. Because of the different expansion rates, the composite will curve toward the side that is shorter due to less expansion or more contraction. This is useful for temperature gauges, or to simply activate or deactivate an electric circuit when the temperature goes beyond certain limits.

Some alloys are specially designed to have low thermal expansion coefficients. The most well-known of these low-expansion alloys is FeNi36, also known by the tradename Invar® (which is a trademark of Aperam Alloys Imphy). When near room temperature, this alloy has a CTE an order of magnitude lower than most metals (shown on Table 1, below). Table 1. Coefficient of Thermal Expansion for Various Metals at 20-300°C. Shown in ppm/°C. Monel® and Inconel® are registered trademarks of Special Metals Corporation

Other alloys have CTEs that are matched to other materials, such as Alloy 42 (FeNi42) and the FeNi29Co17 alloy known as Kovar. Both alloys have thermal expansion coefficients that are matched to glass and ceramic materials. This makes them ideal for hermetically sealed processor or sensor packages, where it is important that the package remain sealed no matter how hot or cold it gets. A differential thermal expansion could easily break such a seal, so CTE-matched alloys are usually used for these purposes.

Under certain conditions, the coefficient of thermal expansion can be negative. For instance, when between 0°C and 4°C, water will expand as it cools and contract as it warms. Other compounds also show negative thermal expansion coefficients over wide temperature ranges, such as cubic zirconium tungstenate ( ZrW2O8). There are some artificially constructed metamaterials that show this behavior as well.

The thermal expansion coefficient provides useful information for several applications, although it varies with temperature. In order to do meaningful calculations, you will need to know the coefficient that is appropriate for the temperature range you are working in. Make sure to choose the material that is best suited to your particular need and environment.

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