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Tutorial: Coefficients of expansion of materials - Materials sciences theory
The coefficients of expansion of materials play a crucial role in materials science. Coefficients of expansion describe how the dimensions of a material change under the influence of temperature fluctuations. This article provides a comprehensive introduction to the topic, including the basics of thermal expansion, influencing factors, and various measurement methods. We also compare typical expansion coefficients of material classes and explain their uses in various industries. Lastly, we address thermal stresses and ways to prevent damage from temperature differences in materials.
Uses and Importance of Expansion Coefficients
When designing and manufacturing machinery and systems, knowledge of expansion coefficients of materials plays a crucial role as it relates to mechanical engineering and custom machinery construction.
- Precision and dimensional accuracy: In mechanical engineering, precision is of great importance, especially for tool building for stamping and injection molding applications, and for precision equipment. The expansion coefficients of the materials used must be taken into account to ensure that the components maintain their dimensional accuracy as temperatures change. Understanding the thermal expansion properties helps engineers take into account the precise dimensions and tolerances and to ensure the dimensional accuracy of components and machines.
- Fits and assembly: When designing machines, parts often need to be joined using various materials, such as screws, bolts, or press fits. The expansion coefficients of the materials must be compatible to avoid unwanted stresses and deformations during operation. Proper fit design and assembly, taking into account thermal expansion, is critical to ensure a secure and permanent connection.
- Linear guides and plain bearings: In machines using linear guides or plain bearings, it is important to consider the thermal expansion of the guide rails or guide elements. The expansion coefficients of the guides must be matched with those of the surrounding housing or structure to ensure smooth movement and consistent functionality.
- Temperature compensation: In some mechanical engineering applications, operating temperatures can change significantly, such as in smelter or high temperature processes. Knowledge of the expansion coefficients of the employed materials makes it possible to develop thermal compensation systems that minimize deformation or damage to the machines and components at changing temperatures.
Typical material length expansion coefficients
The coefficients of expansion of materials can vary greatly depending on the material class.
Metal
Metals typically have a relatively high linear expansion coefficient.
- Coefficient of expansion aluminum: 23 x 10-6 K-1
- Coefficient of expansion copper: 16.5 x 10-6 K-1
- Coefficient of expansion iron: 12 x 10-6 K-1
- Coefficient of expansion steel: 12 x 10-6 K-1
- Coefficient of expansion nickel: 13 x 10-6 K-1
Note: This information is for reference only.
Polymers
Polymers typically have a significantly higher coefficient of expansion compared to metals.
| Material | Coefficient of expansion [x 10-6 K-1] |
|---|---|
| Polyethylene (PE) | 100 to 250 |
| Polypropylene (PP) | 100 to 200 |
| Polystyrene (PS) | 50 to 100 |
| Polyvinyl chloride (PVC) | 50 to 80 |
Note: This information is for reference only.
Composites
Composites consist of multiple material components that are bonded together in a matrix. The coefficients of expansion of composite materials can vary greatly and depend on the proportions of the individual components.
It is important to note that plastics can also have negative expansion coefficients. Temperature changes create stresses in these components, which can have advantages and disadvantages.
Fundamentals of thermal expansion
Thermal expansion is a fundamental physical phenomenon that causes the dimensions of a material to change when it is exposed to temperature changes. This effect occurs due to the movement of the atoms or molecules in the material. As the temperature increases, the average kinetic energy of the particles increases, resulting in expansion of the material.
The linear expansion coefficient is a physical parameter that indicates how much the length of a material changes per unit length when exposed to a change in temperature. It quantifies the linear expansion of a material along a particular direction, such as along an axis, a wire, or rod.
Formally speaking, the linear expansion coefficient (αL) is defined as the relative change in length (ΔL/L0) of a material per temperature change (dT). The definition is:
\alpha_L = \frac{\frac{\Delta L}{L_0}}{d T}
- αL ist der lineare Ausdehnungskoeffizient in der Einheit 1/K (Kelvin).
- ΔL ist die Änderung der Länge des Materials in einer bestimmten Richtung.
- L0 ist die ursprüngliche Länge des Materials, bevor es der Temperaturänderung ausgesetzt wurde.
- dT ist die Temperaturänderung in Kelvin (K) oder Grad Celsius (°C)
A positive value of the linear expansion coefficient means that the material expands and elongates as the temperature increases. A negative value would mean that the material shrinks and contracts as the temperature increases. Most materials have a positive linear expansion coefficient, which means they expand when heated and contract again when cooled.
The linear expansion coefficient typically varies with temperature. An expansion anomaly can occur in the vicinity of phase transformations – for example, during melting or solidification. This anomaly is caused by changes in the structure or aggregate state of the material and is also known in thermodynamics as a critical point or conversion point. This critical point can be crucial when it comes to the design of components or equipment, as extra care must be taken to avoid damage caused by thermal stress.
Calculation example for the coefficient of thermal expansion
A dilatometer is used to examine a test specimen with a length of 0.1 m made of steel at a reference temperature of 20°C. At 35°C, the material has a length of 0.100018 m.
\alpha_L = \frac{\frac{\Delta L}{L_0}}{d T}
\alpha_{L,Stahl} = 12 \times 10^{-6} K^{-1}
Linear expansion
Linear expansion describes the change in length of a material in a particular direction when subjected to a change in temperature. This type of expansion occurs in materials that have a particular geometric shape, such as rods, wires, or pipes. Linear expansion means that the material expands along an axis or line.
- L0 - Original length
- L1 - Expanded length
- ΔL - Change in length
The linear expansion coefficient (αL) quantifies this type of expansion and indicates how much the length of a material changes per unit of length per Kelvin (K) or degrees Celsius (°C). A positive αL means that the material expands as temperature increases, while a negative αL indicates that the material shrinks as temperature increases.
Surface expansion of materials during temperature changes
The surface expansion describes the change in the surface area of a material during temperature changes. This occurs with materials that have flat, two-dimensional shapes, such as plates or sheet metals.
The area expansion is a combination of the linear expansion in two orthogonal directions.
- W0 - Original width
- W1 - Expanded width
- ΔW - Change in length
- L0 - Original length
- L1 - Expanded length
- ΔL - Change in length
The area expansion coefficient (αA) indicates how much the area of a material changes per unit of area per Kelvin or degrees Celsius. It is associated with the linear expansion coefficient (αL) and the dimension of the material.
The area expansion coefficient is calculated from the area change, the starting area and the temperature change:
Area expansion coefficient αA
\alpha_A = \frac{\frac{\Delta A}{A_0}}{d T}
- αA ist der lineare Flächenausdehnungskoeffizient in der Einheit 1/K (Kelvin).
- ΔA ist die Änderung der Fläche des Materials in einer bestimmten Richtung.
- A0 ist die ursprüngliche Fläche des Materials, bevor es der Temperaturänderung ausgesetzt wurde.
- dT ist die Temperaturänderung in Kelvin (K) oder Grad Celsius (°C).
\alpha_A = 2 \times \alpha_L
Calculation example for the coefficient of thermal expansion of copper
A copper sheet with a negligible thickness has an original area A0 of 0.5 m x 1 m. The coefficient of expansion αL is 16.5 x 10-6 K-1. The temperature is increased by 20 K.
A_1 = A_0 + \alpha_A \times A_0 \times \Delta T
A_1 = 0.50027 m^2
Volume expansion of materials
Volume expansion describes the change in the volume of a material during temperature changes. This is relevant for materials that have a three-dimensional shape, such as cubes or spheres. Volume expansion is a combination of linear expansion in three orthogonal directions.
The volume expansion coefficient (αV) indicates how much the volume of a material changes per unit of volume per Kelvin or degrees Celsius. It is associated with the linear expansion coefficient (αL) and the dimension of the material.
The relationship between the volume expansion coefficient and the linear expansion coefficient is:
Volume expansion coefficient αV
\alpha_V = \frac{\frac{\Delta V}{V_0}}{d T}
- αV ist der lineare Volumenausdehnungskoeffizient in der Einheit 1/K (Kelvin).
- ΔV ist die Änderung des Volumens des Materials in einer bestimmten Richtung.
- V0 ist das ursprüngliche Volumen des Materials, bevor es der Temperaturänderung ausgesetzt wurde.
- dT ist die Temperaturänderung in Kelvin (K) oder Grad Celsius (°C).
For small temperature changes, αV can be simplified as follows:
\alpha_V = 3 \times \alpha_L
in most cases, materials have a positive linear expansion coefficient and therefore expand when heated. However, there are certain materials that have a negative coefficient of expansion and contract as temperature increases, known as anomalous or negative thermal expansion.
Knowledge of the different types of thermal expansion is of great importance for designing and selecting materials in different applications to account for, and minimize, potential problems due to temperature changes.
Calculation example for the expansion coefficient of aluminum
An aluminum block has an original volume V0 of 0.25 m x 0.25 m x 1 m. The coefficient of expansion αL is 23.1 x 10-6 K-1. The temperature is increased by 30 K.
V_1 = V_0 + \alpha_V \times v_0 \times \Delta T
V_1 \approx 0.06263 m^3
Factors affecting thermal expansion
The material composition of a material is a critical factor affecting the coefficient of expansion. The material composition determines how the atoms or molecules in the material move and interact with each other during temperature changes, which ultimately determines the thermal expansion properties of the material.
- Homogenous materials: Materials made of a single element or compound typically have a well-defined and constant coefficient of expansion over a specific temperature range.
- Composites: Materials that consist of different phases or components can have more complex thermal expansion properties. The different coefficients of expansion of the components can result in internal stresses, especially when the phases expand at different temperatures.
Influence of mechanical stress
The influence of mechanical stress on the thermal expansion of a material is an important aspect that must be taken into account when analyzing and designing structures and components. Mechanical stress refers to the exposure of a material to external forces that cause deformation or stress in the material. This can result in a change in the thermal expansion properties of the material.
- Elastic materials exhibit reversible deformation when subjected to mechanical stress. This means that they return to their original shape after the load is removed, such as rubber bands or seals.
- Plastic materials, on the other hand, exhibit irreversible deformations under mechanical stress that do not fully return when stress is removed, such as bending a metal wire or pressing a piece of metal.
Mechanical stress can result in internal stresses in the material that can affect thermal expansion. When the material is subjected to mechanical stress, the atoms may shift or rearrange in certain areas of the material, which can lead to local differences in expansion characteristics. These internal stresses can cause a non-uniform expansion of the material during temperature changes.
Measurement methods for determining the coefficient of expansion
Dilatometry is a commonly used method for measuring thermal expansion of materials. A sample of the material under test is inserted into a dilatometer test setup, which is equipped with a heating and cooling device. The sample is subjected to a defined temperature change while the length change of the material is accurately measured. The linear expansion coefficient can be calculated by measuring the change in length and knowing the change in temperature.
Interferometry is an optical method based on the interference pattern of light waves. This method uses an interferometer to illuminate a sample. The interference pattern is then observed as the temperature of the sample is changed. The linear expansion coefficient is determined by mapping the change in the interference pattern to the change in the length of the sample. Interferometry provides precise measurements over a wide temperature range.
