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coefficient of thermal expansion

November 11, 2025
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The coefficient of thermal expansion (CTE) is an important physical property of a material. It describes the change in volume or length of a material when its temperature changes. This concept is widely used in engineering, physics, and various industries. In this article, we will examine the coefficient of thermal expansion, its types, calculation methods, and practical applications in detail.

Determination of the coefficient of thermal expansion

The coefficient of thermal expansion (CTE) measures how much a material expands or contracts with a change in temperature. It is usually expressed in degrees K⁻¹ (inverse of degrees Kelvin) or °C⁻¹ (inverse of degrees Celsius), as both units measure temperature changes in the same way.

Types of coefficients of thermal expansion

  1. linear coefficient of thermal expansion (α)     : the change in length of a material when its temperature changes.

  2. Coefficient of thermal expansion of the surface (β)     : the change in the surface area of ​​a material with temperature.

  3. volumetric coefficient of thermal expansion (γ)     : the change in volume of a substance as a function of temperature.

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Calculation of the coefficient of thermal expansion

linear expansion coefficient

The linear expansion coefficient is calculated as follows:

α = (ΔL / L₀) / ΔT

Where:

  • α: Coefficient of linear expansion

  • ΔL: Change in length (L – L₀)

  • L₀: Initial length

  • ΔT: Temperature change (T – T₀)

coefficient of surface expansion

For isotropic materials, the following relationship exists between the linear modulus and the surface modulus:

β ≈ 2α

Coefficient of volume expansion

   For isostatic materials, the following  relationship exists between the linear modulus of elasticity and the global modulus of elasticity:

γ ≈ 3α

Factors that influence the coefficient of thermal expansion

The coefficient of thermal expansion of a material is influenced by several factors:

  1. Molecular structure     : Materials with stronger bonds generally have lower coefficients of expansion.

  2. Ambient temperature     : The coefficient of thermal expansion can change at different temperatures.

  3. States of matter     : A substance has different coefficients of expansion in different states of matter (solid, liquid, gaseous).

  4. Chemical composition     : Different alloys and compounds behave differently.

  5. Crystalline orientation: In    anisotropic   materials, the coefficient of thermal expansion changes in different directions.

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Coefficients of thermal expansion of different materials

metal

  • Aluminum: 23.1 × 10⁻⁶ K⁻1

  • Steel: 11-13 × 10⁻⁶ K⁻¹

  • Copper: 16.5 × 10⁻⁶ K⁻¹

  • Tungsten: 4.5 × 10⁻⁶ K⁻⁹

Ceramics     and glass

  • Normal glass: 9×10⁻⁶ K⁻¹

  • Pyrex glass: 3.3 x 10⁻⁶ K⁻¹

  • Aluminum oxide (Al₂O₃): 8×10⁻⁶ K⁻¹

polymer

  • Polyethylene: 100-200 × 10⁻⁶ K⁻¹

  • PTFE (Teflon): 100-135×10⁻⁶ K⁻¹

  • Epoxy resin: 45-65 × 10⁻⁶ K⁻¹

Practical application of the coefficient of thermal expansion

1. Construction

  • Design of expansion joints for bridges and skyscrapers

  • Choose compatible materials to avoid cracks.

2. Electronic Products

3. Airspace

  • Choose materials that can withstand extreme temperature fluctuations.

  • Design of components for satellites and spacecraft

4. Automotive industry

  • Design of engine and exhaust system taking thermal expansion into account

  • Selection of materials for plastic and metal components

5. Medicine

  • Manufacturing dental implants with the same coefficient of expansion as teeth

  • Design of precision surgical instruments

Methods for measuring the coefficient of thermal expansion

1. Expansion method

  • The most accurate method for measuring linear expansion

  • Tensile testing devices are used to measure changes in length.

2. Intervention

  • Measurement     of length changes using optical interference patterns

  • Very high accuracy

3. Heat capacity method

  • Relationship between heat capacity and coefficient of thermal expansion

4. Visual methods

  • Use of laser and diffraction technology

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Coefficient of thermal expansion of composite materials

Due to the heterogeneous composition of composite materials, their behavior under temperature changes is quite complex. The effective coefficient of thermal expansion of these materials depends on the following factors:

  1. Matrix phase expansion coefficient

  2. Phase expansion coefficient of the amplifier

  3. Volume fraction per stage

  4. Amplifier design and  phase assignment

Take thermal expansion into account during the design process.

Engineers must consider the effects of thermal expansion when designing mechanical systems. Common solutions include:

  1. Use of expansion joints

  2. Choose materials with similar coefficients of thermal expansion.

  3. Design of sensitive  components  with appropriate tolerances

  4. Use of thermal compensators

Problems caused by thermal expansion

  1. Thermal stresses     : when expansion or contraction is limited

  2. Communication failure  :     in multi-material systems 

  3. Critical dimensional change     : Precision instruments

  4. Creep phenomenon     : at high temperatures

Materials with specific coefficients of thermal expansion

Materials with low thermal expansion

  • Infrared (nickel-iron alloy): ~1.2 × 10⁻⁶ K⁻¹

  • Special ceramics: virtually zero

Materials with negative thermal expansion

  • Some materials shrink in    certain   directions.

  • Application for extension compensation

Mathematical modeling of thermal expansion

Modeling the thermal behavior of materials requires a deep understanding of thermodynamics and solid mechanics. The equations underlying this modeling include the following:

  1. Generalized Hooke’s Law

  2. Structural equation of matter

  3. Relationship between stress and thermal deformation

Standards regarding the coefficient of thermal expansion

Many standards organizations have developed methods for testing and reporting data on the coefficient of thermal expansion:

  1. ASTM E228: Standard test method for linear expansion

  2. ISO 11359: International standard for plastics

  3. DIN 51045: German ceramic standard

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In conclusion

The coefficient of thermal expansion (CTE) is a fundamental   criterion    in material selection for technical applications. A comprehensive understanding of this concept and the ability to predict a material’s behavior under temperature fluctuations can prevent many structural failures. New technological advances enable the development of materials with controlled or even negative CTE, thus opening up new possibilities in structural engineering.

Frequently Asked Questions

1. Is the coefficient of thermal expansion always constant?

No, for many materials this coefficient changes slightly depending on the temperature.

2. Why is the coefficient of thermal expansion of metals generally higher than the coefficient of thermal expansion of ceramics?

Due to differences in atomic bonds and crystal structure.

3. How can the negative effects of thermal expansion be reduced?

By choosing the right materials, designing flexible connections, and using expansion joints.

4. Do gases also have a coefficient of thermal expansion?

Yes, but they behave differently than solids and liquids and are usually described by the ideal gas law.

5. What is the main difference between the linear expansion coefficient and the volumetric expansion coefficient?

The linear coefficient only considers the change in length, while the volume coefficient considers the total change in volume of the material.