Coefficient of thermal expansion - A272

Coefficient of thermal expansion
Foundational knowledge article
Document Type	Article
Document Identifier	272
Themes	Residual stress and dimensional control management
Relevant Class	Material
Tags	Material parameters Matrix Process modelling Thermoset polymers
Prerequisites	Basic Definitions of Material Properties

Introduction

The coefficient of thermal expansion (CTE) is a material property representing the extent to which a material will contract or expand under a given temperature change . It can be used to predict changes in length, area, and volume due to changes in temperature. Although generally reported as a single value, the CTE for most materials varies with both temperature and direction. The direction-dependent effect is most noticeable with anisotropic materials like carbon fiber.

Scope

This page defines the coefficient of thermal expansion, explains its significance in composites processing, and provides some typical values. Measurement and processing effects are briefly discussed. Links to ASTM measurement techniques are provided, but are not discussed in great detail on this page, as this is covered in CMH-17 and the provided ASTM links.

Significance

In composite processing, the coefficient of thermal expansion (CTE) is an important material property that is heavily considered in both part and tooling design. Expansion of the composite part and the tooling during the cure cycle can have large effects on the final geometry of the part. The mismatch between the CTEs of the composite material and the tooling can introduce residual thermal stresses upon cooling. These residual stresses can show up as localized stress concentrations, warping, and distortion, which may compromise dimensional tolerances and structural integrity, and eventually lead to premature failure of the composite part. Accurate consideration and compensation for CTE mismatches are therefore essential in process development.

Significance for Tooling

Effect of tooling in a thermal management system

The mismatch between the CTEs of the composite part and the tooling is one of the biggest factors in tooling design and tool material selection, as it can significantly affect the formation of defects during processing. Metal tooling with a high CTE (such as aluminum) will experience greater dimensional changes throughout the cure cycle and is more likely to warp, inducing stresses in the lower CTE part . This results in a common defect known as “spring-in,” where the final angle of the part is more acute than the angle formed by the tooling. This is due to residual stresses caused by differences in CTEs during cooling. With enough data analysis, some of this effect can be mitigated through tooling design; however, the introduction of stresses in general is not desirable. Most often, this results in poor dimensional accuracy of the part. To reduce this effect, when high-dimensional accuracy or complex geometry is required, material with a low CTE (such as Invar or CFRP) is more often used . Other considerations for tooling material selection include tool lifetime, cost, and heat capacity. Metal tooling is often more robust and can withstand many more cycles than a composite tool, although they generally have a much higher initial cost. Materials such as Invar have a high heat capacity, meaning they take more time and energy to heat up. This can lead to a lower part temperature and must be considered during the cure cycle. CFRP has a much lower heat capacity, meaning that it heats up faster, but results in much more uneven temperature distribution. A case study on different tooling materials can be seen here:Conducting a thermal tooling survey on three complex tools of different materials.

Significance for Assembly

Macro-Mechanics

It is important to consider CTEs at a higher level than the laminate and component level. Composite parts are frequently combined with other materials and joined together through bonding and bolted connections. CTE mismatch can play a huge role in the quality of a bonded connection, especially when the connection is through a metal and a composite, such as the bonding of aluminum to carbon fiber. Since the CTE difference is so high for these materials, it is important that testing is performed on the bonded part to ensure that stresses induced at the extreme ends of the operating conditions do not result in part failure or too small of a safety factor . To model this effect, Laminated Plate Theory and Finite Element Analysis (FEA) can be used.

Level I

Level II

Definition

The linear coefficient of thermal expansion is defined as a value relating the change in length of a material to the change in temperature. It is the slope of a strain versus temperature graph between two temperatures [4]. For materials where this curve is non-linear, separate CTEs can be calculated for each approximated linear region. Changes in the slope (and therefore the CTE) can point towards other thermal events, such as the Glass transition temperature (Tg). The linear coefficient of thermal expansion \(\alpha_l\) is expressed as:

\( \alpha_l = \frac{\Delta l}{l_0 \Delta T} \)

With the following variable definitions:

• \(\Delta l\) – The change in length [mm]
• \(l_0\) – initial length [mm]
• \(\Delta T\) – change in temperature \([\mathrm{K}]\)

The area coefficient of thermal expansion \(\alpha_v\) is expressed as:

\( \alpha_a = \frac{\Delta A}{A_0 \Delta T} \)

With the following variable definitions:

• \(\Delta A\) – change in area \([\mathrm{mm}^2]\)
• \(A_0\) – initial area \([\mathrm{mm}^2]\)
• \(\Delta T\) – change in temperature \([\mathrm{K}]\)

\( \alpha_v = \frac{\Delta V}{V_0 \Delta T} \)

With the following variable definitions:

• \(\Delta V\) – change in Volume \([\mathrm{mm}^3]\)
• \(V_0\) – initial Volume \([\mathrm{mm}^3]\)
• \(\Delta T\) – change in temperature \([\mathrm{K}]\)

where:

\(\alpha_a = 2\,\alpha_l\) \(\alpha_v = 3\,\alpha_l\)

This is not true for orthotropic materials such as composites, where their CTE is highly dependent on the direction of the fiber. For example, carbon fiber typically has a negative axial CTE and a positive transverse CTE .

Terminology and Notation

The general coefficient of thermal expansion is often referred to in literature as its abbreviation, CTE. In both equations and literature, it is also frequently represented by the Greek letter \(\alpha\). Depending on the type of thermal expansion (linear, volume, axial, transverse), \(\alpha\) will include a relevant subscript representing the specific property. For example, the linear coefficient of thermal expansion is denoted as \(\alpha_l\) while the volume coefficient of thermal expansion is denoted as \(\alpha_v\).

\( \alpha = \frac{Length Unit}{Length Unit* Temperature Unit} \)

Typical Property Values

Composite Use	Material	CTE (×10⁻⁶/K)	Reference
Matrix Materials	Epoxy	45–65
	Polyester	55–100
Reinforcement Fibers	E-Glass	5.04
	S-Glass	5.58
	GY70	−1.08 (axial)
Tooling Material	Aluminum (Alloy 2014)	22.5
	Steel (High-carbon 1080)	11.0
	Invar	1.5

Measurement

Thermomechanical Analyzer (TMA) CMH-17, Volume 1 provides a detailed list of which ASTM standards to follow to obtain the desired coefficients of thermal expansion. The standards are listed below and make use of Thermomechanical Analysis, Dilatometers, and Interferometers to measure the coefficient of thermal expansion. All methods are capable of monitoring dimensional changes over changes in temperature, with different levels of precision. Interferometry is capable of the highest precision, followed by the push-rod dilatometer and thermomechanical analysis.

Standard	Description
ASTM E228	Standard Test Method for Linear Thermal Expansion of Solid Materials with a Push-Rod Dilatometer
ASTM E831	Standard Test Method for Linear Thermal Expansion of Solid Materials by Thermomechanical Analysis
ASTM E289	Standard Test Method for Linear Thermal Expansion of Rigid Solids with Interferometry

Material Microstructure Dependence

At the atomic level, expansion due to a thermal change represents an increase in the average interatomic distance, whereas contraction represents a decrease in the average interatomic distance. This is better visualized through the use of a potential energy vs interatomic distance graph, seen in Figure 1 .

Figure 1 Potential energy vs interatomic distance of atoms.

Thermal expansion and contraction are mainly due to the asymmetric nature of this graph. Since the center point of the energy level shifts towards the right as vibrational energy increases, the distance between atoms increases as well. The symmetry of this graph is affected by the strength of the bonds in the material. As a material’s bond strength increases, the trough deepens and becomes more symmetric. With weaker bonds, the trough is shallow and leads to quick and drastic changes in the interatomic distance .

Macrostructure Dependence

Composite materials are highly anisotropic, meaning that their material properties are strongly direction-dependent. This anisotropy significantly influences the coefficient of thermal expansion (CTE), mostly due to fibre architecture, fibre volume fraction, and fibre orientation. Unidirectional prepreg material that is aligned in the longitudinal direction will experience a very low CTE longitudinally as the fibre itself has a low CTE. In the transverse direction, the CTE will be much higher as the behaviour is dominated by that of the matrix . Woven prepreg is less anisotropic since there are fibres in both directions; however, since the fibre bends slightly during the weave, it is less stiff, leading to a higher CTE. Woven pre-preg also tends to have a lower fibre volume fraction, which contributes to the higher CTE. Unidirectional material tends to have a higher fibre volume fraction, resulting in a lower CTE as the properties are more heavily dominated by the fibre .

Prcoess and Environmental Dependence

The CTE of a composite material is affected by many factors. Fiber and void volume, layup angle and sequence, thermal cycles, and moisture content can all have drastic effects . Temperature and architecture of a part both have large effects on the final part as they both have strong effects on the part’s CTE.

Temperature

Values of CTE are typically reported as a single number; however, they can change quite drastically across different temperature ranges. As seen in the figure representing the relationship between energy and interatomic distance, there is an inflection point at Tg where the slope of the graph representing CTE changes. It is linear and therefore has a constant value before and after this change. Due to the large temperature ranges followed during a cure cycle, the CTEs of the tooling and constituent materials will vary to some degree.

Figure 2 Ideal results from a TMA test of a polymer, showing two CTEs and the Tg at the inflection point.

Residual Stress

Stresses can be introduced into prepreg material during its manufacturing process, caused by the different shrink rates caused by the difference between the CTEs of the fibre and the matrix, as well as the temperature gradient over the material . These residual stresses can result in curling under heating, which can affect CTE measurements. It has been observed that this effect can result in negative CTE measurements in certain temperature ranges of a material that would normally have a positive CTE. This effect can be mitigated by annealing the material above Tg and below Tm, to avoid inducing permanent deformation .

Procedure

Due to the anisotropic nature of composite materials, the CTE varies strongly depending on the fiber orientation. This is especially noticeable for carbon fiber laminates, as the CTE is slightly positive in the transverse direction and slightly negative in the axial direction. Since the material properties of composites are determined through manufacturing, the CTE can be designed within the limits of the material by varying the orientations of the fibers during the layup . By alternating 0, 45, and 90-degree plies, a more uniform CTE can be obtained to mimic the effects of a more isotropic material.

Related pages

Page type	Links
Introduction to Composites Articles
Foundational Knowledge Articles	Coefficient of cure shrinkage - A273 Coefficient of thermal expansion - A272 Conduction - A118 Convection - A106 Cure of thermosetting polymers - A162 Degree of cure - A104 Glass transition temperature (Tg) - A210 Heat of reaction - A114 Specific heat capacity - A117 Thermal conductivity - A116 Thermal diffusivity - A143 Thermal phase transitions of polymers - A102 Thermoplastic polymers - A161 Thermoset polymers - A105 Viscosity (resin) - A203
Foundational Knowledge Method Documents	How to measure curing time and degree of cure - M100 How to measure gel time - M101 How to measure reinforcement content (and corresponding matrix content) - M109
Foundational Knowledge Worked Examples
Systems Knowledge Articles
Systems Knowledge Method Documents
Systems Knowledge Worked Examples
Systems Catalogue Articles
Systems Catalogue Objects – Material
Systems Catalogue Objects – Shape
Systems Catalogue Objects – Tooling and consumables
Systems Catalogue Objects – Equipment	Rheometer - A357
Practice Documents	Ply Draping Modelling for Structural Analysis and Manufacturing - P174 Practice for developing a thermal transformation process step - P105
Case Studies	Improving Quality of a Truck Fender Using Ply Draping Modelling - C122 Troubleshooting of room temperature processes for large recreational and industrial parts - C100
Perspectives Articles	Case Study: Optimizing a Press Moulding Process - A324 Composite materials engineering webinar session 1 - Introduction - A120 Composite materials engineering webinar session 3 - Constituent materials - Resin - A122 Composite materials engineering webinar session 4 - Thermal management and resin cure - A123 Composites Process Simulation: A Review of the State of the Art for Product Development - A283 Dr. Anoush Poursartip's cdmHUB global composites expert webinar - A136 Fabric Forming: how it affects design and processing, and how simulation can address this - A310 Interview with Prof. Kevin Potter - A134 Simulation models for rapid liquid composite molding - A333.0 Understanding Polyester Resin Processing: The Effect of Ambient Temperature on the Final Part - A286

References

About	Help

Engineered materials (designed to have specific properties) made from two or more constituent materials with different physical or chemical properties. The constituents remain separate and distinct on a macroscopic level within the finished structure.

Fibre architecture refers to how the fibre is assembled.

Most common architectures:

• Unidirectional
• Chopped strand mat (CSM) and chopped fibre (chopper gun)
• Continuous filament mat (CFM)
• Woven fabric
• Non crimp fabric (NCF)
• Braided
• Prepreg

Volume fraction of either matrix or fibres with respect to total composite volume (matrix + fibre).

Pre-impregnated (prepreg) material refers to fibre that is already combined with resin. It is the most common material form used in aerospace.

During prepreg production, (e.g. fibres are run through a resin bath), prepreg is heated and partially cured to B Stage (< 5 % degree of cure). Thermoset prepregs (e.g. epoxy prepreg) have to be kept in a freezer at around -20 °C. At room temperature, the epoxy starts to cure.

The continuous material phase that binds the reinforcement together, maintains shape, transfers load, protects the reinforcement from environment and damage, and provides the composite support in compression.

Desirable characteristics:

• Moisture/chemical resistance
• Low density
• Processability

The individual materials that combine to form the composite material. The constituent materials are separate and distinct on a macroscopic level.