thermal expansion - asm international

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THE COEFFICIENT OF LINEAR thermal expansion (CTE, α, or α 1 ) is a material property that is indicative of the extent to which a mate- rial expands upon heating. Different substances expand by different amounts. Over small tem- perature ranges, the thermal expansion of uni- form linear objects is proportional to tempera- ture change. Thermal expansion finds useful application in bimetallic strips for the construc- tion of thermometers but can generate detrimen- tal internal stress when a structural part is heated and kept at constant length. For a more detailed discussion of thermal expansion including theory and the effect of crystal symmetry, the reader is referred to the CINDAS Data Series on Material Properties, Volumes 1 to 4, Thermal Expansion of Solids (Ref 1). Definitions Most solid materials expand upon heating and contract when cooled. The change in length with temperature for a solid material can be ex- pressed as: (l f l 0 )/l 0 = α 1 (T f T 0 ) l/l 0 = α 1 T α 1 = 1/l(dl/dT) where l 0 and l f represent, respectively, the origi- nal and final lengths with the temperature change from T 0 to T f . The parameter α 1 CTE and has units of reciprocal temperature (K –1 ) such as µm/m · K or 10 –6 /K. Conversion factors are: To convert To Multiply by 10 –6 /K 10 –6 /°F 0.55556 10 –6 /°F 10 –6 /K 1.8 ppm/°C 10 –6 /K 1 10 –6 /°C 10 –6 /K 1 (µm/m)/°F 10 –6 /K 1.8 (µm/m)/°C 10 –6 /K 1 10 –6 /R 10–6/K 1.8 The coefficient of thermal expansion is also often defined as the fractional increase in length per unit rise in temperature. The exact definition varies, depending on whether it is specified at a precise temperature (true coefficient of thermal expansion or α or over a temperature range (mean coefficient of thermal expansion or α). The true coefficient is related to the slope of the tangent of the length versus temperature plot, while the mean coefficient is governed by the slope of the chord between two points on the curve. Variation in CTE values can occur ac- cording to the definition used. When α is con- stant over the temperature range then α = α . Finite-element analysis (FEA) software such as NASTRAN (MSC Software) requires that α be input, not α . Heating or cooling affects all the dimensions of a body of material, with a resultant change in volume. Volume changes may be determined from: V/V 0 = α V T where V and V 0 are the volume change and original volume, respectively, and α V represents the volume coefficient of thermal expansion. In many materials, the value of α V is anisotropic; that is, it depends on the crystallographic direc- tion along which it is measured. For materials in which the thermal expansion is isotropic, α V is approximately 3α 1 . Measurement To determine the thermal expansion coeffi- cient, two physical quantities (displacement and temperature) must be measured on a sample that is undergoing a thermal cycle. Three of the main techniques used for CTE measurement are dilatometry, interferometry, and thermomechani- cal analysis. Optical imaging can also be used at extreme temperatures. X-ray diffraction can be used to study changes in the lattice parameter but may not correspond to bulk thermal expansion. Dilatometry. Mechanical dilatometry tech- niques are widely used. With this technique, a specimen is heated in a furnace and displace- ment of the ends of the specimen are transmitted to a sensor by means of push rods. The precision of the test is lower than that of interferometry, and the test is generally applicable to materials with CTE above 5 × 10 –6 /K (2.8 × 10 –6 /°F) over the temperature range of –180 to 900 °C (–290 to 1650 °F). Push rods may be of the vitreous sil- ica type, the high-purity alumina type, or the isotropic graphite type. Alumina systems can ex- tend the temperature range up to 1600 °C (2900 °F) and graphite systems up to 2500 °C (4500 °F). ASTM Test Method E 228 (Ref 2) cove the determination of linear thermal expan- sion of rigid solid materials using vitreous silica push rod or tube dilatometers. Interferometry. With optical interference techniques, displacement of the specimen ends is measured in terms of the number of wave- lengths of monochromatic light. Precision is sig- nificantly greater than with dilatometry, but be- cause the technique relies on the optical reflectance of the specimen surface, interferom- etry is not used much above 700 °C (1290 °F). ASTM Test Method E 289 (Ref 3) provides a standard method for linear thermal expansion of rigid solids with interferometry that is applicable from –150 to 700 °C (–240 to 1290 °F) and is more applicable to materials having low or neg- ative CTE in the range of <5 × 10 –6 /K (2.8 × 10 –6 /°F) or where only limited lengths of thick- ness of other higher expansion coefficient mate- rials are available. Thermomechanical analysis measurements are made with a thermomechanical analyzer consisting of a specimen holder and a probe that transmits changes in length to a transducer that translates movements of the probe into an elec- trical signal. The apparatus also consists of a furnace for uniform heating, a temperature-sens- ing element, calipers, and a means of recording results. ASTM Test Method E 831 (Ref 4) describes the standard test method for linear thermal expansion of solid materials by thermo- mechanical analysis. The lower limit for CTE with this method is 5 × 10 –6 /K (2.8 × 10 –6 /°F), but it may be used at lower or negative expan- sion levels with decreased accuracy and preci- sion. The applicable temperature range is –120 Chapter 2 Thermal Expansion © 2002 ASM International. All Rights Reserved. ASM Ready Reference: Thermal Properties of Metals (#06702G) www.asminternational.org

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Page 1: Thermal Expansion - ASM International

THE COEFFICIENT OF LINEAR thermalexpansion (CTE, α, or α1) is a material propertythat is indicative of the extent to which a mate-rial expands upon heating. Different substancesexpand by different amounts. Over small tem-perature ranges, the thermal expansion of uni-form linear objects is proportional to tempera-ture change. Thermal expansion finds usefulapplication in bimetallic strips for the construc-tion of thermometers but can generate detrimen-tal internal stress when a structural part is heatedand kept at constant length.

For a more detailed discussion of thermalexpansion including theory and the effect ofcrystal symmetry, the reader is referred to theCINDAS Data Series on Material Properties,Volumes 1 to 4, Thermal Expansion of Solids(Ref 1).

Definitions

Most solid materials expand upon heating andcontract when cooled. The change in length withtemperature for a solid material can be ex-pressed as:

(lf – l0)/l0 = α1 (Tf – T0) ∆l/l0 = α1∆Tα1 = 1/l(dl/dT)

where l0 and lf represent, respectively, the origi-nal and final lengths with the temperaturechange from T0 to Tf. The parameter α1 CTE andhas units of reciprocal temperature (K–1) such asµm/m · K or 10–6/K. Conversion factors are:

To convert To Multiply by

10–6/K 10–6/°F 0.5555610–6/°F 10–6/K 1.8ppm/°C 10–6/K 110–6/°C 10–6/K 1

(µm/m)/°F 10–6/K 1.8(µm/m)/°C 10–6/K 1

10–6/R 10–6/K 1.8

The coefficient of thermal expansion is alsooften defined as the fractional increase in lengthper unit rise in temperature. The exact definition

varies, depending on whether it is specified at aprecise temperature (true coefficient of thermalexpansion or α− or over a temperature range(mean coefficient of thermal expansion or α).The true coefficient is related to the slope of thetangent of the length versus temperature plot,while the mean coefficient is governed by theslope of the chord between two points on thecurve. Variation in CTE values can occur ac-cording to the definition used. When α is con-stant over the temperature range then α = α−.Finite-element analysis (FEA) software such asNASTRAN (MSC Software) requires that α beinput, not α−.

Heating or cooling affects all the dimensionsof a body of material, with a resultant change involume. Volume changes may be determinedfrom:

∆V/V0 = αV∆T

where ∆V and V0 are the volume change andoriginal volume, respectively, and αV representsthe volume coefficient of thermal expansion. Inmany materials, the value of αV is anisotropic;that is, it depends on the crystallographic direc-tion along which it is measured. For materials inwhich the thermal expansion is isotropic, αV isapproximately 3α1.

Measurement

To determine the thermal expansion coeffi-cient, two physical quantities (displacement andtemperature) must be measured on a sample thatis undergoing a thermal cycle. Three of the maintechniques used for CTE measurement aredilatometry, interferometry, and thermomechani-cal analysis. Optical imaging can also be used atextreme temperatures. X-ray diffraction can beused to study changes in the lattice parameter butmay not correspond to bulk thermal expansion.

Dilatometry. Mechanical dilatometry tech-niques are widely used. With this technique, aspecimen is heated in a furnace and displace-ment of the ends of the specimen are transmitted

to a sensor by means of push rods. The precisionof the test is lower than that of interferometry,and the test is generally applicable to materialswith CTE above 5 × 10–6/K (2.8 × 10–6/°F) overthe temperature range of –180 to 900 °C (–290to 1650 °F). Push rods may be of the vitreous sil-ica type, the high-purity alumina type, or theisotropic graphite type. Alumina systems can ex-tend the temperature range up to 1600 °C(2900 °F) and graphite systems up to 2500 °C(4500 °F). ASTM Test Method E 228 (Ref 2)cove the determination of linear thermal expan-sion of rigid solid materials using vitreous silicapush rod or tube dilatometers.

Interferometry. With optical interferencetechniques, displacement of the specimen endsis measured in terms of the number of wave-lengths of monochromatic light. Precision is sig-nificantly greater than with dilatometry, but be-cause the technique relies on the opticalreflectance of the specimen surface, interferom-etry is not used much above 700 °C (1290 °F).ASTM Test Method E 289 (Ref 3) provides astandard method for linear thermal expansion ofrigid solids with interferometry that is applicablefrom –150 to 700 °C (–240 to 1290 °F) and ismore applicable to materials having low or neg-ative CTE in the range of <5 × 10–6/K (2.8 ×10–6/°F) or where only limited lengths of thick-ness of other higher expansion coefficient mate-rials are available.

Thermomechanical analysis measurementsare made with a thermomechanical analyzerconsisting of a specimen holder and a probe thattransmits changes in length to a transducer thattranslates movements of the probe into an elec-trical signal. The apparatus also consists of afurnace for uniform heating, a temperature-sens-ing element, calipers, and a means of recordingresults. ASTM Test Method E 831 (Ref 4)describes the standard test method for linearthermal expansion of solid materials by thermo-mechanical analysis. The lower limit for CTEwith this method is 5 × 10–6/K (2.8 × 10–6/°F),but it may be used at lower or negative expan-sion levels with decreased accuracy and preci-sion. The applicable temperature range is –120

Chapter 2

Thermal Expansion

© 2002 ASM International. All Rights Reserved.ASM Ready Reference: Thermal Properties of Metals (#06702G)

www.asminternational.org

Page 2: Thermal Expansion - ASM International

to 600 °C (–185 to 1110 °F), but the temperaturerange may be extended depending on instrumen-tation and calibration materials.

Application Considerations

With respect to temperature, the magnitude ofthe CTE increases with rising temperature.Thermal expansion of pure metals has been wellcharacterized up to their melting points, but datafor engineering alloys at very high temperaturesmay be limited. In general, CTE values for met-als fall between those of ceramics (lower values)and polymers (higher values). Common valuesfor metals and alloys are in the range of 10 to 30 × 10–6/K (5.5 to 16.5 × 10–6/°F). The lowestexpansion is found in the iron-nickel alloys suchas Invar. Increasing expansion occurs with sili-con, tungsten, titanium, silver, iron, nickel, steel,gold, copper, tin, magnesium, aluminum, zinc,lead, potassium, sodium, and lithium.

Low-expansion alloys are materials with di-mensions that do not change appreciably withtemperature. Alloys included in this category arevarious binary iron-nickel alloys and several ter-nary alloys of iron combined with nickel-chromium, nickel-cobalt, or cobalt-chromiumalloying. Low-expansion alloys are used in ap-plications such as rods and tapes for geodeticsurveying, compensating pendulums and bal-ance wheels for clocks and watches, movingparts that require control of expansion (such aspistons for some internal-combustion engines),bimetal strip, glass-to-metal seals, thermostaticstrip, vessels and piping for storage and trans-portation of liquefied natural gas, superconduct-ing systems in power transmissions, integrated-circuit lead frames, components for radios andother electronic devices, and structural compo-nents in optical and laser measuring systems.

Aluminum and Aluminum Alloys. The di-mensional change of aluminum and its alloyswith a change of temperature is roughly twicethat of the ferrous metals. The average CTE forcommercially pure metal is 24 × 10–6/K (13 ×

10–6/°F). Aluminum alloys are affected by thepresence of silicon and copper, which reduce ex-pansion, and magnesium, which increases it. Itshigh expansion should be considered when alu-minum is used with other materials, especially inrigid structures, although the stresses developedare moderated by the low elastic modulus of alu-minum. If dimensions are very large, as for ex-ample in a light alloy superstructure on a steelship or where large pieces of aluminum are set ona steel framework or in masonry, then slip joints,plastic caulking, and other stress-relieving de-vices are usually needed. In the aluminum inter-nal-combustion engine piston that works in aniron or steel cylinder, differential expansion iscountered by the employment of low-expansioniron cylinder linings, or by split piston skirts andnonexpanding struts cast into the piston.

Steels. Plain chromium stainless steel gradeshave an expansion coefficient similar to carbon(mild) steels, but that of the austenitic grades isabout 11⁄2 times higher. The combination of highexpansion and low thermal conductivity meansthat precautions must be taken to avoid adverseeffects. For example, during welding ofaustenitic grades use low heat input, dissipateheat by use of copper backing bars, and use ad-equate jigging. Coefficient of thermal expansionmust be considered in components that use amixture of materials such as heat exchangerswith mild steel shells and austenitic grade tubes.

Welding. The coefficient of thermal expan-sion is an important factor when welding twodissimilar base metals. Large differences in theCTE values of adjacent metals during coolingwill induce tensile stress in one metal and com-pressive stress in the other. The metal subject totensile stress may hot crack during welding, or itmay cold crack in service unless the stresses arerelieved thermally or mechanically. This factoris particularly important in joints that will oper-ate at elevated temperatures in a cyclic tempera-ture mode. A common example of this isaustenitic stainless steel/ferritic steel pipe buttjoints used in energy-conversion plants.

Data Tables

Table 2.1 lists ferrous and nonferrous metaland alloy groups in increasing order of CTE alongwith the range of CTE values from approximatelyroom temperature to 100 °C (212 °F). Table 2.2list CTE values for specific metals and alloysalong with temperature, density, reference, andqualifying information where available. Table2.2 is ordered according to material hierarchy.Refer to Appendix A.1 for a complete hierarchy.

REFERENCES

1. R.E. Taylor, CINDAS Data Series on Materi-als Properties, Thermal Expansion of Solids,Vol 1–4, ASM International, 1998

2. “Standard Test Method for Linear ThermalExpansion of Solid Materials with a VitreousSilica Dilatometer,” E 228-95, Annual Bookof ASTM Standards, ASTM, 1995

3. “Standard Test Method for Linear ThermalExpansion of Rigid Solids with Interferome-try,” E 289-99, Annual Book of ASTM Stan-dards, ASTM, 1999

4. “Standard Test Method for Linear ThermalExpansion of Solid Materials by Thermome-chanical Analysis,” E 831, Annual Book ofASTM Standards, ASTM, 2000

SELECTED REFERENCES

● J.E. Eltherington & Son (Aluminium) Ltd,http://www.eltherington.co.uk/, 2002

● Interdisciplinary Research Centre in Com-puter Aided Materials Engineering, MaterialsEngineering Department, University of Wales,Swansea, U.K., http://irc.swansea.ac.uk/, 1998;revised 2001

● Metals Handbook Desk Edition, 2nd ed., ASMInternational, 1998

● R. Nave, Hyperphysics, Georgia State Univer-sity, http://hyperphysics.phy-astr.gsu.edu, 2002

● Welding, Brazing, and Soldering, Vol 6, ASMHandbook, ASM International, 1993

10 / Thermal Properties of Metals

© 2002 ASM International. All Rights Reserved.ASM Ready Reference: Thermal Properties of Metals (#06702G)

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Page 3: Thermal Expansion - ASM International

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