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Thermal Expansion Chapter 2 Thermal Expansion THE COEFFICIENT OF LINEAR thermal varies, depending on whether it is specified at a to a sensor by means of push rods. The precision expansion (CTE, α, or α ) is a material property precise temperature (true coefficient of thermal of the test is lower than that of interferometry, 1 − that is indicative of the extent to which a mate- expansion or α or over a temperature range and the test is generally applicable to materials rial expands upon heating. Different substances (mean coefficient of thermal expansion or α). with CTE above 5 × 10–6/K (2.8 × 10–6/°F) over expand by different amounts. Over small tem- The true coefficient is related to the slope of the the temperature range of –180 to 900 °C (–290 perature ranges, the thermal expansion of uni- tangent of the length versus temperature plot, to 1650 °F). Push rods may be of the vitreous sil- form linear objects is proportional to tempera- while the mean coefficient is governed by the ica type, the high-purity alumina type, or the ture change. Thermal expansion finds useful slope of the chord between two points on the isotropic graphite type. Alumina systems can ex- application in bimetallic strips for the construc- curve. Variation in CTE values can occur ac- tend the temperature range up to 1600 °C tion of thermometers but can generate detrimen- cording to the definition used. When α is con- (2900 °F) and graphite systems up to 2500 °C − tal internal stress when a structural part is heated stant over the temperature range then α = α. (4500 °F). ASTM Test Method E 228 (Ref 2) and kept at constant length. Finite-element analysis (FEA) software such as cove the determination of linear thermal expan- For a more detailed discussion of thermal NASTRAN (MSC Software) requires that α be sion of rigid solid materials using vitreous silica − expansion including theory and the effect of input, not α. push rod or tube dilatometers. crystal symmetry, the reader is referred to the Heating or cooling affects all the dimensions Interferometry. With optical interference CINDAS Data Series on Material Properties, of a body of material, with a resultant change in techniques, displacement of the specimen ends Volumes 1 to 4, Thermal Expansion of Solids volume. Volume changes may be determined is measured in terms of the number of wave- (Ref 1). from: lengths of monochromatic light. Precision is sig- nificantly greater than with dilatometry, but be- ∆ α ∆ V/V0 = V T cause the technique relies on the optical Definitions reflectance of the specimen surface, interferom- ∆ where V and V0 are the volume change and etry is not used much above 700 °C (1290 °F). α Most solid materials expand upon heating and original volume, respectively, and V represents ASTM Test Method E 289 (Ref 3) provides a contract when cooled. The change in length with the volume coefficient of thermal expansion. In standard method for linear thermal expansion of α temperature for a solid material can be ex- many materials, the value of V is anisotropic; rigid solids with interferometry that is applicable pressed as: that is, it depends on the crystallographic direc- from –150 to 700 °C (–240 to 1290 °F) and is tion along which it is measured. For materials in more applicable to materials having low or neg- α ∆ α ∆ α < × –6 × (lf – l0)/l0 = 1 (Tf – T0) l/l0 = 1 T which the thermal expansion is isotropic, V is ative CTE in the range of 5 10 /K (2.8 α α –6 1 =1/l(dl/dT) approximately 3 1. 10 /°F) or where only limited lengths of thick- ness of other higher expansion coefficient mate- where l0 and lf represent, respectively, the origi- rials are available. nal and final lengths with the temperature Measurement Thermomechanical analysis measurements α change from T0 to Tf. The parameter 1 CTE and are made with a thermomechanical analyzer has units of reciprocal temperature (K–1) such as To determine the thermal expansion coeffi- consisting of a specimen holder and a probe that µm/m · K or 10–6/K. Conversion factors are: cient, two physical quantities (displacement and transmits changes in length to a transducer that temperature) must be measured on a sample that translates movements of the probe into an elec- To convert To Multiply by is undergoing a thermal cycle. Three of the main trical signal. The apparatus also consists of a 10–6/K 10–6/°F 0.55556 techniques used for CTE measurement are furnace for uniform heating, a temperature-sens- 10–6/°F 10–6/K 1.8 dilatometry, interferometry, and thermomechani- ing element, calipers, and a means of recording –6 ppm/°C 10 /K 1 cal analysis. Optical imaging can also be used at results. ASTM Test Method E 831 (Ref 4) 10–6/°C 10–6/K 1 (µm/m)/°F 10–6/K 1.8 extreme temperatures. X-ray diffraction can be describes the standard test method for linear (µm/m)/°C 10–6/K 1 used to study changes in the lattice parameter but thermal expansion of solid materials by thermo- 10–6/R 10–6/K 1.8 may not correspond to bulk thermal expansion. mechanical analysis. The lower limit for CTE Dilatometry. Mechanical dilatometry tech- with this method is 5 × 10–6/K (2.8 × 10–6/°F), The coefficient of thermal expansion is also niques are widely used. With this technique, a but it may be used at lower or negative expan- often defined as the fractional increase in length specimen is heated in a furnace and displace- sion levels with decreased accuracy and preci- per unit rise in temperature. The exact definition ment of the ends of the specimen are transmitted sion. The applicable temperature range is –120 10 / Thermal Properties of Metals to 600 °C (–185 to 1110 °F), but the temperature 10–6/°F). Aluminum alloys are affected by the Data Tables range may be extended depending on instrumen- presence of silicon and copper, which reduce ex- tation and calibration materials. pansion, and magnesium, which increases it. Its Table 2.1 lists ferrous and nonferrous metal high expansion should be considered when alu- and alloy groups in increasing order of CTE along Application Considerations minum is used with other materials, especially in with the range of CTE values from approximately rigid structures, although the stresses developed room temperature to 100 °C (212 °F). Table 2.2 With respect to temperature, the magnitude of are moderated by the low elastic modulus of alu- list CTE values for specific metals and alloys the CTE increases with rising temperature. minum. If dimensions are very large, as for ex- along with temperature, density, reference, and Thermal expansion of pure metals has been well ample in a light alloy superstructure on a steel qualifying information where available. Table characterized up to their melting points, but data ship or where large pieces of aluminum are set on 2.2 is ordered according to material hierarchy. for engineering alloys at very high temperatures a steel framework or in masonry, then slip joints, Refer to Appendix A.1 for a complete hierarchy. may be limited. In general, CTE values for met- plastic caulking, and other stress-relieving de- als fall between those of ceramics (lower values) vices are usually needed. In the aluminum inter- REFERENCES and polymers (higher values). Common values nal-combustion engine piston that works in an for metals and alloys are in the range of 10 to iron or steel cylinder, differential expansion is 1. R.E. Taylor, CINDAS Data Series on Materi- 30 × 10–6/K (5.5 to 16.5 × 10–6/°F). The lowest countered by the employment of low-expansion als Properties, Thermal Expansion of Solids, expansion is found in the iron-nickel alloys such iron cylinder linings, or by split piston skirts and Vol 1–4, ASM International, 1998 as Invar. Increasing expansion occurs with sili- nonexpanding struts cast into the piston. 2. “Standard Test Method for Linear Thermal con, tungsten, titanium, silver, iron, nickel, steel, Steels. Plain chromium stainless steel grades Expansion of Solid Materials with a Vitreous gold, copper, tin, magnesium, aluminum, zinc, have an expansion coefficient similar to carbon Silica Dilatometer,” E 228-95, Annual Book lead, potassium, sodium, and lithium. (mild) steels, but that of the austenitic grades is of ASTM Standards, ASTM, 1995 1 Low-expansion alloys are materials with di- about 1 ⁄2 times higher. The combination of high 3. “Standard Test Method for Linear Thermal mensions that do not change appreciably with expansion and low thermal conductivity means Expansion of Rigid Solids with Interferome- temperature. Alloys included in this category are that precautions must be taken to avoid adverse try,” E 289-99, Annual Book of ASTM Stan- various binary iron-nickel alloys and several ter- effects. For example, during welding of dards, ASTM, 1999 nary alloys of iron combined with nickel- austenitic grades use low heat input, dissipate 4. “Standard Test Method for Linear Thermal chromium, nickel-cobalt, or cobalt-chromium heat by use of copper backing bars, and use ad- Expansion of Solid Materials by Thermome- alloying. Low-expansion alloys are used in ap- equate jigging. Coefficient of thermal expansion chanical Analysis,” E 831, Annual Book of plications such as rods and tapes for geodetic must be considered in components that use a ASTM Standards, ASTM, 2000 surveying, compensating pendulums and bal- mixture of materials such as heat exchangers ance wheels for clocks and watches, moving with mild steel shells and austenitic grade tubes. SELECTED REFERENCES parts that require control of expansion (such as Welding.
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