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Measurement of Thermal Expansion Coefficient Using Strain Gages

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Measurement of Thermal Expansion Coefficient Using Strain Gages MICRO-MEASUREMEntS Strain Gages and Instruments Tech Note TN-513-1 Measurement of Thermal Expansion Coefficient Using Strain Gages The thermal expansion coefficient is a very basic physical task of determining directional expansion coefficients of property which can be of considerable importance in materials with anisotropic thermal properties. mech an ical and structural design applications of a Because typical expansion coefficients are measured material. Al though there are many published tabulations in terms of a few parts per million, close attention to of expansion coefficients for the common metals and procedural detail is required with any measurement standard alloys, the need occasionally arises to measure method to obtain accurate results; and the strain gage this property for a specific material over a particular method is not an exception to the rule. This Tech Note has temperature range. In some cases (e.g., new or special been prepared as an aid to the gage user in utilizing the full alloys, composites, etc.), there is apt to be no published precision of the modern foil strain gage for determining data whatsoever on expansion coefficients. In others, data expansion coefficients. Given in the first of the following may exist (and eventually be found), but may encompass sections is an explanation of the technical principles the wrong temperature range, apply to somewhat different underlying the method. The next section describes, material, or be otherwise unsuited to the application. in some detail, the strain-gage-related materials and Historically, the classical means for measuring expansion procedures in making the measurement. Basically, the coefficients has been the “dilatometer”. In this type of latter consists of essentially the same techniques required instru ment, the difference in expansion between a rod for any high-precision strain measurement in a variable made from the test material and a matching length of thermal environment. Suggested refinements for achieving quartz or vitreous silica is compared1,2. Their differential maximum accuracy are then given in the following section; expansion is measured with a sensitive dial indicator, after which, the principal limitations of the method are or with an electrical dis place ment transducer. When described. necessary, the expansion properties of the quartz or silica can be calibrated against the accurately known expansion Principle of The Measurement Method of pure platinum or copper. The instrument is normally inserted in a special tubular furnace or liquid bath to obtain When a resistance strain gage is installed on a stress-free the required temperatures. Making measurements with the specimen of any test material, and the temperature of dila tometer is a delicate, demanding task, however, and the material is changed, the output of the gage changes is better suited to the materials science laboratory than correspondingly. This effect, present in all resistance strain to the typical experimental stress analysis facility. This gages, was formerly referred to as “temperature-induced Tech Note provides an alternate method for easily and apparent strain”, but is currently defined asthermal output3. quite accur ately measuring the expansion coefficient of a It is caused by a combination of two factors. To begin with, test material with respect to that of any reference material in common with the behavior of most conductors, the having known expansion characteristics. resistivity of the grid alloy changes with temperature. An additional resistance change occurs because the thermal The technique described here uses two well-matched strain expansion coefficient of the grid alloy is usually different gages, with one bonded to a specimen of the reference from that of the test material to which it is bonded. material, and the second to a specimen of the test material. Thus, with temperature change, the grid is mechanically The specimens can be of any size or shape compatible with strained by an amount equal to the difference in expansion the available equipment for heating and refrigeration (but coefficients. Since the gage grid is made from a strain- speci mens of uniform cross section will minimize potential sensitive alloy, it produces a resistance change proportional NO TECH prob lems with temperature gradients). Under stress-free to the thermally induced strain. The thermal output of the conditions, the differential output between the gages on gage is due to the combined resistance changes from both the two specimens, at any common temperature, is equal to sources. The net resistance change can be expressed as the dif ferential unit expansion (in/in, or m/m). Aside from the sum of resistivity and differential expansion effects as the basic simplicity and relative ease of making thermal follows: expansion measurements by this method, it has the distinct ΔR advantage of requiring no specialized instruments beyond T =+βαGs()− αGGFT Δ (1) those normally found in a stress analysis laboratory. This R E technique can also be applied to the otherwise difficult Document Number: 11063 For technical support, contact www.micro-measurements.com Revision: 01-Nov-2010 [email protected] 119 TN-513-1 Micro-Measurements Measurement of Thermal Expansion Coefficient Using Strain Gages where: TEMPERATURE IN º CELSIUS Lot No. A38AD497 ΔR/R = unit resistance change -50 0+50 +100 +150 -200 +250 +400 βG = thermal coefficient of resistivity of A STD grid material +300 +200 24ºC N +100 α – α = difference in thermal expansion coefficients of 2.00) s G F. between specimen and grid, respectively 0 -100 F = gage factor of the strain gage 75ºF G -200 B ΔT = temperature change from arbitrary initial ARENT MICROSTRAI -300 reference temperature APP -400 (Based on Instrument G. -500 -100 0+100+200+300+400+500 The indicated strain due to a resistance change in the gage TEMPERATURE IN º FARENHEIT is: Figure 1 – Rotation of the thermal output from ΔRR/ ε = (2) a strain gage when installed on materials with i F I differing thermal expansion coefficients. where: F = instrument gage factor setting I other material with a different coefficient of expansion, the result is to effectively rotate the curve in Figure 1 about its reference point at +75°F [+24°C]. Installation on a material Then, the thermal output in strain units can be expressed with a higher coefficient of expansion than steel will rotate as: the curve counterclockwise, while a material with a lower expansion coefficient than steel will cause clockwise rotation. For example, the broken curve labeled A in the βαGS+−()αGGFT Δ ε = (3) figure illustrates the general effect of installing a gage from TO/(GS/) F I the subject lot on a beryllium alloy having an expansion coefficient of about 9 x 10-6/°F [16 x 10-6/°C]. Similarly, where: if a gage from this lot were bonded to a titanium alloy with a somewhat lower expansion coefficient than steel, εT/O(G/S) = thermal output for grid alloy G on specimen material S the thermal output would be shifted in the manner of the broken curve labeled B. The principle of measuring expansion coefficients with Or, in the usual case, with the instrument gage factor set strain gages then becomes evident from Figure 1, since equal to that of the strain gage, so that FI = FG, the rotation from one thermal output curve to the other is due only to the difference in thermal expansion properties between the materials represented by the two curves. An βG algebraic demonstration of the principle can be obtained εTO//GS =+()ααSG− ΔT (4) () F G by rewriting Equation (4) twice; once for the gage installed on a specimen of the test material of unknown expansion It should not be assumed from the form of Equation (4) coefficientα S, and again for the same type of gage installed that the thermal output is linear with temperature, since on a standard reference material with a known expansion all of the coefficients within the brackets are themselves coefficientα R: functions of temperature. As an example, typical thermal output characteristics for a Micro-Measurements A-alloy E gage (self-temperature-compensated constantan grid), βG εTO//()GS =+ ()ααSG− ΔT (5a) T bonded to steel, are represented by the solid curve in FG Figure 1. The lot of foil identified in the upper right corner of the graph was specially processed to minimize the thermal output over the temperature range from about –50° βG to +300°F [–45° to +150°C]. Strain gages fabricated from εTO//()GR =+ ()ααRG− ΔT (5b) FG this lot of foil are intended for use only on material such as steel with a coefficient of expansion of approximately -6 -6 TECH NO TECH 6 x 10 /°F [11 x 10 /°C]. If the gages are installed on some www.micro-measurements.com For technical questions, contact Document Number: 11063 120 [email protected] Revision: 01-Nov-2010 TN-513-1 Micro-Measurements Measurement of Thermal Expansion Coefficient Using Strain Gages Subtracting Equation (5b) from (5a), and rearranging, An excellent reference material with these and the other desirable properties is ULE™ Titanium Silicate Code 7972, available from Corning Glass Company, Corning, NY εε− ()TO/(GS/) TO/(GR/) 14831.* As illustrated in Figure 2, this special glass has an ααSR−= (6) ΔT extremely low thermal expansion coefficient, particularly Thus, the difference in expansion coefficients, referred over the temperature range from about –50° to +350°F to a particular temperature range, is equal to the unit [–45° to +175°C)]. It should be noted, however, that the difference in thermal output for the same change in material has a low coefficient of thermal conductivity, temperature. Although this technique for measuring making it slow to reach thermal equilibrium. For optimum expansion coefficients is widely applicable, and often results, a dwell time of at least 45 minutes should be used at the most practical approach, there is relatively little each new temperature point before taking data.
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