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. mech­an­ical and structural design applications of a Because typical expansion coefficients are measured material. Al­though 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 instru­ment, 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 dis­place­ment 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 dila­tometer 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 accur­ately 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- speci­mens of uniform cross section will minimize potential sensitive alloy, it produces a resistance change proportional TechNo prob­lems 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 dif­ferential 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 e =+βαGs()− αGGFT Δ (1) those normally found in a stress analysis laboratory. This R   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 Tech Note TN-513-1 120 www.micro-measurements.com

equal to that of the strain gage, so that F that so gage, set strain of the factor that to equal gage instrument the with case, usual the in Or, F is: gage the in change aresistance to due strain indicated The Δ

ε α where: where: where: as: expressed be can units strain in output thermal the Then, as steel with a coefficient of expansion of approximately approximately of 10x 6 expansion of coefficient a with such steel as material on only use for intended are foil of lot this from fabricated the gages +150°C].Strain to [–45° +300°F minimize to to processed output thermal over range from the specially about temperature –50° was graph the of in corner right curve upper the in solid 1. lot ofidentified foil The Figure the grid), by represented are constantan steel, to bonded (self-temperature-compensated gage A-alloy Micro-Measurements a for characteristics output thermal typical example, an As themselves temperature. are of functions brackets the within coefficients the since of temperature, all with linear is output (4) thermal Equation the of that form the from assumed be not should It where: F where: ε ε T/O(G/S) TO TO /( s // -6 – Δ Measurement ofThermal ExpansionCoefficient UsingStrainGages /°F [11 /°F 10x () R/R GS GS α β /) G T = G ε G = = i = gage factor of the strain gage strain of the factor =gage I = change resistance =unit = instrument gage factor setting factor gage =instrument = =+ = between specimen and grid, respectively grid, and specimen between coefficients expansion thermal in difference Micro-Measurements reference temperature reference initial arbitrary from change temperature specimen material S material specimen Gon alloy for output grid thermal grid material grid of resistivity of coefficient thermal      Δ βα β F RR GS F G G I / -6 +− /°C]. If the gages are installed on some some on installed are gages the If /°C]. () () αα

SG F − I α GG FT    Δ T   Δ

I =F [email protected] G , For technical questions,contact (3) (4) (2)

coefficient of about 9 x 10 x 9 about expansion of an coefficient having alloy beryllium a on lot subject the clockwise a gagefrom ofinstalling effect thegeneral cause illustrates the figure in A will labeled curve steel broken the example, than For rotation. coefficient expansion lower a with material a while counterclockwise, curve the rotate will steel than ofexpansion coefficient a higher with at point +75°Freference [+24°C]. on a material Installation its 1about Figure in curve the rotate effectively to is result the ofexpansion, coefficient a with different material other coefficient coefficient expansion known a with material reference standard a on coefficient coefficient expansion unknown of material test the of specimen a on installed for gage (4)the once Equation twice; by rewriting obtained be can principle the of demonstration An algebraic curves. two the by represented materials the between properties expansion thermal in difference the to only due since 1, is other the to Figure curve output from thermal one from rotation the evident becomes then gages with strain coefficients expansion measuring of principle The B labeled curve broken the of manner the in shifted steel, be would output than thermal the alloy coefficient titanium expansion a lower to somewhat a bonded with were lot this from gage a if

APP ARENT MICROSTRAIN (Based on Instrument G.F. of 2.00) ε ε +400 +100 +200 +300 -100 -500 -400 -300 -200 TO TO 0 -100 // // Figure 1–R Figure a strain gage when installed on materials with with materials on installed when gage a strain () () GR GS α α differing thermal expansion coefficients. expansion thermal differing -50 S R , and again for the same type of gage installed , forand of again the type gage installed same : =+ =+ 0+       β F β F 0+ G G G G otation of the thermal output from from output thermal the of otation 24ºC 75ºF () . () TEMPERA αα 100+ αα TEMPERA SG RG 50 − − -6 TURE INºF /°F [16 x 10 x [16 /°F TURE INºCELSIUS 200+ +100       Δ Δ T ARENHEIT T Document Number:11063

A

+150 Revision: 01-Nov-2010 300+ -6 /°C]. Similarly, Similarly, /°C]. (5b) (5a) . Lot No. -200 400+ A38AD497 +250 STD B 500 Tech Note 121 +140 +120 ompany). +100 TN-513-1 +80 +60 orning Glass C www.micro-measurements.com +40 +20 emperature (ºC) T 0 –20 Thermal Expansion: –80º to +150ºC Micro-Measurements –40 . The subject Tech Note should serve as the 6 –60

8 6 4 2 0 –80

-2 An excellent reference material with these TitaniumdesirableSilicate 7972, propertiesandULE™ Code is the other available from Corning As illustratedGlass 14831.* in Figure Company,2, this specialCorning, glass has an NY extremely low thermal expansion coefficient, particularly over the temperature range from[–45° about to It +175°C)]. –50° should to be +350°F noted, however,material that thehas a low coefficientreachtomaking thermal slow it equilibrium. optimum For of thermal conductivity,minutesused beleastshould at 45 results, at timedwell of a each new temperature point before taking data. Another potential disadvantage of titanium silicate as materiala referenceis its brittleness, since it droppedwill fractureon a readilyhard if surface. Becauselow-expansion metal as(such Invar or a similarof may alloy) the foregoing, a offer a preferable alternative if andthe alloyaccurately has repeatable knowntemperature range interest. of expansion properties over the (ppm) ∆L/L Strain Gage Selection The type of strain expansiongage coefficients selected is also an important consideration,for usejust as it is infor stress analysis measuringand transducer applications. Gage selection usually requires weighing directlyindirectlya canwhichfactorsor suitability affectthevariety of of a particular gage type to a specified measurement task. assist gageTo users in this process, Noteour TN-505 Tech provides extensive background dataalongwith procedures, recommendations, for gageandapplication selection, examples primaryreference gageon selection, supplemented here by special considerations applicable to the measurementexpansion coefficients. of For good accuracy, combined with ease of installation, gagea from Micro-Measurements CEA Series is ordinarily a suitable choice. Thisextremes for the assumesmeasurements fall thatwithin the therange temperatureof ­ ­re 150 Code 7972 For technical questions, contact 130 [email protected] (6) 110

/) 90 GR /( TO 70 T − Δ emperature (ºC) T /) GS 50 /( TO εε Thermal Expansion: 0º to 150ºC () . Measurement Procedures 30 4,5 −= SR 10 αα Measurement of Thermal Expansion Coefficient Using Strain Gages Using Strain Expansion Coefficient of Thermal Measurement 0 Figure 2 – Thermal expansion characteristics of the titanium silicate reference material (data source: C

6 4 2 0

-2 ∆L/L (ppm) (ppm) ∆L/L Also available from Micro-Measurements as Part No. TSB-1. No. as Part Micro-Measurements from Also available specimen dimensions. for See Appendix Thus, the difference in expansion coefficients,to a referred particular temperature range, is differenceequal to in the thermalunit output for the sametemperature. change in Althoughexpansion thiscoefficients istechnique widely applicable, and theforoften mostmeasuring practical information approach,about it in the theretechnical literature.is Rep relatively little Selectionthematerial of beusedto reference a as standard is naturally an important factor in the the method, accuracy of as it is dilatometry. for In anyprinciple, otherthe reference formmaterial be of could any differentialsubstance for which the expansion accuratelyproperties known are over the temperature range of interest. In practice, however, it is often advantageous a to materialselect with expansion properties as possible.close to Doing zero this as will provide an output closelycorresponds expansion“absolute”signalthecoefficient to that of the test material, and permits a test more procedure.straightforward The thermal materialexpansion should also be highlyof repeatable,the and stablereference with time at any constant temperature. In addition, the elastic modulus of the material should be great mechanicalenough reinforcementthat the by strain gage is negligible. Reference Material

Subtracting Equation (5b) from (5a), andSubtracting rearranging, from (5a), Equation (5b) sentative applications are described in the bibliography to this Note Tech Document Number: 11063 Revision: 01-Nov-2010 *  Tech Note (or temperature deviation between the reference and test test and reference the between temperature deviation in temperature error (or small a circumstances, temperatures. such extreme Under both or one at steep very become can curve output thermal the of slope the specimen, coefficient the of expansion the and gage the of the number S-T-Cbetween mismatch excessive with that, 1 evident is Figure It from accuracy. measurement best the obtain to ex an ex over made be must measurements expansion When S-T-C above for the of the to one number. specify therefore, expedient, be often will It applications. transducer and analysis stress forwidely most compensations the used are these since groups, S-T-C 13 and 06 the in available is gages of selection greatest the rule, a As resistance. and pattern, gage selected series, the desired the in of gage availability the is these of One Practically, choice. the calculations. may influence which expansion two considerations are however, there the in involved is materials, different two on type gage thermal same the for in output, difference the Only selected. what is matter number S-T-Cnot should it (6), Equation by indicated as principle, In self- number. (S-T-C) the is temperature-compensation specified be to parameter gage Another the to substrate. transfer heat improved and operation stable more a [ in 1/8 employ say, — to length gage feasible, medium when practice, good is In it temperature. addition, with leadwires the in changes resistance unsymmetric to due occur may which imbalances small of effects the current. reducing in advantageous excitation also is 350Ω gage The the by self-heating minimize to order in preferable is gage 350Ω a cases, foregoing the of each In narrow. and thin is specimen test or the of elasticity, a low modulus has material test the if necessary be may effects of reinforcement consideration and however, stiffer, somewhat is gage latter type The choice. preferred the becomes Series the WK from gage a involved, is range temperature wider a +65°C)]. If to +150°F(–45° to –50° [about gage of in foil type this constantan the for precision and stability greatest TN-513-1 122 www.micro-measurements.com tended temperature range, or at high or low temperature or or low at temperature high range, temperature ­tended tremes, the S-T-C number should be carefully selected selected carefully be should number S-T-C the ­tremes, Measurement ofThermal ExpansionCoefficient UsingStrainGages type 125MG dual-grid strain gage pattern. gage strain 125MG dual-grid type Micro-Measurements Figure 3 – Micro-Measurements 3–Micro-Measurements Figure ~3.5x actual size actual ~3.5x 3 mm 3 ] or larger — for — larger or ] [email protected] For technical questions,contact smooth, nonporous surfaces, and should not be used where where and not should be used surfaces, nonporous smooth, foron relatively use intended are adhesives These gage. the maximum for to surface from the specimen strains in transmitting fidelity “gluelines” hard thin, forming of are capable adhesives Both 610. or 600 M-Bond as such adhesive should be done bonding with a performance accuracy, high step procedures for a wide variety of materials of variety wide a for procedures step Instruc in and Micro-Measurements for as prepared bonding described cleaned thoroughly be should surfaces Specimen together. cooled or heated are specimens both when achieving temperature quickly same most the in helpful be will same inertia foregoing, the about thermal the for dimensions Beyond specimen the of stiffness. selection section overall compared the negligible to is stiffness gage strain the that so section cross in enough large be also should specimens The installations. sensor and temperature easier and for gage make higher-quality will specimens flat of use the and cooling; or heating during induced gradients temperature however, minimize general, to section In cross in so. uniform be should them specimens have to if reason shape a or is size there in different be even can the specimens the fact, two to In equipment. suitable refrigeration or heating configuration available or size be can convenient materials any test of and reference the of that is specimens method the this of advantages the of one noted, As Installation Gage other properties). other all in (as characteristics output thermal in closest match the possible provide will and twins, identical effect, in are, gages resulting The gages. to form two individual cut apart as and the such grids the 125MG 3) (Figure be selected, can gage a pattern accuracy, dual-grid for measurement greater desired is relationship closer still a When outputs. their thermal in close equally be will number, lot but same the packages, having different from taken type identical the of simply Gages package. by same the met from taken gages be of pair a can using requirements Both characteristic. output thermal same related closely the — assure to lot from be manufacturing must and material type, same the test must identically gages the be the on is, That one well-matched. be and always must the on specimen, one — gages reference two the however, earlier, As indicated coefficients. expansion measuring for can employed pattern be grid “linear” single-element any Almost range of interest. temperature the in control good reasonably materials thermal under reference the and of test the slopes both for the curves output keep simultaneously to be can used mismatch S-T-C the of selection Judicious signal. output thermal the in error large a produce can materials) tion Bulletin B-129, which includes specific step-by- specific B-129, includes which Bulletin ­tion Document Number:11063 Revision: 01-Nov-2010 7 . For best best For . Tech Note 123 TN-513-1 www.micro-measurements.com train gage (half dual-gage of the 125MG Micro-Measurements pattern, at and top) resistance temperature sensor, Figure 4 – S installed side-by-side on a specimen of test material ­factory instruments for this purpose include the The process of gage installation has beenvery summarizedbriefly here, since detailed instructionselsewhere are supplied in our appreciated,technical however, thatpublications. proper basicgage requirement installation for accurateIt measurementshould of isexpansion a be coefficients. In general, gage installations should beof the highest quality — comparable to those found in precision straingage transducers. Care should also be taken that the gageinstallations,two referencetheon testandspecimens, are as uniform as possibledifferences to minimizewhich could smallresponse.affect If physicalinstallationthe differentialquestions or problemsuser thermal arise, should the consult theDepartment our for assistance.Applications Figure 4 is Engineeringa a photographproperly of installed strain gage on a thermalmetal expansionspecimen measurements. for A bondable resistancetemperature sensor (see Figure 6) is installedto the adjacent gage to monitor the specimen photographtemperature. shows the installation just priorThis to application of the protective coating over the sensor. gage and temperature Strain and Instrumentation Temperature Basically, any stable precision strain indicator can be used for the strain measurementsSatis needed in this procedure.Model P3 and Model 3800 Strain Indicators produced by theInstruments Division Micro-Measurements. of Beyond the necessity for instrument precision and importantstability, it is that the gage enoughexcitation to voltageavoid thebe effectskept Bothlow ofthe Modelsself-heating P3 and 3800 in are thehigh-gain with low excitationgage. instrumentsvoltages. Using these strain indicators, there is ordinarily no self-heatingsuch as the 125MG pattern probleminstalled on witha metal specimena gage

TV For technical questions, contact [email protected] oat A orC C W-1 Wax W-1 Coating M- 3140 or 3145 R or 3145 3140 ­tion against moisture, Two coats M-Bond 43BTwo tec ­mally produced resistance C ° Range [–20 +65] to [+15 to +120] to [+15 [–75 to +260] to [–75 [–269 to+200] Temperature Temperature PROTECTIVE COATING PROTECTIVE °F Measurement of Thermal Expansion Coefficient Using Strain Gages Using Strain Expansion Coefficient of Thermal Measurement 0 to +150 0 to

+60 +250 to Operating –100 to +500 to –100 –452 +400 to Document Number: 11063 Revision: 01-Nov-2010 dew point condensation in cold tests andmaximum minimum/ operating recommendationstemperature in the following tablerange. also takeconsideration The into low reinforcementcoating of the specimen. Fur- changes in the leadwires will generate which circuit are outputs indistinguishable from the thermal beingoutputs measured. If these differreference in andany testway betweenspecimens, theexpansion data thewill indicatedbe in error accordingly.differential To minimizesuch effects, leadwire resistance should be kept possibleemploying asby generous a wire size, low keeping andby as bothleads wiringtheTheshort. samefor the alsobe should specimens — in size, length, and routing. If measurements are to be made on both specimens in the same chamber or liquid bath at the same time, the leadwires should be kept physically together throughout as much of their as lengthpractical. Leadwire insulation must be course,selected, for compatibility of with the temperature range environmentand encountered in the measurements. In attaching leadwires to the gage solder tabs or to solder terminals, the solder joints should be smooth, bright, and free of spikes or excess solder. The joints should also be as uniform as possible; and the leadwires should thebe dressedsame on both specimens. After gage lead installationsattachment, must thebe thoroughly cleaned solvent to remove allwith traces soldering of rosin flux and residues. The final step in the installationcoating issystem to whichapply ais protectiveappropriate to the testexpected environment.conducted under Since short-termcoating is theseselectedlaboratory for basic pro­ testsconditions, a are normally ther details on these and other coatings Micro-Measurementscan be found in Book. Strain Gage Accessories Data the adhesive is required to fill surfaceseal irregularities pores. For orthe to latter conditions, the Inall cases, adhesiveAE-15. or recommendedAE-10 completeisM-Bond instructions for applyingincluded in the package withand the material. curing the adhesive areExtra care is required in the selectiontheir of leadwiresattachment and to mostthe accurategages, results.in order Ther to obtain the Tech Note Tech TN-502 Note exci setting for guidelines be reduced. Compre reduced. be self-heating must gage the to applied conductivity, voltage the and excessive, be may thermal low of specimens installed on gages with or voltages, excitation higher having characteristics. instruments other with made are measurements When heat-dissipating good reasonably with TN-513-1 124 www.micro-measurements.com to the difference in the individual thermal outputs. The The outputs. thermal individual the in difference the equal is to output instrument the circuit, bridge the of arms adjacent as connected are gages two the When subtraction electrically. the perform to circuit half-bridge properties the of the uses 5b) (Figure arrangement second The a or indicator) strain instrument. two-channel single-channel a with used (when this of ap disadvantage A test. the conducting when occur may which readings strain anomalous or cause improper any the of identify to simple relatively is gages it the independently, monitoring permit circuits separate the Since (6). Equation with use for strain differential the to determine subtracted subsequently and individually, outputs read gage are the arrangement, this test With and reference specimens. the on gages the for circuits quarter- bridge three-wire, separate, employs 5a, Figure shown in these, of One coefficients. in expansion used be measuring can arrangements circuit basic two of Either Reference Mat’l Reference Mat’l proach is that it requires a switch-and-balance unit unit switch-and-balance a requires it that is ­proach Test Material Test Material Measurement ofThermal ExpansionCoefficient UsingStrainGages quarter-bridge circuits; (b) half-bridge circuit. half-bridge (b) circuits; quarter-bridge Figure 5–S Figure thermal expansion coefficients: (a) separate separate (a) coefficients: expansion thermal Micro-Measurements 3 2 1 3 2 1 8 . train gage circuits for measuring measuring for circuits gage train 3 2 1 ­hen sive background information and and information background ­sive ­ta tion voltages are provided in in provided are voltages ­tion 3 2 1 e e o o [email protected] For technical questions,contact has essentially the same construction except that the the that installed is It except foil. nickel high-purity from construction made is same grid the and essentially gage, strain has a like looks sensor (Figure temperature The 6). TG-Series temperature Micro-Measurements as resistance such use sensors to is approach alternate An surface. specimen the to wires extension the be of mm] 75 to can [50 in 3 to 2 junction first the the taping by to improved selected. specimen be the from should transfer wire Heat thermocouple grade premium AWGof range AWGthe to leadwires 30 (in and 0.4 [0.25 mm]), 26 to the should as small, The be should type. junction this sensing with compatible the is that range assuming temperature test preferred, is (iron-constantan) J type specimen, each on employed is thermocouple a If sensors. temperature resistance with or thermocouples with either and preference can availability, be temperatures instrumentation measured personal on primarily Depending made. are readings strain paired whenever must be the of same, course, The temperature be measured. at both sites gage that the temperatures specific it heat, is necessary and conductivity thermal their in specimens differ test and normally chamber. reference test the in the in materials the Since equilibrium thermal of conditions under temperature specimen uniform assure to necessary, as specimen the on measurements temperature by multiple procedure assumes that previous hasverification been made, spec the indicate specimen to the with surface, contact intimate in and gage, the to probe is a adjacent placed immediately temperature-sensing data. Typically, expansion to accurate obtain consideration and care requires also measurement Temperature cause a false output of about 17 of about output afalse cause will AWG in — size wire jumper [0.25 30 mm] the #3, in or #1 and leadwires in whether — wiring the in [~150-mm] dissymmetry 6-in a that noting worth is Itin wire. change jumper the resistance induced thermally the to due signal output false a avoid thus and arm, bridge respective its in gage each with series in resistance jumper the of half place to done is This gages. the between jumper the of midpoint at the #2 connected be leadwire that 5b, it necessary is also Figure in shown as such circuit instrument half-bridge a With the output. in appear could which changes resistance differential physically minimize to lengths, their throughout maintained together and well-matched particularly be should they circuit, bridge the of arms adjacent in always #1the same and size wire leadwires Since length. and #3 are to should and be the as should be gages of short as possible, In the both ofleadwires the arrangements, foregoing circuit gage the primary suspected. Its isolating is of operation improper when malfunctioning be difficulty may which the direct-reading. in lies is and disadvantage and wiring both instrumentation, of terms in simpler obviously is circuit ­i men/gage temperature. This This temperature. ­­men/gage με per 100°F [per 55°C]. 100°F [per per Document Number:11063 Revision: 01-Nov-2010 Tech Note 125 TN-513-1 www.micro-measurements.com Micro-Measurements is essentially uniform — at least in the specimens.theregion Temperaturestabilitycontaining chambertheinalso is necessarypermitto measuring specimen temperatures and strains under static, nonvarying conditions. Thermal equilibrium in the specimen cana chamberbe achievedequipped in with a forced convection vigorouslysystem to circulate the beheat-transfer alsospecimenshould coolingrates surfaces.and Heating medium past minimizeto low keptthe temperature gradients perpendicular specimenthe to surface.required The uniform condition of temperature throughout the specimen is difficult to judge, however, and is not equalnecessarily temperature assured readings surface. by One of observingtheat most effectivedifferent ways to test for controlpoints over the uniformity of onspecimen thetemperature is to make a continuous plot of strain gage output versus temperature over the working temperature range — in both the heating and cooling directions. In this is process,changed incrementally;the temperature and, at eachafter test the temperature,specimen is evidentlythe temperature inand thermalthermal output areequilibrium, recordedplotted. If uniformity and of specimen temperature is actually achieved, the heating and cooling legs of the plotted curve should very nearly coincide. If, on the other hand, the two portions of the curve are significantly separated toform a hysteresis loop, a likely cause is nonuniform temperaturedistribution through the thickness of the specimen. In the latter case, the heating and cooling rates must be lowered, thermalor stabilization times increased, othermeasuresor taken to essentially eliminate thetemperature gradients. Means must be provided for supporting the specimens the in chamber so that friction cannot impede expansioncontraction. or In some cases, a simple way thisto is accomplishto suspend the specimens from one the end. specimenAlthough may be strained slightly by the its strain own is weight, constant (as long as the essentiallyelastic constant),modulus and is does not affect the thermalchange outputin with temperature. If the ofelastic the modulus test material changes of significantlytemperatures toover be theencountered, range the erroreffect due must to be thisevaluated to determine thethe method.suitability Another approach of is to lay the specimens on the floorof the chamber or compartment, supportedby a fiberglassother low-friction of some or layer medium.cloth When this method is verifiedused,by observing its theeffectiveness behaviorof theasthespecimen thermal cycledis through theworking should temperature output be range. Erratic output, hysteresis, or lack may indicateof excessiverepeatability friction. Before performing actual measurements to determine the coefficientof expansion, the entire system, including both specimensgages(with installed applied),powerandshould be stabilized by cycling several times leastto temperatures10°F [5°C] above the at highest, and below the lowest, For technical questions, contact [email protected] eries S TG-50B/W bondable temperature sensor. Figure 6 – Micro-Measurements TG- E Making Expansion Measurements Measurement of Thermal Expansion Coefficient Using Strain Gages Using Strain Expansion Coefficient of Thermal Measurement Document Number: 11063 Revision: 01-Nov-2010 For any method of dilatometry,that the referenceit and is test alwaysspecimens necessaryleastbe exposed to at two differentexpansion coefficient. The actual temperatures meansof desired achieving temperatures the in a inparticular case measuringtemperaturesdepends on the involved, the andThese may consist,on for instance,the of ovens,available or liquid baths, facilities.or various other forms of strainenvironmental gage method imposes chamber.no special restrictionsThe on naturethe or design of the chamber. On the contrary, the size and shape of the specimen can usually be adapted the toexisting suit facilities. Since the available equipment varies widely from one laboratoryremarks toare limitedthe next,to the thegeneraldilatometric temperaturefollowingrequirements chamber. for any Two of the most desirable features of a measuring chamber for expansion coefficients are uniformity stability and of temperature.development of To thermalavoid temperature errors stressesshould be uniform due inthroughout the tothe specimen at thespecimen, the timethe ofestablished onlychambertheif temperature equilibriummeasurement. at This condition can be with standard strain gage installation procedures,shouldbe mounted side-by-side and with the strain gage onthe specimen surface. Because it is physically gage, like andthe isstrain attached to thethe temperaturespecimen sensor in has theabout thesame same characteristics way, heat-transfer and thermal time constant gage.as When the used strainin conjunction with a specially designed passive resistance network scalingfor (Micro-Measurements linearizationpermits it LST), Type direct and signalmeasurement of temperature with any conventional strain indicator.The small size stiffness andlow theTG-Series of temperaturesensor present minimum mechanical restraint to the free thermalspecimen. expansion and contraction of the Tech Note cases) is the error due to transverse sensitivity. This error error This sensitivity. transverse to due error the is cases) certain (in inaccuracy correctable readily a of example An refinement — i.e., by removing or minimizing all of the the of all minimizing known or removing by i.e., — refinement by technique primarily and error is reduction accomplished nebulous, more is relationship In cause-and-effect the others, elimination. or correction for procedures routine to may which second-order in systematic nature,errors and are effects responsive well-defined, these instances, smaller some In ever errors. introduce examine to is it method), necessary any with (or method gage strain the with accuracy greater and greater achieve to attempting When indi strain reproducible by verified stabilization, Following our from En obtained Applications be can needed, if assistance, Further trouble. the correcting and may clue forthe provide finding Tech Note of this re-reading Careful both. or strain, the or such the with the problem temperature, may In cases, be associated found. be must variability the of sources the test, the from required accuracy the to repeatability compared of lack significant is the if be and cycle, should not, If temperature repeatable. highly given stabilizing any at third output or thermal second the the after Normally, yielding, cycling. of the purpose the cause defeat thus and may stress, residual stress, the on thermal the superimposed Otherwise, gradients. temperature due to specimens the in stresses thermal minimize to temperature change of rates enough low at performed be should procedure cycling The nonrepeatable. be to data the cause and test the during change otherwise might which stresses any residual redistribute and/or to relax is intended cycling Thermal etc. leadwires, the installed, and as manufactured gages the the specimens, of test and all reference the — in components present generally are stresses procedure residual this that for is reasons the of One temperatures. test TN-513-1 126 www.micro-measurements.com or the differential thermal output can be read directly as as directly 5b. Figure in shown read the be 5a, can output Figure. thermal chamber, in as differential the the or separately in made be can together measurements specimens both ar having preferable the With coefficient. thermal expansion differential the give to (6) Equation by indicated as change) temperature the by divided subtracted difference are the (and data output The thermal 5a. of sets Figure two of resulting circuit the using one-at-a-time, tested ac be can material. specimen test the single of a only that such is chamber other or oven the properties When expansion thermal the per to ready ­ca tions throughout the temperature range, the user is is user the range, temperature the throughout ­tions Measurement ofThermal ExpansionCoefficient UsingStrainGages possible Special Precautions and Refinements Refinements and Precautions Special form the final measurements for determining determining for measurements final the ­form Micro-Measurements sources of error. sources for Improving Accuracyfor Improving ­gi neering Department. ­neering commodated, the two specimens are are specimens two the ­commodated, [email protected] rangement, ­rangement, For technical questions,contact employed in gage factor calibration factor gage in that from employed different generally (1)] is [Equation grid and specimen the between expansion thermal in by grid difference gage the the in induced field strain the because arises (6)] by the factor (1 – 0.285 – (1 factor the by (6)] the multiplying by made be can correction error, here, derived not Although easily. for rather corrected be can transverse-sensitivity small, quite ordinarily is the which properties, thermal expansion their in isotropic are materials test and reference When expansion measurements are made incrementally incrementally made are measurements expansion When to remove most of simple. error, the is relatively by correction, first-order introduced but feasible, always error not is variation small factor gage the of elimination Complete type. gage that to applicable variation factor gage the for data includes package gage each in contained sheet data technical The gages. of in types both for 7 Figure illustrated are temperature with variation factor gage Represen 100°C]. per –1.8% to [–0.9 100°F per –1.0% to –0.5 from range the in the of generally is but number S-T-C gage, the on depends change of rate The K-alloy temperature. with inversely of varies gages factor Karma) (modified gage the contrast, In 100°C]. per [0.9% at a rate of with varies temperature, about 0.5%directly per 100°F factor gage the example, for gages, With different. constantan slightly is it temperature other any At +75°Fat [+24°C]. measured is gages strain Micro- for Measurements specified factor gage The temperature. with factor gage of variation the is source error minor Another a sensitivity. for transverse correct to required are between gages outputs specimen oriented thermal perpendicularly two and gage reference differential case which for materials, to is factor not orthotropic This applicable correction use. in gage the of sensitivity transverse decimalized

GAGE F ACTOR V ARIA TION (Typical) +2.0% +1.0% –2.0% –1.0% Figure 7 – Gage factor variation with temperature temperature with variation factor 7–Gage Figure 0 (typical) for A for (typical) 0+ 0+ difference 24ºC 75ºF 100+ - and K-alloy strain gages. strain K-alloy - and TEMPERA 50 TEMPERA in thermal outputs [Equation [Equation outputs thermal in TURE INºF K TURE INºCELSIUS 200+ +100 t )/(1 + )/(1 Document Number:11063 ARENHEIT +150 K 300+ Revision: 01-Nov-2010 9 t . When both the the both When . ), where where ), tative plots of plots ­tative -200 400+ K t 09 03 is the the is A-alloy K-alloy +250 500 Tech Note 127 TN-513-1 www.micro-measurements.com Micro-Measurements stable, accurate instrumentation, both for temperature and strain. high-quality, stable gage installations, exhibiting negligible drift the over operating temperature range. gage excitation a level at enough low self- to avoid heating effects. thermal stabilization specimens, of gages, and wiring prior to making expansion measurements. assurance thermal of equilibrium in the specimens when measurements are made. avoidance significant of thermal stressesduring heating and cooling. elimination frictional of effects preventing free expansion and contraction. is obvious, therefore, that achievedthe highest by minimizing accuracy all differenceswill For thisin be reason,gage behavior.as noted earlier, the thermalcharacteristics output the of gages should be as nearly the same as possible. two nominallyHowever, identical gages from the same manufacturing lot do not especiallythermal have identicaloutputs. Instead,is a as toleranceshown on inthe thermalFigure tolerance output.* can 8,Almostbe removed by splittingthereall a dual-element of gage the (such as thepattern) 125MG to make a pair of twin gages, and this procedure is always recommended accuracywhen high is the goal. The repeatedsame emphasis inreasoning this Tech Note on the underliesuniformity of gageinstallations.the Identical installation procedures should be used for both gages; and, ideally, therevisible differencesshould in thebe completed installations. no The remaining areas of possible refinementfor accuracy improved are primarily associated with the measurementsprocedures.followingchecklist theitemsinthe Each of can be considered, and steps taken as necessary conditions: to desired satisfy the a) b) c) d) e) f) g) Except for the absolute accuracy of the degreetheinstrumentation, whichtoconditionsforegoingthe been have met can be judged quite well by the repeatabilityHighly reproducible datagenerally of indicatesystemthe the that data. is functioning properly, and that random error sources are well-controlled. After it has been demonstrated that the system measurement and procedures are suitable for reproducibleobtaining data closelyfrom a single specimen, shouldconsideration be given to the questionproperties from ofspecimen variation to specimen. The usualin purposethermal of expansion-coefficient measurementsthe nominal value is which tois representativedetermine of material. a particularBut the thermal and other physical of any propertiesmaterial tend to vary randomly from specimen to

G is F G 500 ∆ F +250 For technical questions, contact 400+ differential -200 ), where ), [email protected] L ). Theterm ). R G F + 300+ +150 ∆ G A R is the gage resistance,gagethe is /( ARENHEIT G G R R +100 200+ x TURE IN º CELSIUS TURE IN º F G F 50 TEMPERA 100+ TEMPERA 75ºF 24ºC )/RG, where )/RG, L 0+ R 0+ + G for the mid-range temperature, but it will ­tween the reference and test specimens. It R G -50 F -alloy strain gages from the same manufacturing lot. ∆ -100

0

is the leadwire resistance in series with the gage in

-200 -300 -400 -500 -100

+300 +200 +100 +400

Measurement of Thermal Expansion Coefficient Using Strain Gages Using Strain Expansion Coefficient of Thermal Measurement ·

(µm/m) (µm/m)

C º

L ±0.27 Uncertainty:

-1

F. G. Instrument on (Based

R 2.00) of

µm/m – OUPUT THERMAL Figure 8 – Tolerance band for the thermal output of randomly See Tech Note TN-504. See Tech selected A be much less effective nonlinearbecause function temperature. of the thermal output is a When the leadwireas recommendedresistance incan attenuation the (“desensitization”)be preceding causedkept by very the section,resistance inert seriesnegligible.inbeshould gageon thewith If, low, the signal theother hand, the series resistance isgreater than about1 percent of the gage resistance, the user who is striving for maximumaccuracy wishmay performto correction.a For this purpose, the indicated thermal outputs are multiplied ( factorthe by the same arm of the bridge circuit. directAn reading alternative, of corrected strains, for is to set the gage factor thecontrolinstrument of at and the specified gagefactor of the gages in use. The supposition is made, in the strain measuringgage method expansion of coefficients, that ifdifferenceidentically,circuits)any thenbehavegagein the (and two gages differencethetheironlyto outputsduecanbe expansionin properties be Document Number: 11063 Revision: 01-Nov-2010 *  thermal output for each increment in temperature can correctedbe individually. This differenceis in indicateddone thermal outputsby from the specimenmultiplying + referenceandthe factor1/(1 thegages by across the working temperature range, the in the foregoing is the decimalized change (with insign) gagecorresponding factor to the middle temperatureeachmeasurement of increment. Sometimes, the average differential expansion coefficient is to be determined over makingonly thetwo sets measurements, of fullthe at temperature temperature rangeextremes. Theby same correction procedure can be applied,using the Tech Note accurate than the instrumentation used to indicate the the indicate to used strains. and more temperatures no instrumentation be the can than measurements accurate the Similarly, be material. never reference the of can that than accuracy material greater the to test determined the all example, of For to coefficient common dilatometry. expansion those differential of generally are methods limitations Other stiffness. section overall of the fraction significant a represents stiffness gage the that narrow and thin so is specimen the unless negligible ordinarily is a reinforcement effect the introduce specimens, and metal field With error. strain sizeable local the perturb the of may gage stiffness the elasticity, of modulus low very a with plastic, as such material a from made When is specimen instances. test some the in limitation a be also strain can the gage by specimen the of reinforcement Mechanical for recommendations. Department Engineering Users Appli with the should Micro-Measurements consult circumstances. the on depending extended, be ranges these can temperature sometimes techniques, from special With +205°C). to (–45° measurements +400°F to –50° strain approximately accurate provide gages, can K-alloy of which use the require normally temperatures Higher +65°C). to +150°F(–45° temperature to –50° a about from range within only measurements high-accuracy temperature allowable Con range. the be may studies of types some for one principal the these, Of has limitations. few special very dilatometry differential of method gage strain The to value. average or integrated arough, obtain coefficient, expansion the of distribution to the angles of define range the wide a over measurements determine make to to impossible it may necessary be axes, is material of natural the it directions When to is measured. coefficient be expansion directional the if critical otherwise) is or orientation fiber direction, rolling the by determined as material, the of axes natural the to respect (with specimen the on gage strain the of such orientation In cases, directional. highly are etc.) reinforcement, fiber oriented with 6A14V,composites titanium graphite, (e.g., materials some of properties thermal and mechanical The composites. and plastics as such particularly materials in great be to apt is deviation. properties standard thermal in and Variability mean the of estimate adequate an provide to enough large size sample a sampling with statistical techniques, use to necessary of becomes control it the user, to the subject not is to variation lot from such widely Since lot. more still and lot, a within specimen TN-513-1 128 www.micro-measurements.com Measurement ofThermal ExpansionCoefficient UsingStrainGages stantan gages, for instance, should be used for used be should instance, for gages, ­stantan Micro-Measurements Limitations [email protected] For technical questions,contact cations ­cations accuracy for many engineering purposes. engineering for many sufficient accuracy with coefficients expansion thermal measure easily and quickly to used be can method the rigorous procedures, less somewhat dictates environment. expedience when thermal Even variable a in measurements precision strain making to procedures, accustomed is which stress laboratory a for recommended practices from the standard represent of should accuracy however, Most utmost the method. the extracting facility. at a aimed such in details procedural to here available given been has attention already Considerable not techniques, materials or instrumentation, special no requires it since usually laboratory, analysis stress the to particularly well-suited is method The properties. expansion having known material reference any of that to test relative a of material coefficient expansion the measuring of straightforward means simple, a described has Note Tech This  2. 1. 9. 8. 7. 6. 5. 4. 3. Silica Dila Silica “Linear Materials, Ther and Testing for Society American No. B95-39.Standard ASTM Metals”, of Expansion Linear for Method Test “Standard Materials, and forTesting Society American to Transverse Sensitivity in Strain Gages”, 1982. Strain in Sensitivity Transverse to Due “Errors TN-509, Note Tech Micro-Measurements, Levels”, 1979. Excitation Gage Strain “Optimizing TN-502, Note Tech Micro-Measurements, Pre “Surface 1976. B-129, Bonding”, Gage for Strain paration Bulletin Micro-Measurements, 1989. Selection Gage Tech TN-505,Note “Strain Micro-Measurements, Gages”, Strain 1978),(March pp. 98-102. with Solids Evaluation and Testing of of Journal the Expansion “Measuring Thermal Kesterson, F. K. and W., M. Poore, XXV, No. 1, 1978, pp. 155-158. Using Metals of Gages”, Strain Characteristics Expansion of Thermal “Determination Heberling, G. T. and E., T. Finke, “Strain Temperature”, 1989. TN-504, with Variation Factor Gage Note and Output Tech Thermal Gage Micro-Measurements, mal Expansion of Rigid Solids with a Vitreous Vitreous a with Solids Rigid of Expansion ­mal ASTM ­tometer”, Criteria, Procedures,RecommendationsCriteria, Proceedings,SESA References Summary Standard No. E228-71.Standard Document Number:11063 , ASTM, Vol. 6, No. 2 No. Vol. 6, ASTM, , Revision: 01-Nov-2010 nw SM, Vol. SEM), (now, ”, ­ Tech Note 129 /ºC]

/ºC] /ºC] -6 m/m/ºC] -6 -6 /ºC] -6 TN-513-1 www.micro-measurements.com in/in/ºF µ [±0.05 [–0.0c x 10 ±0.03 [0.03 ±0.03 x 10 ±0.03 [0.03 [0.00 ±0.03 x 10 ±0.03 [0.00 ] in Equation by (6), /ºF /ºF -6 ] in Equation by (6), /ºF /ºF [0.00 ±0.015 x 10 /ºF ±0.015 [0.00 -6 T/O(G/R) -6 m/m/ºC] -6 ε – T/O(G/R) ε Micro-Measurements – T/O(G/S) ε 0.00±0.008 x 10 0.00 ±0.017 x 10 0.00±0.017 0.017 ±0.017 x 10 ±0.017 0.017 –0.017 ±0.017 x 10 ±0.017 –0.017 T/O(G/S) in/in/ºF [±0.10 µ in/in/ºF [±0.10

ε , it is, it much simpler to adjust the gage factor setting of G R )/ L m/m/ºC] m/m/ºC] + R G APPENDIX R For technical questions, contact . ) [email protected] L ) for a Single) for Gage in a Three-Wire Configuration L in/in/ºF µ [±0.45 in/in/ºF µ [±0.27 to +35ºC] [+5º to+35º] [+5º [0º to +200ºC] [–100º to [–100º +200ºC] REFERENCE INFORMATION R + R G R /( G in decimal form, corresponds to the midpoint the of temperature increment which over or 95% confidence or 95% overlevel the +175ºC] temperature[0º +350ºF to +32º to rangeof x R G σ G F Δ for isotropic) — for materials only. t = F +40º to +95ªF +40º to +95ºF +32º to +390ºF to +390ºF +32º –150º to +390ºF to +390ºF –150º 1 1 + K ( / ) t ). The). term G , in decimal form, multiply the parenthetic expression [ F t Δ K is the resistance a single of leadwire in the three-wire connection to the instrument. the avoid tedious To task of L Measurement of Thermal Expansion Coefficient Using Strain Gages Using Strain Expansion Coefficient of Thermal Measurement Correction Transverse for Sensitivity WK-XX-125MG-350 usedas described the for EA gage: µ WK-XX-125MG-350 ±0.06 EA-XX-125MG-120 with one grid and on Code other 7971 on unknownEA-XX-125MG-120 material: µ ±0.03 R 1/(1 + 1/(1 the instrument to F To evaluate theTo need this for correction, the approximate lead resistances typical for Micro-Measurements cables are: ohms/m] ohms/ft 0.043 326-DTV: [0.141 326-DFV, ohms/ft ohms/m] 330-FTE: [0.354 0.108 330-FFE, 330-FJT, 330-DFV, Tolerance within one specimen purchased from Micro-Measurements (Part TSB-1): No. thermal output measurements are made. Correction Leadwire for Resistance ( Specificationfor CORNING GLASS WORKS Titanium Silicate, Code 7972 ULE™ thermalexpansion coeffecient Control limit: This tolerance also applies to typical values noted above. Micro-Measurements specimen 30 x mm] x 6.5 size: in 6 x 1 x 0.25 [155 Micro-Measurements specimen finish: 80 grit Thermal Output Scatter Micro-Measurements of Strain Gages All data are based on a 2 Typical values: Correction vs. Gage for Factor Temperature any temperatureFor increment, multiply the parenthetic expression [ correcting all individual readings the by factor ( Single-element µ A-alloy gages: ±0.15 Single-element gages: K-alloy µ ±0.25 (1 – 0.285 K – 0.285 (1 III. W i t h



V. Document Number: 11063 Revision: 01-Nov-2010 I. II.

IV.