11.3471Resistance StrainGaugeswhereA=D=(1×10-3)*= 7.85×10-7m2The resistance is then(1.7 × 10-8 m)(5 × 10-2 m)=1.08×10-3 nR7.85×107m2COMMENT If the material were nickel (Pe= 7.8 × 10-8 m) instead of copper, the resist-ancewouldbe5×10-3forthe samediameterand lengthofwire.Example11.2Avery common material for theconstruction of strain gauges is thealloy constantan (55% copperwith 45% nickel), having a resistivity of 49 x 10-8 m. A typical strain gauge might have aresistanceof1202.Whatlength of constantan wireofdiameter0.025mmwould yield a resistanceof120Q?KNOWNTheresistivityof constantan is 49×10-8m.FINDThe lengthof constantan wire needed to producea total resistance of 1202SOLUTION From Equation 11.6, we may solve for the length,which yields in this caseRA。 (120 02)(4.91 × 1010 m2)L0.12m49×10-80mPeThe wire would then be 12 cm in length to achieve a resistance of 120.COMMENTAs shownby thisexample,a single straightconductor is normallynotpractical fora local strain measurement with meaningful resolution.Instead, a simple solution is to bend the wireconductor so that several lengths of wire are oriented along the axis of the strain gauge, as shown inFigure 11.4.Solder connectionsGridLead wiresFigure11.4Detailof a basic strain gaugeconstruction. (Courtesy of Micro-MeasurementsBottom layelDivision, Measurements Group, Inc., Raleigh,Top layer(backing)(encapsulating layer)NC.)
E1C11 09/14/2010 13:14:2 Page 471 where Ac ¼ p 4 D2 ¼ p 4 1 103 2 ¼ 7:85 107 m2 The resistance is then R ¼ 1:7 108 V m 5 102 m 7:85 107 m2 ¼ 1:08 103 V COMMENT If the material were nickel (re ¼ 7:8 108 V m) instead of copper, the resistance would be 5 103 V for the same diameter and length of wire. Example 11.2 A very common material for the construction of strain gauges is the alloy constantan (55% copper with 45% nickel), having a resistivity of 49 108 V m. A typical strain gauge might have a resistance of 120 V. What length of constantan wire of diameter 0.025 mm would yield a resistance of 120 V? KNOWN The resistivity of constantan is 49 108 V m. FIND The length of constantan wire needed to produce a total resistance of 120 V SOLUTION From Equation 11.6, we may solve for the length, which yields in this case L ¼ RAc re ¼ ð Þ 120 V 4:91 1010 m2 49 108 V m ¼ 0:12 m The wire would then be 12 cm in length to achieve a resistance of 120 V. COMMENT As shown by this example, a single straight conductor is normally not practical for a local strain measurement with meaningful resolution. Instead, a simple solution is to bend the wire conductor so that several lengths of wire are oriented along the axis of the strain gauge, as shown in Figure 11.4. Lead wires Bottom layer (backing) Top layer (encapsulating layer) Grid Solder connections Figure 11.4 Detail of a basic strain gauge construction. (Courtesy of Micro-Measurements Division, Measurements Group, Inc., Raleigh, NC.) 11.3 Resistance Strain Gauges 471
472Chapter 11StrainMeasurementGaugeEnd loopwidth1GaugelengthOverallMatrixpatternlengthlengthSoldertabsFigure11.5Constructionofatypical metallicfoilOverallstrain gauge.(Courtesy of Micro-MeasurementspatternwidthDivision,MeasurementsGroup,Inc.,Raleigh.MatrixwidthNC.)Strain Gauge Construction and BondingFigure11.5 illustrates the constructionof atypical metallic-foil bonded strain gauge.Such a straingaugeconsists of ametallicfoil patternthat is formed ina mannersimilartotheprocess usedtoproduceprinted circuits.This photoetched metal foil pattern is mounted on a plastic backingmaterial.The gauge length, as illustrated in Figure 11.5, is an important specification for aparticular application. Since strain is usually measured at the location on a component where thestress is a maximum and the stress gradients are high, the strain gauge averages the measuredstrain over the gauge length. Because the maximum strain is the quantity of interest and thegauge length is the resolution, errors due to averaging can result from improper choice of a gaugelength (5).Thevariety ofconditions encountered inparticularapplications require special constructionand mounting techniques,including design variations in the backing material, the grid con-figuration, bonding techniques,and total gauge electrical resistance.Figure 1l.6 shows avariety of strain gauge configurations. The adhesives used in the bonding process and themounting techniques for a particular gauge and manufacturer vary according to the specificapplication.However, there are some fundamental aspects that are common to all bondedresistance gauges.The strain gauge backing serves several important functions. It electrically isolates themetallic gauge from the test specimen, and transmits the applied strain to the sensor.A bondedresistance strain gauge mustbe appropriately mounted to the specimen for which the strain is tobe measured. The backing provides the surface used for bonding with an appropriate adhesive.Backing materials are available that are useful over temperatures that range from -270° to290°C.The adhesive bond serves as a mechanical and thermal coupling between the metallic gauge andthetest specimen.As such,the strengthof the adhesive should be sufficientto accuratelytransmitthe strain experienced by the test specimen,and should have thermal conduction and expansion
E1C11 09/14/2010 13:14:2 Page 472 Strain Gauge Construction and Bonding Figure 11.5 illustrates the construction of a typical metallic-foil bonded strain gauge. Such a strain gauge consists of a metallic foil pattern that is formed in a manner similar to the process used to produce printed circuits. This photoetched metal foil pattern is mounted on a plastic backing material. The gauge length, as illustrated in Figure 11.5, is an important specification for a particular application. Since strain is usually measured at the location on a component where the stress is a maximum and the stress gradients are high, the strain gauge averages the measured strain over the gauge length. Because the maximum strain is the quantity of interest and the gauge length is the resolution, errors due to averaging can result from improper choice of a gauge length (5). The variety of conditions encountered in particular applications require special construction and mounting techniques, including design variations in the backing material, the grid con- figuration, bonding techniques, and total gauge electrical resistance. Figure 11.6 shows a variety of strain gauge configurations. The adhesives used in the bonding process and the mounting techniques for a particular gauge and manufacturer vary according to the specific application. However, there are some fundamental aspects that are common to all bonded resistance gauges. The strain gauge backing serves several important functions. It electrically isolates the metallic gauge from the test specimen, and transmits the applied strain to the sensor. A bonded resistance strain gauge must be appropriately mounted to the specimen for which the strain is to be measured. The backing provides the surface used for bonding with an appropriate adhesive. Backing materials are available that are useful over temperatures that range from 270 to 290C. The adhesive bond serves as a mechanical and thermal coupling between the metallic gauge and the test specimen. As such, the strength of the adhesive should be sufficient to accurately transmit the strain experienced by the test specimen, and should have thermal conduction and expansion Solder tabs End loop Overall pattern width Matrix length Overall pattern length Gauge length Matrix width Gauge width Figure 11.5 Construction of a typical metallic foil strain gauge. (Courtesy of Micro-Measurements Division, Measurements Group, Inc., Raleigh, NC.) 472 Chapter 11 Strain Measurement
47311.3ResistanceStrainGaugesT(b)(c)(d)(a)(e)(r)(g)(h)Figure11.6 Strain gaugeconfigurations. (a)TorqueRosette: (b)LinearPattern; (c)DeltaRosette; (d)ResidualStress Pattern; (e)Diaphragm Pattern; (f)TeePatterm; (g)Rectangular Rosette; (h)Stacked Rosette.(CourtesyofMicro-MeasurementsDivision,Measurements Group,Inc.,Raleigh,NC.)characteristics suitable for the application. If the adhesive shrinks or expands during the curingprocess,apparent strain can becreated in thegauge.A wide array of adhesives are availableforbonding strain gauges to a test specimen. Among these are epoxies, cellulose nitrate cement, andceramic-based cements.Gauge FactorThe change in resistance of a strain gauge is normally expressed in terms of an empiricallydetermined parameter called the gauge factor (GF).For a particular strain gauge, the gauge factor issupplied by the manufacturer.The gauge factor is defined asSR/RSR/R(11.11)GF=SL/LEaRelating this definition to Equation 11.10, we see that the gauge factor is dependent on thePoisson ratiofor the gauge material and its piezoresistivity.For metallic strain gauges,the Poissonratio is approximately0.3and theresultinggaugefactoris~2.The gauge factor represents the total change in resistance for a strain gauge, under acalibration loading condition.The calibration loading condition generally creates a biaxial strainfield, and the lateral sensitivity of the gauge influences the measured result. Strictly speakingthen, the sensitivity to normal strain of the material used in the gauge and the gauge factor are notthe same. Generally gauge factors are measured in a biaxial strain field that results from thedeflection of a beamhaving a value of Poisson's ratioof 0.285.Thus, for any other strain fieldthere is an error in strain indication due to the transverse sensitivity of the strain gauge. The
E1C11 09/14/2010 13:14:2 Page 473 characteristics suitable for the application. If the adhesive shrinks or expands during the curing process, apparent strain can be created in the gauge. A wide array of adhesives are available for bonding strain gauges to a test specimen. Among these are epoxies, cellulose nitrate cement, and ceramic-based cements. Gauge Factor The change in resistance of a strain gauge is normally expressed in terms of an empirically determined parameter called the gauge factor (GF). For a particular strain gauge, the gauge factor is supplied by the manufacturer. The gauge factor is defined as GF dR=R dL=L ¼ dR=R ea ð11:11Þ Relating this definition to Equation 11.10, we see that the gauge factor is dependent on the Poisson ratio for the gauge material and its piezoresistivity. For metallic strain gauges, the Poisson ratio is approximately 0.3 and the resulting gauge factor is 2. The gauge factor represents the total change in resistance for a strain gauge, under a calibration loading condition. The calibration loading condition generally creates a biaxial strain field, and the lateral sensitivity of the gauge influences the measured result. Strictly speaking then, the sensitivity to normal strain of the material used in the gauge and the gauge factor are not the same. Generally gauge factors are measured in a biaxial strain field that results from the deflection of a beam having a value of Poisson’s ratio of 0.285. Thus, for any other strain field there is an error in strain indication due to the transverse sensitivity of the strain gauge. The Figure 11.6 Strain gauge configurations. (a) Torque Rosette; (b) Linear Pattern; (c) Delta Rosette; (d) Residual Stress Pattern; (e) Diaphragm Pattern; (f) Tee Pattern; (g) Rectangular Rosette; (h) Stacked Rosette. (Courtesy of Micro-Measurements Division, Measurements Group, Inc., Raleigh, NC.) 11.3 Resistance Strain Gauges 473
474Chapter11StrainMeasurementpercentage error due to transverse sensitivity for a strain gauge mounted on anymaterial at anyorientation in the strain field isK,(eL/ea + Upo)× 100(11.12)eL=1 -UpoK,whereEa,eL=axial and lateral strains, respectively (with respect the axis of the gauge)Upo =Poisson's ratio of the material on which the manufacturer measured GF(usually 0.285 for steel)ez=error as a percentage of axial strain (with respect to the axis of the gauge)K,=lateral (transverse) sensitivity of the straingaugeThe uncorrected estimate can be used as the uncertainty.Typical values of the transverse sensitivity for commercial strain gauges range from -0.19 to0.05.Figure 11.7 shows a plot of the percentage error for a strain gauge as a function of the ratio oflateral loading to axial loading and the lateral sensitivity.It is possible to correct for the lateralsensitivity effects (6).60505=erle403020ens101eixe+1JO%OE102033040Figure 11.7 Strain measurement50error due to strain gaugetransverse sensitivity.5010987654321012345678910(Courtesyof MeasurementsLateral sensitivity, K, (%)Group,Inc.,Raleigh,NC.)
E1C11 09/14/2010 13:14:2 Page 474 percentage error due to transverse sensitivity for a strain gauge mounted on any material at any orientation in the strain field is eL ¼ Kt eL=ea þ yp0 1 yp0Kt 100 ð11:12Þ where ea; eL ¼ axial and lateral strains, respectively (with respect the axis of the gauge) yp0 ¼ Poisson’s ratio of the material on which the manufacturer measured GF (usually 0.285 for steel) eL ¼ error as a percentage of axial strain (with respect to the axis of the gauge) Kt ¼ lateral (transverse) sensitivity of the strain gauge The uncorrected estimate can be used as the uncertainty. Typical values of the transverse sensitivity for commercial strain gauges range from 0.19 to 0.05. Figure 11.7 shows a plot of the percentage error for a strain gauge as a function of the ratio of lateral loading to axial loading and the lateral sensitivity. It is possible to correct for the lateral sensitivity effects (6). 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Lateral sensitivity, Kt (%) Error % of axial strain 60 50 40 30 20 10 0 10 + – 20 30 40 50 60 4 3 2 1 5 0 –3 –2 –1 0 1 2 3 4 5 –4 –5 –2 –3 –4 –1 –5= L/ a L/ a = L a Figure 11.7 Strain measurement error due to strain gauge transverse sensitivity. (Courtesy of Measurements Group, Inc., Raleigh, NC.) 474 Chapter 11 Strain Measurement
47511.3Resistance Strain GaugesSemiconductor StrainGaugesWhen subjected toa load, a semiconductormaterial exhibits a change in resistance, and thereforecanbeusedforthemeasurementofstrain.Siliconcrystalsarethebasicmaterialforsemiconductorstrain gauges; the crystals are sliced into very thin sections to form strain gauges. Mounting suchgauges in a transducer, such as a pressure transducer, or on a test specimen requires backing andadhesivetechniques similar to those used formetallic gauges.Because of the large piezoresistancecoefficient, the semiconductor gauge exhibits a very large gauge factor, as large as 200 for somegauges.These gauges also exhibit higher resistance,longerfatigue life, and lower hysteresis undersome conditions than metallic gauges.However, the output of the semiconductor strain gauge isnonlinear with strain, and the strain sensitivity or gauge factor may be markedly dependent ontemperatureSemiconductor materials for strain gauge applications have resistivities ranging from 10- to10-22-m. Semiconductor strain gauges may have a relatively high or low density of charge carriers(3, 7). Semiconductor strain gauges made of materials having a relatively high density of chargecarriers (~102° carriers/cm) exhibit lttle variation of their gauge factor with strain or temperature.On the other hand, for the case where the crystal contains a low number of charge carriers (<1o17carriers/cm),thegaugefactormaybeapproximated asToGF=GFo+(11.13)7where GFo is thegaugefactor at the reference temperatureTo,under conditions of zero strain (8)and C, is a constant for a particular gauge.The behavior with temperature of a high-resistivity P-type semiconductoris shown inFigure11.8.Semiconductor strain gauges find their primary application in the construction of transducers.such as load cells and pressure transducers.Because of the capability for producing small gaugeTemperature (°C)025255075100125150175100111Carriers/cm380H=2×1016G=5×1017ss60K=1×1020L=7.5×10194F=1.5×101840E=3x1018C=2x101920D=1×1019-Figure 11.8 Temperature effectonresistanceforvariousimpurity20concentrations for P-type semi-conductors (reference resistance-40o4080-40120160200240280320360at81°F).(Courtesy of KuliteTemperature (°F)Semiconductor Products, Inc.)
E1C11 09/14/2010 13:14:2 Page 475 Semiconductor Strain Gauges When subjected to a load, a semiconductor material exhibits a change in resistance, and therefore can be used for the measurement of strain. Silicon crystals are the basic material for semiconductor strain gauges; the crystals are sliced into very thin sections to form strain gauges. Mounting such gauges in a transducer, such as a pressure transducer, or on a test specimen requires backing and adhesive techniques similar to those used for metallic gauges. Because of the large piezoresistance coefficient, the semiconductor gauge exhibits a very large gauge factor, as large as 200 for some gauges. These gauges also exhibit higher resistance, longer fatigue life, and lower hysteresis under some conditions than metallic gauges. However, the output of the semiconductor strain gauge is nonlinear with strain, and the strain sensitivity or gauge factor may be markedly dependent on temperature. Semiconductor materials for strain gauge applications have resistivities ranging from 106 to 102 V-m. Semiconductor strain gauges may have a relatively high or low density of charge carriers (3, 7). Semiconductor strain gauges made of materials having a relatively high density of charge carriers (1020 carriers/cm3 ) exhibit little variation of their gauge factor with strain or temperature. On the other hand, for the case where the crystal contains a low number of charge carriers (<1017 carriers/cm3 ), the gauge factor may be approximated as GF ¼ T0 T GF0 þ C1 T0 T 2 e ð11:13Þ where GF0 is the gauge factor at the reference temperature T0, under conditions of zero strain (8), and C1 is a constant for a particular gauge. The behavior with temperature of a high-resistivity Ptype semiconductor is shown in Figure 11.8. Semiconductor strain gauges find their primary application in the construction of transducers, such as load cells and pressure transducers. Because of the capability for producing small gauge –40 0 –25 0 25 50 75 100 125 150 175 40 80 120 160 200 240 280 320 360 H Temperature (°F) Temperature (°C) Percent resistance change –40 –20 0 20 40 60 80 100 Carriers/cm3 H = 2 × 1016 G = 5 × 1017 K = 1 × 1020 L = 7.5 × 1019 F = 1.5 × 1018 E = 3 × 1018 C = 2 × 1019 D = 1 × 1019 G K L F E D C Figure 11.8 Temperature effect on resistance for various impurity concentrations for P-type semiconductors (reference resistance at 81F). (Courtesy of Kulite Semiconductor Products, Inc.) 11.3 Resistance Strain Gauges 475