《工程测试与信号处理》课程教学资源（文献资料）Strain Measurement

11ChapterStrain Measurement11.1 INTRODUCTIONThe design of load-carrying components for machines and structures requires information con-cerning the distribution offorces within the particular component.Proper design of devices such asshafts,pressurevessels,and support structures must considerload-carryingcapacityand allowabledeflections.Mechanicsofmaterials provides a basisforpredictingtheseessential characteristicsofamechanical design, and provides the fundamental understanding of the behavior of load-carryingparts.However,theoretical analysis is often not sufficient,and experimental measurements arerequired to achieve a final design.Engineering designs are based on a safe level of stress within a material. In an object that issubjecttoloads,forceswithintheobjectactto balancetheexternal loads.As a simple example, consider a slender rod that is placed in uniaxial tension, as shown inFigure 11.1.If the rod is sectioned at B-B,a force within the material at B-B is necessary tomaintain static equilibriumfor the sectioned rod. Such a force within the rod, acting per unit area, iscalled stress.Design criteria are based on stress levels within a part.In most cases stress cannot bemeasured directly.But the length of the rod in Figure 11.1 changes when the load is applied, andsuch changes in length or shape of a material can be measured. This chapter discusses themeasurement of physical displacements in engineering components. The stress is calculatedfromthesemeasured deflections.Upon completion of this chapter, the reader will be able to.define strain and delineatethe difficulty in measuring stress,? state the physical principles underlying mechanical strain gauges,.analyze strain gauge bridge circuits, and. describe methodsfor optical strain measurement.11.2STRESSANDSTRAINBeforeweproceedtodeveloptechniquesforstrainmeasurements,webrieflyreviewtherelationshipbetween deflections and stress.The experimental analysis of stress is accomplished bymeasuringthedeformation of apart underload,and inferringtheexisting stateof stressfromthemeasureddeflections.Again, consider therod in Figure 11.1.If therod has a cross-sectional area of A,and the466

E1C11 09/14/2010 13:14:1 Page 466 Chapter 11 Strain Measurement 11.1 INTRODUCTION The design of load-carrying components for machines and structures requires information con￾cerning the distribution of forces within the particular component. Proper design of devices such as shafts, pressure vessels, and support structures must consider load-carrying capacity and allowable deflections. Mechanics of materials provides a basis for predicting these essential characteristics of a mechanical design, and provides the fundamental understanding of the behavior of load-carrying parts. However, theoretical analysis is often not sufficient, and experimental measurements are required to achieve a final design. Engineering designs are based on a safe level of stress within a material. In an object that is subject to loads, forces within the object act to balance the external loads. As a simple example, consider a slender rod that is placed in uniaxial tension, as shown in Figure 11.1. If the rod is sectioned at B–B, a force within the material at B–B is necessary to maintain static equilibrium for the sectioned rod. Such a force within the rod, acting per unit area, is called stress. Design criteria are based on stress levels within a part. In most cases stress cannot be measured directly. But the length of the rod in Figure 11.1 changes when the load is applied, and such changes in length or shape of a material can be measured. This chapter discusses the measurement of physical displacements in engineering components. The stress is calculated from these measured deflections. Upon completion of this chapter, the reader will be able to
define strain and delineate the difficulty in measuring stress,
state the physical principles underlying mechanical strain gauges,
analyze strain gauge bridge circuits, and
describe methods for optical strain measurement. 11.2 STRESS AND STRAIN Before we proceed to develop techniques for strain measurements, we briefly review the relationship between deflections and stress. The experimental analysis of stress is accomplished by measuring the deformation of a part under load, and inferring the existing state of stress from the measured deflections. Again, consider the rod in Figure 11.1. If the rod has a cross-sectional area of Ac, and the 466

11.2467Stress and StrainFNFNCross-sectionalBareaA.BFNFigure 11.1 Free-body diagramillustrating internal forcesforarod inBuniaxial tension.load is applied only along the axis of the rod, the normal stress is defined as(11.1)a=FN/Acwhere A。is the cross-sectional area and F is the tension force applied to the rod normal to the areaAc. The ratio of the change in length of the rod (which results from applying the load) to the originallengthistheaxial strain,defined as&a = 8L/L(11.2)whereis theaveragestrain overthelengthL,Lis the change in length,andListheoriginal unloadedlength.For mostengineeringmaterials, strain is a small quantity; strain is usuallyreported in units of10- in./in.or 10-m/m. These units are equivalent to a dimensionless unit called a microstrain (μ).Stress-strain diagrams are very important in understanding the behavior of a material under load.Figure 11.2 is such a diagram for mild steel (a ductile material). For loads less than that required topermanently deform the material,most engineering materials display a linear relationship betweenstress and strain.Therange of stress over whichthis linear relationship holds is called the elasticregion.The relationshipbetween uniaxial stress and strain for this elastic behavior is expressed as(11.3)Oa=Em&awhere Em is the modulus ofelasticity,or Young'smodulus, and the relationship is called Hooke'slaw.Hooke'slawapplies onlyoverthe rangeof applied stress wherethe relationshipbetween stress andstrain is linear.Different materialsrespond in a variety of waystoloadsbeyond the linear range, largelydepending on whether the material is ductile or brittle.For almost all engineering components,stresslevels aredesigned to remain well belowthe elastic limit of thematerial; thus, a direct linearrelationshipmaybeestablishedbetweenstressandstrain.Underthisassumption,Hooke'slawformsthe basis for experimental stress analysis through the measurement of strain.Lateral StrainsConsider the elongation of therod shown in Figure 11.1 that occurs as aresult of the load Fn.As therod is stretched in the axial direction,the cross-sectional area must decrease since the total mass (or

E1C11 09/14/2010 13:14:1 Page 467 load is applied only along the axis of the rod, the normal stress is defined as sa ¼ FN=Ac ð11:1Þ where Ac is the cross-sectional area and FN is the tension force applied to the rod normal to the area Ac. The ratio of the change in length of the rod (which results from applying the load) to the original length is the axial strain, defined as ea ¼ dL=L ð11:2Þ where ea is the average strain over the length L, dL is the change in length, andL is the original unloaded length. For most engineering materials, strain is a small quantity; strain is usually reported in units of 106 in./in. or 106 m/m. These units are equivalent to a dimensionless unit called a microstrain (me). Stress–strain diagrams are very important in understanding the behavior of a material under load. Figure 11.2 is such a diagram for mild steel (a ductile material). For loads less than that required to permanently deform the material, most engineering materials display a linear relationship between stress and strain. The range of stress over which this linear relationship holds is called the elastic region. The relationship between uniaxial stress and strain for this elastic behavior is expressed as sa ¼ Emea ð11:3Þ where Em is the modulus of elasticity, or Young’s modulus, and the relationship is called Hooke’s law. Hooke’s law applies only over the range of applied stress where the relationship between stress and strain is linear. Different materials respond in a variety of ways to loads beyond the linear range, largely depending on whether the material is ductile or brittle. For almost all engineering components, stress levels are designed to remain well below the elastic limit of the material; thus, a direct linear relationship may be established between stress and strain. Under this assumption, Hooke’s law forms the basis for experimental stress analysis through the measurement of strain. Lateral Strains Consider the elongation of the rod shown in Figure 11.1 that occurs as a result of the load FN. As the rod is stretched in the axial direction, the cross-sectional area must decrease since the total mass (or FN FN FN B B B B FN Ac σa = Cross-sectional area Ac Figure 11.1 Free-body diagram illustrating internal forces for a rod in uniaxial tension. 11.2 Stress and Strain 467

468Chapter 11StrainMeasurement300-40,00025030,000200(wNW)es(sd) ssal15020,000品10010,00050-111100.0010.0020.0040.003in./in. or m/m0.10.20.30.4percentFigure 11.2 A typical stress-Strainstrain curve for mild steel.volumefor constant density)mustbe conserved.Similarly.if the rod were compressed in the axialdirection, the cross-sectional area would increase. This change in cross-sectional area is mostconveniently expressed in terms of a lateral (transverse)strain.For a circularrod, the lateral strain isdefined as the change in the diameter divided by the original diameter. In the elastic range, there is aconstant rate of change in the lateral strain as the axial strain increases. In the same sense that themodulus ofelasticity is a property of a given material, the ratio oflateral strain to axial strain is also amaterial property. This property is called Poisson's ratio, defined as[Lateral strain]_EL(11.4)Up:[AxialstrainGaEngineering components are seldom subject to one-dimensional axial loading.The relationshipbetween stress and strain mustbegeneralizedtoamultidimensionalcase.Considera two-dimensionalgeometry, as shown in Figure 11.3, subject to tensile loads in both the x and y directions, resulting innormal stresses x and y.In this case,for a biaxial state of stress, the stresses and strains areOxoy-UpEmdxQy-UpEm&x=Ey=EmEmEm(ex +upe)Em(ey + Upex)(11.5)Ox=Oy =1-51-%Txy = GYxyIn this case, all of the stress and strain components lie in the same plane.The state of stress in theelastic condition for a material is similarly related to the strains in a complete three-dimensionalsituation (1, 2). Since stress and strain are related, it is possible to determine stress from measuredstrains under appropriate conditions. However, strain measurements are made at the surface of anengineering component.Themeasurement yields information about the state of stress on the surface

E1C11 09/14/2010 13:14:1 Page 468 volume for constant density) must be conserved. Similarly, if the rod were compressed in the axial direction, the cross-sectional area would increase. This change in cross-sectional area is most conveniently expressed in terms of a lateral (transverse) strain. For a circular rod, the lateral strain is defined as the change in the diameter divided by the original diameter. In the elastic range, there is a constant rate of change in the lateral strain as the axial strain increases. In the same sense that the modulus of elasticity is a property of a given material, the ratio of lateral strain to axial strain is also a material property. This property is called Poisson’s ratio, defined as yP ¼ jLateral strainj jAxial strainj ¼ eL ea ð11:4Þ Engineering components are seldom subject to one-dimensional axial loading. The relationship between stress and strain must be generalized to a multidimensional case. Consider a two-dimensional geometry, as shown in Figure 11.3, subject to tensile loads in both the x and y directions, resulting in normal stresses sx and sy. In this case, for a biaxial state of stress, the stresses and strains are ey ¼ sy Em yp sx Em ex ¼ sx Em yp sy Em sx ¼ Em ex þ ypey
1 y2 p sy ¼ Em ey þ ypex
1 y2 p txy ¼ Ggxy ð11:5Þ In this case, all of the stress and strain components lie in the same plane. The state of stress in the elastic condition for a material is similarly related to the strains in a complete three-dimensional situation (1, 2). Since stress and strain are related, it is possible to determine stress from measured strains under appropriate conditions. However, strain measurements are made at the surface of an engineering component. The measurement yields information about the state of stress on the surface 0.001 0.1 0.002 0.2 0.003 0.3 0.004 0.4 in./in. or m/m percent 0 50 100 150 200 250 300 Strain Stress (psi) Stress (MN/m2) 10,000 20,000 30,000 40,000 Figure 11.2 A typical stress– strain curve for mild steel. 468 Chapter 11 Strain Measurement

11.3469ResistanceStrainGaugesoyFigure11.3Biaxial state of stress.of the part.The analysis of measured strains requires application of the relationship between stressand strain at a surface.Such analysis of strain data is described elsewhere (3),and an exampleprovided in this chapter.11.3RESISTANCESTRAINGAUGESThemeasurementof thesmalldisplacementsthatoccurinamaterial orobjectundermechanicalload canbeaccomplished bymethods as simple asobservingthe change in thedistancebetween twoscribe marks on the surface of aload-carrying member, or as advanced as optical holography.In anycase,the ideal sensorfor the measurement of strain would (1) have good spatial resolution,implyingthat the sensor would measure strain at a point; (2) be unaffected by changes in ambient conditions;and (3) have a high-frequencyresponse for dynamic (time-resolved) strain measurements. A sensorthat closely meets these characteristics is the bonded resistance strain gauge.In practical application, the bonded resistance strain gauge is secured to the surface of the testobjectbyanadhesive so that it deforms as thetest object deforms.The resistance of a strain gaugechanges when it is deformed, and this is easily related to the local strain.Both metallic andsemiconductor materials experience a change in electrical resistance when they are subjected to astrain.The amount that the resistance changes depends on how the gauge is deformed, the materialfrom which it is made, and the design of the gauge.Gauges can bemade quite small forgoodresolution and with a low mass to provide a high-frequency response.With some ingenuity,ambienteffectscanbeminimizedoreliminatedIn an1856publication in thePhilosophicalTransactions of theRoyal Society in England,LordKelvin (William Thomson)(4)laid the foundations for understanding the changes in electricalresistance that metals undergo when subjected to loads, which eventually led to the strain gaugeconcept.Twoindividuals began themodern developmentof strain measurement in thelate1930s-Edward Simmons attheCalifornia Instituteof TechnologyandArthurRugeattheMassachusettsInstitute of Technology.Their development of the bonded metallic wire strain gauge led tocommercially available strain gauges.The resistance strain gauge also forms the basis for a varietyof other transducers,such as load cells,pressure transducers,and torquemeters

E1C11 09/14/2010 13:14:1 Page 469 of the part. The analysis of measured strains requires application of the relationship between stress and strain at a surface. Such analysis of strain data is described elsewhere (3), and an example provided in this chapter. 11.3 RESISTANCE STRAIN GAUGES The measurement of the small displacements that occur in a material or object under mechanical load can be accomplished by methods as simple as observing the change in the distance between two scribe marks on the surface of a load-carrying member, or as advanced as optical holography. In any case, the ideal sensor for the measurement of strain would (1) have good spatial resolution, implying that the sensor would measure strain at a point; (2) be unaffected by changes in ambient conditions; and (3) have a high-frequency response for dynamic (time-resolved) strain measurements. A sensor that closely meets these characteristics is the bonded resistance strain gauge. In practical application, the bonded resistance strain gauge is secured to the surface of the test object by an adhesive so that it deforms as the test object deforms. The resistance of a strain gauge changes when it is deformed, and this is easily related to the local strain. Both metallic and semiconductor materials experience a change in electrical resistance when they are subjected to a strain. The amount that the resistance changes depends on how the gauge is deformed, the material from which it is made, and the design of the gauge. Gauges can be made quite small for good resolution and with a low mass to provide a high-frequency response. With some ingenuity, ambient effects can be minimized or eliminated. In an 1856 publication in the Philosophical Transactions of the Royal Society in England, Lord Kelvin (William Thomson) (4) laid the foundations for understanding the changes in electrical resistance that metals undergo when subjected to loads, which eventually led to the strain gauge concept. Two individuals began the modern development of strain measurement in the late 1930s— Edward Simmons at the California Institute of Technology and Arthur Ruge at the Massachusetts Institute of Technology. Their development of the bonded metallic wire strain gauge led to commercially available strain gauges. The resistance strain gauge also forms the basis for a variety of other transducers, such as load cells, pressure transducers, and torque meters. σx σx σy σy Figure 11.3 Biaxial state of stress. 11.3 Resistance Strain Gauges 469

470Chapter11StrainMeasurementMetallicGaugesTo understand how metallic strain gauges work,consider a conductor having a uniform cross-sectional area A。 and length L made of a material having an electrical resistivity, Pe. For thiselectrical conductor, the resistance, R, is given by(11.6)R = pL/A.If the conductor is subjected to a normal stress along the axis of the wire, the cross-sectionalarea and the length change resulting in a change in the total electrical resistance, R. The total changein R is due to several effects, as illustrated in the total differential:dR = A.(pdL + Ldp.) -p,LdA.(11.7)A2whichmaybeexpressed interms ofPoisson'sratioasdRdLdp(1 + 2u) +(11.8)R/PeHence,the changes in resistance are caused by two basic effects: the change in geometry as thelength and cross-sectional area change, and the change in the value of the resistivity,Pe.Thedependence of resistivity on mechanical strain is called piezoresistance,and maybe expressed interms ofapiezoresistancecoefficient,definedbyI dpe/pe(11.9)TI = Em dL/LWith this definition, the change in resistance may be expressed(11.10)dR/R=dL/L(1+2Up+TEm)Example 11.1Determine the total resistance of a copper wire having a diameter of 1 mm and a length of 5 cm. Theresistivity of copper is 1.7 × 10-8m.KNOWND=1mmL=5cmP,=1.7x10-80mFIND Thetotal electrical resistanceSOLUTION Theresistancemaybecalculated fromEquation 11.6 asR=pL/Ac

E1C11 09/14/2010 13:14:1 Page 470 Metallic Gauges To understand how metallic strain gauges work, consider a conductor having a uniform cross￾sectional area Ac and length L made of a material having an electrical resistivity, re. For this electrical conductor, the resistance, R, is given by R ¼ reL=Ac ð11:6Þ If the conductor is subjected to a normal stress along the axis of the wire, the cross-sectional area and the length change resulting in a change in the total electrical resistance, R. The total change in R is due to several effects, as illustrated in the total differential: dR ¼ Ac redL þ Ldre ð Þ reLdAc A2 c ð11:7Þ which may be expressed in terms of Poisson’s ratio as dR R ¼ dL L 1 þ 2yp
þ dre re ð11:8Þ Hence, the changes in resistance are caused by two basic effects: the change in geometry as the length and cross-sectional area change, and the change in the value of the resistivity, re. The dependence of resistivity on mechanical strain is called piezoresistance, and may be expressed in terms of a piezoresistance coefficient, p1 defined by p1 ¼ 1 Em dre=re dL=L ð11:9Þ With this definition, the change in resistance may be expressed dR=R ¼ dL=L 1 þ 2yp þ p1Em
ð11:10Þ Example 11.1 Determine the total resistance of a copper wire having a diameter of 1 mm and a length of 5 cm. The resistivity of copper is 1:7  108 V m. KNOWN D ¼ 1 mm L ¼ 5 cm re ¼ 1:7  108 V m FIND The total electrical resistance SOLUTION The resistance may be calculated from Equation 11.6 as R ¼ reL=Ac 470 Chapter 11 Strain Measurement

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 resist￾ance 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 P￾type 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 semi￾conductors (reference resistance at 81F). (Courtesy of Kulite Semiconductor Products, Inc.) 11.3 Resistance Strain Gauges 475