Strain Gage

Prévia do material em texto

The Bonded Electrica l Resistanc e
Strain Gag e
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The Bonde d
Electrical Resistanc e
Strain Gag e
An Introduction
WILLIAM M . MURRA Y
Professor Emeritus
Massachusetts Institute of Technology
WILLIAM R . MILLE R
Professor Emeritus
The University of Toledo
New Yor k Oxfor d
OXFORD UNIVERSIT Y PRES S
1992
Oxford Universit y Pres s
Oxford Ne w Yor k Toront o
Delhi Bomba y Calcutt a Madra s Karach i
Kuala Lumpu r Singapor e Hon g Kon g Toky o
Nairobi Da r e s Salaam Cap e Tow n
Melbourne Aucklan d
and associate d companie s i n
Berlin Ibada n
Copyright ; 199 2 b y Oxfor d Universit y Press , Inc .
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Library o f Congres s Cataloging-in-Publicatio n Dat a
Murray , Willia m M .
The bonde d electrica l resistanc e strai n gag e :
an introductio n / b y Willia m M. Murra y an d Willia m R . Miller.
p. cm . Include s bibliographica l reference s an d index .
ISBN 0-19-507209- X
1. Strai n gages . 2 , Electri c resistanc e Measurement .
I. Miller . Willia m R . (Willia m Ralph) , 1917 - . II . Title .
TA413.5.M87 199 2 624.1'76'028 7 dc2 0 91-4136 9
2 4 6 8 9 7 5 3 1
Printed i n th e Unite d State s o f Americ a
on acid-fre e pape r
PREFACE
Experimental stres s analysi s i s an importan t too l i n th e overal l desig n an d
development o f machinery an d structures . While analytica l technique s an d
computer solution s ar e available during th e design stage, the results are stil l
dependent o n many assumption s tha t mus t be made i n order t o adap t the m
to th e problem s a t hand . Onl y whe n th e desig n i s fixed, the prototype s ar e
constructed, and testin g is underway, can th e proble m area s b e realistically
determined, and thi s must b e done throug h experimenta l means .
One metho d o f findin g th e weaknesses , an d a metho d whic h i s use d
extensively, i s through th e us e o f the electrica l resistanc e strai n gage . Strai n
gages ar e relativel y lo w i n cost , easil y applie d b y a reasonabl y skille d
technician, d o no t requir e extensiv e investment i n instrumentatio n (fo r th e
general user) , and ye t they yield a wealth o f information in a relatively short
time. The information and it s validity is, of course, dependent o n the trainin g
and knowledg e o f th e enginee r wh o plan s th e test s an d reduce s th e data .
The latter statemen t become s painfull y apparen t whe n one finds a user trying
to interpre t dat a fro m a singl e strai n gag e applie d i n a n unknow n biaxia l
stress field.
In 1988 , th e author s decide d t o edi t Dr . Murray' s notes , whic h wer e
developed ove r hi s extensiv e career , an d t o writ e a n introductor y tex t o n
electrical resistanc e strai n gages. Th e tex t is directed a t senio r an d first-yea r
graduate student s i n th e engineerin g disciplines , althoug h student s fro m
other field s (geology , engineering physics , etc. ) wil l als o benefit .
The prerequisite s fo r a strai n gag e cours e ar e th e following : (1 ) Th e
basic courses in resistance o f materials. (2) An elementary course in electrical
circuits. (3) At least one course in mechanical or structural design is desirable.
It follow s tha t the more experienc e student s have in analysis and design , the
more the y wil l benefi t fro m a n experimenta l course . I t i s in th e laborator y
and i n experimental courses tha t student s reall y develop a sens e o f security
in, an d a bette r understandin g of , the theor y the y have bee n expose d t o i n
their analytica l studies .
The development o f stress an d strai n transformatio n equations an d th e
corresponding Mohr' s circles , a s wel l a s the stress-strai n relationships , ar e
covered in Chapter 2. Depending o n the student's preparation , th e instructo r
may us e this chapter fo r a rapid revie w or eliminate i t entirely. The authors ,
however, hav e foun d i t beneficia l t o spen d a t leas t severa l period s o n th e
material.
Basic electrica l circuit s ar e examine d i n Chapter s 3 throug h 5 . A n
elementary circui t consisting of a single strain gage and its response t o strai n
is first considered, followe d b y the potentiometric circui t and the Wheatston e
vi PREFAC E
bridge. In the development of the expressions for output voltage, as the strain
gage's resistanc e change s wit h increasin g loading , i s th e effec t o f circui t
nonlinearity. Th e equation s ar e develope d s o tha t th e studen t ca n easil y
handle the intervening algebra between steps and thereb y see the nonlinearity
terms unfold . I t i s importan t tha t student s recogniz e thi s an d understand ,
when recordin g larg e strains , how t o correc t th e indicated strain s to obtai n
the actua l strains . Th e effec t o f resistanc e i n bot h th e powe r suppl y an d
indicating mete r i s also accounte d for .
Lead-line resistance is considered i n the Wheatstone bridg e circuits. The
circuits ar e th e ful l bridge , th e hal f bridg e wit h fou r wires , th e hal f bridg e
with three wires, the quarter bridg e wit h three wires , and th e quarte r bridg e
with tw o wires . The equation s ar e developed s o tha t th e nonlinearit y effect s
are apparent .
Sensitivity variation in order to obtain a desired output is next discussed
in Chapter 6 . Equations ar e developed , including nonlinearity effects, fo r th e
desensitization o f single gages , half-bridg e circuits, and full-bridg e circuits.
Chapter 7 is devoted t o th e lateral , or transverse , effect o n strai n gages ,
along wit h a discussio n o f th e method s use d t o determin e th e gag e facto r
and th e transvers e sensitivit y factor o f strai n gages . Thi s i s followe d b y
Chapters 8 and 9 o n strai n gag e rosette s an d dat a reduction . I t i s shown
how t o reduc e rosett e dat a b y bot h analytica l method s an d graphica l
methods. This is followed b y considering transverse effects, usin g information
from Chapte r 7 , in rosett e dat a reduction.
Chapter 1 0 discusses ho w strai n gage s ma y b e use d t o measur e bot h
normal stresse s an d shearin g stresse s directly , while Chapte r 1 1 consider s
the effec t o f temperatur e o n strai n gag e readings . Temperature-induce d
strains ar e discussed , followe d b y a n examinatio n o f self-temperature -
compensated gage s an d thei r therma l outpu t curve s whe n th e gage s ar e
bonded t o severa l differen t materials . On e ca n se e ho w t o correc t th e
indicated strain not onl y for the temperature-induced strain , but als o fo r the
gage facto r variatio n resultin g from temperatur e change .
Several type s o f strain-gag e transducer s ar e covere d i n Chapte r 12 .
Among them ar e th e axial-force load cell , the torque meter , the shear meter ,
and th e pressur e transducer . Th e purpos e i s t o introduc e th e studen t t o
several type s o f transducer s tha t coul d b e mad e an d calibrate d fo r hi s us e
in th e laboratory .
At the time of Dr. Murray' s death on August 14, 1990, the major portio n
of th e manuscrip t ha d bee n completed . I f there are error s o r discrepancies ,
the faul t i s not hi s bu t mine . In completin g th e text , I gathered togethe r al l
of th e sourc e materia l i n orde r t o giv e proper credit ; I sincerel y hope non e
has bee n overlooked .
A textboo k i s not th e wor k o f one o r several people alone . Al l of us ar e
influenced no t onl y by our contemporarie s bu t b y those wh o hav e precede d
us (one has only to thin k of Professor Ott o Moh r t o realiz e this). Therefore,
I want to acknowledg e our deb t t o al l of these people, no t th e least o f whom
PREFACE vi i
were our students . I wan t especially t o than k Marth a Watso n Spaldin g of
Measurements Group, Inc. fo r her cooperation in furnishing a considerable
amount o f material . I als o wan t t o acknowledg e th e assistanc e o f th e
following companies: BLH Electronics, Inc.; Eaton Corporation, Transducer
Products; Electri x Industries , Inc. ; Hartru n Corporation ; Measurement s
Group, Inc. ; Stein Engineering Services, Inc.; and Texa s Measurements, Inc.
W. R . Miller
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CONTENTS
1. Fundamenta l Concept s fo r Strai n Gages , 3
1.1 Introduction, 3
1.2 Characteristics Desired in a Strain Gage, 4
1.3 General Considerations, 5
1.4 Analysis of Strain Sensitivity in Metals, 14
1.5 Wire Strain Gages, 24
1.6 Foil Strain Gages, 29
1.7 Semiconductor Gages, 32
1.8 Some Other Types of Gages, 33
1.9 Brittle Lacquer Coatings, 36
2. Stress-Strai n Analysi s and Stress-Strai n Relations , 42
2.1 Introduction, 42
2.2 Basic Concepts of Stress, 43
2.3 Biaxial Stresses, 45
2.4 Mohr's Circle for Stress, 54
2.5 Basic Concepts of Strain, 61
2.6 Plane Strain, 62
2.7 Mohr's Circle for Strain, 68
2.8 Stress-Strain Relationships, 72
2.9 Application of Equations, 77
2.10 Stress and Strain Invariants, 81
3. Elementar y Circuits , 90
3.1 Introduction, 90
3.2 Constant- Voltage Circuit, 91
3.3 Constant-Current Circuit, 94
3.4 Advantages of the Constant-Current Circuit, 96
3.5 Fundamental Laws of Measurement, 97
x CONTENT S
4. Th e Potentiometri c Circuit, 100
4.1 Introduction, 100
4.2 Circuit Equations, 101
4.3 Analysis of the Circuit. 106
4.4 Linearity Considerations, 119
4.5 Temperature Effects, 129
4.6 Calibration, 141
5. Wheatston e Bridge , 146
5.1 Introduction, 146
5.2 Elementary Bridge Equations, 149
5.3 Derivation of Elementary Bridge Equations, 157
5.4 General Bridge Equations, 172
5.5 Effect o f Lead-Line Resistance, 18 0
5.6 Circuit Calibration, 193
5.7 Comments, 195
6. Sensitivit y Variation , 205
6.1 Introduction, 205
6.2 Analysis of Single Gage Desensitization, 207
6.3 Analysis of Half-Bridge Desensitization, 218
6.4 Analysis of Full-Bridge Sensitivity Variation, 227
1. Latera l Effect s i n Strai n Gages , 23 4
7.1 Significance of Strain Sensitivity and Gage Factor, 234
7.2 Basic Equations for Unit Change in Resistance, 236
7.3 Determination of Gage Factor and Transverse
Sensitivity Factor, 242
7.4 Use of Strain Gages Under Conditions Differing from those
Corresponding to Calibration, 246
7.5 Indication from a Pair of Like Strain Gages Crossed at
Right Angles, 248
8. Strai n Gage Rosette s and Dat a Analysis , 253
8.1 Reason for Rosette Analysis, 253
8.2 Stress Fields, 253
8.3 Rosette Geometry, 256
8.4 Analytical Solution for the Rectangular Rosette, 258
CONTENTS
8.5 Analytical Solution for the Equiangular or Delta Rosette, 267
8.6 Rosettes with Four Strain Observations, 275
8.7 Graphical Solutions, 281
9. Strai n Gag e Rosette s an d Transvers e Sensitivit y Effect , 29 1
9.1 Introduction, 291
9.2 Two Identical Orthogonal Gages, 291
9.3 Two Different Orthogonal Gages, 294
9.4 Three-Element Rectangular Rosette, 296
9.5 The Equiangular or Delta Rosette, 301
10. Stres s Gages , 310
70.7 Introduction, 31 0
10.2 The Normal Stress Gage, 310
10.3 The SR-4 Stress-Strain Gage, 316
10.4 Electrical Circuit for Two Ordinary Gages to Indicate
Normal Stress, 320
10.5 The V-Type Stress Gage, 321
10.6 Application of a Single Strain Gage to Indicate
Principal Stress, 326
10.7 Determination of Plane Shearing Stress, 327
11. Temperatur e Effect s o n Strai n Gages , 337
11.1 Introduction, 337
11.2 Basic Considerations of Temperature-Induced Strain, 337
11.3 Self-Temperature-Compensated Strain Gages, 343
11.4 Strain Gage-Test Material Mismatch, 349
11.5 Compensating Gage, 353
12. Transducers , 36 0
72.7 Introduction, 36 0
12.2 Axial-Force Transducers, 363
12.3 Simple Cantilever Beam, 368
12.4 Bending Beam Load Cells, 372
12.5 Shear Beam Load Cell, 375
12.6 The Torque Meter, 378
12.7 The Strain Gage Torque Wrench, 380
12.8 Pressure Measurement, 382
xi
xii CONTENT S
13. Strai n Gag e Selectio n and Application , 390
13.1 General Considerations, 390
13.2 Strain Gage Alloys. 391
13.3 Grid Backing Materials, 393
13.4 Gage Length, Geometry, and Resistance, 394
13.5 Adhesives , 39 6
13.6 Bonding a Strain Gage to a Specimen, 398
Answers t o Selecte d Problems, 402
Index, 405
The Bonded Electrical Resistance
Strain Gage
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1
FUNDAMENTAL CONCEPTS FOR STRAIN GAGES
1.1. Introduction
The constan t deman d fo r improvemen t i n th e desig n o f machin e an d
structural part s ha s le d t o th e developmen t o f various experimental techni-
ques fo r determinin g stres s distributions . These experimenta l method s ar e
employed for both the checking of theoretical predictions, and the evaluation
of stresse s i n situation s wher e mathematica l approache s ar e unavailabl e o r
unsuited.
However, sinc e stres s canno t b e measure d directly , th e experimenta l
procedures, o f necessity, make thei r approach throug h som e typ e o f strain
measurement. The measured strain s are then converted into their equivalent
values i n term s o f stress . I n orde r t o achiev e thi s ultimat e objective , som e
type o f strain-indicating device o r measurin g device i s required.
In additio n t o thei r use s fo r stres s analysis , strai n gage s als o fin d
wide applicatio n i n sensin g device s an d contro l devices . I n thes e applica -
tions, th e strai n i n som e mechanica l par t i s use d a s a n indicatio n o f force ,
bending, torque , pressure , acceleration , o r som e othe r quantit y relate d t o
strain.
Even th e mos t casua l surve y of the literatur e relatin g t o th e measure -
ment of mechanical strain wil l yield information on a wide variety of devices
which have been developed fo r this purpose . I n addition to photoelasticity ,
brittle lacquer (1 , 2, 3),1 and X-rays , one finds all sorts o f mechanical, optical ,
and electrica l strai n gage s an d extensometers , an d variou s combination s
thereof, whic h have bee n develope d fo r on e purpos e o r another , frequentl y
with regar d t o som e ver y specifi c application . I t i s ver y obviou s tha t th e
development o f a single instrument possessin g al l the optimu m characteris -
tics, fo r al l applications , i s unlikely . However , a goo d approac h t o th e
ultimate i s stil l possible .
The brittl e lacque r markete d a s Tens-La c (1 , 2 ) i s n o longe r avail -
able, althoug h Stresscoa t (3 ) ca n b e obtained . Thes e references , however,
give a goo d descriptio n o f the us e o f brittle lacquer s i n experimenta l stres s
analysis.
1 Numbers in parentheses refe r t o Reference s a t th e en d o f a chapter.
4 TH E BONDE D ELECTRICA L RESISTANCE STRAIN GAG E
1.2. Characteristics desired in a strain gage
If w e se t ou t t o devis e a general-purpos e strai n gage , w e woul d probabl y
make a lis t o f all possible desire d characteristics . Some o f these include , no t
necessarily i n thei r order o f importance , th e following :
1. Abilit y t o measur e strain s precisel y unde r stati c an d dynamic
conditions.
2. Smal l siz e and weight . The smal l size permits mounting th e instrument
in confine d locations , o r t o obtai n reasonabl y precis e indication s
in region s o f hig h stres s gradient . Smal l weigh t i s require d s o tha t
the inerti a effect s i n th e gag e wil l b e negligibl e unde r dynami c
conditions.
3. Th e possibility of remote observation and recording . This is very much
a relativ e requirement, sinc e remote migh t mean anythin g fro m a few
feet i n the laborator y t o thousand s o f miles, as in th e cas e of a rocke t
or missil e with radi o transmissio n (telemetering ) of th e signa l t o th e
location o f the observer .
4. Independenc e o f th e influenc e o f temperature . Thi s i s probabl y th e
most difficul t requiremen t o f all . Ver y satisfactor y result s ca n b e
achieved ove r smal l temperatur e excursions , bu t whe n th e tempera -
ture ma y fluctuat e u p o r dow n i n th e rang e fro m abou t — 400°F
to +1500° F (-24 0 t o 815°C) , th e proble m become s exceedingl y
difficult.
5. Eas y installation . In order to b e commercially attractive, a strain gag e
should b e sufficientl y eas y t o instal l so that relatively unskilled people
can b e trained , i n a shor t spac e o f time , t o perfor m thi s operatio n
satisfactorily an d reliably .
6. Stabilit y o f calibration . I t i s extremely desirable tha t th e calibratio n
should b e stable ove r th e entir e range o f operating conditions .
7. Linea r respons e t o strain . Althoug h no t absolutel y essential , thi s i s
very desirable . Smal l deviation s fro m linearit y ca n frequentl y b e
brought withi n tolerable limit s b y combinatio n (opposition ) wit h the
inherent nonlinearit y of the electrica l circui t of which the gag e forms
a part . Fo r large r departure s fro m linearity , the electrica l circui t can
be specially designed t o provid e automatic compensation (4 , 5). When
large-scale computer s ar e employe d t o conditio n an d proces s th e
strain gag e indications , provide d tha t th e relatio n betwee n strai n an d
gage indicatio n i s known, thi s functio n ca n b e directl y programme d
into th e machine .
8. Lo w cost . Thi s i s anothe r relativ e consideratio n tha t depend s upo n
the work a t hand. Generally speaking, the cost of modern strain gages
is relatively insignificant in comparison wit h the other cost s associate d
with a n importan t project .
9. Dependability . Unles s th e strai n gag e indication s ca n b e depende d
upon, it s us e become s ver y limited . Fortunately , th e strai n gage s
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 5
available toda y ar e ver y dependabl e whe n used unde r th e conditions
for whic h they were intended.
10. Th e possibility of operation as an individual strain gage , or in multiple
arrangements, t o determin e quantitie s tha t ar e indicate d b y th e
simultaneous observatio n o f strains a t mor e tha n on e location . Thi s
means that , fo r certain applications , w e should b e abl e t o us e strai n
gages in multiple arrangements to perfor m automatic computation of
some quantit y tha t i s related t o strain s a t severa l locations .
No on e ha s ye t developed a strai n gag e possessin g al l of these desire d
characteristics. However , on e ca n generall y sa y tha t bonde d electrica l
resistance strain gages (wire, foil, o r semiconductor ) come much nearer than
any othe r devic e to satisfyin g al l these requirements.
1.3. General considerations
Basic principle
In commo n wit h photoelasticit y an d stresscoat , th e basi c principl e under -
lying th e operatio n o f electrica l resistanc e strai n gage s ha s bee n know n
for a long time. However, the application o f the principl e to strai n measure-
ment (on a commercial scale ) is much more recent . In 185 6 Lord Kelvi n (6)
reported hi s observation s tha t certai n electrica l conductor s h e ha d bee n
studying exhibite d a chang e i n electrica l resistanc e wit h chang e i n
strain.
The chang e o f electrica l resistanc e resultin g fro m mechanica l strai n
represents th e basi c principl e upo n whic h electrical resistanc e strai n gage s
operate. Fo r semiconducto r gages , the detai l o f the mean s b y whic h strai n
changes th e resistanc e seem s t o b e wel l understood , bu t fo r metalli c
conductors (wir e or foil), we are still a long way from a complete understand-
ing o f what takes place within the material .
Definition of strain sensitivity
When a conducto r i s trained i n th e axia l direction , it s lengt h wil l change ,
and, i f unrestrained laterally , it s cross-sectiona l are a wil l als o chang e (th e
Poisson effect) . Th e increas e in length, shown in Fig . 1.1 , is accompanied by
a decrease i n the cross-sectional area , and vic e versa. In addition, the specifi c
resistivity o f the materia l ma y change . These thre e influences, the chang e i n
length, th e chang e i n cross-sectiona l area , an d th e chang e i n specifi c
resistivity, combin e t o produc e a chang e i n th e overal l electrica l resistanc e
of th e conductor . Th e amoun t o f the resistanc e change , i n relatio n t o th e
change i n lengt h o f th e conductor , i s an inde x o f wha t i s calle d th e strai n
sensitivity of the materia l o f the conductor . This relationship is expressed a s
a dimensionles s rati o calle d th e strain sensitivity factor. Fo r a straigh t
THE BONDE D ELECTRICA L RESISTANC E STRAIN GAG E
FIG. 1.1 . Schemati c diagra m o f strained conducto r (tensil e effec t shown) .
conductor o f uniform cros s section , thi s is expressed a s
unit change in resistanc e
Strain sensitivity factor =
unit change i n length
unit change i n resistanc e
strain
In symbols , thi s can b e written a s
where S , = strai n sensitivit y (factor) of the conducto r an d i s
dimensionless; thi s is a physica l property o f the materia l
R = resistanc e i n ohm s
L = length i n inches
R, L = corresponding changes i n resistance and length, respectively,
in ohm s an d inche s
E = L/ L = strai n alon g th e conducto r (dimensionless )
Examination o f Eq. (1 .1 ) and th e definition s of the symbol s wil l rais e a
question regardin g th e values that should b e used fo r R an d L i n calculatin g
the strai n sensitivity . Do thes e symbol s represen t th e following?
6
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 7
1. Th e initial resistance, R0, and the initial length, L0, when the conductor
is stress free? In whic h case the denominator , E, corresponds to nomina l
strain based o n L 0.
1. An y corresponding value s o f resistance an d lengt h which may prevai l
after a certain amoun t o f initial load ha s bee n applied?
3. Th e instantaneous values of resistance and lengt h which prevail during
infinitely smal l change s o f lengt h an d resistance . I n whic h case , a s
L 0 , in the limit,
In Eq . (1.2) the denominator, e = dL/L, i s what is sometimes called th e
true strain (a s contrasted wit h th e nomina l strain) , and th e valu e of S,
obtained i n thi s manne r i s sometime s calle d th e instantaneou s sen -
sitivity factor, since it refers to the resistance and length in the stretched
condition fo r which both R an d L ar e variabl e (7).
Except fo r th e specia l cas e in whic h R happen s t o b e directly proportional
to L , theoretically , these thre e mode s o f interpretation wil l yiel d differen t
results for the value of S,, the strain sensitivity factor. This means that we are
confronted wit h th e proble m o f havin g t o decid e upo n whic h particula r
procedure w e should follow . Fo r th e specia l case in which the resistanc e is
directly proportiona l t o th e length , R = KL, wher e K i s a constant . Thus ,
R = K ( L) , and hence
Since R = pL/A, therefore K = p/A, whic h means tha t t o fulfil l thi s condi-
tion, the specific resistivity , p, will have to be proportional t o the area o f the
cross section .
Elastic strains in metals
For smal l strain s with correspondingl y smal l changes i n resistance , such as
might b e expected i n metal s whe n strained withi n the elasti c limit , there is
no problem. Here L0 an d L wil l be nearly equal and, likewise , R0 an d R wil l
be s o nearl y alik e i t wil l mak e n o noticeabl e differenc e i n th e valu e of S t,
whether i t i s computed o n th e basi s o f L0 an d R 0, o r fro m th e value s of L
and R whic h correspond t o th e elasti c limit. This i s a great convenience for
the followin g reasons :
1. Th e initia l resistance , R 0, an d th e initia l length , L 0, provid e goo d
references from whic h the changes R and L ca n be readily determined.
THE BONDE D ELECTRICA L RESISTANC E STRAIN GAG E
2. Th e strai n sensitivity, S,, can b e determined fro m th e slop e o f the curve
which i s established by plotting R/R 0 agains t L/L 0.
3. Th e analyse s o f the basi c electrica l circuits which are use d wit h strai n
gages, develope d i n followin g chapters , sho w tha t th e output , o r
indication, i s given in term s of R/R 0.
Plastic strains in metals
When a meta l conducto r i s strained beyon d th e elasti c limi t int o th e plasti c
range, th e change s i n resistanc e an d lengt h (fro m th e initia l values ) wil l
ultimately becom e s o larg e tha t ther e wil l b e a considerabl e differenc e
between R an d R (), an d als o betwee n L an d L 0.
When this happens, the previous approximate metho d o f determining St
from th e value s o f R 0 an d L 0 wil l n o longe r b e satisfactory . I t wil l b e
necessary t o comput e th e instantaneou s valu e of S , from th e instantaneou s
values o f R an d L , accordin g t o Eq . (1.2) .
At first glance, this might appear to be a formidable task, but fortunately
this i s not so . W e determin e a serie s of corresponding value s of R an d L a s
the conductor i s being stretched (o r compressed), an d then plot the logarithm
of the dimensionless ratio, R/R 0, agains t th e logarith m o f the dimensionless
ratio, L/L 0. Th e slop e o f the lin e thu s draw n represent s th e instantaneou s
value of the strain sensitivit y factor, St. Furthe r discussion wil l be found late r
in th e chapter .
Semiconductor materials
The relativel y hig h strai n sensitivit y o f silico n an d germaniu m ha s mad e
these semiconducto r material s attractiv e fo r strai n gag e sensin g elements .
For silicon , whic h is the preferre d material , the theoretica l valu e of St lies in
the rang e betwee n —15 0 and abou t +175 . Furthermore , b y suitabl e
processing (doping) , silico n can b e produce d wit h an y arbitraril y specified
value of S, within this range. For commercia l strain gages, in order t o achieve
a suitable compromise betwee n respons e t o strai n and respons e t o tempera -
ture, i t i s usual t o proces s th e materia l fo r strai n sensitivities in th e rang e of
about -10 0 t o abou t + 120.
The resistance-strai n relatio n fo r silico n i s somewha t mor e elaborat e
than tha t fo r metalli c conductors . I t i s nonlinear , an d ver y noticeabl y
influenced b y temperature . Dorse y (8 , 9) give s the followin g expressio n fo r
unit chang e i n term s o f strain:
8
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 9
where R = chang e in resistanc e fro m R O(TO> (ohms )
R0(To} = resistance (ohms) of the unstressed material (prior to being
mounted a s a strai n gage ) a t temperatur e T 0, in Kelvin
T0 = temperatur e a t whic h R O(TO) wa s determined (Kelvin)
T = temperatur e (Kelvin)
e = strai n (dimensionless )
GF', C'2 = constant s fo r the particula r piec e o f material
(dimensionless)
Equation (1.4 ) indicate s th e followin g characteristic s regardin g th e
relation betwee n uni t chang e i n resistanc e an d strai n fo r silicon:
1. Th e strain sensitivity factor , which corresponds to the slope of the curve
of R/R 0(:ro) vs . e, will be a variable whose value will depend upo n bot h
the strai n leve l and th e temperature.
2. Sinc e th e relationshi p expresse d i n Eq . (1.4 ) represent s a parabola ,
one ca n expec t th e degre e o f nonlinearit y t o var y wit h strai n an d
temperature.
3. A t constan t temperature , T 0, Eq. (1.4 ) reduces t o
Hence, for this special condition show n in Fig. 1.2 , GF' correspond s t o
the slop e o f th e curve , o r th e sensitivit y factor , fo r e = 0 , an d C' 2
represents th e nonlinearit y constan t whic h determine s th e degre e o f
departure o f the curv e fro m th e slop e a t th e poin t R = 0 , e = 0 , for
which th e resistanc e equal s Ro(r 0i- Bake r (10 ) als o expresse s Eq . (1.5 )
in essentiall y th e sam e form .
Over a limite d rang e o f strain , fo r exampl e abou t 60 0 microstrai n ( 1
microstrain = 1 uin/in), an d particularl y at strai n level s wher e th e slop e of
the curv e change s mor e gradually , th e variabl e strai n sensitivit y ca n b e
approximated b y a constant tha t corresponds t o the average value, and good
results ma y b e expected fro m this . For large r range s o f strain , o r fo r mor e
precise indications , mor e elaborat e method s mus t be employed .
When th e temperatur e varies , the whol e problem o f relating resistance
changes t o strai n become s mor e complicated . Thi s i s du e t o th e fac t tha t
changes i n temperature , a s indicate d i n Eq . (1.4) , produc e change s i n th e
sensitivity. I n addition , th e valu e of R 0(To-> wil l als o chang e wit h variation s
in th e referenc e temperature, T 0.
10 THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E
FIG. 1.2. Schemati c diagra m fo r R/R n(Ta
R/R0(Tat whe n R = e = 0.)
al constan t temperature , T 0. (Resistance =
Desired properties of strain-sensitive materials
1. Linea r relation between unit chang e i n resistance an d chang e i n strain
(i.e., constant sensitivity) .
2. Negligibl e effec t fro m temperature .
3. Hig h strai n sensitivit y factor .
4. Moderatel y hig h resistance.
5. Abilit y t o b e connected t o lea d wire s easily.
6. Lo w cost .
7. Availability.
8. Absenc e of creep and hysteresis .
One canno t expec t t o fin d al l th e desirabl e characteristic s i n an y
particular materia l withou t som e advers e properties , too . I n general , th e
selection o f a materia l fo r th e sensin g element o f a strai n gage wil l resul t in
a compromis e dependin g upo n th e intende d use o f the gage .
Properties of some metals
In vie w o f th e previou s discussio n o f strai n sensitivity , and th e propertie s
desired in strain sensing materials, let us look a t som e typica l characteristics
as represented b y a few metals. These are indicated i n Figs. 1. 3 and 1.4 , taken
from th e wor k o f Jones an d Masle n (11) . In eac h case , th e percen t chang e
in resistance , base d o n R 0, ha s bee n plotte d agains t percen t strain , o n th e
basis of L/L 0. Th e slope s of the line s represent S, . and th e differen t genera l
relationships ar e indicate d as follows :
vs
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 1 1
1. Th e same linear relatio n betwee n R/R 0 and L/L 0 i n both th e elastic
and plasti c ranges . Thi s conditio n i s represented b y anneale d copper ,
as well as annealed copper-nicke l alloys like Ferry. This means that the
strain sensitivit y factor wil l b e th e sam e i n th e plasti c range a s i t i s in
the elasti c range . Thi s characteristi c i s highl y desirabl e because i t
eliminates al l concern abou t th e possibilit y o f a change in gag e factor
in th e even t th e sensin g elemen t o f a strai n gag e migh t b e straine d
beyond it s elasti c limit . In consequence, thi s typ e o f materia l i s wel l
suited fo r gages whic h will be required t o measur e high elastic strains ,
or bot h elasti c and plasti c strains .
2. Nonlinea r relationshi p such a s exhibited by nickel.
3. Relationshi p approximated b y two straigh t lines indicating a change of
strain sensitivit y with the transitio n from elasti c to plasti c conditions .
Some materials , suc h a s minalpha , manganin , an d har d silver -
palladium, sho w a lowe r strai n sensitivit y at lo w strain s tha n a t hig h
strains.
4. Th e sam e genera l relationshi p a s indicate d i n Ite m (3) , bu t wit h th e
difference tha t th e highe r strai n sensitivit y corresponds t o th e lowe r
strains, a s shown by rhodium-platinum .
For th e relation s indicate d i n Items (3 ) and (4) , the chang e i n slope a s
yielding set s i n i s no t abrupt , a s suggeste d b y th e graphs , bu t follow s a
smooth transitio n fro m th e elasti c t o th e pasti c range .
Numerical values of the strain sensitivity factor
Table 1. 1 presents typica l strain sensitivit y values for a number o f metals a t
low strain , togethe r wit h correspondin g informatio n wit h respec t t o th e
effects o f temperature change s (12).
A mor e elaborat e tabulation , whic h includes some o f the pur e metal s
and a numbe r o f alloy s (wit h approximat e compositions) , i s give n i n th e
Appendix o f thi s chapter . Wher e possible , informatio n fo r sensitivitie s i n
both the elastic and plastic strain ranges, and for material in the cold worked
and anneale d conditions , has bee n included .
Approximate composition s o f some o f the alloy s in Tabl e 1. 1 are given
in Tabl e 1.2 .
A stud y of the literatur e an d o f the tabulate d dat a i n th e Appendi x a t
the en d o f the chapte r yield s the followin g observations regardin g materia l
properties:
1. Differen t value s o f strain sensitivit y for har d an d anneale d condition s
of the same materia l suggest s that th e degree o f cold working , and th e
heat treatment , hav e a n influence . This i s o f particular importanc e i n
relation t o th e effect s o f temperature an d temperatur e compensation .
THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E
FIG. 1.3. Typica l example s o f resistanc e chang e vs . strai n (Fro m ref. 11 wit h permissio n o f
HMSO.)
2. Difference s i n sensitivit y fo r differen t lot s o f nominall y th e sam e
material sugges t tha t difference s i n impurities , and i n trac e elements ,
exert an influenc e o n th e physica l properties. This i s also of importance
with respec t t o temperatur e effects .
3. Fo r nearl y al l th e metal s investigated , th e strai n sensitivit y facto r
appears t o approac h a valu e of 2.0 in th e plasti c range .
For larg e strain s (u p t o 3 0 percent), Weibul l (13 ) has reporte d som e ver y
interesting detailed experimenta l results on th e relatio n betwee n changes i n
length an d resistanc e for 0.45-mm diamete r Cope l wire . This i s a 5 5 percent
copper, 4 5 percent nicke l alloy.
From the data in Table 1.3 , the values of R/R0, L/L 0, R/R0, an d L/L 0,
have been computed . Plot s of \n(R/R0) vs . ln(L/L0) an d R/R0 \sAL/L 0 ar e
shown i n Fig . 1. 5 fo r comparativ e purposes . Fro m th e slop e o f th e
logarithmic plot , whic h i s represente d b y a straigh t line , th e valu e o f th e
12
FIG. 1.4. Resistanc e chang e vs . strai n fo r anneale d Ferr y wir e (60/4 0 cupronickel) . (Fro m
ref. 12. )
Table 1.1. Typica l strain sensitivit y factor s
Material
Strain sensitivity factor
(for small strains)
Stress in Ib/in equivalent to influence
of temperature change of 1°C for
installation on steel material"
Manganin
Nickel
Nichrome
Phosphor bronz e
5% Iridium-Platinu m
Advance
Copel
Monel
Isoelastic
0.47
— 12.1 (nonlinear)
2.1
1.9
5.1
2.1 (selected material )
2.4
1.9
3.6
-400
-13500
2100
7800
11600
±30
-200
8000
5000
Source: reference 12 .
" One shoul d not e tha t thes e figures can onl y be considered a s semiquantitative indications because they will
vary wit h hea t treatmen t an d col d workin g of the materia l an d als o wit h temperature level.
Table 1.2. Compositio n o f alloys
Material Composition
Advance an d Cope l
5% Indium-platinu m
Isoelastic
Manganin
Nichrome V
45% Ni; 55 % Cu
5% Ir ; 95 % P t
36% Ni; 8 % Cr; 52 % Fe;
0.5% Mo; + (Mn, Si, Cu, V) = 3.5%
4% Ni ; 12 % Mn; 84 % Cu
80% Ni; 20% Cr
14 TH E BONDE D ELECTRICA L RESISTANC E STRAI N GAGE
Table 1.3. Weibull' s observation s fro m stati c tes t
on Cope l wir e
Initial diameter = 0.45 mm; initial length = 125 mm
\L (mm) R (ohms)
0.00
6.25
12.50
18.75
25.00
31.25
37.50
0.376
0.414
0.455
0.497
0.542
0.588
0.635
Source: referenc e 13 . Reprinte d b y permission , r 194 8 Mac -
millan Magazine s Ltd .
strain sensitivit y facto r i s found t o b e
Weibull does not stat e the metallurgical condition o f the wire , but fro m
the magnitud e (6 0 percent ) o f th e elongatio n reporte d fo r on e o f hi s
specimens, i t is assumed tha t th e material wa s in the annealed condition . H e
also report s essentiall y comparabl e result s for a dynami c tes t o n 0.45-m m
diameter wir e wit h a lengt h o f 10 1 mm. Th e maximu m strai n reache d 3 4
percent wit h a velocity of 6.2 m/sec for the moving head of the testing device.
The 0.45m m (0.017 7 in) wir e diamete r whic h Weibul l investigate d i s
somewhat large r tha n the 1-mi l (0.001-in) size normally employed for bonde d
strain gages . Wit h th e smalle r diameter , smalle r ultimat e elongatio n i s
expected becaus e mino r variation s i n diamete r wil l have , relatively , much
greater influence . Shou b (14 ) report s elongation s u p t o 2 2 percen t fo r
specially anneale d constanta n wir e of 0.001 i n diameter . His result s indicat e
a straight-lin e relationship , wit h a slop e o f 2.02 , fo r th e plo t o f log (R/R 0)
vs. log (L/L 0). Thi s confirm s Weibull's observations .
1.4. Analysis of strain sensitivity in metals
The general case
Figure 1. 6 shows a metal conductor o f uniform cross sectio n (no t necessaril y
rectangular, althoug h thi s i s shown) referre d t o th e axe s X , Y , and Z . W e
want t o establis h a n expressio n fo r the rati o o f unit chang e i n resistanc e i n
the X directio n t o th e uni t chang e i n length , in term s o f strains e x, e y, an d
e. (in th e direction s o f the X , Y , and Z axes , respectively ) and th e materia l
property o f the conductor .
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 15
FIG. 1.5. Weibull' s experimenta l result s from 0.45-m m diameter Copel wire. (From ref . 13. )
The expressio n fo r th e resistanc e in th e X directio n ca n b e written as
where R = resistance i n length L (ohms )
p = specific resistivity of the materia l (ohms-in )
L = length (in)
A = area o f the cross section (in2)
16 THE BONDE D ELECTRICA L RESISTANC E STRAIN GAG E
Fie. 1.6. Meta l conductor referre d t o X , Y , and Z axes .
By multiplying the numerato r an d denominato r o f the right-han d ter m
by th e lengt h L , Eq . (1.7) can b e rewritte n as
where V — LA = volum e (in3). By takin g th e logarith m o f bot h sides , Eq .
(1.8a) become s
Differentiation o f Eq . (1.8b ) results in
Equation (1.9 ) expresses th e uni t chang e i n resistanc e i n term s o f th e uni t
changes i n resistivity , length, an d volume .
We no w postulat e tha t th e uni t change i n resistivit y ca n b e relate d t o
the uni t chang e i n volum e a s follows :
where m = a functio n o f th e materia l propertie s an d th e tw o ratio s o f th e
transverse t o the longitudinal strain. Fo r th e elastic strains , an d fixed values
of th e tw o strai n ratios , som e material s exhibi t a constan t valu eo f th e
function m . This relatio n i s stated b y Biermas z e t al . (15) , who give s credi t
for i t t o Bridgeman . Meie r (16 ) uses th e sam e relatio n i n a slightl y differen t
form.
FUNDAMENTAL CONCEPT S FO R STRAI N GAGES 1 7
By substituting th e valu e of dp/p give n by Eq . (1.10 ) into Eq . (1.9) , we
may write
or
Dividing al l terms o f Eq. (1.11 ) by dL/L, w e obtai n
Equation (1.12 ) indicates that , for plastic deformation (which takes place a t
constant volume , s o tha t d V = 0), th e valu e o f th e instantaneou s strai n
sensitivity ca n b e expected t o b e 2 for an y strai n condition.
Since dL/L = ex, an d because dV/V = (sx + sy + ez), Eq. (1.12) can be
expressed i n term s o f the strain s a s follows :
Special case of a uniform straight wire
For th e specia l cas e o f a straigh t wir e o f any unifor m cross section , which
is free t o contrac t or expan d laterall y due t o th e Poisso n effect , th e ratio s of
lateral t o axia l strain ar e give n by the expressio n
where v = Poisson' s ratio o f the material .
When th e value s o f the strai n ratios , give n for thi s specia l cas e b y Eq .
(1.14), ar e substitute d into Eq . (1.13 ) for strain sensitivity , we arrive a t
For smal l changes , suc h a s encountere d withi n th e elasti c range s o f
metals, Eq . (1.15 ) can b e modified to rea d
18 TH E BONDE D ELECTRICA L RESISTANCE STRAIN GAG E
Equations (1.15 ) and (1.16 ) indicat e tw o interestin g characteristic s i n
regard t o th e strai n sensitivit y of a wire .
1. I f the materia l property i s such that m = 1 , then, regardless o f the valu e
of Poisson' s ratio , th e strai n sensitivit y factor o f th e meta l wil l b e 2 .
This mean s th e strai n sensitivit y will b e th e sam e i n th e elasti c an d
plastic ranges , eve n thoug h ther e wil l be a chang e i n v as on e proceed s
from elasti c to plastic conditions . Conversely, this also tells us that onl y
those materials whose strain sensitivity is 2 can hav e the same sensitivit y
in bot h th e elasti c an d plasti c ranges .
2. Fo r perfectl y plasti c deformation, which takes place at constant volume,
dV - 0 and v = 0.5 . Therefore , n o matte r wha t th e valu e of m is, the
strain sensitivit y factor fo r plasti c deformation wil l b e 2 , as previousl y
indicated b y Eq . (1.12) . Thi s mean s that , fo r plasti c deformation , al l
metals shoul d exhibi t a strai n sensitivit y factor o f 2 . Thi s i s substan -
tiated b y th e result s o f tests, a s indicate d i n th e tabulatio n presente d
in the Appendi x of this chapter, for which, in almost al l cases, th e strai n
sensitivities i n th e hig h strai n range s approximat e a valu e o f 2.
The sligh t deviation o f some o f the value s from 2 i s probably du e
to th e effec t o f a certai n amoun t o f elastic strain whic h wil l b e presen t
during th e plasti c deformation . The fe w cases involvin g larger devia -
tions fro m 2 likel y correspon d t o rathe r incomplet e o r gradua l plasti c
deformation, and possibl y the influence o f some typ e of work hardening.
Equations (1.15 ) an d (1.16 ) can no w b e converted int o a mor e familia r
form customaril y foun d i n th e literature . Expansio n o f the secon d ter m o n
the right-han d sid e o f these equation s result s i n th e expressio n
In order to write Eq . (1.17) in a differen t form , the change i n the volum e
of th e wir e a s i t i s straine d axiall y can b e considered . Th e unstraine d wir e
volume i s
Taking th e logarith m o f both side s an d the n differentiatin g yield s
As th e wir e i s strained , it s lengt h increase s b y dL , bu t du e t o th e Poisso n
effect it s diamete r decrease s b y ( — v dL/L)D, wher e D i s th e wir e diameter .
FUNDAMENTAL CONCEPT S FO R STRAI N GAGES 1 9
The fina l wir e diameter i s
The chang e i n area ca n no w b e written as
If th e higher-orde r ter m i n Eq . (d ) i s neglected, the n w e can writ e
Substituting the valu e o f dA/A give n b y Eq . (e ) into Eq. (b ) give s
Thus, Eq . (f ) can b e expressed a s
From Eq . (1.10 ) we can write
If th e value s o f ( 1 — 2v) an d m fro m Eqs . (g ) an d (h) , respectively , ar e
substituted i n Eq . (1.17) , the n
or
20 TH E BONDE D ELECTRICA L RESISTANC E STRAI N GAGE
For smal l changes , a s encountered wit h elasti c strains, we can write
Equation (1.18 ) is of particular interest , not jus t becaus e i t represents a
more familia r form o f the expressio n for the strai n sensitivit y factor , but fo r
two othe r reason s a s well .
1. Th e relationshi p give n i n Eq . (1.18 ) ca n b e derive d independentl y o f
the relatio n give n by Eq . (1.10) .
2. Fo r an y particula r metal , Eq. (1.18) indicates the portion s o f the strain
sensitivity facto r whic h ar e th e resul t o f geometrica l chang e an d
resistivity change , respectively . The valu e (1 + 2v ) corresponds t o th e
geometrical change , whil e (dp/p)/(dL/L) correspond s t o th e resistivity
change.
We see that whe n plastic deformation takes place, since v = 0. 5 and d p = 0,
Eq. (1.18 ) als o indicate s a valu e of 2 fo r S t.
Small strain vs. large strain
Let u s now loo k int o the detai l o f the differenc e betwee n the expression s fo r
the instantaneou s an d approximat e value s o f th e strai n sensitivit y factors .
The instantaneou s valu e o f S , is
while th e approximat e valu e of S , is
For smal l strains (less than 1 percent), a s developed i n the elastic rang e
of metals , bot h expression s wil l yield , fo r al l practica l purposes , th e sam e
result. However , sinc e i t wil l b e mor e convenien t t o evaluat e th e strai n
sensitivity, an d subsequentl y t o comput e strains , o n th e basi s o f change s
from th e initia l condition , w e wis h t o kno w th e magnitud e o f th e larges t
strain tha t ca n b e handled i n thi s manner withou t running int o intolerabl y
large errors .
Returning t o Fig . 1.5 , w e se e a comparison , base d o n Weibull' s
experimental observations , betwee n th e plo t o f AK/R 0 vs . L/L0 an d th e
logarithmic plo t o f \n(R/R 0) vs . ln(L/L 0). Th e logarithmi c plo t show s a
FUNDAMENTAL CONCEPT S FO R STRAI N GAGES 2 1
straight lin e wit h a slope , S t, of 2.0, wherea s the plo t of R/R0 vs. L/L 0
gives a long radiu s curv e whose initia l slope (fo r R = L = 0) is 2.0, but
for whic h the slope increases slightl y as the changes in length an d resistanc e
build up .
Examination o f Fig. 1. 5 reveals that, for a graph o f this size and withi n
the limit s of error i n plotting th e points, the curve of R/R0 vs . L/L0 ca n
be represented b y a straigh t lin e u p t o value s o f about 1 0 to 1 5 percent o f
L/L0. Fo r larger strains the departure fro m linearity , although not serious,
can be noticed. However, we observe that the slope of the line (the indicated
value of Sr) is slightly greater tha n that o f the logarithmic plot. This explains
why one can use post-yield gages up to strain levels in the range of 10 percent
or more , on the basis of R/R0 an d L/L0, withou t introducing noticeabl e
errors a s a resul t of making a linea r approximation .
As thes e comment s hav e bee n develope d fro m experimenta l observa -
tions, w e ca n no w examin e th e situatio n fro m a theoretica l poin t o f view .
We star t by developing the relatio n betwee n resistance an d lengt h fro m Eq .
(1.20) o n th e assumptio n tha t S t is a constant . W e can rewrit e Eq. (1.20) in
the followin g form :
Equation (1.22 ) can als o b e expressed a s
Integrating Eq . (1.23 ) results in
where C = constant o f integration.
Since th e initia l value s o f resistanc e an d length , R 0 an d L 0, wil l b e
known, the constan t o f integration can b e written as
Substitutingthe valu e of C from Eq . (1.25 ) into Eq . (1.24 ) gives us
This expressio n ca n b e modified to rea d
22 TH E BONDE D ELECTRICA L RESISTANC E STRAIN GAG E
Equation (1.26 ) tell s u s tha t th e plo t o f ln(R/R0) vs . ln(L/L0) wil l give
a straigh t lin e whos e slop e i s equa l t o S t. Thi s ha s bee n verifie d experi -
mentally b y bot h Weibul l (13 ) and Shou b (14) .
From Eq . (1.26 ) w e ca n expres s th e relatio n betwee n resistanc e an d
length o f a meta l conducto r tha t ha s bee n straine d i n the plasti c range a s
Since th e valu e of S t fo r plasti c strai n ha s bee n predicte d theoreticall y
as 2.0, as shown b y Eq. (1.12), and becaus e thi s value has bee n corroborated
by th e experiment s o f Weibull (13) and Shou b (14) , thi s is the numbe r tha t
will b e use d fo r th e exponen t i n Eq . (1.27) . Thus , Eq . (1.27 ) ca n no w b e
written a s
Because R = R0 + R an d L = L0 + L, Eq . (1.28) can be converted int o
terms of R , L , R 0, and L0. Thus ,
Expanding th e right-han d sid e o f Eq . (1.29 ) result s in
Equation (1.30 ) presents the theoretical relationship between R/R 0 an d
L/L0 fo r a meta l conducto r subjecte d t o plasti c strain . I t provide s th e
following information :
1. R/R 0 i s a nonlinear functio n a t L/L 0.
2. Fo r positive value s of L (tension) , R/R0 wil l alway s be larger tha n
2( L/L0).
3. Th e slop e o f the curv e a t th e origi n i s 2.
4. Th e deviatio n fro m th e tangen t (slop e = 2 ) through th e origi n i s given
by ( L/L0)
2.
or
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 23
Item 4 indicate s bot h th e deviatio n fro m linearit y an d th e deviatio n fro m
the relatio n involvin g the instantaneous value s of R an d L .
It i s noteworth y tha t whe n L/L 0 i s 1 0 percent, th e deviatio n fro m
linearity i s only 5 percent. Thi s i s illustrated i n Fig . 1.7 , which shows a plo t
of theoretica l value s of R/R0 vs . L/L0, a s computed fro m Eq . (1.30).
If an approximat e linea r relatio n i s set up b y using the secan t fro m th e
origin t o som e poin t o n th e curve , then th e erro r wil l b e zero a t th e poin t
of intersection wit h the curve , and a t al l othe r point s th e erro r wil l b e less
than that represented b y the deviation o f the secant fro m th e initial tangent .
This i s due to the fac t tha t the curve lies between the secant an d th e tangent
0
FIG. 1.7 . Theoretica l relation between R/R0 an d L/L 0 fo r large strains.
24 TH E BONDE D ELECTRICA L RESISTANCE STRAIN GAGE
through the origin. For example , when L/L0 equal s 10 percent, the expected
error, a t an y point , wil l neve r b e mor e tha n 5 percent , a s a maximum . In
general i t will probably no t excee d 2.5 percent, except for relatively low strain
values where the numerica l magnitude of the error wil l be of less importance .
Examination o f Fig . 1. 7 will hel p t o clarif y thes e points .
From Eq . (1.30) an expressio n can b e written for th e valu e of the strai n
sensitivity factor :
The value of S, varies in accordance wit h the value of L/L0 an d correspond s
to th e slop e o f th e secan t fro m th e origi n t o th e poin t whos e coordinate s
are ( R/R0, L/L 0) o n the curve .
1.5. Wire strain gages
The unbonded wire strain gage
One o f th e earl y wir e gage s wa s th e unbonde d type . I n thi s typ e o f
instrument, the strain-sensitiv e wire i s mounted, unde r tension , on mechani -
cal support s (pins ) i n suc h a manne r tha t a sligh t relativ e motio n o f th e
supports wil l caus e a chang e i n strain . This, i n turn , produce s a chang e i n
electrical resistance . This resistanc e change i s then a measur e o f the relativ e
displacement o f th e support s and , i n turn , ma y represen t a strai n o r som e
other quantity.
With th e unbonde d typ e of gage, th e fac t tha t th e strain-sensitiv e wires
must b e carrie d o n som e sor t o f mechanica l moun t give s ris e t o certai n
difficulties i n connection wit h attachment . Discrepancies , due t o inertia , may
be introduce d whe n dynami c observation s ar e made . Th e procedur e o f
making observation s a t a n appreciabl e distanc e fro m th e surfac e o n which
strain i s to b e determined may sometime s b e ope n t o question .
The bonded wire strain gage
The firs t majo r improvemen t i n th e wir e resistanc e strai n gag e cam e wit h
the realizatio n tha t man y o f th e difficultie s wit h th e unbonde d wir e gag e
could b e eliminate d b y bondin g a ver y fine strain-sensitive wire directly t o
the surfac e o n whic h strai n i s t o b e measured . Th e filamen t ha s t o b e
electrically insulated an d th e bondin g perfec t fo r the strain-sensitive element
to follo w th e strai n o n th e surfac e to whic h i t i s attached. Onl y conductor s
of smal l diameter ar e suitable , since the force necessary t o strai n th e sensin g
element mus t b e transmitte d throug h it s surfac e by shea r i n th e cement , o r
bonding agent . Unles s th e surfac e are a pe r uni t lengt h i s larg e relativ e t o
the cross-sectiona l area , th e shearin g stres s i n th e cemen t wil l b e to o hig h
FUNDAMENTAL CONCEPT S FO R STRAI N GAGES 2 5
to permi t faithfu l followin g o f th e strain s i n th e surfac e t o whic h th e
conductor i s attached .
Since th e surfac e are a (pe r uni t length ) o f small-diamete r wire s i s
enormously greater tha n th e cross-sectional area (for 0.001-in diameter wire,
the rati o is 4000 to 1) , the bonding agen t i s able to forc e th e filament t o tak e
up th e necessar y strai n withou t excessiv e stres s i n itself . Suitabl e cement s
can actually force the small conductor into the plastic range (and back again )
when necessary.
Chronologically, th e secon d majo r development , an d tha t whic h ha s
actually bee n responsibl e for makin g th e bonde d strai n gag e commercially
attractive, i s represente d b y th e concep t o f premounting th e strain-sensin g
element o n some suitabl e carrier tha t ca n be attached t o a surfac e relatively
easily. Originally , the strai n gag e wir e was cemented directl y to th e surface
on whic h strai n wa s t o b e measured , an d th e glu e o r cemen t acte d a s
insulation. A s fa r a s operatio n wa s concerned , thi s procedur e wa s satis -
factory, bu t fro m th e poin t o f view o f gage installation , i t was inconvenient.
The attachment o f the gage require d an inordinat e amoun t o f skill and time
on th e par t o f th e installe r i f consisten t result s wer e t o b e obtained . Th e
introduction o f a paper , plastic , metal , o r othe r typ e of carrier upo n whic h
the strain-sensin g wir e coul d b e premounted , unde r controlle d factor y
conditions, represente d a tremendou s improvement . Wit h thi s for m o f
premounted filamen t strai n gage , muc h les s skil l and tim e ar e require d t o
achieve satisfactor y installations givin g good an d consisten t results .
Most bonde d wir e strai n gage s ar e mad e fro m wir e o f approximately
0.001 in diameter, o r less , and i n resistances varying from abou t 5 0 ohms t o
several thousan d ohms . Th e filament s ar e mounte d o n carrier s mad e o f
materials selected fo r th e particula r application s fo r which the gage s ar e t o
be employed.
Since a length of several inches o f wire is usually needed to produce the
necessary tota l resistance , an d becaus e th e desire d gag e lengt h i s almos t
always les s tha n th e require d lengt h o f wire , i t i s necessar y t o arrang e th e
wire i n som e for m o f grid i n orde r t o economiz e o n space , an d thereb y t o
permit reductio n o f th e gag e lengt h t o a suitabl e size . Figur e 1. 8 shows
diagrams o f typical grid configurations for wir e gages. There are , o f course,
variations of these typical designs,as manufacturers' literature shows (17,18).
The fla t gri d i s probably th e mos t usefu l form . When th e gag e i s on a
flat surface, the centre line of the entire sensing element lies in one plane that
is parallel t o th e surfac e of attachment. Du e t o th e end loops , ther e is some
response t o strai n a t righ t angle s t o th e directio n o f the gri d axis . Usually
the filamen t consist s o f on e continuou s lengt h o f wire ; however, for som e
self-temperature-compensated gages , two elements , which possess opposing ,
or compensating , temperatur e characteristic s ar e joined together .
An alternat e typ e o f constructio n originate d a s a n expedien t fo r
manufacturing gage s o f shor t gag e lengt h (0.25 0 in o r less ) prio r t o th e
development o f the technique s now use d t o mak e shor t fla t gri d gages . I n
26 THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E
FIG. 1.8 . Typica l wir e strai n gages , (a , b) Singl e elemen t gages , (c , d) Two-elemen t stacke d
rectangular rosettes , (e , f) Three-elemen t stacke d rectangula r rosettes , (g ) Two-element rectan -
gular rosette , (h ) Three-elemen t rectangula r rosette . (Fro m ref . 18.).
the wrap-around construction , the sensin g element is wound tightl y aroun d
a smal l flat carrier whic h i s then encased betwee n two cover sheet s providing
insulation an d protection . A n alternativ e procedure i s t o win d th e sensin g
element on a small tubular mandrel (like a soda straw ) that is then flattened
and encase d betwee n th e cove r sheets .
For th e variou s type s o f bonde d wir e strai n gages , th e strai n i s
determined fro m th e relatio n
where e = strain i n th e directio n o f the gag e axis
R/R = unit chang e in resistance
GF = manufacturer's gage factor
Due t o th e geometrical difference s betwee n a straigh t wire and a strain
gage grid, the value of the manufacturer's gage factor, GF, is generally slightly
lower tha n th e strai n sensitivit y factor , S, , o f th e wir e fro m whic h the gri d
FUNDAMENTAL CONCEPT S FO R STRAI N GAGES 2 7
is constructed . Furthermore , th e magnitud e o f G F wil l var y slightl y with
variations i n grid design .
Gages containin g a singl e continuou s filamen t whic h i s woun d bac k
and fort h wil l respond slightl y to th e effec t o f lateral strai n whic h is sensed
by the end loops. This means tha t Eq . (1.32), although generally applicable ,
is subjec t to som e erro r whe n th e strai n field in whic h the gag e i s actually
used differ s fro m tha t of calibration. Usually the error caused by the response
to latera l strai n ca n b e neglected , bu t ther e ar e a few situations i n whic h it
becomes appreciable . Th e magnitud e o f the erro r cause d b y latera l effect s
and, wher e necessary, the mean s o f correcting for thi s error , ar e discussed
in detai l i n a later chapter .
Some specifi c example s o f the relatio n betwee n strain an d uni t change
in resistance for complete wire gages are show n in Fig . 1.9 . In eac h case th e
slope o f line relating the percen t chang e i n resistanc e to th e percen t strai n
represents the gag e factor. One wil l note tha t the advanc e wir e (constanta n
type) gag e ha s th e sam e gag e facto r fo r bot h elasti c an d plasti c strains ,
whereas the isoelastic and nichrome gages both show a change in gage factor
as one proceeds from elasti c to plastic conditions. One should not be alarmed
about thi s chang e i n gag e facto r because w e ar e usuall y intereste d i n
measuring elastic strain s i n metals , an d thes e occu r wel l below the chang e
points show n i n th e diagrams . Thi s i s especially so i n th e cas e o f isoelastic
wire (whos e chang e poin t occur s a t approximatel y 0.7 5 percen t strain) ,
because thi s material i s usuall y chosen t o tak e advantag e o f it s hig h gag e
factor fo r measurin g very smal l strains.
Wire gage s wer e use d unti l th e earl y 1950s , whe n foi l gage s wer e
introduced. Some wire gages are stil l used today and ca n be purchased fro m
several manufacturers.
Weldable wire gages
The first weldable wire gage was developed in the mid-1950s (19). Subsequent
development fo r a quarter-bridg e circui t used a singl e filamen t o f nickel -
chromium wir e tha t wa s chemicall y etche d s o tha t it s cente r lengt h wa s
approximately 1 mil i n diameter . Th e wir e wa s the n folde d i n hal f an d
inserted into a stainless stee l tube . The tube wa s filled with a metallic oxide
powder whic h wa s compacte d s o tha t i t no t onl y electricall y isolated th e
filament but mechanicall y coupled i t to th e tube in order t o transmi t strain.
The constructio n i s shown in Fig . 1.10 .
In orde r t o minimiz e the apparen t strai n du e t o temperatur e changes ,
the nickel-chromiu m filamen t i s hea t treated . Sinc e differen t level s o f heat
treatment resul t in differen t value s of the therma l coefficien t o f resistivity, it
is possible t o make thi s change equal in magnitude but o f opposite polarity
to th e therma l coefficien t o f expansion .
To achiev e temperatur e compensation , a separat e compensating , o r
dummy, gage can be mounted o n a stress-fre e piec e o f material identica l to
FIG. 1.9. Typica l gag e characteristic s i n tension . (Fro m ref . 11 , with permission o f HMSO. )
FIG. 1.10. Singl e active gag e construction . (From ref . !9. )
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 2 9
FIG. 1.11. 'True ' dummy gage construction. (From ref . 19.)
FIG. 1.12. Ni-C r half-bridg e gag e construction. (From ref . 19.)
the materia l o n whic h th e activ e gag e i s mounted. Th e tw o gage s ar e the n
arranged int o a half-bridg e circuit. This i s a satisfactor y metho d providin g
the materia l o n whic h the dumm y gage i s mounted i s completely stress fre e
and tha t th e dumm y gage' s temperatur e i s identica l t o th e activ e gage .
Because thes e condition s d o no t alway s prevail , a 'true ' dumm y gag e wa s
developed. The dumm y gag e filament , identica l t o th e activ e gag e filament ,
is woun d i n a tigh t heli x of the prope r pitc h angle . Sinc e i t i s embedded i n
a strai n tub e wit h compacte d magnesiu m oxid e powder , th e sam e a s th e
active gage , i t ha s th e sam e heat-transfe r characteristics . Therefore , th e
dummy gag e ca n b e use d wit h a compensate d activ e gage t o minimiz e the
apparent strain . Th e dumm y gage i s shown i n Fig . 1.11 .
The nex t ste p wa s to incorporat e th e singl e activ e gage an d th e 'true '
dummy gage into one strain tube and mounting flange assembly. This results
in a half-bridge gage rather than a quarter-bridge gage. The half-bridge gage
is shown i n Fig . 1.12.
The earl y weldabl e wir e strai n gag e ha s resulte d i n a lin e o f bot h
quarter- and half-bridg e gages (20). Two wire types are used for the filament.
The firs t i s a nickel-chromiu m tha t i s temperature compensate d an d use d
for stati c measurements up to 600°F (315°C). Because of excessive drift abov e
600°F, th e gages are use d only for dynamic test s between 600°F and 1500° F
(815°C). Th e secon d wir e typ e i s platinum-tungste n tha t ca n b e use d fo r
static measurement s u p t o 1200° F (650°C) . Sinc e thi s wir e cannot b e hea t
treated for temperature compensation, th e half-bridge gage is recommended .
1.6. Foil strain gages
General characteristics
The foi l gag e operate s i n essentiall y th e sam e manne r a s a wir e gage .
However, the sensing element consists of very thin metal foi l (about 0.0002 i n
30 TH E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E
thick) instea d o f wire . I n contras t t o th e wir e gage , i n whic h th e sensin g
element possesse s a unifor m cross sectio n throughou t it s entir e length , th e
crosssection of the sensing element of the foi l gage may be somewhat variabl e
from on e en d t o th e other . On e o f the mos t importan t advantage s o f the foi l
gage i s that th e rati o o f contact surfac e area t o th e volum e o f the resistanc e
element i s relatively high, whereas in the wir e gage, du e t o th e circula r cros s
section, thi s rati o i s a minimum.
The earl y foi l gages , introduce d i n Englan d i n 1952 , were mad e fro m
foil cemente d t o a lacque r sheet . The desire d gri d desig n fo r th e strai n gag e
was printe d o n th e foi l wit h a n acid-resistin g in k an d th e shee t wa s the n
subjected t o a n aci d bat h whic h removed al l metal excep t wher e th e printe d
design protecte d it . Durin g th e intervenin g years, a tremendou s amoun t o f
very fruitfu l researc h ha s bee n carrie d o n wit h respec t t o foi l gages . Th e
well-established alloy s hav e bee n improve d an d ne w one s developed . I n
addition, ther e has been a vast improvement i n the photographic technique s
currently use d i n th e photoetchin g proces s employe d t o manufactur e foi l
gages. Th e degre e o f precision wit h whic h gages ca n no w b e produced , an d
the sharpnes s o f definitio n o f the boundarie s o f line elements , hav e made i t
possible t o mak e gage s possessin g a unifor m gage facto r fo r a larg e rang e
of gage length s (previously, gage facto r varied slightly with gage length) . The
result o f these improvement s ha s bee n t o exten d th e advantage s o f th e foi l
gage t o a muc h wide r variet y of applications , includin g those a t ver y lo w
and ver y high temperatures , an d especiall y for ver y precis e transducers .
Foil gage s ar e availabl e i n variou s gag e length s fro m 1/6 4 in t o 6 in,
and i n a wid e variet y o f gri d configurations , including singl e gages , two- ,
three-, an d four-elemen t rosettes , hal f bridges , an d ful l bridges . Figur e 1.13
shows a fe w o f th e availabl e designs . Standar d alloy s suc h a s constantan ,
isoelastic, nichrome , karma , an d platinum - tungsten, as wel l as a numbe r of
special proprietar y alloys , ar e use d i n th e sensin g elements .
In general , foi l gage s exhibi t a slightl y highe r gag e facto r an d lowe r
transverse response than their equivalent in wire. Since they are thinner, they
conform mor e easily to surface s with smal l radius of curvature, which means
they ar e easie r t o instal l i n fillets . A s a resul t of thei r greate r contac t area ,
they ca n dissipat e hea t mor e readil y and , i n consequence , i t i s possibl e t o
use higher operating current s (applied voltage) with foi l gages . The relatively
large contac t area , especially a t th e end s o f the grid , reduce s shearin g stres s
in the bondin g agent , an d consequently , foil gage s show comparatively littl e
creep an d hysteresis . Dependin g upo n th e carrier , th e alloy , and it s metal -
lurgical condition , foi l gage s (generall y the large r sizes ) wil l measur e strain s
precisely into the rang e o f 1 0 to 1 5 percent. In term s of fatigue, suitabl e gage s
have exhibited life i n exces s of ten millio n cycle s at strain s of + 150 0 uin/in.
Foil gage s ca n b e obtaine d o n carrier s o f paper , epoxy , phenolic , glas s
reinforced resins , an d othe r plastics .
By judicious choic e o f alloy and b y carefu l contro l o f the metallurgica l
condition (col d workin g and hea t treatment) , i t i s possibl e t o produc e foi l
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 31
FIG. 1.13. Foi l strai n gages, (a, b) Single-elemen t gages, (c) Stacked two-elemen t rectangular
rosette, (d ) Stacke d three-elemen t rectangula r rosette , (e ) Three-elemen t delt a rosette , (f )
Two-element rectangula r rosette torque gage. (Courtesy of Measurements Group, Inc .
with it s coefficien t o f linear expansio n an d resistance-temperatur e charac -
teristic ver y closel y matche d t o th e coefficien t o f linear expansio n o f som e
arbitrarily selecte d material . B y this means, i t ha s bee n possibl e t o produc e
temperature-compensated foi l gages whose response (within certain limits) is,
for practica l purposes, independent of temperature, within a given tempera-
ture range .
Weldable foil strain gages
For situation s i n which the conventional installatio n technique s may not be
applicable, weldable foil gages are available (18 , 20, 21). Single-element gages
and T-rosette s (two-element ) are mad e b y premountin g gage s o n a carrie r
of stainless steel shim stock approximately 0.005 in thick. Surface preparation
of the specime n requires solven t cleanin g and abrasio n wit h silicon-carbide
paper o r a smal l han d grinder . Th e uni t i s then attache d t o th e specime n
with a smal l spo t welde r designed specificall y for thi s purpose .
Sensing elements of constantan, nichrome , and high-temperatur e alloys
are available . Th e norma l operatin g temperatur e range s fro m — 320°F t o
570°F (-19 5 t o 300°C ) fo r stati c observations , althoug h unde r som e
conditions a single-loo p wir e (typically nichrom e V ) encased i n a stainles s
steel tub e may b e used t o 925° F (495°C ) o r higher .
32 TH E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E
1.7. Semiconductor gages (4 , 8, 9 , 22-25)
Within certai n limitations , semiconductor gage s ca n b e use d i n th e sam e
manner as metallic gages. However , the semiconductor gage i s really a much
more elaborat e devic e whose optimu m us e require s a knowledg e o f al l th e
variables involved , and th e degre e t o whic h they influenc e th e performanc e
of th e instrument . Th e compariso n betwee n th e use s o f meta l an d semi -
conductor gage s i s somewha t paralle l t o th e differenc e betwee n playin g
checkers an d playin g chess. Bot h ar e goo d games , bu t ches s ha s a muc h
broader rang e o f opportunitie s fo r makin g move s and , correspondingly ,
many mor e possibilitie s of getting into troubl e unless one consider s al l th e
variables carefully .
The mai n attractio n o f th e semiconducto r is , of course, th e hig h strain
sensitivity o f silicon , which i s th e favore d materia l fo r th e sensin g element .
This mean s a relativel y larg e resistanc e chang e pe r uni t o f strain , which
characteristic i s helpfu l fo r bot h hig h and lo w value s o f strain.
1. Fo r hig h strains , th e larg e respons e enable s on e t o driv e indicatin g
devices directl y withou t intermediat e amplification . This provide s a
simplification whic h is accompanied b y reduce d weigh t and expense .
2. Fo r lo w strains , which produce exceedingl y small changes i n resistance
of metal gages, the semiconductor gages wil l develop unit changes abou t
50 time s greater , wit h th e resul t tha t th e indication s o f R/R ca n b e
measured convenientl y an d precisely .
As contrasted wit h th e abov e advantages , one mus t also recognize , and
be able t o cop e with , certain disadvantages .
1. Th e uni t chang e i n resistanc e (whic h i s based o n th e initia l resistance,
R0, o f the unstresse d senso r a t temperatur e T 0) is a nonlinea r functio n
of th e strain , althoug h fo r som e specia l condition s i t ca n b e take n a s
linear fo r smal l strai n excursions.
2. Th e larg e resistanc e chang e pe r uni t o f strain , which i s the ver y thin g
that makes the semiconductor gage attractive, may also present a minor
problem du e to the fact that , in the process o f installation, the resistanc e
of the gage may b e altered considerabl y from th e value which prevailed
in th e unstresse d conditio n o f the sensin g element. O n thi s account, i t
is necessar y t o determin e th e gag e resistanc e followin g installation s o
that, i f necessary, an appropriat e correctio n ca n b e mad e fo r th e gag e
factor.3. Th e resistanc e o f the gag e wil l chang e wit h chang e i n temperature .
4. Th e strai n sensitivity , o r gag e factor , wil l chang e wit h chang e i n
temperature.
Investigation o f silico n reveal s tha t bot h th e strai n sensitivit y an d th e
temperature sensitivit y (change o f resistance with temperature) vary consider -
ably wit h th e quantit y of impurity whic h i s present. I t i s also observe d tha t
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 3 3
high sensitivit y to strai n i s accompanie d b y hig h sensitivit y to change s i n
temperature. This suggests that som e compromise betwee n strain sensitivity
and temperature response may be desirable, and perhaps essential, dependin g
upon th e particular application .
Fortunately, b y suitable doping (introductio n of controlled amount s of
impurities) durin g th e manufacturin g process , th e strai n an d temperatur e
sensitivities can be varied and adjusted (although not independently) to meet
specified requirements . Therefore, by suitable procedures i n the manufactur-
ing process , i t i s possible t o achiev e a desired compromise whic h wil l result
in muc h improve d temperatur e characteristic s a t th e expens e o f a modes t
reduction i n strain sensitivity . Practical consideration s indicate tha t a goo d
balance i s achieved when the gag e facto r is about 120 .
Since semiconductor gages are available with both positive and negative
gage factors, another approach , althoug h perhaps a more difficul t one , i s to
take advantage of the characteristics of the electrical circuit of which the gage
forms a part , and t o emplo y two simila r gages with gage factor s o f opposite
sign.
Due t o th e relativel y larg e numbe r o f variable s involved , an d con -
sequently th e somewha t mor e comple x procedur e require d fo r convertin g
resistance chang e int o term s o f strain , i t seem s unlikely , a t leas t fo r th e
present, tha t semiconducto r gage s wil l replac e metalli c gage s fo r purpose s
of stress analysis, excep t perhaps, unde r specia l circumstances involving th e
determination o f very smal l strains.
However, for transducers, in which gages can be installed under carefull y
controlled factor y conditions , an d subsequentl y calibrate d i n complet e
bridges, th e hig h outpu t o f th e semiconductor s make s the m exceedingl y
attractive. I t seem s tha t semiconducto r strai n gage s wil l achiev e greates t
success and optimu m utilit y i n thi s type of application.
1.8. Some other types of gages
Temperature gages
Examination o f the characteristic s o f metal an d semiconducto r strai n gage s
reveals tha t change s i n resistanc e occu r no t onl y a s a resul t o f changes i n
strain, bu t als o fro m change s i n temperature . Althoug h th e respons e t o
temperature ma y complicat e th e determinatio n o f strain , i t nevertheles s
provides th e possibilit y o f making , an d using , temperatur e sensor s wit h
essentially th e same technique s as those which are employe d i n the makin g
and usin g of strain gages .
The choice of material for the sensing element, of course, will be differen t
for thes e tw o applications . Whe n i t i s desire d t o measur e strain , wit h a
minimum influenc e fro m temperatur e changes , a copper-nicke l allo y o f
the constanta n typ e i s frequentl y employe d fo r temperature s i n th e rang e
from abou t -250° F t o abou t SOO T (155-260°C) . Fo r lowe r o r highe r
34 TH E BONDE D ELECTRICA L RESISTANCE STRAIN GAG E
temperatures, i t i s necessary t o selec t anothe r typ e o f allo y (26) . However ,
for a temperature sensor , i t is preferable to choose a material, such as nickel,
platinum, o r a n iridium-platinu m alloy , whic h possesse s a muc h greate r
response t o change s i n temperature . Fo r semiconducto r materials , th e
processing i s varied t o produc e th e preferre d characteristics for either strai n
or temperatur e sensing.
For a numbe r o f years , bonde d wir e temperatur e sensor s hav e bee n
commercially available , followe d mor e recentl y b y foi l temperatur e gage s
(27, 28) . Foi l temperatur e gage s hav e severa l advantage s ove r wire-wound
sensors i n tha t the y ar e les s expensive , no t a s fragile , an d thei r time -
temperature response i s similar to tha t of a strain gage. Standar d strai n gage
instrumentation ma y als o b e use d wit h them .
For convenienc e i n makin g observations , sensor s an d thei r signal -
conditioning networks have been designed t o produce signal s correspondin g
to indication s o f 1 0 or 10 0 microstrain pe r degre e Fahrenheit . Therefore ,
when th e strai n indicato r i s referenced t o som e temperature , on e i s able t o
obtain a direc t readin g o f al l other temperature s withi n th e workin g range
of the system. For example , if a temperature sensor an d networ k is used tha t
provides a n indicatio n o f 1 0 microstrain pe r degre e Fahrenheit , th e initia l
balance o f th e indicato r ma y b e adjuste d s o tha t th e readin g wil l b e 75 0
microstrain whe n th e senso r i s actuall y 75° F (24°C) . Then , fo r an y sub -
sequent observation , th e temperatur e i n Fahrenhei t wil l b e represente d b y
the indicato r readin g divided by 10 . If a subsequen t readin g turns out t o b e
830, then th e temperatur e at th e senso r is 83 F (28 0C).
The obviou s advantage o f this method o f determining temperature lies
in th e fac t tha t a standar d strai n indicatin g (an d recording ) syste m ca n b e
employed, without an y modificatio n at all , for the measuremen t o f tempera-
ture a t strai n gag e locations , o r elsewhere , b y th e simpl e procedur e o f
switching the temperature sensor (wit h it s conditioning network) in and ou t
of th e indicatin g circuit just a s i f it wer e another strai n gage .
Crack measuring gages
Another instrument incorporating certain features of the strain gage is known
commercially a s th e Kra k Gage . It s mai n purpos e i s t o monito r th e
progression o f cracks whic h usually develop as a resul t o f fatigue cause d b y
repeated stressing . If the progres s o f a crack i s watched, a part can b e take n
out o f service before a disaster occurs , which is a very valuable consideratio n
in th e aircraf t an d man y othe r industrie s (29) .
A schemati c diagra m o f th e gage , show n i n Fig . 1.14 , i s produce d b y
Hartrun Corporatio n i n a variet y of different size s (30). I t possesse s certai n
characteristics whic h ar e lik e thos e o f th e strai n gage , bu t it s us e i s very
different. Basically , the Kra k Gag e consist s o f a constantan foi l senso r 5 urn
thick mounte d o n a n epoxy-phenoli c o r cas t epox y carrier , dependin g o n
the operating temperature . The carrier an d th e gage ar e cemented t o the tes t
FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 35
FIG. 1.14 . Schemati c diagram o f a crack measuring gage. (From ref . 30.)
piece, o r machin e part , b y th e usua l strai n gag e bondin g procedur e a t a
location wher e a crack i s expected t o start , o r may already have started. The
positioning o f the gag e i s such tha t i t wil l be cracked unde r it s centerline in
step wit h the materia l underneat h it . The gag e is energized wit h a constan t
current, usually in the range between 0 and 10 0 milliamperes, and the change
in potentia l dro p betwee n it s tw o inne r leads i s a measur e o f the distanc e
by whic h th e crac k ha s advanced . Sinc e thes e gage s hav e a resistanc e o f
about 1 ohm before the crack commences, they cannot be used with ordinary
strain gag e equipment.
Another crac k detectio n gag e i s th e CD-Serie s produce d b y Micro -
Measurements (31). This gage is used to indicate the presence of a crack, an d
crack growth rate may be monitored b y using several CD