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Theses and Dissertations
1914
The change in the elasticity of a copper wire with
current and external heating
Homer Levi Dodge
State University of IowaThis work has been identified with a
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Recommended Citation
Dodge, Homer Levi. "The change in the elasticity of a copper wire with current and external heating." PhD (Doctor of Philosophy) thesis, State University of Iowa, 1914.
Thesis presented
to
the
FACULTY OF THE GRADUATE COLLEGE
of the
STATE UNIVERSITY OF IOWA
in partial fulfillment of the
requirements
for the
Degree
of Doctor of Philosophy
July,
1914
Part I
The Change
in the Elasticity of a Copper Wire
with Current and External Heating.
Part
II
The Change
in the Elasticity of an
Iron Wire
T H E C H A N G E IN T H E E L A S T IC IT Y O F A C O P P E R W I R E W IT H C U R R E N T A N D E X T E R N A L H E A T IN G .
By H . L . Do d g e.
(R e p rin te d fro m the Physical Review, N .S ., V ol. II., N o . 6, D e c e m b e r, 1 9 1 3.]
T H E C H A N G E IN T H E E L A S T IC IT Y O F A C O P P E R W I R E W IT H C U R R E N T A N D E X T E R N A L H E A T IN G .
By H . L . Do d g e.
' I ^HE present investigation consists of determ inations of Y oung’s m odulus of a specimen of copper wire a t various tem p eratu res between 20° C. an d 150° C. w ith th e object of ascertaining w hether or not heating by an electric cu rren t has an y effect o th er th a n th a t caused b y th e accom panying increase of tem p eratu re, an d also to learn certain facts regarding th e tem p eratu re effect itself, concerning which there seems to be considerable doubt.
Pr e v i o u s Re s u l t s u p o n t h e Ef f e c t o f Cu r r e n t He a t i n g. W ertheim 1 reported a decrease in Y oung’s m odulus caused by cu rren t and believed this decrease to be independent of an y te m p eratu re effect, a conclusion n o t sufficiently justified as he assum ed th e h eatin g effect of the current negligible. E d lu n d 2 m ade allowance for th e tem p eratu re effect, and concluded t h a t cu rren t, except for th e accom panying te m perature change, does n o t affect Y oung’s m odulus. S trein tz3 m ade observations of th e torsion m odulus a t room tem p eratu re an d a t 55.50 C. T h e following figures represent th e percentage change of elasticity caused by th e current, correction being m ade for th e tem p eratu re effect: Brass (a). - 5-9: Q>), + 12.8; copper, 0.0; silver, + 3.9; iron, + 3.1; steel, — 12.2. M ebius4 found th e effect due to c u rre n t in th e case of steel, iron, brass, and silver rods to be extrem ely small. M iss Noyes5 studied steel, silver, an d copper wires an d found t h a t th e c u rre n t h eatin g caused a uniform decrease in Y oung’s m odulus which could be entirely accounted for as a tem p eratu re effect. W a lk er6 em ployed th e sam e experim ental m ethods as Miss Noyes, b u t found v ery irregular changes w ith steel, soft iron, platinum , an d copper, which could n o t be accounted for by tem perature and which differed greatly w ith increasing an d decreasing
4 3 2 H. L. DODGE. [ Se c o n d LSe r i e s.
current. A survey of previous w ork leaves a serious d o u b t as to the effect of cu rren t heating, an d this d o u b t a critical stu d y serves to increase. Pr e v i o u s Re s u l t s u p o n t h e Ch a n g e o f Yo u n g’s M o d u l u s w i t h
Te m p e r a t u r e.
T h e investigations th a t have been carried on to determ ine the change of Y oung’s m odulus w ith tem p eratu re are so extensive th a t th e results w ith copper only can be considered. These are however representative of th e results for o th er m etals. S hakespeare1 has found a decrease of Y oung’s m odulus for copper of 3.6 per cent, on heating from 130 C. to ioo° C. M iss Noyes2 m ade observations a t various tem peratures up to 150° C. and reports a uniform decrease of th e m odulus of 0.13 per cent, an d 0.07 per cent, per degree for tw o sam ples of wire. G ray, Blyth, and D unlop3 also were able to m ake observations a t in term ed iate tem pera tu res up to ioo° C. T h e y also re p o rt a uniform decrease, which for two sam ples was 0.015 per cent, an d 0.04 per cent, per degree respectively. On th e o th er h an d S lo tte’s4 observations a t intervals over a range of io° C. to 70° C. upon tw o sam ples of copper wire showed a decrease of Y oung’s m odulus w ith increase of te m p e ra tu re which was n o t uniform b u t became less rap id w ith one sam ple an d m ore rap id w ith th e o th er as th e tem pera tu re increased. A n increasing ra te of change is also indicated b y the results of W ertheim 6 a t 150 C., 100° C., an d a t 200° C., the change being m uch greater for th e upper interval. In th is connection th e work of P isa ti,6 K ohlrausch an d Loomis,7 and of S lotte8 upon th e torsion modulus is of interest. T h e y found th e ra te of decrease of th e m odulus greater as th e tem p eratu re was raised. T h is is suggestive as it is highly probable t h a t th e general n a tu re of th e elastic changes is sim ilar for the two m oduli. T hese results leave no d o u b t th a t increase of tem perature causes a decrease of Y oung’s m odulus, b u t th ere is no conclusive evidence as to th e exact n a tu re of th e decrease.
Sc o p e' o f W o r k.
In view of th e facts which have been cited it was decided to a tte m p t an investigation of th e effect of c u rre n t upon Y oung’s modulus, accompanied b y a parallel investigation of th e tem p eratu re change produced by an
4 3 3
Vo l. I I . 1
N o . 6. J CH ANGE I N THE E L A S T I C I T Y OF A C O P P E R WIR E.
external source of heat. E xam ination of th e previous w ork led me to believe th a t th e d esideratum was n o t a h astily m ade series of observations upon a n um ber of m etals b u t ra th e r thorough an d extended observations upon a single m etal. C onsequently this re p o rt is confined to results w ith copper although w ork is under w ay w ith o th er m etals. A review of th e lite ratu re m ade clear t h a t th e causes of inaccurate results in th e p ast have been chiefly a lack of m echanical perfection in th e ap p a ra tu s and insufficient a tte n tio n to th e distrib u tio n of tem p eratu re. In none of th e w ork upon th e effect of cu rren t has th ere been an y knowledge of th e differences in tem p eratu re along th e wire or an y effective a tte m p t m ade to secure a uniform distribution. T h e a p p a re n t cru d ity of th e ap p a ratu s th a t is here described is th e result of a n a tte m p t to keep every p a rt of th e a p p a ra tu s as sim ple as possible and to refrain from u n w arranted refinements. E ach source of error has been considered in relation to th e others and th e effectiveness' of th e a p p a ra tu s is best judged from th e consistency of th e results th a t have been obtained.
De s c r i p t i o n o f t h e Ap p a r a t u s.
In order to prevent convection currents and consequent non-uniform ity of heating necessarily accom panying vertical suspension th e wire was m ounted horizontally. T h e m echanical features of th e a p p a ra tu s (Fig. i) m ay be considered independently of th e h ea tin g box. T h e wire is
suspended between a b a r (b) an d a pulley (p) which are in tu rn supported b y a heavy cast iron base, bolted to th e carriage of th e dividing engine. T he wire is stretched b y th e w eights W i an d W 2. C u rre n t is led in a t th e points (d) through th e m ercury cups (c), th e wire being insulated a t (a) and a t (b). T hree copper-constantan therm o-couples (t) of No. 36 wire are for tem p eratu re m easurem ents. T h e w eight W \ is k e p t upon th e wire continually an d is heavy enough to stretc h th e wire alm ost straight. T he weight W 2 is added gradually an d w ith o u t ja r b y a simple
F ig . 1.
4 3 4 H. L. DODGE.
E C O N D Se r i e s.
a p p a ra tu s o perated b y a foot lever. T h e elongation of th e wire is m easured w ith tw o m icrom eter microscopes atta c h e d to th e bed of the dividing engine. These are focused upon b rig h t scratches on th e wire, m ade visible by incandescent lamps.
T h e heating a p p a ra tu s (Fig. 2), although supported by the cast-iron base, is m echanically en tirely independent of th e wire. I t m u st not only furnish a m eans of h eatin g th e wire externally b u t also m ust retain the heat when th e cu rren t is passed through th e wire itself. T h e base (0), bottom piece (5), an d back (z>) are covered w ith asbestos an d fastened together. T h e asbestos-covered fro n t (h), which has two windows of mica (m) for observing th e wire, an d th e glass cover (n ) are held to g eth er by clamps easily rem oved. T h e glass tu b es (k) are large enough to give sufficient clearance to th e wire (w) b u t sm all enough to prevent convection or oth er a ir currents. Zigzagging back an d forth across th e b o tto m piece (s) is a h eatin g wire of G erm an silver through which a cu rren t can be
sent. T his h eatin g box proved to be one of th e three im p o rtan t features of th e a p p a ra tu s a n d its sim ple form in no w ay indicates th e difficulty of securing a reasonably even tem p eratu re distribution. T his fact is v ery suggestive in connection w ith th e results of form er investigations in which average te m p e ra tu re only was known, there being no inform a tio n concerning th e te m p e ra tu re distribution. T h e use of therm o couples which could be slipped to different portions of the wire revealed surprising inequalities of tem p eratu re. E ven afte r uniform ity of heating had been secured w ith th e external source it was necessary to still further a lte r th e box before a satisfactory d istribution could be obtained w ith internal or cu rren t heating. A t th e highest tem perature em ployed the m axim um varia tio n of te m p eratu re over the portion of th e wire under observation was less th a n 150 C.
A second feature of im portance is th e free suspension of th e wire. T h e possibility of error due to co n ta ct of th e wire w ith rigid portions of th e a p p a ra tu s is elim inated b y perm itting it to touch n o thing except th e therm o-couples which are of fine flexible wire. C onsequently the specim en hangs freely in th e form of a flat catenary. T h e elongation
F ig . 2.
CH ANGE I N T H E E L A S T I C I T Y OF A C O PPER W IRE. 4 3 5
caused b y th e straightening of th e ca ten ary on th e application of th e additional stretching w eight is negligible, being a b o u t i/ioo of th e sm allest reading of th e m icrom eter microscope. T h e th ird feature is also m echan ical. I t is th e pulley which changes th e direction of th e applied force, which m u st be tra n sm itte d w ith o u t gain or loss resulting from friction or change of leverage. Fig. 3 shows its construction an d suggests th e m ethod of centering th e axis. T h e pulley was tested a t different
posi-F i g . 3.
Side view of pulley and section along A - B .
tions a t various tim es during th e progress of th e w ork an d found to introduce no appreciable error.
M e t h o d o f De t e r m i n i n g Te m p e r a t u r e o f Sp e c i m e n. W hen th e wire is h eated b y th e external source it is of uniform te m p eratu re th ro u g h o u t its cross section and of th e sam e tem p eratu re as th e surrounding air. U nder these conditions th e therm o-couples which have already been m entioned are a perfectly satisfactory m eans of m easuring its tem perature. In th e case of internal or cu rren t h eatin g there is a n increase of tem p eratu re tow ard th e center, resulting in a num ber of effects, change of resistance, therm al conductivity, etc., each of which has its influence upon th e cu rren t d istribution and in tu rn upon th e tem p eratu re distribution. T h e average tem p eratu re of th e wire is of course a function of th e surface te m p eratu re and of th e current, b u t for evident reasons th e therm o-couples do n o t give th e surface te m perature of the wire when th e wire itself is th e source of heat.
M easurem ent of tem p eratu re b y change of length caused by therm al expansion has proved to be th e m ost accurate an d convenient m ethod. B londlot1 has shown conclusively for several pieces of brass and one of G erm an silver t h a t cu rren t has no direct effect upon th e dim ensions of
1 J o u r n a l d e P h y s ., [i], 8, 122, 1879.
4 3 6 H. L. DODGE. [Se c o n d
LSe r i e s.
a m etal carrying a current. T his has also been shown b y Righi1 for four wires whose com position is n o t given. T here is no reason to believe th a t copper should be peculiar in showing th is effect. In fact our own experience an d th e persistency w ith which our results for Y oung’s m odulus a t an y te m p eratu re as determ ined b y this m ethod rem ain the same, no m a tte r w h a t m ay be th e source of heat, is in itself proof th a t c u rre n t has no direct effect upon th e length of th e wire. In using change of length as an indication of tem p eratu re account m ust be tak e n of the change caused indirectly by th e effect of tem p eratu re upon Y oung’s m odulus. T herm o-couple readings an d length observations were taken for all w ork w ith external heating. W ith internal heating length obser vations were m ade an d th e te m p eratu re determ ined b y com parison with th e o th er results. T h e therm o-couples were connected to a sensitive Siemens an d H alske galvanom eter b y a three-w ay switch and were calibrated b y d irect com parison w ith an accurate therm om eter.
W ith c u rre n t heating th e central portions of th e wire tend to lengthen m ore th a n th e o u ter portions which are a t a lower tem p eratu re and it is assum ed t h a t th e ac tu al lengthening so averages this effect as to be an indication of th e average tem p eratu re. I t is also w orthy of notice th a t th e contribution to Y oung’s m odulus will be different for th e different portions, th e v alue observed being an average of th e moduli of th e various circular layers m aking up th e cross section of th e wire. T here is of course a sim ilar averaging of tem p eratu re an d modulus along th e length of th e wire.
Na t u r e o f t h e Te s t s.
T h re e sections of com m ercial copper wire were tested, all from the sam e piece obtained from th e D river-H arris W ire Co., H arrison, N . J. T h e p u rity of th e specimen was determ ined electrolytically, w ith results of 99.88 per cent, an d 99.92 per cent. As th e second was th e more accu ra te we m ay ta k e 99.91 ± .01 as th e p u rity of th e specimen. There was no trace of eith er silver, lead, or iron. T h e coefficient of linear expansion was .0000169. Specim en No. 2 was 0.81 m m. in diam eter. T h e portion under observation was 50.7 cm. long. T h e other samples w ere of sim ilar dimensions. A w eight of 2,109 gram s was k ept upon the wires continually. T h e modulus was determ ined by m easuring the elongation produced b y a n additional w eight of 2,252 grams. T he elongation was observed w ith a microscope, th e m icrom eter head of which can be read to 1/50 revolution, corresponding to .00093 mm. stre tc h . A t room tem p eratu re th e stretc h am ounted to ab o u t .02 mm.
Vo l. 11.1
N o . 6. 1 CH ANGE I N T HE E L A S T I C I T Y OF A C O PPER W IRE. 4 3 7
C urrents as high as 15 am peres were used. T h e accuracy of th e w ork is. of a high degree. T h e error in th e determ ination of Y oung’s m odulus is ab o u t 1 per cent. A greater accuracy m ay prove desirable if small peculiar effects appear w ith oth er m etals, b u t th e present ap p a ra tu s is sufficiently accurate to determ ine th e general n atu re of th e changes in the modulus.
Each determ ination of Y oung’s m odulus as represented b y a p oint on th e curves or a value in th e tables is th e result of te n or more sep arate m easurem ents taken so as to elim inate an y error caused by a slight d rift in tem perature. G enerally a half hour or more was required for th e tem perature to reach a sufficiently stead y state. A fter th e prelim inary work th e greater p a rt of th e observations consisted of series of d e te r m inations w ith increasing or decreasing te m p eratu re over a range of 20° C. to 150° C.
Pr e l i m i n a r y Te s t s.
T he possible ways in which cu rren t m ay affect elasticity are so n um er ous th a t a determ ination of Y oung’s m odulus a t a given tem p eratu re is of no value unless in terp reted in connection w ith th e whole therm al history of th e wire. Since th e essential p a rt of this investigation is th e comparison of observations tak e n under as nearly sim ilar therm al condi tions as possible, except for th e difference in tim e and in th e source of heat, an elim ination as far as possible of an y o th er th a n pure tem p eratu re effects is desirable. If this is impossible a thorough understanding of th e n atu re of th e oth er effects is essential. In order to elim inate th e well-known p erm anent changes accom panying extrem e tem p eratu res th e tre a tm e n t of th e wire after its first heating and stretching was confined to a tem p eratu re range of 20° C. to 150° C. In order to find o u t how consistent was its behavior w ithin this range an d under th e various conditions of tre a tm e n t a great deal of prelim inary w ork was necessary. This showed th a t there are no sudden changes in th e elastic sta te of th e wire, th a t th e ra p id ity of heating and cooling had little if an y effect upon th e changes in elasticity, and t h a t th e length of tim e th e wire was held a t a given tem p eratu re h ad no g re at effect. As explained later, perm a n en t or history effects were found, b u t for an y series of readings these changes were insignificant, th e value of Y oung’s m odulus retu rn in g a t th e close of an y d a y ’s observations to practically its original value.
4 3 8 H. L. DODGE. [Se c o n d
LSe r i e s.
In order to learn w hether or not a t any given tem p eratu re th e value of Y oung’s m odulus is dependent upon th e m anner of heating, th e observa tions of T ab le I. were made. T h e shifting from one m ethod of heating to an o th e r was done w ith o u t p erm ittin g m uch alteratio n in th e tem p era tu re of th e wire.
Ta b l e I.
Showing Constancy of Young's M odulus under Different Conditions of H eating, Temperature R em aining the Same.
M e t h o d o f H e a t i n g . V a r i a t i o n i n L e n g t h , M m .
V a r i a t i o n i n S t r e t c h , M m .
•
1 2 .8 a m p e res, a lte r n a tin g c u r r e n t ... .0000 .00 0 0 12.8 a m p e res, d ir e c t c u r r e n t ... .0065 .00 0 9 E x te r n a l, d irect c u r r e n t ... .0059 .0 009 E x te r n a l, a lte r n a tin g c u r r e n t ... .0032 .0014 12.2 am pe res, d irect c u r r e n t ... .0018 .0009
T h e greatest varia tio n in length is .0065 mm. or 0.14 revolution of th e m icrom eter head. T his corresponds to a tem p eratu re variation of less th a n one degree C entigrade. T h e g reatest variation in stretch is .0014 mm. T his corresponds to only 0.03 revolution and is about the experim ental error. These results show clearly th a t a t this tem perature, which was approxim ately n o ° C., th e stretch m odulus is independent of th e source of heat. T hese prelim inary investigations and others which need n o t be described showed th e behavior of copper to be very consistent and are th e justification for the tak in g of more extended series of observa tions and th e com parison of results secured a t different tim es.
Te s t s u p o n Sp e c i m e n N o. 2 .
A fter specimen No. 2 was placed in th e ap p a ra tu s it was heated to a high tem p eratu re an d stretched to rem ove th e kinks. A fterw ard it was never heated above 150° C. T he four series of readings composing Fig. 4 were ta k e n a fte r the wire had been heated by both m ethods a total of a b o u t 50 hours and after a b o u t 500 applications of th e stretching weight. If such a th ing were possible as a state in which all changes in elasticity, except those caused by tem p eratu re could be eliminated, this tre a tm e n t should have secured th e result.
Vo l. I I . l
N o . 6 . J CH ANGE I N T HE E L A S T I C I T Y OF A C O P P E R W IRE. 4 3 9
F ig . 4.
R e s u l t s o f t e s t s u p o n s p e c im e n N o . 2, s h o w in g e ffe c t o f in t e r n a l a n d e x te r n a l h e a tin g .
Ta b l e II.
D a ta fo r Series 5, W ire 2, E xternal Heating.
4 4 0 H. L. DODGE. [Se c o n d
LSe r i e s.
Ta b l e II I.
D ata fo r Series 7, W ire 2, Internal Heating.
Obs. N o. T i m e . T e m p . A m p e r e s .C u r r e n t , S t r e t c h (R e v s, of Micr. S c r e w ) . X io-11 D y n e s p e r C m 2.Y o u n g ’s M o d u lu s 1 1 0 :3 0 P .M . 21° C. 0 . 0 4 .1 0 11.42 2 11:15 A .M . 32 3 .8 4.0 8 5 11.45 3 11:50 50 7.0 4.1 6 5 1 1 .2 2 4 1:05 P .M . 77 9.8 4 .1 8 1 1 .2 0 5 1:30 10 2 1 2 .0 4 .2 8 10.93 6 2:30 128 14.1 4.4 9 10.43 7 3:05 105 11.9 4.31 10.85 8 3 :5 0 80 9.5 4 .2 0 11.13 9 4 :2 0 53 6 . 8 4.1 3 11.33 1 0 5 :1 0 27 0 . 0 4.07 11.50 11 6:30 27 OJO 4 .0 8 11.46 12 7:25 43 5 .0 4.0 9 11.44 13 8 : 0 0 6 6 8 . 0 4 .1 5 11.26 14 8:45 97 1 1 .2 4 .2 4 5 1 1 .0 2 15 9:15 129 13.9 4.42 10.60 16 9 :4 0 28 0 . 0 4 .1 0 11.42
tem p eratu re was being raised by steps an d th e crossed points those when th e te m p eratu re was being lowered. P articu lar a tte n tio n is called to points 2 an d 3, th e second of which was tak e n afte r th e tem p eratu re had been m aintained for tw elve hours. T h e y show t h a t th e length of tim e th e wire was k e p t a t th e high tem p eratu re h ad no effect upon its elasticity. T h e proxim ity of points 8 and 1 shows th e elim ination of history effects and perm anent changes.
Tw o days la te r Series 5 was secured, which yields results identical w ith those of Series 4, showing th a t th e w ork of one d ay can be repeated a t a later tim e. F u rth e r proof of th e absence of history effects is afforded b y th e positions of points 14, 15, 16, 17, an d 18 relative to observations ta k e n a t earlier tim es a t th e sam e tem peratures. T he fact th a t points 5, 6, and 7 ta k e n w ith increasing tem p eratu re fall on the same curve as points 9, 10, an d 12 ta k e n w ith decreasing tem p eratu re shows th e absence of a n y hysteresis effects.
441 Vo l. I I . 1
N o . 6. J CH A N G E I N T H E E L A S T I C I T Y OF A C O PPER W IRE.
be expected from th e experim ental error shows t h a t one is justified in tak in g these results as a n indication of tem p eratu re effect alone and is justified in com paring th e results of different days w ith different m ethods of heating.
Te s t s u p o n Sp e c i m e n N o . . 3.
A th ird sam ple of th e sam e wire was n ex t placed in th e ap p a ra tu s in order to confirm th e results obtained w ith th e form er specimen, to a tte m p t still other variations in th e th erm a l tre a tm e n t an d to secures the complete elastic history of a specimen. T h e results are given in Figs. 5, 6, and 7. U pon these curves ap p e ar every observation t h a t was made w ith th e th ird specimen, n o t one having been discarded.
First Changes in E lasticity.—T h e specimen was m ade as free from kinks
as possible and placed in th e a p p a ra tu s. O bservation i , Fig. 5, was a t room tem perature. T h e wire was th e n h eated externally to a b o u t 1250 C. and cooled. O bservation 2 shows no change. H e atin g b y 17 am peres current to ab o u t th e sam e te m p eratu re b ro u g h t a slight increase in the m odulus as indicated by observation 3. T here was v ery little increase in length during this tre a tm e n t, as th e additional w eight was n o t applied a t high tem peratures.
F i g . 5.
442 H. L. DODGE. [Se c o n d
LSe r i e s.
I t was an ticip a te d t h a t continued application of the additional weight while th e wire was h o t would remove an y kinks in the wire, cause an increase in length an d a change in th e modulus. T he wire was heated b y 17 am peres cu rren t an d th e w eight applied a num ber of times. On
F ig . 6.
4 4 3
Vo l. II.1
N o . 6 . J C H AN G E I N T H E E L A S T I C I T Y OF A C O P P E R W IRE.
cooling an increase of length of ab o u t 0.2 mm. was found and a n increase of Y oung’s modulus as indicated b y observation 4. T h e wire was again heated to 1250 C. and Y oung’s m odulus m easured a t this tem perature, resulting in observation 5. T h e te m p e ra tu re was th e n increased to ab o u t 150° C. and th e additional w eight applied a b o u t th irty times, resulting in observation 6. Cooling to 1220 C. gave observation 7, and to 28° C., observation 8. I t was believed t h a t a sufficiently stead y sta te had been secured to justify th e trial of a series of readings an d a com parison of points 6, 7, an d 8 w ith reference to la te r observations shows this to have been th e case. I t is highly probable t h a t th e increase in Y oung’s modulus due to th is p articu lar tre a tm e n t was caused as m uch by th e m echanical stretching and straightening of th e wire as b y th e therm al tre a tm e n t, although th e la tte r was u n d o ubtedly essential to th e securing of th e stead y state.
Discussion of Fig. 6 .-—Following th e w ork ju s t described cam e eight
d ays of testing under a v arie ty of conditions. Fig. 6 shows th e results and has been m ade practically self explanatory. T h e order in which th e observations a t the different tem p eratu res were ta k e n and th e m anner of heating were varied in a num ber of w ays in order to catch an y possible changes caused by trea tm e n t. Series 1 is w ith external heating, increas ing tem perature. Series 2, first p art, was ta k e n in exactly th e same m anner and th e results show th a t for external h eatin g a sufficiently stead y sta te had been reached so t h a t only small p erm an en t changes were to be anticipated. T h e second p a r t of Series 2 was ta k e n in exactly th e sam e m anner except for th e change in th e m anner of heating, th e external heating having been changed to internal by m eans of th e current. No difference is observed. T h e following d ay a series was ta k e n in a sim ilar way except th a t th e observations were m ade w ith decreasing tem p eratu re. The n atu re of th e tem p eratu re change rem ains th e sam e an d a slight perm anent increase appears.
In Series 4 th e system atic heating and cooling over th e whole tem p era tu re range was n o t followed, b u t no peculiar effects were observed. D uring the following night the wire lay unstretched an d it is possible t h a t this had some influence in causing th e p erm an en t increase of elas ticity observed th e next day. Series 5 is th e result of heating and cooling w ith internal heating and Series 6 is for th e sam e tre a tm e n t w ith external heating. N o hysteresis effect appears nor does th ere appear any difference due to m ethod of heating.
Discussion of Series 7.—Series 7 is th e m ost interesting single set of
4 4 4 H. L. DODGE. [Se c o n d
LSe r i e s.
appreciable change of tem p eratu re. A t each tem p eratu re a t which observations were tak e n , th e change in heating was m ade gradually, w ith o u t perm ittin g th e te m p eratu re of th e wire to change. O bservation I was a t room tem p eratu re. O bservation 2 was w ith current heating. W ith o u t p erm ittin g m ore th a n a few degrees variation in tem perature th e c u rre n t in th e wire was decreased while t h a t in th e heater was increased. O bservation 3 was w ith external heating. T he tem perature w as th e n increased and observation 4 taken. Again th e m ethod of heating was gradually changed until observation 5 was secured with in tern al heating a t practically th e sam e tem perature. In th e sam e way 6 an d 7; 8 an d 9; 10 and 11 were taken. T h e results show in a striking m anner th a t th e changes accom panying cu rren t heating are only te m p eratu re effects. T h e tests of the te n th d ay w ith internal heating gave Series 8.
In an y single series of this group th e perm anent changes w ith trea tm e n t do n o t ap p e ar prom inent. B u t a comparison of the eight curves, Fig. 5, reveals a gradual increase of Y oung’s m odulus from d a y to day. T his was caused u n d o ubtedly b y th e continued application of the additional w eight which caused a very small gradual stretching.
Effect of the Period of R est.—Tw o days afte r th e observations of Series
8 a m easurem ent was ta k e n a t room tem p eratu re afte r which the wire was u n disturbed for 106 days. T his last observation appears w ith Series 9, Fig. 7, and a com parison w ith point 1 shows a slight recovery of
th e wire during th e long rest. N ine o th er observations were then taken w ith a n idea of determ ining w hether or n o t th e m odulus would rem ain low upon heating. T his proved to be th e case both for external and cur re n t h ea tin g as is clearly bro u g h t o u t by comparison w ith th e d o tted line which is th e Series 8 curve. Tw o d ays later, as shown by Series 10, fu rth e r h eatin g an d testing b ro u g h t th e specimen back to its former state
F ig . 7.
4 4 5
Vo l. I I . 1
N o . 6. J CH ANGE I N T HE E L A S T I C I T Y OF A C O PPER WIR E.
for low tem peratures although a t high tem p eratu res greater values of Y oung’s m odulus were observed th a n ever before. Too m uch stress m u st n o t be laid upon these slight changes in th e wire. T h e difference between Series 9 and Series 8 is small an d m ay h ave its origin in a slight change in th e working condition of th e a p p a ra tu s caused by its period of idleness. T he im p o rtan t fact is t h a t no g re at change occurred an d the wire was in practically th e sam e condition as when left. T h e explanation of th e difference in slope in some of th e curves is t h a t th e experim ental error is of a b o u t th e am o u n t of th e variation. T h e to tal change in Y oung’s modulus over th e entire tem p eratu re range is only 10 per cent, and th e difference in stretc h which this represents corresponds to less th a n one revolution of th e m icrom eter head. In order to show this variatio n th e scale to which Y oung’s modulus is plo tte d has to be so large th a t th e errors of observation become appreciable. W hen the m agnitude of th e probable error is ta k e n into consideration th e con sistency of th e curves tak e n under such a v arie ty of conditions is rem arkable.
Co n c l u s i o n s.
T h e following conclusions apply to the sam ple of copper wire th a t has Teen studied. I am of the opinion th a t o th er specimens of copper will yield results differing only in th e m agnitude of th e changes, an d believe th a t the results are very suggestive of th e effects th a t can be expected in o th e r m etals. B u t work w ith iron wire now u nder w ay and certain results of other investigations to which reference is m ade la te r m akes me feel t h a t too g reat caution cannot be exercised in m aking general conclusions from observations upon particular samples.
From th e prelim inary observations and the tests upon specimens Nos. 2 and 3 we reach these conclusions:
4 4 6 H. L. DODGE.
[Se c o n d -[Se r i e s.
2. H eatin g by an electric cu rren t has no effect other th a n th a t caused by th e accom panying tem p eratu re. T his was proved for one tem pera tu re b y th e results of T ab le I. and a t a n um ber of different tem peratures b y th e w ork of Series 7, W ire 2, shown in Fig. 6, in which th e heating was alte rn a te ly b y cu rren t and th e external source.
F u rth e r proof w ith step b y step heating is also furnished b y th e identical n a tu re of th e results recorded in th e four curves of Fig. 4, two curves of which are for cu rren t and tw o for external heating. Con clusive evidence is afforded by th e long series of experim ents carried out upon th e th ird specimen, in which every possible w ay of varying th e m ethods of h eatin g was a tte m p te d w ith o u t detecting an y difference in th e results.
3. Y oung’s m odulus of a copper wire decreases w ith increase of tem p eratu re a t an increasing rate. In th e case of th e specimen studied, the am o u n t of decrease a t different tem peratures in term s of per cent, of the value of th e m odulus a t 20° C. is as follows: 40° C., 0.6 per cent.; 60° C., 1.7 per cent.; 8o° C., 2.9 per cent.; xoo° C., 4.6 per cent.; 120° C., 7.0 per cent.; 140° C., 10.0 per cent. A t 20° C. th e m odulus of the sample in th e stead y elastic sta te was 11.5 X i o 11 dynes per cm2.
T h e tru e n a tu re of th e te m p eratu re change was first shown b y the w ork w ith specim en 2 as represented b y th e curves of Figs. 4 and was fully su b sta n tia te d by th e fu rth e r w ork w ith specimen 3 which yielded identical results.
4. H eating an d repeated stretching increases Y oung’s modulus. As is shown b y th e results w ith specim en 3 there is a n a p p a re n t increase of a b o u t 5 per cent, caused by th e first four heatings and th e accompanying stretching, which is th o u g h t to be due in p a rt to th e straightening out of th e kinks. B u t there was a fu rth e r increase of 2 per cent., which was a tru e h istory effect, although caused probably as m uch by th e repeated stretching as by th e heating.
Co m p a r i s o n w i t h Re s u l t s o f Ot h e r s.
As has already been sta te d previous results upon th e effect of current do n o t agree. A com parison of these w ith th e present w ork will n o t be a tte m p te d as th e degree of accuracy in th e la tte r is far greater. T h e conclusion t h a t th ere is no effect of cu rren t upon copper other th a n t h a t caused b y tem p eratu re can be accepted to an accuracy of less th a n 1 per cent.
A n interesting com parison regarding th e history effect and th e m agni tu d e of change w ith tem p eratu re is furnished by th e w ork of Shakespeare,1
4 4 7
Vo l. II.1
No. 6. J CH ANGE I N T HE E L A S T I C I T Y OF A C O PPER W IR E.
who used interference m ethods to determ ine th e m odulus a t 130 C. and a t xoo° C. He does n o t com pute th e value of th e m odulus b u t gives th e change from 130 C. to ioo° C. as 3.6 per cent. F o r this range th e present specimen shows a change of 5 per cent. Shakespeare also found th a t th e modulus increased w ith continued h eatin g an d cooling, until a perm anent sta te was reached. T h e first h eatin g produced a sudden increase of several per cent, which was followed b y a gradual change. Nine heatings produced a to ta l increase of 12 per cent. T h e perm anent change described in this p aper was of th e sam e kind an d am ounts to 7 per cent.
In th e following com parison of results upon th e te m p e ra tu re effect th e bracketed figures are tak e n from our own curves. A t 20° C. o u r value of Y oung’s m odulus was 11.5 X i o 11 dynes per cm 2. M iss N oyes1 found th e m odulus of a specimen of copper to be 12.02 a t 20° C. T his and another specimen tested a t tem p eratu res up to 150° C. gave a uniform decrease of the m odulus which com puted for 20° C. to 140° C. gives a change of 8.5 per cent, an d 16 per cent, respectively [10 per cent.]. A t 17.40 C. G ray, B lyth, and D unlop2 found th e m odulus of a sam ple of commercial copper to be 11.15 and of h ard draw n electrolytic copper a t 19.5° C. to be 12.9. T hey m ade tests over a range of ioo° C. an d reported a uniform ra te of decrease which com puted for 20° C. to ioo° C. gives for the two samples changes of 1.2 per cent, an d 3.4 per cent, respectively [4.6 per cent.].
S lotte3 found for two specimens changes of 6.6 per cent, an d 4.2 per cent, respectively [1.5 per cent.], for a tem p eratu re increase from 20° C. to 6o° C. T he m oduli a t 20° C. were respectively 12.4 and 12.7. T he more im p o rtan t point is th a t he did n o t find a uniform ra te of change. T he m odulus of th e form er specimen decreased a t a n increasing ra te while th e la tte r shows a decrease a t a decreasing rate. Decrease of the m odulus a t a n increasing ra te is also indicated b y th e w ork of W ertheim .4 If his results a t 150 C., 100° C., an d a t 200° C. are p lotted an d a sm ooth curve draw n it is alm ost identical w ith our own curves, except th a t th e values are ab o u t 10 per cent, lower. R egarding th e n a tu re of th e change observations on th e torsion m odulus are valuable although too g re at stress m ust n o t be placed on them . K ohlrausch an d Loomis5 m ade observations upon copper a t various tem peratures. T h e y found an increasing ra te of change as th e tem p eratu re increased. C om puted for 20° C. to ioo° C. th e change is 4.2 per cent. [4.7 per cent.], and for 20° C. to 140° C. 7.5 per cent. [10 per cent.]. S lo tte6 has also m ade a stu d y
4 4 8 H. L. DODGE. [SLSe c o n de r i e s.
of th e torsion m odulus of copper and finds th e same increasing rate of change. C orresponding values from his w ork are 5.9 per cent, and 8.3 per cent.
T h e results th a t have been cited m ust be considered from two s ta n d points, first as to th e general n a tu re of th e changes of Young’s modulus, w hether uniform or otherw ise, and secondly, regarding the numerical values.
E x am ination of th e m ethods of those who have reported a uniform decrease shows t h a t th eir w ork was n o t of , sufficient accuracy to have d etected th e varia tio n from a linear relation found by others. We are n o t certain how m uch im portance should be attac h ed to th e work of S lotte upon Y oung’s m odulus. He worked also w ith alum inum , iron, and platinum as well as copper and found a decreasing ra te of change. N one of these results seem to have been repeated.. T h e torsion modulus can be determ ined to a m uch higher degree of accuracy and S lotte’s recent w ork upon th e torsion modulus of all these m etals shows an increasing ra te of change. Since th e evidence t h a t is available points to w ard a sim ilarity in th e n a tu re of th e tem p eratu re change of the two m oduli th is raises a question as to w hether S lo tte’s former results upon Y oung’s m odulus are a tru e tem p eratu re effect.
W ith th e results of W ertheim an d of S lotte upon one specimen agreeing w ith our own an d th e w ork of K ohlrausch and Loomis and of S lotte upon th e torsion m odulus showing th e sam e general type of change, I believe th a t th e conclusion t h a t Y oung’s modulus of copper decreases w ith in crease of te m p eratu re a t a n increasing ra te has sufficient corrobora tion.
4 4 9
Vo l. I I . l
No. 6. J C H AN G E I N T H E E L A S T I C I T Y OF A C O P P E R W I R E .
In conclusion I wish to acknowledge indebtedness to the staff of the physical laboratory of the State University of Iowa for their interest in the work and especially to Professor G. W. Stewart for suggesting the problem and to Professor J. N. Pearce, of the department of chemistry, for the chemical analysis.
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A ddress correspondence to
Part
II
To the Members of the Examining Committee:
As it is expected that Mr. Dodge's
thesis will be published in the near future, I
have permitted him to submit it to you in
manuscript form.
The original drawings will
of course be sent to the printer and cannot be
bound with the manuscript.
1
THE CHANGE
IN THE ELASTICITY OE AN
IRON
WIRE WITH CURRENT AND EXTERNAL HEATING.
By
H.
L.
Dodge.
The effect of current and external heating upon the
Young's m&dulus of a copper wire was described in a previous
number of the Physical Review1 .
The present paper deals with a
similar investigation of an iron wire, the more ffl&rked changes
being an increased temperature range, namely 20°C. to 475°C.
and slightly greater accuracy.
For a detailed description of the
apparatus and method og making the determinations the former
paper should be consulted.
Improvements in Apparatus
and Method.
Mention cannot however be omitted of certain changes
which have resulted in greater accuracy.
Attention was called in
the previous paper to the desirability of having the specimen
of wire free from mechanical interference between the two points
of suspension.
The only interference in the case of copper wire was
the three thermo-couples of fine wire which did ndit seem to cause
any inaccuracy in the results.
Some changes in the method of
measuring the tempefature have made these unnecessary and in the present
apparatus tjje specimen of wirehangs absolutely free.
As before, the temperature of the wire is determined by
its change in length with increase of temperature.
In the former
paper the justification and advantages of this method were
explained.
The same statements apply in the present instance as the
only changes have been in the process of finding the relation between
temperature and length.
This was found as follows.
2
A thermo-couple was carefully calibrated by means of a
thermometer.
It was then attached to the wire at the middle point
and the relation between the temperature of this point and the length
determined. As all £Se points of the wire were not at the same tem
perature it wt?.s necessary to find the relation between the average
temperature and the temperature of the middle point.
The heating box
was maintained at a constant temperature of about 350°C. and a large
number of thermo-couple readings taken at various points on the wire.
This resulted Ant only in an established relation between the average
temperature of the wire and its length but also gave complete
information concerning the temperature distribution.
Uhder the above
conditions the maximum variation of the temperature along the wire
was about 40°C. while the error in the measurement of the average
temperature was probably not more than 10°C,
At lower temperatures
the accuracy is of course greater, the error amounting to but one
or two gegrees at or near room temperature and perhaps five degrees
at 150°C, At higher temperatures the possibility of error is of
course greater amounting to about 20°C. at 475°C.
The curve
representing the relation between change of length and temperature
was plotted and found to be a straight line within the limits of
accuracy of the temperature measurements.
The increase in length
between 20°C. and 475®C., corrected for the stretch caused by the
decrease in the modulus, was 3.67 mm.
As the wire was 57.6 cm.
between points of observation the mean temperature coefficient of
expansion proves to be .00014 with a possible error of five per cent.
Considerably increased accuracy has been secured in the
measurement of the stretch and consequently the modulus with the
present apparatus by the use of new microscopes and micrometer
3
^he Specimen.
The sample of wire upon which the tests were made was
obtained from the Driver-Earris Wire Co., Harrison, N. J.
4
Th e
ynean ^
coefficient of expansion between 20°C. and
*
—
c
475°C. is
14 x 10
per degree Centigrade to an accurac y of
about
5 per cent.
The wire is
.82 mm. in diameter.
The portion
between the points of observation was
57.6 cm. long.
A weight
of 2,109 grams was kept unon the wire continually, the modulus
being determined by measuring the elongation produced by an
additional weight of 2,252 grams.
Certain additional facts
regarding the loading, current, number of readings, time between
readings, duration of tests, etc. appear in Table I which
contains the complete data of a characteristic series of
observations with current heating.
The other series differ
materially only in method of heating or temperature range.
Eature
of the
Tests.
The preliminary observations made while the wire was
being straightened and brought to a cyclic state revealed striking
irregularities in its behavior.
At first increase of temperature
increased the modulus, but this effect soon disappeared.
Continued heating and testing gradually increased the modulus.
This was probably due largely to stretching ana straightening.
Both current and external heating were employed and no effect
characteristic of the method of heating could be detected.
After a number of cycles of heating and cooling accompanied by
stretching the wire was brought to a cyclic state and the more
extended tests made.
The first group of observations were with a temperature
range of 20°C. to £>00°C.
The tests were with both external and
5
temperature was reached.
later the temperature range was extended
to 450°Q. and to 475°G. and similar series of tests made. A few
characteristic results are shown in Figs. 1, 2 and 3, Lr:
explanation^)! which follow*
’ (
Explanations
of the Curves.
Figs. 1, 2 and
3
represent graphically certain of the
results that have been obtained.
Table I contains the complete
data for Series 9, Fig.2.
The dat^ for the other curves do*.
not differ materially in general character.
In every case th§
dotted points represent determinations of the modulus with
increasing temperature and the crosses those with decreasing tem
perature.
Nearly all the points are the fesult of ten or more
observations.
±sx±hx When the terms "external'1
and "internal"
heating are employed the former indicates that the wire was heated
by means of a heating element of Uichrome wire extending along
the bottom of the enclosing box, while the latter term refers
to heating by means of an electric current passing through the
specimen itself.
In all of the statements regarding the absence of certain
effects it should be understood that this refers to effects
amounting to one per cent or more.
The presant apparatus cannot
detect variations in the modulus of a less amount.
The accurac y
of the work can best be judged from the curves.
change in the
modulus
from 18 to 19 corresponds to a difference in stretch
of only .007 mm.
I, .
xhe first four series cover^a temperature range of
20°c . to 300°C. with both external and internal heating and
6
increasing and decreasing temperature.
Series 6 and 8 are plotted
together,
^hey were w i t h internal heating, increasing and decreasing
temperature.
f
i'he results of the first eight series show that for
temperatures below 300°C. the elasticity is a definite function of
temperature, independent of manner of heating and free from any
hysteresis effects.
It was next thought desirable to extend the temperature range
by a considerable amount and to repeat the tests in various orders to
ascertain whether the same relations held for a greater range of tem«* .
*
perature.
Upon heating above 300°S. a very ragid decrease of the r
1
modulus was
discovered, the rate of decrease becoming very marked at
about 450°C.
The results are shown in Fig. 1, Series 9.
At
this
temperature the wire stretched about one fourth of a millimeter,
approximately .05 per cent of its length.
When the elastic limit is
exceeded there should be an increase of the modulus for all temperatures.
The next observation at room temperature showed an increase,from 18.5 to
19.2 or 2.6
per cent.
The increase of the modulus for other
temperatures
as well can be aeen from Fig. 2.
The first three series of observations of Fig. 2 give the
temperature variation of Y o u n g ’s modulus with external heating. Series
11 shows the type of curve, and the return of the modulus to the same
value after a day of heating and stretching. Series 12 was taken under
similar conditions.
By yielding the same results it proves the total
disappearance of all history effects except the very gradual increase of
the modulus with continued heating and stretching.
As the observations
for decreasing temperature in the last two series were few, ^eries 13
was made in order to learn whether the curve with decreasing temperature
is exactly the stone in every case the only difference being that
it is gradually raised to correspond to the general increase of
the modulus as the work progressed.
All of the observations so far had been made with the
wire enclosed in an asbestos board heating box so that the currents
used in the wire and the heating element might be as low as possible.
Walker'*' has found a great difference between the effects of
internal and external heating, dependent in part upon the current
density.
He interprets these results as caused by the magnetic
effect of the current, the maximum field which he secures having
an average value of 31 gausses.
Then Series 19 was taken an attempt was made to find
this effect in case it were associated with residual magnetism.
Between the first two observations of this series ( Pig. 2)
the wire was subjected for an instant to a current corresponding
to an average fiftld of 50,gausses.
The difference in the two plotted
points is but a fraction of the experimental error,
ileither this
test nor any of the other results up to this time gave any
suggestion of the effect Walker reports,
ft
,
*
It seemed desirable however to employ as large 'currents
as possible in an attempt to bring out the effect.
Accordingly
the cover of the box enclosing the wire was removed giving the
air free access.
Series 21, Pig. 3, is the result, the maximum
current being 9 amperes, corresponding to a current density
of 17 ampefes per square millimeter and an average field
of 29 gausses,
“
“or purposes of comparison Series 22 was taken
with external heating, the slight tendency of the readings to
fall lower ( less than .001 mm. difference in the actual
measurements) being probably due ir entirely to the experimental
errors caused by the fact that the main part of the apparatus
was much hotter in the latter case.
iJ0 indication of the effects
reported by Walker is to be found.
Comparison With the Results of Others.
The earlier work1 upon the effect of an electric current
upon Young's modulus
is of very questionable value. Reference
will be
made only to
the more recent work of Walker and Lliss Iloyes.
, , 2
Miss Noyes
found no effect peculiar to current in the case
of piano wire but her work was not of sufficient accurac y
to have revealed even a relatively large difference.
3
Walker
on the otherhand, taking measurements with what
appears to be a considerably greater accurac y -^6
u a<4 striking
differehces of large magnitude.
There are several points
concerning even his most recent work that make one seriously
question whether the results which he secured are attributable
to the causes to which they are assigned,
A lengthy discussion
of kizxxsx Walker's work is out of place at this time but two
or three xxiixts differences in our results may be of interest.
Walker worked over a temperature range of 20°8. to
125°C., finding in the oase of iron wire a maximum value of the
modulus at about 50°C., the value at 125°C. being about the
same as at 20°C. Employing different loads he finds that the
is
z
c
ss
maximum treoroacoo-
hoi-ght with the heavier loads amountixng
p
9
which I have employed is only 8.2 kg. per square millimeter.
It would seem as if some trace of '"alker's maximum ought to
have appeared hut in my results there has been no indication of
any such effect.
Walker ^ also studied the effect of variation ,of load
while keeping the current constant at a certain value,
^or some
reason the results are now interpreted in terms of field intensity
produced in the wire although formerly the effect of the current
was regarded as one of temperature.
The fields employed vary
from .7 p u M to 31.9 gausses, skB*±xgxiHrger The modulus is
larger for small loads, the difference being the greatest
for fields of intermediate values, amounting to 12 per cent
for a field strength of 9 gausses.
In my own work the field
strength is 3.25 times the current in amperes. It -var&ed
therefore from 0 to 29 gausses and since the external heating
was sometimes used in conjunction it was-not always the same
for the same temperature.
In no instance have I been able to
detect any effect peculiar to the current or its accompanying
magnetis field qnd any variation amounting to more than one
per cent would have been observed.
Concerning the effect of magnetization upon
Young’s modulus a great deal of work has been carried on.
The more important results have been those of
2 3 a
Honda and Terada,
Rensing
,
Stevens ,
2
Phil. M ag. , 13, 36; 1907.
1 0
1 ^
Brackett
, and Bock
.
An effect
has been observed but in every case it was small amounting
to less than one half per cent.
As far as these results.xxshcsoy
roomcgga which are for longitudinal and not circular
magnetisation, and my own can be compared to those of »*alker
there is flat contradiction.
In the near future
x
hope to have
more data as a basis for comparison.
The differences may of
course be due entirely to an inherent difference in the samples
of wire.
11
-he results of the Curlier investigations of the
temperatute coefficient of Young's modulus of an iron wire
were over rather limited temperature ranges and were subject
to considerable errear.
Wertheim 1
found a maximum in the
neighborhood of 100°Q. followed by a rapid decrease up to
200°C.
Kupfer
1
employing a method of bending rods
1 2
nore
recently Gray, Blyth, and Dunlop,'*'
2
3
Katzenelsohn
, and
Shakespeare
have worked at room
temperature and at or about 100°G. and found a decrease
in the modulus of iron wire per 100cC. of
1.36, 2.33, and 1.6 per cent respectively,
H
•
The bracketed figures here and following are results taken
from my own work for corresponding temperatures.
But little work has been done at intermediate temperatures.
4
1 3
On he o herhand Pisati
finds for both iron and
steel a decrease of the modulus at an increasing rate for
a temperature range up to 300°C.
His value of the modulus
for steel is
18.47. Heating from 25°C. to 300°C. produces
a decrease of 6.3 per cent.
Por iron the modulus was
21.43, the decrease, 12 per cent,
j
f
2
Walker
found the modulus of a piece of soft iron wire
to be
18.22 at 17.5 °C. and with
ordinary heating
h x e x xup to 129°C.^a uniform decrease amounting
to 3.7 per cent,
*
The results of Pisati
, Gray, Blyth, and Dunlop
,
5 6
Kohlrauoch and Loomis
, and Sfotte
upon the torsion
modulus of iron are of interest showing as they do a decrease
of the modulus with increase of temperature, the decrease
in nearly every case becoming more rapid at the higher temperatures.
4 lluovo Cimento, (3), 4, 152; 1878.
2 Proc. Roy. Soc. Edin., 28, 652; 1907.
3 loc. cit.
4
loc. cit.
3 Annal. d. Phys., 141, 481; 1871.
0
