ME 354, MECHANICS OF MATEIALS LABOATOY MECHANICAL POPETIES AND PEFOMANCE OF MATEIALS: TOSION TESTING* MGJ/08 Feb 1999 PUPOSE The pupose of this execise is to obtai a umbe of expeimetal esults impotat fo the chaacteizatio of mateials. I paticula, the esults fom the tosio test will be compae to the esults of the egieeig tesile test fo a paticula alloy usig the effective stess-effective stai cocept. EQUIPMENT Costat-iamete gage sectio tosio specime of 6061-T6 alumium Tosio test machie with gips, toptomete, a foce seso. POCEDUE Measue the iamete (D) of the gage sectio fo each test specime to 0.0 mm. Istall the bottom e of the tosio test specime i the lowe gip of the test machie. otate the leve am as fa to the ight as possible. (Note: uscew the hoizotal theae ive o as much as possible). otate the top gip as fa as possible i the iectio ecessay to emove the 'slack' fom the eactio cables a istall the top e of the tosio test specime i the top gip of the test machie. Zeo the output of the foce seso. Use the theae ive o to apply toque to the base of the test specime a eco the applie toque, T, vesus agula otatio,, at icemets util 30 of otatio. emove the hoizotal theae ive o a fi the toque afte 90 a 360 of otatio, beig caeful ot to allow elastic uloaig. Afte 360 of otatio, uloa a emove the specime. Measue the gage legth L (gip to gip legth) of the istalle specime to 0.1 mm. ESULTS Plot measue toque, T, vesus agula isplacemet pe uit legth, l. Usig liea egessio, fit the cuve to 30 of elative otatio. (It is assume that T is popotioal to fom 0 to 30 ). (Note that must be i aias, i.e. π aias 180 ). Calculate the shea moulus, G, fom the liea potio of the T- usig liea egessio to fi T/ fom 0 to 30. Compae this value of G to the shea E moulus etemie fom the tesile test esults (i.e. G ) usig ν0.345 fo (1+ ν) alumium. Usig the stegth coefficiet coefficiet, K (o H), a the stai haeig expoet,, etemie fom the tesile test fo the appoximate costitutive elatio σ Kε Hε, itegate the peicte shea stess,, vesus aial istace,, to obtai the peicte toque, T, afte 90 a afte 360 of otatio. Compae these values of T to those measue expeimetally. (Note that must be i aias fo the calculatios, i.e. π aias 180 ). Use the attache "cook book" metho to facilitate you wok. O the same gaph, plot shea stess,, a egieeig shea stai, as fuctios of aial istace,, at 30 of otatio. Costuct simila plots a vesus fo 90 a 360 of otatio. (Note that must be i aias fo the calculatios, i.e. π aias 180 ).
LABOATOY EPOT 1. As a miimum iclue the followig ifomatio i the laboatoy epot. a. aw ata (type i tabula fom) b. Two values fo the shea moulus, G (tesio a tosio) c. Two values of the tosioal yiel stess, o (tesio a tosio). "" a "K" fom the tesio test (use these i the calculatios) e. Total toque as equie i the table: Agle of otatio... 90 360 Peicte Toque ( ) Measue Toque ( ). % Diffeece... f. Plot of Toque vs. Agula isplacemet pe uit legth (T vs. ') g. Oe gaph each of a as fuctios of aial istace,, fo 30, 90, a 360 ( plots o each gaph fo a total of 3 gaphs) Shea stess, 0 y K /L f( ) G aial istace, h. Discuss compaisos of basic mechaical popeties as etemie fom tesio a tosio tests. Compae esults of these tests fo each alloys to 'book' values fom such souces as the ASM Metals Habook. Commet o ay iffeeces. Compae the shapes of the stess vs. aial istace cuves a the magitues of the plastic a elastic toques.. Iclue the followig ifomatio i the appeix of the laboatoy epot. THIS MAY NOT BE ALL THAT IS NECESSAY (i.e., o't limit youself to this list.) a. Oigial ata sheets a/o pitouts b. All suppotig calculatios. Iclue sample calculatios if usig a spea sheet pogam. c. All "cookbook" calculatios fom the Tosio Test Solutio Path. * EFEENCES Aual Book o ASTM Staas, Ameica Society fo Testig a Mateials, Vol. 3.01 E143 Staa Test Methos fo Shea Moulus at oom Tempeatue. ME354 NOTES o Tosio Testig Shea stai,
ME 354, MECHANICS OF MATEIALS LABOATOY MECHANICAL POPETIES AND PEFOMANCE OF MATEIALS: TOSION TESTING* MGJ/08 Feb 1999 DATA SHEET NAME DATE LABOATOY PATNE NAMES EQUIPMENT IDENTIFICATION Note: Be sue to eco uits fo each quatity. Specime measumets D ( ) L ( ) 17.61 mm 84.95 mm J ( ) Fo Agle(egees) Foce ( ) Foce 0 CoupleT(Fo) x Dg 4 5.3 mm Gip Diamete, Dg 6 8 10 1 14 16 18 0 4 6 8 30 90 360
ME 354, MECHANICS OF MATEIALS LABOATOY MECHANICAL POPETIES AND PEFOMANCE OF MATEIALS: TOSION TESTING* TOSION TEST Tosio Test Solutio Path The iitial set of calculatios has iput paametes obtaie oly fom the tosio test. The esults of these calculatios will late be compae to esults calculate with ifomatio obtaie fom the tesio test. 1. eco the tosio specime iamete (D) a the legth of the gippe sectio of the tosio specime, L. Calculate the pola momet of ietia fo a soli o, J πd4 3. D L mm mm J mm 4. Fom the measue toque, T, vesus agula otatio,, ata poits, plot T vesus elative agula eflectio, l betwee two coss sectios (i.e. L ). Obtai the "best fit" of the liea potio of the T vesus ata usig liea egessio. (It is assume that T is popotioal to fom 0 to 30 ). Detemie the slope, T/ fom 0 to 30. (Note that must be i aias fo the calculatios, i.e. π aias 180 ). T/ () / (a/mm) 3. The shea moulus, G, fom the tosio test ca ow be calculate fom the elatio: G T (0 30 ) 1 (0 30 ) J (N/mm MPa) 4. Fially, eco the measue toques a calculate fo 90 a 360. l (Note that must be i aias fo the calculatios, i.e. π aias 180 ). T 90 90 T 360 360 /mm /mm
TENSION TEST This set of calculatios has iput paametes obtaie oly fom the tesio test. The esults of these calculatios will late be compae to esults calculate with ifomatio obtaie fom the tosio test. 1. eco the uiaxial elastic moulus, E, uiaxial yiel stess, σ o, the stai haeig coefficiet, K, a the stai haeig expoet,, etemie fom the tesio test. E N/mm σ o N/mm K N/mm. Calculate the value of the shea moulus fom the esults of the tesio test: G E (1+ ν) usig ν0.345 fo alumium. 3. Usig the effective stess cocept, calculate the shea stegth iicate by the tesio test ata such that: [( ) + ( σ y σ z ) + ( σ z σ ) + 6 + x ( + )] xy yz xz σ _ 1 σ x σ y 1 a settig σ σ o a solvig fo xy stess equal to zeo. y (yiel stess i shea) with all othe y N/mm
EVALUATION OF TOSION TEST ESULTS FO YIELDING This set of calculatios has iput paametes obtaie oly fom the tosio a tesio tests. The esults of these calculatios ae use to evaluate the shea stesses a shea stais acoss the aius of the tosio specime as it yiels. 1. Fi the aius of the tosio specime at yielig, y fo 90 (Note that must be i aias fo the calculatios, i.e. π aias 180 ). y y G whee max ( 90 ) max y 90 mm. Withi the elastic omai, the shea stess is a liea fuctio of aial istace,, such that ( ) y y 3. The shea stess as a fuctio of aial istace,, ca ow be multiplie by iffeetial aea elemet, π a a momet am,, a itegate to fi the toque ove the elastic omai. (i.e. M 0 ). T e y o y ( )π y 4. I the plastic omai, oly the shea stai emais a liea fuctio of aial istace,. Theefoe, it is avatageous to chage the itegatio vaiable to. I oe to accomplish this vaiable chage, the shea stess,, momet am,, a iffeetial aea of itegatio, π, must be expesse as fuctio of. I the tesio test the uiaxial stess, σ, was expesse as a fuctio of uiaxial stai, ε, though the steght coefficiet, K (o H), stai haeig expoet,, such that: σ Kε Hε Sice the uiaxial stess is ietical to the effective stess, a the uiaxial stai is ietical to the effective stai, the equatio elatig effective stess to effective stai woul be exactly the same. σ Kε Hε Whe the effective stess a effective stai ae evaluate fo the case of pue tosio, the shea stess ca be fou as a fuctio of the shea stai.
( ) 1 3 K 3 Sice it is also tue that a, sice is a costat. Substitutig these elatios ito the basic tosio itegal yiels: T e T p y o y ( )π fo the elastic toque y max 1 3 K 3 π fo the plastic toque. y Note that the limits of itegatio ae y y G a max. (Note that must be i aias fo the calculatios, i.e. π aias 180 ). The total toque, T, is fou as the sum of the elastic a plastic toques such that: T T e +T p This toque value is the compae to the value measue i the tosio test. Fo 90, calculate toques ae: T e T p T Fo 90, measue toque is: T 90
5. Steps 1 to 4 ae epeate fo 360 (Note that must be i aias fo the calculatios, i.e. π aias 180 ). Fo 360, calculate toques ae: T e T p T Fo 360, measue toque is: T 360 6. Fially, plot a as fuctios of afte 30 fo elative otatios of 90 a 360. (Note that must be i aias fo the calculatios, i.e. π aias 180 ).
ME 354, MECHANICS OF MATEIALS LABOATOY MECHANICAL POPETIES AND PEFOMANCE OF MATEIALS: TOSION TESTING* STESSES IN THE ELASTIC ANGE NOTES o Tosio Testig I the elastic age, stesses i the shaft will emai less tha the popotioal limit a less tha the elastic limit as well. Fo this case Hooke s law will apply a thee will be o pemaet efomatio. Hooke s Law fo shea stess is as follows: G G Moulus of igiity (shea moulus) Shea Stess Shea Stai max Figue 1. Distibutio of Stess i the Elastic age. The elemetay foces exete o ay coss sectio of the shaft must be equal to the magitue T of the toque exete o the shaft: o ( A) T whee A π max G G max max
T A max A J 1 π4 T max J A max T J T J The last two equatios ae kow as the elastic tosio fomulas. ANGLE OF TWIST IN THE ELASTIC ANGE Fo this sectio the etie shaft will agai be assume to be i the elastic age. Theefoe Hooke s Law applies. max φ T L Figue. Demostatio of the Agle of Stess a the Sheaig Stai. max max max G ø L ø TL JG T JG
The agle of twist, Ø, is expesse i aias. The agle of twist is popotioal to the toque T applie to the shaft. The above equatio povies a coveiet metho fo etemiig the moulus of igiity, G. Toques of iceasig magitue T ae applie to the specime, a the coespoig values of the agle of twist i a legth L of the specime ae ecoe. As log as the yiel stess of the mateial is ot exceee, the poits obtaie by plottig Ø agaist T will fall o a staight lie. The slope of the lie epesets the quatity JG/L fom which the moulus of igiity, G, may be compute. T, SlopeJG/L ø,a PLASTIC DEFOMATIONS IN CICULA SHAFTS If the yiel stegth is exceee i some potio of the shaft the elatios iscusse i the ealie sectios cease to be vali. The pupose of this sectio is to evelop a moe geeal metho fo etemiig the istibutio of stesses i the soli cicula shaft, a fo computig the toque equie to pouce a give foce. max O Figue 3. Stess Distibutio i a Shaft fo Plastic Defomatio. As the toque iceases, max evetually eaches the sheaig yiel stess, y, of the mateial. Solvig fo the coespoig value of T, we obtai the value of T y at the oset of yiel: T Y J Y T y is efee to as the maximum elastic toque, sice it is the geatest toque fo which efomatio emais fully elastic. ecallig that, fo a soli cicula shaft, J/1/ (π 3 ) we have:
T Y 1 π 3 Y The Y ca be fou usig the ata fom the tesio test a the iea of effective stess. Usig the Distoitoal Eegy (vo Mises) citeio a the yiel stess fom the tesile test laboatoy Y ca be etemie. [( ) + ( σ y σ z ) + ( σ z σ ) + 6 + x ( + )] xy yz xz σ _ 1 σ x σ y _ σ 1 6 [ xy ] 1 3 xy 1 σ _ σ 0 yiel stess fom tesio test xy Y σ_ 3 0.577 σ_. y y G y,measue G fom the elastic pat of the tosio test. y l, whee l π L π (at 90 ) a L (at 360 ) fo this lab. The total toque is a fuctio of the toque i the elastic age a the toque i the plastic age. T total T elastic +T plastic y T total π + 0 ( ) y fo the elastic age ( ) π y y () Telas π 3 y y 0 y
fo the plastic age σ ε ε π ( )( ) + K K K K T K T cost T plastic plastic y 3 3 3 3 3 3 0 577 3 3 3,,., l l plastic plastic G plastic K T K L y y ( ) 3 3 1 3 3 3 π π l l l max, O