A Method to Calculate the True Stress and True Strain for Tensile Test of Plastic

Transcription

1 Ke Egieerig Materials Vols (4) pp olie at 4 Tras Tech Publicatios, Switzerlad Method to Calculate the True Stress ad True Strai for Tesile Test of Plastic X. W. Du, a,g. Su, b,c. Nie, c Departet of Egieerig Mechaics,Shaghai Jiao Tog Uiversit, Shaghai, 3, Chia GPDaiko Corp. Shaghai Office, Shaghai, 336, Chia a duxw@sjtu.edu.c, b gjsu@sjtu.edu.c, c cie@gpdaiko.co.jp Kewords: egieerig plastic; ulti-phases asseblig ethod; eckig; true stress-true strai curve; load-elogatio curve. bstract. The ulti-phases asseblig ethod is proposed to calculate the true stress true strai relatio for plastic i stadard tesile test. B this ethod, the easured load-elogatio curve is divided ito three phases. ccordig to characteristic of each phase, the true stress-true strai relatio is obtaied b distict aalzig ethod. The true stress-true strai curve for the whole test is extracted fro the data of load-elogatio record ad assebled. esults fro experiets of four plastics verif the efficiec of this ethod. Itroductio I structure aalsis, the selectio of costitutive equatio of aterials is iportat. Oe-diesioal costitutive relatio is ofte deduced fro the stress-strai curve of tesile test of the aterial. dvaced egieerig aalsis requires high accurac to predict the respose of structures or achie parts. No-liear fiite eleet theor ad ethods fid ore ad ore applicatio i egieerig aalsis. Coercial fiite eleet softwares such as BQUS, MC, NSYS, are provided with various oliear aterial odels to ru with directl iputted stress-strai curve. Curret data treatig for stadard tesile tests are cofied to calculatios of coo egieerig costats such as Youg s odulus, ield stress, ield strai. There is lackig i ethods to calculate stress-strai relatio i ieldig ad eckig stages. The desig of experiet for the easureet of oliear aterial properties ad appropriate data treatig ethods has attracted cotiual attetio i the literature. Zhag[] et al uses coercial fiite eleet software BQUS to siulate the tesile test of plastic specie which cotais a predisposed defect. The research shows that the true stress-true strai curve of the aterial ca be obtaied b the easured load-thickess curve at the defect of the rectagular specie. I efs [,3] a procedure is proposed b Zhag et al to easure the stress-strai relatio for species with rectagular cross-sectio of aisotropic aterial. I the tests ot ol the variatio of load ad thickess, but also the variatio of width of the specie ust be recorded. I ef.[4], Zhag et al propose a testig ethod usig the specie with a otch. This ethod applies to species ade of various aterials. The otch akes sure that eckig occurs at the aterial iterested. The testig ethod also applies to specie ade of sole aterial, however, with less accurac. Brueig[5] proposes a sei-iplicit uerical ethod to siulate the eckig pheoeo of tesile test. He poited out that the aterial properties tested is affected b the predisposed defect. I order to get the stress-strai curve uder various strai rates, it is reported i Proceedigs of Japaese utoobile ssociatio that the tesile tests for plastics with speeds fro/s to /s were coducted. The stadard JIS species were used i the tests. dese grid is disposed o the surface of the specie. The deforatio of the grid is photographed with high-speed caera. The variatio of the cross-sectio i eckig is deduced fro the ll rights reserved. No part of the cotets of this paper a be reproduced or trasitted i a for or b a eas without the writte perissio of the publisher: Tras Tech Publicatios Ltd, Switzerlad, (ID: //4,::5)

2 78 dvaces i Egieerig Plasticit ad Its pplicatios deforatio of the grid b eas of coputer software to calculate the true stress ad true strai. Copariso of the siulatio o the etire structure with test data shows that grid-photographig ethod ca reach a relative high accurac. Luo ad Zhao[6] propose a trucated coe odel for siulatio of eckig of etal tesile test uder high teperature. The true stress ad true strai i eckig ca be calculated b eas of the load ad elogatio relatio. The odel ca siulate the eckig process ad the result copares well with test whe strai is less tha.3. Suig up the above researches, the calculatio ethod for siulatig the stress-strai curve of the plastic ieldig stage as well as the eckig stage is still lackig. I the testig stadards (such as JIS,ISO,STM), the data treatet is cofied to coo egieerig costats such as elastic odulus, ield stress, ield strai etc.. Curretl the predisposed defect ethod ad the grid photographig ethod are frequet used. The predisposed defect ethod has ore applicatios. However, the fabricatio of specie is ot eas. High accurac testig equipet is eeded ad the cost of testig is high. The grid photographig ethod has relativel high accurac i dealig with the stress-strai relatio i large strai; however, the accurac for sall strai is low. dditioal grid o the specie is eeded. The coputig cost for graph treatet is high. Based o the test data of plastics, the true stress-true strai relatio of the etire testig process is studied i this work. ovel test data treatig ethod, the ulti-phases asseblig ethod is proposed. Multi-phases sseblig Method for Calculatio of True Stress-True Strai Curve Cocepts of Neckig. Soe cocepts cocerig eckig are explaied as follows: Neckig:Durig plastic large deforatio the local strai at soe locatio of the specie icreases uch faster tha the rest, because at this locatio the cross-sectio of the specie is slightl saller tha the rest, so that the stress is higher. The local stress reaches the ieldig poit first ad eckig starts. Neckig evolutio:fter eckig iitiates, the local cross-sectioal area becoes saller ad saller, the cotractive deforatio cocetrates at the locatio of eckig. The tesile load decreases with elogatio of the specie. Neckig diffusio:fter eckig iitiates, the cross-sectioal area at the locatio of eckig does ot reduce further while the eighborig cross-sectio reduces. The eckig exteds alog the axial directio to the etire legth of the testig regio of the specie. The tesile load does ot chage with the elogatio of the specie. The Three Phases of the Load-Elogatio Curve of Tesile Test. The tesile test curves for the four tpical egieerig plastics are show i igs.-4. ccordig to the load-elogatio curve ad the status of eckig evolutio or eckig diffusio, the curve is divided ito three phases. Phase : the load asceds. This stage eds whe eckig occurs. Phase : the load desceds. This is the stage of eckig evolutio, startig as eckig starts ad edig as aterial fractures or eckig diffusig starts. Phase 3: The load is stable. It is the stage whe eckig diffusio occurs. The tesile test curve of plastics a be siplified as a copositio of the three phases. s show i igs.-4, the curve of the first aterial cotais phases ad 3; the secod aterial has phases ad ; the third aterial has all three phases ad the fourth aterial has ol phase. ccordig to characteristic of each phase, distict ethod is used to calculate the true stress ad true strai. iall the are assebled to for the etire stress-strai curve. This is aed as ulti-phases asseblig ethod. Judget of Neckig ad the Neckig Critical Curve. Sice the load desceds whe eckig iitiates ad develops, the start, evolutio or diffusio could be deteried b the treds of the load-elogatio relatio. The followig coditio is satisfied whe eckig starts: d( ) <,

3 Ke Egieerig Materials Vols specie specie specie specie specie specie3 Tesile Load(KN).5..5 Tesile Load(KN) ig. Load-elogatio curve of the first aterial Elogat io() ig. Load-elogatio curve of the secod aterial Tesile Load(KN) specie specie specie3 Tesile Load(KN) specie specie specie ig.3 Load-elogatio curve of the third aterial ig. 4 Load-elogatio curve of the fourth aterial where is the tesile force, i.e. the load; is the true strai. Whe eckig diffuses, the followig coditio holds: d( ) =. Sice dσ d dσ dl dσ dσ d( ) = σ ( + ) = σ ( ) = σ ( d) = σ d( ) σ σ L σ σ d where σ is the true stress, is the cross-sectio area ad L is the arked legth, the followig equatio should be satisfied whe eckig diffuses: dσ σ d =. () The solutio of Eq. is: σ σ e ( ) =. () Eq. is the eckig critical equatio, which is adopted as stress-strai curve of phase 3; σ ad are iitial values of the phase 3. True Stress-True Strai elatio for Phase. Phase is the stage of uifor tesile

4 8 dvaces i Egieerig Plasticit ad Its pplicatios deforatio with stable aterial properties. The data before the load desceds or keeps costat is obtaied fro the load-elogatio curve. The strai ad stress are expressed as L dl = = l λ (3) L L ad σ =. (4) or icopressible aterial,we have: σ = λ (5) where λ = ( L + δ ) / L ad δ is the elogatio of the arked legth. The true stress ad true strai result fro Eqs.3 ad 5. egressig the stress-strai curve with several straight-lie segets, the axiu slope of these straight-lie segets is adopted as elastic odulus. The ield stress σ, the ield strai, ad the stress ad strai at the ed poit of phase ca also be obtaied. fter deteriatio of elastic odulus ad ield stress, the true stress give b Eq. 5 eeds to be aeded, sice icopressible assuptio is used i this equatio. I elastic stage, plastic is ot icopressible so that Eq.4 applies. ssue that strai is coposed of elastic strai e ad plastic strai p : = e + p. Whe, we have vl λ = ( v ) ( v ) e = (6) v λ where ν is Poisso s ratio, ad σ v = = λ. (7) Whe >, we have v p ( v + p ) = e e = e (8) ad v e + p σ = =. (9) Eqs.7 ad 9 are used to re-calculate the true stress; the stress-strai curve for the phase ca be obtaied. s for plastics with p >> e ad ν is rather large (ν =.4, for istace), the stress after aedet differs ot uch fro that before aedet. True Stress-True Strai elatio for Phase. Phase occurs after phase, followed either b phase 3, or directl b fracture. Phase is the stage whe eckig develops. Large deforatio takes place i the eckig area ad elogatio is ot uifor alog the specie axis. Calculatio of true strai a ot use Eq.3, which is based o uifor elogatio assuptio. ather the followig expressio applies: = l () where is the area of cross-sectio at the locatio of eckig. The chage of cross-sectio area at the locatio of eckig is difficult to easure directl i tesile test. The true strai ad true stress caot be calculated directl. The tred of the load is cosidered ad soe assuptios are ade to obtai the stress-strai curve for phase.

5 Ke Egieerig Materials Vols Multi straight-lie segets are used to fit the stress-strai curve. The strai rage of each seget is assued as k, where k is a coefficiet ad is the strai at the ed of phase, i.e., the strai whe load reaches the axiu. The ethod is based o observatio that the status of stress ad strai at the start poit of eckig affects the aterial properties sigificatl. k is geerall take as.5, or less if the load desceds slowl. The slope of the straight-lie seget assues to be the iitial stress of the seget ultiplied b a coefficiet k, which is to be deteried. It is assued that the start poit ad the ed poit of the i-th seget of the phase are i ad i+ respectivel. We have i = l, i i i e =, = σ = σ, i i i i i e + k + k i + = σ i+ = ( σ i + kσ i ) = σ i = i. () i+ i + i e e e e e I the above equatios, = k, i ad i + are loads at poit i ad poit i + respectivel. i is the cross-sectio area at poit i. ssuig that the strai of the ed poit of phase is, we have = + + () where is a iteger, ca be deteried. ro Eqs. ad, we have is the strai rage of the last straight-lie seget. Give, ad + kk + k = k e e (3) where ad are loads at the start poit ad ed poit of phase, respectivel; k is solved fro Eq.3 ad the stress-strai curve for phase is deteried. Sice the start poit of phase is the ed poit of phase, the curve is cotiuous at the coectio. The value of k is chose epiricall,ragig fro. to.5 depedig upo the speed of descedig of the load. iterative procedure is used to ake the siulatio approach closel to the test data. s show i ig.5 the true stress-true strai curve of the secod aterial is obtaied with the above-etioed ethod. Iputtig the costitutive relatio ito the fiite eleet software for the tesile specie, the siulated load-elogatio curve is copared with the test curve i ig.6. It is foud that the EM curve coicides closel with the test oe. Stress-strai relatio for the phase 3. I phase 3, the load keeps costat, d =. The stress-strai curve of this phase is called eckig critical curve. I Eq.5, σ ad are the stress ad strai of the ed poit of previous phase so that the curve is cotiuous at the coectio. The first aterial has phase ad phase 3 ol. The true stress-true strai relatio is obtaied fro Eq.5 ad depicted i ig.7. Iputtig the costitutive relatio ito the fiite eleet software for the tesile specie, the siulated load-elogatio curve is copared with the experietal curve i ig.8. It is see that the EM curve coicides closel with the test oe. Coclusio Based o the load-elogatio test curves of several egieerig plastics, the ulti-phases asseblig ethod for true stress-true strai curve of plastic is proposed. I this ethod, the load-elogatio curve of the specie is divided ito three phases accordig to the treds of loadig

6 8 dvaces i Egieerig Plasticit ad Its pplicatios ad eckig evolutio or eckig diffusio. The true stress-true strai curve is obtaied for each phase with distict techique. The are assebled to get a cotiuous, sooth stress-strai curve for the whole test. Copariso of the results fro EM aalsis with tests verifies the applicabilit of this ethod Tesile Load(N) Experiet EM esult ig.5 True stress-true strai curve of the secod aterial ig. 6 Load-elogatio curve of the secod aterial 5 Experiet EM esult Tesile Load(N) ig.7 True stress-true strai curve of the first aterial ig. 8 Load-elogatio curve of the first aterial efereces [] Z. L. Zhag, M. Hauge, J. Odegard ad C. Thaulow. It. J. Solids Struct. Vol. 36(997), p.3497 [] Z. L. Zhag, J. Odegard, O.P. Sovik, C. Thaulow. It. J. Solids Struct. Vol. 38(), p [3] Z. L. Zhag, J. Odegard, O.P. Sovik. Coput. Mater. Sci., Vol. (), p.77 [4] Z. L. Zhag, M. Hauge, C. Thaulow, J. Odegard. Eg. ract. Mech., Vol. 69(), p.353 [5] M. Brueig. iite Ele. al. Des., Vol. 8(998), p.33 [6] H. W. Luo ad P. Zhao, J. Steel esearch, Vol. (998), p.6 (i Chiese).