Pressure-sensitivity Effects on Toughness Measurements of Compact Tension Specimens for Strain-hardening Solids

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1 Amerin Journl of Alied Sienes (9): , 5 ISSN Siene ublitions ressure-sensitivity Effets on Toughness Mesurements of Comt Tension Seimens for Strin-hrdening Solids Abdulhmid A. Al-Abduljbbr Dertment of Mehnil Tehnology Riydh College of Tehnology, Riydh, Sudi Arbi Abstrt: The th-indeendent J-integrl is used s frture redition riterion for loding beyond the elsti limit s liner elsti frture mehnis (LEFM) n not be lied. While J ws originlly defined from energy ersetive, it ws demonstrted tht it ould be inferred from loddislement digrms. Therefore, the omt tension (CT) seimen testing is used to omute J for mterils. In this work, the -ftor, n imortnt rmeter in the omuttion of the J-integrl, is investigted for omt tension seimen for mterils with ressure sensitive yielding. This is hieved by using lower bound roh to derive the rorite exression for from the test geometry nd mteril roerties. The seimen is onsidered t fully lsti loding where it is t the stte of ollse. The effet of ressure sensitivity is ounted for by using Druker-rger yield riterion for solid mterils. Sine CT testing is usully onduted on metlli mterils, strinhrdening behvior of the mteril is inororted in this nlysis. This is done by ssuming simle liner hrdening urve of the mteril. Numeril results omuted for different ses show tht s the mteril strin hrdening inreses. Key words: J-Integrl, lsti yielding, toughness, omt tension testing, ressure-sensitive yielding. INTRODUCTION The Liner Elsti Frture Mehnis (LEFM) theory is useful for estimtion of frture behvior of mterils in the elsti rnge. However, s lrge inelsti strin behvior is enountered, the LEFM is not dequte to desribe the toughness nd frture behvior. Attemts to model suh behvior go s erly s Rie [1,] who introdued th-indeendent line integrl round the ti of the frture noth to desribe the elsti-lsti frture behvior. Lter on, Begley nd Lndes [3,4] demonstrted tht it is ossible to estimte the J integrl from the lod-dislement urves of vrious test seimen geometries. Bui et l. [5] nd Rie et l. [6] roosed ossible estimte methods of J for omt tension seimens mong other seimen geometries Lter on, Merkle nd Corten [7] imroved the estimtion of J by onsidering the effets of the bending moment resulting from the lied lod to the remining ligment of the omt tension seimen. They used lower-bound roh to derive the ftor for the J integrl estimtion. The ftor roh hs been used to estimte the J integrl from the re under the lod-dislement (- ) urve of test seimen or struture The Von-Mises yield riterion ssumes n idel mteril yield tht only deends on the equivlent yield stress. While suh ssumtion is etble for mny lsses of mterils eseilly metls, it is not the se for other lsses of mterils suh s olymers, hsetrnsformtion ermis, st irons nd even some lsses of steels. In suh mterils, yielding behvior is ffeted by the stte of hydrostti stress nd the yield enveloe on the equivlent stress versus the hydrostti stress lne ( e m ) is not horizontl, but rther hving negtive sloe. Suh henomenon, whih is lso known s ressure-sensitivity, is lso demonstrted by the vrition between the vlues of yield stress in tension nd omression The effet of ressure sensitivity on yield n be ounted for using the Druker-rger yield riterion, where, generlized equivlent yield stress is exressed s funtion of the effetive yield stress nd the hydrostti stress using ressure sensitivity ftor. The ressure sensitivity ftor is mteril mehnil roerty deendent on the differene between yield stresses in tension nd omression. Reorted vlues of ressure sensitivity show wide rnge for different engineering mterils. For exmle, for olymers it ws reorted by Sternstein nd Onghin [8], Sitzig nd Rihmond [9], nd Kinloh nd Young [1], tht the ressure sensitivity ftor, µ, hs rnge of.1 ~.5. In st irons it hs vlue round., ording to Dong et l. [11]. It is lso Corresondene Address: Abdulhmid A. Al-Abduljbbr, O Box 19 Riydh 11455, Sudi Arbi, Tel/Fx , 19

2 Am. J. Alied Si., (9):19-195, 5 resent, to lesser degree, in some steels s shown by Rihmond nd Sitzig [1] with µ hs quite low vlue of.64. hse trnsformtion in some ermi mterils is highly deendent on ressure sensitivity s reorted by exerimentl investigtions of Yu nd Shetty [13] nd Chen [14] where µ hs lrge vlue u to.93 for zironi-ontining hse-trnsformtion ermis. Al-Abduljbbr nd n [15] inororted the mteril ressure-sensitivity, into the lultion of to rodue better estimte of the toughness roerty, using the sme roh nd ssumtion of rigid erfetly-lsti mteril behvior used in revious works. Sine most metls exhibit strin hrdening in vrious degrees, inororting the effet of strin hrdening on the estimte of frture toughness s desribed by J will indeed result in n imrovement of this mesure of mteril frture behvior. Al- Abduljbbr [16] onsidered strin hrdening effet on J for norml solids without ressure sensitivity In this work, roedure similr to those doted in revious works in order to rodue exressions for the ftor of the J integrl for mteril with ressure sensitivity nd strin hrdening. The ressure sensitivity is modeled using Druker-rger yield riterion for solid mterils, while strin hrdening is deited by ssuming simle liner hrdening model for the mteril. nd the effetive omressive stress ( ) n be obtined s follows: 1 t 1+ µ 1 1 µ where, µ µ / ; () ; (3) 3. To desribe the mteril stress strin reltion, we dot the Ludwik's exression [19] + n Hε ; (4) where, is the initil yield stress, H is the hrdening onstnt, ε is the strin nd n is the hrdening exonent. The exression is further simlified by ssuming liner hrdening se with n 1. This reltion n be used to desribe old worked metls suh s high rbon nd lloy steels where there is high mount of restrining [19, ]. LASTIC LIMIT ANALYSIS The Druker-rger yield riterion [17,18] is generliztion of the Coulomb rule is solid mehnis where the sher stress required for simle sli is linerly deendent uon the norml ressure on the sli surfe. In the D- riterion, generlized yield funtion is roosed s ombintion of the effetive yield stress ( e ) nd the men stress ( m ) by using the ressure sensitivity ftor (µ) s follows: f ( ~ ) + 3µ e m. (1) In eqution (1), e (S ij S ij ) 1/, where S ij re the devitori stress omonents defined by S ij ij - m ij. Here, ij reresent stress omonents nd ij is the Kroneker delt. The summtion onvention is doted for reeted indies. The men stress ( m ) is defined by the reltion m kk /3. The stress quntity is the generlized effetive tensile stress t the onset of yield. Considering the unixil tensile nd omressive loding onditions nd lying the yield riterion in eqution (1) vlues for the effetive tensile stress ( t ) Fig. 1: Stress-strin for different ses: () n idelized erfetly-lsti mteril, (b) n idelized linerly hrdening mteril nd () tyil dutile metl. Shown in Fig. 1 re three urves desribing stressstrin behvior. The first one () is for n idelized rigid erfetly-lsti mteril, where the mteril exhibits no elsti strin nd the stress remins t the onstnt yield vlue of throughout the deformtion nd lsti strining. Suh behvior is the one ssumed in revious works for estimtion of the J integrl [15]. The seond urve is for rigid linerly hrdening mteril s desribed by eqution (4) with n 1. The third urve is for tyil dutile mteril. For metls, the elsti strin is usully very smll when 191

3 Am. J. Alied Si., (9):19-195, 5 omred with the lsti strin, eseilly t high loding levels, s is onsidered here. This ft ermits the ssumtion of negleting the ontribution of the elsti strin in this nlysis. Next, we onsider the omt tension (CT) seimen of rigid strin-hrdening mteril s shown in Fig. [1]. Due to the symmetry of the seimen geometry nd loding ondition long the horizontl xis, only the uer hlf of the seimen is nlyzed here. The totl width of the seimen is W, the rk length is, the remining ligment width is b nd the thikness of the seimen (through the er) is unity. The fully lsti lod vlue lied s shown is. Also shown in the figure, is the stress distribution in the remining ligment, where the ortion ner the ti is under tensile stress resulting from the tension nd bending moment of the lod nd the other ortion is under omressive loding. At the oint of stress reversl, both omressive nd tensile ortions strt with initil yield stresses with stress levels inresing linerly u to their mximum vlues t edges. On the omressive side, the strting stress level is nd the stress t the outer edge of the seimen is A nd on the tensile side, the strting stress level is t nd the stress t the inner edge is B. mteril. Assuming tht the sloe of the liner hrdening urve on the stress digrm is roortionl to γ, the vlues of the two end stresses re determined s [ 1+ γ ] A [ 1+ γ ] B t ; (5). (6) For the lsti limit lod, we onsider the xil fore nd the bending moment equilibrium due to the in-lne fore nd stresses. The fore equilibrium in the vertil diretion requires tht A + B + t + (1 α ) (1 + α). (7) Hene, eqution (7) is used with equtions (5) nd (6) to get the lsti limit lod s follows ( + λ)( α µ ) 1 µ. (8) Further, the moment equilibrium round stress reversl oint requires tht the internl resisting moment blnes the moment generted by the lied lod M. The internl resisting moment is omosed of two ortions: M t due to the tensile stress nd M due to the omressive stress. This rodues: M t + M M. (9) Fig. : Geometry of the uer hlf of the omt tension seimen nd the stress digrm for the remining ligment. Unit thikness is ssumed. The dimensionless rmeter, α, is introdued s n inditor of the devition of the neutrl xis from the enter of the remining ligment width b, where the tensile side length is (1+α), beuse it hs lower stress levels nd the omressive side length is (1-α). Within the ftor α lies the effet of ressure sensitivity of the mteril resulting from the vrition of yield stresses. Another dimensionless rmeter, γ, is introdued to ount for the hrdening effet in the 19 The vlues of these moment omonents re determined, with the hel of equtions (-3, 5-6), nd Fig., s follows: 1 γ (1 + α) M t µ 1 γ (1 α) M µ M [ + (1 + α) ] ; (1) ; (11). (1) Substituting vlues for moments from equtions (1-1) into eqution (9) nd lso using eqution (8) for the limit lod; qudrti exression for the rmeter α is obtined in the form Aα + Bα + C ; (13)

4 Am. J. Alied Si., (9):19-195, 5 where, A ( λ 1), (14) B + 1 µ λ + µ nd C + 1λµ 1, (15). (16) In equtions (14-16), λ is dimensionless rmeter whih is defined by: + γ λ γ. (17) From the bove reltions, α n be solved for esily using the stndrd qudrti eqution solution. Relling the work for ressure sensitive rigid erfetly-lsti mterils, resented by Al- Abduljbbr nd n [15], we n see tht one we set the strin hrdening oeffiient to zero, the onstnts A, B nd C of eqution (13) bove revert to the result given in eqution no. 14 in tht work, thereby onfirming the vlidity of reltions develoed here. α η 1+ α µ ; () Differentiting eqution (13) with reset to nd using the reltion W-; we get n exression for the lst term in eqution () λ + α α µ α ( λ 1) + λ + 1 µ ( λ ). (1) Eqution (13) is gin used to obtin n exression for λ (/+1) 1+ µ α( λ ) α ( λ 1) λ + 1 α µ. () : Finlly, from equtions (-), we n determine + µ ( λ ) µ λ + (1 µ ) λα η 1+ µ ( λ ) ( λ 1)(µ α α ), (3) If the mteril hrdening is removed by setting the rmeter γ to zero, so tht λ, then eqution (3) redues to THE J INTEGRAL ANALYSIS The exression for the J integrl for mterils with negligible elsti strin n be doted from the work of Merkle nd Corten [7] s follows J η b d * η + b d ; (18) Where, nd * re rmeters deendent on the seimen geometry nd loding onditions. The first nd seond integrls of eqution (18) re the lsti work nd the omlementry lsti work done on the seimen, resetively. The -ftor is defined by: η 1 b d db. (19) As will be shown lter, is deendent on the geometry, loding onditions nd mteril behvior. The -ftor n be determined by first differentiting eqution (8) with reset to b, noting tht d/db 1/. Then, using equtions (8) nd (19) (1 + α)(1 µ ) η 1 µ α + α, (4) This exression for is extly the sme one resented for erfetly lsti mteril [15]. This result for the se without strin hrdening serves s hek for the solution obtined for in the nlysis bove. DISCUSSION The reltions tht were develoed for in the reeding setion will be used to ssess the effets of the strin hrdening nd ressure sensitivity on the frture behvior in metls bsed on the omt tension seimen rmeters. Sine those reltions re bsed on the forementioned ssumtions in revious setion, they serve s inditive mesures of the mteril behvior, not diret evlutors of the rel roerty. The reltions develoed for erlier were resented in terms of /. It is rorite to exress them here in terms of the normlized rk length /W. This is hieved by relling from the geometry of the seimen in Fig., tht W 1 W. (5) 193

5 Am. J. Alied Si., (9):19-195, 5 We first onsider the behvior of the ftor s funtion of the normlized rk length for erfetly lsti mteril nd two ses of strin hrdening mterils (γ,.5 nd 1., resetively) where ll mterils hving no ressure sensitivity(µ ). The results for this se re shown in Fig. 3. The figure shows tht t low vlues of rk length the hnge in the hrdening ftor rodues hnge in the vlue of the ftor in the sme diretion. An inrese in the vlue of γ gives n inrese in, resulting in higher vlue of the J integrl. The se of no hrdening orresonds to the results obtined by revious works [16, 17]. hnge in the hrdening ftor rodues hnge in the vlue of η in the sme diretion with the inrese in γ resulting in n inrese in η, lthough the effet is milder. It n be seen from the figure tht the effet of hrdening dereses s the rk extension beomes lrge nd onverges to the vlue of beuse for deely rked seimens, the ftor rohes regrdless of the mteril onstitutive behvior [6]. Figure 5 is lot of η s funtion of the rk length for higher vlue of the ressure sensitivity; µ.3. The figure shows the hnge in η s the rk length inreses for different ses of strin hrdening. By omrison with Fig. 3, we n see tht the effet of ressure sensitivity is to redue the vlue of η while the min fetures of the deendene on the strin hrdening oeffiient re the sme. It is noted tht the hnge in η due to the hnge in hrdening generlly gets milder s the strin hrdening gets higher nd ressure sensitivity tends to redue the ounter blne effets rodued by hrdening. Fig. 3: The ftor s funtion of the normlized rk length (/W) for different vlues of the strin hrdening oeffiient γ nd no ressure sensitivity (µ ). Fig. 5: The η ftor s funtion of the normlized rk length (/W) for different vlues of the strin hrdening oeffiient γ nd ressure sensitivity ftor µ.3. Fig. 4: The ftor s funtion of the normlized rk length (/W) for different vlues of the strin hrdening oeffiient γ nd ressure sensitivity ftor µ.1. Next, onsider the hnge of η with reset to the normlized rk length for different ses of strin hrdening oeffiient, γ, for low vlue of ressure sensitivity µ.1 s deited in Fig. 4. The sme observtion tht t low vlues of rk length, the 194 CONCLUSION In this work, n imroved estimte of the ftor involved in the evlution of the integrl ws obtined. As frture toughness riterion for the omt tension seimen, J estimtion for strin-hrdening ressure-sensitive mterils ws ossible by using lsti limit lod nlysis nd ssigning liner hrdening behvior to the seimen mteril; whih rovided the bsis for derivtion of nlytil exressions for the relevnt rmeters. The numeril results for different ses of ressure sensitivity nd strin hrdening rovide quntifition of exeted inrese in frture toughness of the mteril due to hrdening when omred with erfetly lsti behvior. The sme

6 Am. J. Alied Si., (9):19-195, 5 result is obtined for the derese due to mteril ressure sensitivity. The ssumtions introdued in the omuttion of ftor limits the vlidity of the results, requiring more rigorous model of hrdening nd lso omrison with exerimentl dt. REFERENCES 1. Rie, J.R., A th indeendent integrl nd the roximte nlysis of strin onentrtion by nothes nd rks. J. Al. Meh., 35: Rie, J.R., Mthemtil Anlysis in Mehnis of Frture, in Frture II. Edited by H. Liebowitz. Ademi ress, : Begley, J.A. nd J.D. Lndes, 197. The J-integrl s frture riterion, frture toughness. ro. Ntl. Sym. Frture Mehnis, rt II, ASTM ST 514, Am. So. Testing nd Mterils, : 1-4. Lndes, J.D. nd J.A. Begley, 197. The effet of seimen geometry on JIC, frture toughness. ro. Ntl. Sym. Frture Mehnis, rt II, ASTM ST 514, Am. So. Testing nd Mterils, : Bui, R.J.,.C. ris, J.D. Lndes nd J.R. Rie, 197. J-integrl estimtion roedures, frture toughness. ro. Ntl. Sym. Frture Mehnis, rt II, ASTM ST 514, Am. So. Testing nd Mterils, : Rie, J.R.,.C. ris nd J.G. Merkle, Some further results of J-integrl nlysis nd estimtes. rogress in Flw Growth nd Frture Toughness Testing. ASTM ST 536, Am. So. Testing nd Mterils, : Merkle, J.G. nd H.T. Corten, A J-integrl nlysis for the omt seimen, onsidering xil fore s well s bending effet. J. ress. Vess. Teh., Trns. ASME, : Sternstein, S. nd L. Onghin, Yield riteri for lsti deformtion of glssy high olymers in generl stress fields. Am. Chem. So. olymers rerint, 1: Sitzig, W. nd O. Rihmond, Effet of hydrostti ressure on the deformtion behvior of olyethylene nd olyrbonte in tension nd omression. oly. Eng. Si., 19: Kinloh, A. nd R. Young, Frture Behvior of olymers. Elsevier Alied Sienes, London. 11. Dong, M., C. iour nd D. Frnois, Dmge influene on the frture toughness of nodulr st iron: rt II, Dmge zone hrteriztion hed of rk ti. Metll. Mter. Trns., 8A, 11: Rihmond, O. nd W. Sitzig, 198. ressure deendene nd diltny of lsti flow. In Intl. Union of Theor. Al. Meh., : Yu, C. nd D. Chetty, Trnsformtion zone she, size nd rk-growth-resistne (R-Curve) behvior of eri-rtilly-stbilized zironi olyrystls. J. Amer. Cerm. So., 7: Chen, I.W., Model of trnsformtion toughening in brittle mterils. J Amer. Cerm. So., 74: Al-Abduljbbr, A. nd J. n, Effets of ressure sensitivity on the -ftor nd the J- integrl estimtion for omt tension seimens. J. Mt. Si., 34: Al-Abduljbbr, A.,. Effet of strin hrdening on the estimtion of J-integrl in omt tension seimens. King Sud Univ. J., 16, Eng. Si. : Druker, D. nd W. rger, 195. Soil mehnis nd lsti nlysis or limit design. Q. Al. Mth., 1: Druker, D., lstiity theory, strengthdifferentil (SD) nd volume exnsion in metls nd lstis. Metll. Trns., 4: Hill, R., Mthemtil Theory of lstiity. Oxford University ress, New York.. Singh, S., 199. Theory of lstiity. Knn ublishers, Delhi. 1. ASTM, Stndrd Test Method for JIC, A Mesure of Frture Toughness. E ε1 Am. So. Testing nd Mterils, A. 195