Comparison of plane-stress, generalized-plane-strain and. and 3D FEM elasticplastic analyses of thick-walled cylinders subjected.

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1 Loughborough University Institutionl Repository Comprison of plne-stress, generlized-plne-strin nd 3D FM elsticplstic nlyses of thick-wlled cylinders subjected to rdil therml grdient This item ws submitted to Loughborough University's Institutionl Repository by the/n uthor. Cittion: KAMAL, S.M....et l., 07. Comprison of plne-stress, generlizedplne-strin nd 3D FM elsticplstic nlyses of thick-wlled cylinders subjected to rdil therml grdient. Interntionl Journl of Mechnicl Sciences, 3-3, pp Additionl Informtion: This is n open ccess rticle under the CC BY license. Metdt Record: Version: Published Publisher: c 07 The Authors. Published by lsevier Ltd. Rights: This work is mde vilble ccording to the conditions of the Cretive Commons Attribution 4.0 Interntionl CC BY 4.0) licence. Full detils of this licence re vilble t: Plese cite the published version.

2 Interntionl Journl of Mechnicl Sciences ) Contents lists vilble t ScienceDirect Interntionl Journl of Mechnicl Sciences journl homepge: Comprison of plne-stress, generlized-plne-strin nd 3D FM elstic plstic nlyses of thick-wlled cylinders subjected to rdil therml grdient S.M. Kml, U.S. Dixit b,, A. Roy c, Q. Liu c, Vdim V. Silberschmidt c Deprtment of Mechnicl ngineering, Tezpur University, Npm, Tezpur , Indi b Deprtment of Mechnicl ngineering, Indin Institute of Technology Guwhti, Guwhti , Indi c Wolfson School of Mechnicl, lectricl nd Mnufcturing ngineering, Loughborough University, Loughborough L 3TU, UK r t i c l e i n f o b s t r c t Keywords: Thick-wlled cylinder lstic plstic Plne stress Generlized plne strin Three-dimensionl finite element method Therml stress In mny industril pplictions, thick-wlled cylindricl components re subjected to high pressure nd/or temperture. During the opertion the cylinder wll my undergo elstic plstic deformtion. This pper presents plne-stress nd plne-strin thermo-elstic plstic stress nlyses of thick-wlled cylinders subjected to rdil therml grdient. A three-dimensionl finite element method 3D FM) nlysis of the thermo-elstic plstic stresses in thick-wlled cylinder is lso crried out. The 3D FM results re compred with the nlyticl plne stress nd the generlized plne strin nlyses in order to study the vlidity of these models on the bsis of length to wll-thickness rtio of cylinders. The plne stress nd generlized plne strin nlyses re bsed on the Tresc yield criterion nd ssocited flow rule. The strin hrdening behvior of the mteril of the cylinder is tken into ccount. It is observed tht for the length to wll thickness rtio of more thn 6, the generlized plne strin nlysis cn provide sufficiently ccurte results. Similrly, for the length to wll thickness rtio of less thn 0.5, plne stress nlysis cn be used. When the length to wll thickness rtio is more thn 0.5 but less thn 6, three-dimensionl nlysis is needed. 07 The Authors. Published by lsevier Ltd. This is n open ccess rticle under the CC BY license. ). Introduction The thick-wlled cylinders subjected to pressures nd tempertures find severl pplictions, e.g., in chemicl industries nd nucler power plnts. In most of the cses, the design ttempts to keep the stresses in the cylinders within elstic limits. However, it is lwys better to crry out n elstic plstic nlysis in order to get n ide bout sfety in cse of untowrd situtions. Moreover, in n utofrettge process, the plstic deformtion is delibertely produced to induce compressive stresses in the inner side of the cylinder. There re lso exmples where plstic deformtion is permitted by design. Thus, the elstic plstic nlysis of the thick-wlled cylinders is n ttrctive reserch re. lstic plstic deformtion of thick-wlled cylinders due to internl pressure loding is well recognized nd hs been investigted by mny reserchers. Mny nlyticl solutions re vilble for stress, strins nd displcement fields during the elstic plstic deformtion of cylinders. An erly nlysis of elstic plstic deformtion in thick tubes ws crried out by Hill et l. [] ssuming non-hrdening mteril, Tresc yield criterion nd plne strin condition. Go [] developed closed form nlyticl solution for the elstic plstic stress, strin nd displcements of n internlly pressurized thick-wlled cylinders under plne stress condition using Hencky s deformtion theory nd von Mises yield criterion. Durbn [3], Go [4], nd Bonn nd Hupt [5] investigted lrge elstic plstic deformtions of thick-wlled cylindricl tube under internl pressure. An nlyticl solution for stress, strin nd displcements in thick cylinder subjected to internl pressure ws developed by Go [6] using strin grdient plsticity theory. An nlyticl solution for elstic plstic stresses considering the Buschinger effect nd Tresc yield criterion in thick-wlled cylindricl vessel mde of elstic liner-hrdening mteril ws given by Drijni et l. [7]. The clssicl solution for thermo-elstic stresses in thick-wlled cylindricl bodies under stedy stte temperture distribution due to temperture grdient is well known. However, only few ppers tret the thermo-elstic plstic nlysis of thick-wlled cylinders nlyticlly. Blnd [8] crried out n elstic plstic nlysis of thick-wlled tubes of work hrdening mteril subjected to internl nd externl pressures with outer nd inner surfces mintined t different tempertures. He considered tht yielding took plce only on the inner side of Corresponding uthor. -mil ddresses: udy@iitg.c.in, usd008@yhoo.com U.S. Dixit). Received 4 Septembe06; Received in revised form June 07; Accepted 0 July 07 Avilble online 0 August / 07 The Authors. Published by lsevier Ltd. This is n open ccess rticle under the CC BY license. )

3 Nomenclture inner rdius of cylinder b outer rdius of cylinder c, d, e, f interfce rdii Young s modulus of elsticity K hrdening coefficient n strin hrdening exponent r rdius of cylinder T temperture t the inner wll of cylinder T b temperture t the outer wll of cylinder u rdil displcement α coefficient of therml expnsion ε r totl rdil strin ε θ totl hoop strin ε 0 constnt xil strin ε e r, ε e, ε e θ z elstic rdil, hoop nd xil strin r, p, ε θ z plstic rdil, hoop nd xil strin uivlent plstic strin ν Poisson s rtio σ r rdil stress σ θ hoop stress σ z xil stress σ uivlent stress σ Y yield stress the cylinder leding to the formtion of n inner plstic zone. Wong nd Simionescu [9] developed n elstic plstic nlyticl model of thickwlled tube subjected to internl heting nd pressure ssuming smll displcements, plne strin condition nd the yield criterion of Tresc without strin hrdening. They did not consider the cse in which there re outer nd inner plstic zones with n intermedite elstic zone. The thermo-elstic plstic deformtion of tubes with inner plstic nd outer elstic zones due to internl het genertion ws investigted by Orçn nsln [0] considering the temperture-dependent mechnicl nd therml properties. Sdeghin nd Toussi [] crried out n xisymmetric thermo-elstic plstic stress nlysis in cylindricl vessels mde of functionlly grded mteril. The stress nlysis of functionlly grded thick-wlled hyperelstic sphericl vessel subjected to internl nd externl pressure ws crried out by Anni nd Rhimi [], lbeit without considering therml stresses. Some ppers in the literture del with the thermo-elstic plstic nlysis of solid cylinders nd sphericl vessels. Orçn [3] crried out thermo-elstic plstic nlysis of elstic-perfectly plstic cylindricl rod with uniform internl het genertion for generlized plne strin condition bsed on Tresc yield criterion. He considered the cse of two plstic zones seprted by n elstic zone. Cowper [4], Johnson nd Mellor [5], nd Drijni et l. [6] nlyzed the elstic plstic stresses in thick-wlled hollow sphere under stedy stte rdil temperture grdient. In recent pper, Zre nd Drijni [7] studied elstic plstic stresses in thick-wlled cylinder rotting t very lrge ngulr velocity. In generl, thick-wlled cylinder my undergo simultneous plstic deformtions emnting from both inner nd outer wlls under the condition of high therml grdient. This cse ws nlyzed by Kml nd Dixit [8,9] for hollow disk nd cylinder ssuming plne stress nd generlized plne strin conditions, respectively. The nlyses re bsed on Tresc yield criterion nd its ssocited flow rule. The effect of strin hrdening ws considered in the plne stress model [8]. However, the generlized plne strin model [9] did not include the strin hrdening, lthough the theoreticl results mtched well with the experiments [0]. In this work, the generlized plne strin model of Kml nd Dixit [9] is extended to incorporte the effect of strin-hrdening. The strinhrdening during plstic deformtion is ssumed to follow Ludwik s hrdening lw. The plne stress nd generlized plne strin models cnnot predict the dimension-rnge in which the respective ssumptions re justified. Hence, three-dimensionl finite element method 3D FM) nlysis is crried out in order to define the pplicbility of the nlyticl models. A criterion bsed on the length to wll thickness rtio of the cylinders is developed to ssess the vlidity of the plne stress nd generlized plne strin ssumptions. During the nlysis, it is ssumed tht the mximum temperture induced in the cylinder is well below the recrystlliztion temperture of the mteril. Hence, it is pproprite to tret the mechnicl nd therml properties of the mteril s temperture-independent.. Problem definition A thick-wlled cylinder with inner rdius nd outer rdius b is considered. The inner wll is subjected to temperture T nd the outer wll is subjected to temperture T b such tht T b > T. The thermoelstic plstic stress nlyses of short cylinder hollow circulr disk) using the ssumption of plne stress σ z = 0) incorporting strin hrdening ws crried out in [8]. Another nlyticl model bsed on generlized plne-strin condition z = constnt) pplicble for long cylinders is vilble in [9]. However, the generlized plne-strin model did not include the effect of strin hrdening. In the present work, the thermo-elstic plstic stress nlysis in the cylinder under the generlized plne-strin ssumption is presented incorporting the strin hrdening in Section 3. The mteril is ssumed to follow the Ludwik s hrdening lw given by []: σ = σ Y + K ) n, ) where σ Y is the yield stress in unixil tension or compression, σ >σ Y ) is the uivlent stress, is the uivlent plstic strin, K is the hrdening coefficient nd n is the strin hrdening exponent. A stedy-stte temperture distribution in the cylinder under rdil temperture difference T b T ) is given by []: T = T + ln r T b T. ) ln b To test the vlidity of plne stress nd generlized plne strin ssumptions, 3D FM nlysis is crried out for different length to wll thickness rtios of cylinder. 3. Generlized-plne-strin nlyticl model considering strin hrdening Under the condition of generlized plne strin condition z = ε 0 = constnt), the thermo-elstic stresses in the cylinder re given by the utions provided in [3], when the cylinder is subjected to sufficiently low temperture. At the inner rdius the yielding of the mteril of the cylinder begins ccording to the Tresc criterion given by σ θ σ r = k σ, σ z σ r = k σ, 3) where σ is given by the Ludwik s hrdening lw q. )). Beyond the inner rdil position, the cylinder yields s per the Tresc yield criterion: σ z σ r = k σ. 4) By substituting the thermo-elstic stresses in q. 3) t r =, nd tking σ = σ Y, the temperture difference ruired for the initition of yielding t the inner rdius cn be obtined. When the mteril of the cylinder yields s per the Tresc yield criterion, during first stge of elstic plstic deformtion the wll of the cylinder consists of three zones plstic zone I r c ), plstic zone II c r d ), nd elstic zone d r b ). During the second stge of elstoplstic deformtion, the cylinder wll gets divided into five zones two 745

4 [ r = ν r k σ Y ln r + C 3 + k K k K r r ε 3 k K r ν 4 k σ Y + ν) α T b T { r + ln ln b { k K ) n + r } ] r d αt + 3 } ε 0 C 4 r. 8) Fig.. lstic nd plstic zones in cylindricl segment during elstic plstic deformtion generlized-plne-strin). inner plstic zones: plstic zone I r c ) nd plstic zone II c r d ), n intermedite elstic zone: d r e nd two outer plstic zones: plstic zone III f r b ) nd plstic zone IV e r f ). The two consecutive plstic zones in the cylinder correspond to two different sides of Tresc yield locus. The generl geometry of the cylinder during elstic plstic deformtion is shown schemticlly in Fig.. In the following subsections the stresses nd plstic strin fields in the first nd second stge of elstic plstic deformtion re obtined by incorporting strin hrdening using Ludwik s hrdening lw. 3.. First stge of elstic plstic deformtion The stresses nd plstic strins in ll the three zones during the first stge of elstic plstic deformtion re obtined in similr mnner s described in Kml nd Dixit [9]. The stresses in the elstic zone, d r b re provided in [9]. However, the stresses nd strins in the plstic zone get modified due to the incorportion of strin-hrdening following the Ludwik s hrdening lw. The resulting fields of stresses nd strins in the plstic zones I, r c nd II, c r e obtined s follows: Plstic zone I, r c : Using q. 3) in the stress uilibrium ution [9], the rdil, hoop nd xil stresses in the plstic zone I re obtined s σ r = k σ Y ln r + k K r + C 3, 5) σ θ = σ z = k σ Y + ln r ) + k K ) r n ε + k K r + C 3. 6) Using the condition of plstic incompressibility, the strindisplcement reltion nd the generlized Hooke s lw, the expression for the rdil displcement, the totl hoop ndil strins cn be obtined [9]. The plstic prts of the hoop, rdil nd xil strins re obtined by subtrcting the elstic prts from the totl strin components. They re given by θ = ν [ k k σ Y ln r + C 3 + K r k K r + 3 k { K r 3 ν 4 k σ Y ν k K ) n + αt α T b T { r + ln ln b r ε ) n } ] r d p 3 } ε 0 + C 4 r. 7) { z = ε 0 ν } k σ Y ln r + C 3 + k K r ν k σ Y ν k K ) n αt α ln r T b T. 9) ln b The uivlent plstic strin is given by = p p ε 3 ij ε ij, 0) where ij denotes the component of the plstic prt of the strin tensor. Plstic zone II, c r d : Using the Tresc yield criterion q. 4)) nd its ssocited flow rule long with the strin comptibility [9], the elstic plstic rdil nd hoop stresses re obtined s σ r = C 5 r + ν) + C 6 r ν) + k σ Y ν ) k K r + { ν) ν + ν } ν) r + ν) c k K r { ν) ν ν } ν) r ν) c + αt ν ) + α T b T ln r ν ) ln b α T b T α T b T ν ) ln b ν ) ln ν) r ) n ν) ε ε 0 b ν ), ) σ θ = C 5 ν) ν) C 6 ν) r ν) + k σ Y ν ) { k K ν) ν) ν + ν } ν) + ν) r ν) r ) n c k K ν) r { ν) ν ν } ν) + ν) r ν) ε + νk K ) n c + αt ν ) + α T b T ln r α T b T ε 0 ν ) ln b ν ) ln b ν ). ) 746

5 The expression for σ z cn be obtined from the Tresc criterion given by q. 4). Using the Tresc ssocited flow rule, the plstic prt of the hoop strin in the plstic zone is obtined s zero implying r = z. In this zone, the plstic strins re given by z = C { 5 + ν) ν ν } ν) + C { 6 ν) ν + ν } ν) }{ ν + ν } ν) p r = ε k K r + { ν) ν ν ν) + ν) r ν) r ) n c k K r { ν) ν + ν }{ ν) ν ν } ν) ν) r ν) ε c + ν ) k K α T b T + ν) + ln b ν ), 3) The constnts C 5 nd C 6 re determined by using the continuity of rdil stresses nd plstic strins t the elstic plstic interfce. The continuity of rdil stresses t the interfce of the plstic zones I nd II provides the constnt C 3 nd the continuity of plstic hoop strin provides the constnt C 4. The constnt xil strin ε 0 is obtined by using the free-end condition, i.e., mking the resultnt of the xil stresses zero. To evlute the unknown boundry rdii, c nd d, numericl procedure needs to be used. The procedure involves n itertive pproch to estimte the vlues of c nd d. The initil estimtes for c nd d cn be obtined from non-hrdening cse K = 0) by using the boundry conditions of vnishing rdil stress t the inner rdius nd plstic zone II ) plstic zone II ) σ θ = σz t r = c nd solving them using FSOLV function in MATLAB. The initil guess vlue of is tken s zero everywhere in the plstic zones I nd II. With these vlues of c nd d, the vlues of p, ε θ r nd re updted in qs. 7) 9) for plstic zone I. 0 q. 0) provides the updted vlues of t ny rdil position in the plstic zone I. Similrly, using q. 3), the vlues of r t different rdil positions in the plstic zone II re updted. The updted vlues of t different rdil positions in the plstic zone II re obtined from q. 0). These vlues of uivlent plstic strin re used to obtin the updted components of plstic strin. From these components, the vlues of re updted further for fixed c nd d in both the plstic zones. This procedure is repeted till the convergence in is chieved. The integrl terms involved in the expressions cn be evluted numericlly by using two-guss-point formul. Now, using the converged vlues of, the boundry conditions of vnishing rdil stress t the inner rdius plstic zone II ) plstic zone II ) nd σ θ = σz t r = c re solved gin to get the new estimtes of c nd d. If these new estimtes of c nd e sme s the previously estimted vlues of c nd d the procedure is stopped, otherwise the whole procedure is repeted till the convergence for c nd d is chieved. 3.. Second stge of elstic plstic deformtion In the second stge of elstic plstic deformtion, the outer plstic zone, i.e., plstic zone III f r b ), develops in the wll of the cylinder ccording to the Tresc yield criterion given by σ θ σ r = k σ, σ z σ r = k σ 4) Another plstic zone, i.e., plstic zone IV e r f ) develops simultneously long with the plstic zone III s per the Tresc yield criterion: σ θ σ r = k σ. 5) The stress nd strin expressions for the plstic zones I nd II during the second stge of elstic plstic deformtion re sme s in the cse of first stge of elstic plstic deformtion. However, the constnts C 3, C 4, C 5 nd C 6 chnge due to chnge of continuity conditions. The stress solutions in the intermedite elstic zone re vilble in Ref. [9]. The utions for stresses nd plstic strins incorporting the Ludwik s hrdening lw in the plstic zones III nd IV re obtined s follows: Plstic zone III, f r b : Using the Tresc yield criterion q. 4)) nd the stress uilibrium ution, the rdil, hoop nd xil stresses re given by σ r = k σ Y ln r k K f r + C 7, 6) σ θ = σ z = k σ Y + ln r ) k K ) r n ε k K f r + C 7. 7) The plstic prts of the hoop, rdil nd xil strins re obtined in similr wy s described in Section 3. ne given by θ = ν [ C 7 k σ Y ln r ) k K r r ε f + k K f r 3 k { K r f f + 3 ν 4 k σ Y + ν k K ) n + αt α T b T { r + ln ln b [ r = ν C 7 k σ Y ln r ) r k K f + k K } ] r d 3 } ε 0 + C 8 r, 8) r r ε + 3 k K r f f 7 6 ν 4 k σ Y ν) α T b T { r + ln ln b { k K ) n + r f } ] r d αt + 3 } ε 0 C 8 r, 9) { z = ε 0 ν k σ Y ln r + C 7 k K f r ) + ν k K } + ν k σ Y ) n αt α ln r T b T. 0) ln b Plstic zone IV, e r f : Using q. 5) in stress uilibrium ution, the rdil stress is given by α σ r = ν ) T + α [ { T b T d ln + ν) ln b ν ) e } ln d { + + e ) + } r ν d ln k e σ Y + ε r 0 ν) k K e ] e p ) n r. ) Knowing the expression for σ r, the expression for σ θ cn be obtined from q. 5). The Tresc ssocited flow rule indictes tht the xil strin in the plstic zone IV is wholly elstic. Hence, the xil stress 747

6 Tble Mteril properties of luminum. Yield stress, σ Y MP) Modulus of elsticity, GP) Poisson s rtio, ν Coefficient of therml expnsion, α / C) Mss density, ρ kg/m 3 ) Therml conductivity, k W/m. K) Specific het, c p J/kg. K) component cn be obtined by using the generlized Hooke s lw [9]. The resulting expression for xil stress is given by [ { α σ z = ν ) T + α T b T ν d ln + ν) ln b ν ) e } d ] r r + ν ln ln e { ν + + e ) } r ν ln k ν d e σ Y + ε 0 ν) νk K e r νk K ) n. ) θ = The plstic prt of the hoop strin, p = ε θ α T b T ν) ln + 4 ν k K r ν ) + ν ) k σ Y b { e + ν r r k K e ν k K e r e } r d r ) is given by r ε + ν ) k K r + C 9 r. 3) The constnts C 5 nd C 6 re obtined by employing the continuity condition of rdil stresses ndil plstic strins t the elstic plstic interfce rdius d. The constnt C 3 is obtined by using continuity of rdil stresses t r = c. The continuity condition of the plstic hoop strins t r = c provides the constnt C 4. To obtin the constnt C 7, continuity of the rdil stresses t r = f cn be employed. The constnts C 9 nd C 8 cn be obtined by using the boundry conditions of continuity of the plstic prt of the hoop strins t the interfce rdii e nd f, respectively. The constnt xil strin ε 0 in the second stge of elstic plstic deformtion cn be obtined by using the free-end condition. The interfce rdii c, d, e nd f re obtined in mnner similr to tht described in Section 3.. Fig.. lstic plstic stresses in luminum cylinder: ) plne-stress, b) generlizedplne-strin. 4. Numericl simultions In this section, numericl simultions with the plne-stress nd generlized-plne-strin models re crried out for typicl cylinder. The objective is to compre the solutions for stresses nd plstic strins of these two models under rdil therml grdient. An luminum cylinder with = 0 mm nd b = 0 mm is considered. The mteril properties of luminum re provided in Tble. The hrdening coefficient K nd strin hrdening exponent n for the cylinder re tken s 58.8 MP nd 0.48 [8]. The temperture difference ruired to initite the yielding t the inner wll of the cylinder is obtined s C for plne stress model nd C for generlized plne strin model. The simultion is crried out for temperture difference, T b T ) = 75 C. With this temperture grdient mechnicl nd therml properties do not vry much [4]. The inner wll is ssumed to be t T = 5 C. According to plne stress model, for T b T ) = 75 C, the cylinder undergoes first stge of elstic plstic deformtion with n inner plstic zone propgting outwrds up to rdius c =.0 mm. However, s per generlized plne strin model, the cylinder undergoes second stge of elstic plstic deformtion dividing the cylinder wll into two inner-plstic, two outer-plstic nd n intermedite-elstic zone. At the inner wll, plstic zone I propgtes outwrds to rdius c =.4973 mm nd plstic zone II propgtes outwrds to rdius d =.05 mm. A plstic zone III propgtes inwrds from the outer rdius to rdius f = mm nd plstic zone IV propgtes inwrds to rdius e = mm. The elstic plstic stresses generted in different zones of the cylinder for plne stress nd generlized plne strin conditions re shown long rdil pth in Fig.. It is observed from Fig. ) tht for plne stress cse, the mximum vlue of hoop stress in the cylinder is generted t the inner rdius of the cylinder nd is tensile in nture. Fig. b) shows tht the mximum vlue of hoop stress exists t the interfce rdius c nd the mximum vlue of xil stress exists t the interfce rdius d s per generlized plne strin model. The mgnitudes of rdil stresses long the rdil pth re smller for plne stress s well s generlized plne strin model. 748

7 Fig. 4. Residul stress distribution in luminum cylinder: ) plne-stress, b) generlizedplne-strin. Fig. 3. Plstic strin distribution in luminum cylinder: ) plne-stress, b) generlizedplne-strin. The plstic prts of rdil, hoop nd xil strins produced in the cylinder re obtined for both plne stress nd generlized plne strin models Fig. 3 ). It is observed from Fig. 3 ) tht the plstic prts of strins re numericlly very smll for plne stress cse. However, Fig. 3 b) shows tht for generlized plne strin cse, lthough the plstic prts of strins re lrger in mgnitude s compred to plne stress cse, their mgnitudes re still smll of the order of 0 3 ). Thus, the mgnitude of uivlent plstic strin generted in the cylinder is lso not very substntil. As the vlue of uivlent stress σ in the plstic zone for yielding depends on the uivlent plstic strin, here the vlue of σ will not devite much from σ Y. The mximum devition of σ from σ Y is only % for the plne stress condition nd 4.87% for the generlized plne strin condition. The mximum uivlent plstic strin occurs t the inner rdius. When the temperture difference T b T ) is removed, i.e., when the cylinder is cooled to room temperture, the residul stresses re set up in the wll of the cylinder. Considering the unloding process to be completely elstic, the residul stresses cn be obtined by subtrcting the thermo-elstic stresses [] from the respective stress utions of elstic plstic zones. The resulting residul stress distribution in the cylinder for the plne stress nd the generlized plne strin condition is shown in Fig. 4. It is observed from Fig. 4 ) nd b) tht the compressive residul hoop stresses re generted t nound the inner rdius of the cylinder. For the sme temperture difference, the gen- erlized plne strin model predicts lrger mgnitude of compressive residul hoop stress t the inner rdius. Fig. 4 b) shows tht for generlized plne strin condition the xil residul stresses generted t nd round the inner rdius of the cylinder re lso compressive. The compressive residul stresses t nound the inner rdius of the cylinder helps in reducing the net mximum stress produced in the cylinder in the next loding stge. This mounts to increse the lod crrying cpcity of the cylinder. The mgnitudes of tensile hoop nd xil stresses re smll t nound the outer wll of the cylinder. 5. Three-dimensionl finite-element modeling of elstic plstic stresses in thick cylinders due to temperture grdient From the numericl simultions of both the plne stress nd generlized plne strin nlyticl models, it is not cler which model is vlid for prticulr length of the cylinder. To ssess the vlidity of the nlyticl models, 3D FM nlysis using ABAQUS 6.0 pckge is crried out. A homogeneous thick-wlled hollow cylinder with inner rdius nd outer rdius b is considered. The inner wll is subjected to temperture T nd the outer wll is subjected to temperture T b such tht T b > T. The problem is solved using ABAQUS stndrd code. Under the rdil temperture difference, T b T ), the stedy stte condition is ssumed in ABAQUS stndrd simultion. The strin hrdening of the mteril during plstic deformtion is considered. For comprison of the FM results with the plne-stress nd generlized-plne-strin models, the nlyticl solution developed in [8] nd the stress solution devel- 749

8 Tble Mesh sensitivity nlysis. Mesh Number of elements in Mximum vlue of Screentime s) Rdil direction Circumferentil direction Axil direction Rdil stress MP) Hoop stress MP) Axil stress MP) Mesh Mesh Mesh Mesh Mesh Mesh Mesh Mesh Mesh Mesh Mesh 9 indicted in the boldfced is the optimum mesh. Tble 3 Comprison of stresses between 3D FM nd nlyticl models. L / b ) L norm of error in stresses L norm of error in stresses between between FM nd plne stress MP) FM nd generlized plne strin MP) σ r σ θ σ z σ r σ θ σ z The boldfced vlues correspond to the cses in which ll stress components show less thn 0% error. oped in Section 3 re used. The detiled nlysis of 3D FM is presented in the following subsections D finite-element modeling A 3D FM model for nlysing thermo-elstic plstic stresses in the cylinder under the rdil temperture difference ws developed using commercil finite element pckge ABAQUS 6.0. A coupled thermomechnicl pproch is used to obtin the stedy-stte therml stress solutions in ABAQUS/Stndrd. In therml utofrettge process, the cylinder is subjected to non-homogeneous elstic plstic deformtion due to rdil therml grdient between the outer nd inner wll of the cylinder. Hence, for the stress solution in therml utofrettge process, coupled temperture-displcement elements of ABAQUS/Stndrd [5] re used. The prt considered for the 3D FM nlysis is thick-wlled solid extruded cylinder with open ends. 5.. Mteril properties An luminum cylinder is considered. The inner nd outer rdii of the cylinder re tken s 0 mm nd 0 mm, respectively. The temperture difference cross the wll thickness of the cylinder is tken s 75 C. The therml stress nlysis during the therml loding nd unloding of cylinder for different lengths is crried out. The mechnicl nd therml properties of luminum re presented in Tble. The model used von Mises criterion with isotropic hrdening Boundry conditions nd mesh genertion The therml stresses in the cylinder re induced due to the rdil therml grdient cross the wll thickness. For nlyzing the therml stress, the Dirichlet temperture boundry conditions re specified t the inner nd outer surfces of the cylinder. The temperture of the inner surfce is prescribed s T = 5 C nd tht of the outer surfce is Fig. 5. Schemtic of 3D prt in ABAQUS long with the boundry conditions. prescribed s T b = 00 C. At ech node, there re three trnsltionl degrees of freedom, viz., rdil displcement u r, circumferentil displcement u θ nd xil displcement u z. In the nlysis, the circumferentil displcement, u θ vnishes. This implies tht the rottion in the circumferentil direction is constrined, but the cylinder is free to expnd xilly ndilly. A schemtic of the 3D prt in ABAQUS long with the therml nd displcement boundry conditions is shown in Fig. 5. An eight-node continuum C3D8T thermlly coupled brick, triliner displcement nd temperture element is used to generte the mesh on the cylinder. The meshed cylinder is shown in Fig. 6. Although the cylinder is modeled s open-ended, the xil stresses re produced in it due to the therml grdient. When the inner surfce of the cylinder is t temperture lower thn tht of the outer surfce, the tendency to expnd in the xil direction is less in the vicinity of the inner surfce thn in the vicinity of the outer one. This cuses tensile xil stresses on the inner side nd compressive xil stresses t the outer side. However, the resultnt xil force due to these stresses is zero. 750

9 Fig. 6. Cylindricl geometry with typicl C3D8T element Mesh sensitivity nlysis A mesh sensitivity nlysis is crried out in order to select n pproprite mesh size. The mesh sensitivity nlysis is presented in Tble bsed on the mximum vlues of rdil, hoop nd xil stresses generted in the cylinder. The length of the cylinder ws tken s 00 mm. It is observed from Tble tht the Mesh 9 with 4 elements in the rdil direction, 3 elements in the circumferentil direction nd 50 elements in the xil direction leds to the convergent mesh. This mesh hs totl of 38,400 elements. Mesh 9 provides less thn % devition in the solution in comprison to Mesh 8 nd Mesh 0. Hence, it is considered s the optimum mesh. The thermo-elstic plstic stresses generted in the cylinder re nlyzed vrying the length of the cylinder using Mesh 9. A comprtive study with the plne stress nd generlized plin strin models is presented for different length to wll-thickness rtios of the cylinder in Section Comprison of 3D finite element elstic plstic therml stresses with plne-stress nd generlized-plne-strin models for different length to wll thickness rtios of cylinder The 3D FM simultions re crried out for different length to wll thickness rtios of the cylinder nd the results re compred with nlyticl plne stress nd generlized plne strin model. The present FM nlysis is crried out in.40-ghz processor nd.60-gb rndom-ccess memory RAM) ACR PC nd it tkes bout h of screen time. However, the nlyticl models provide the solution in less thn 0 min. The comprison of rdil, hoop nd xil stresses between 3D FM nd nlyticl models long the rdil pth for different length to wll thickness rtio is shown in Tble 3 using L norm of error in stresses. The stresses long rdil pth from the 3D ABAQUS finite element model re obtined considering rdil pth t the mid-length of the cylinder. The stresses long rdil pth t the edges become sufficiently low in FM model ne not tken into ccount for the nlysis. It is observed from Tble 3 tht L norm of error in stresses between 3D FM nd generlized plne strin re resonbly smll when the length to wll thickness rtio of the cylinder is greter thn or ul to 6. Thus, the generlized plne strin model is well estblished for L / b ) 6. It is lso observed from Tble 3 tht when L / b ), the L norm of error between 3D FM nd plne stress becomes smller. This shows tht the plne stress nlyticl model is vlid for length to wll thickness rtio, L / b ). The comprison of 3D finite element solutions with the generlized plne strin model for L / b ) = 0 is shown in Fig. 7 ). Fig. 7 b) shows the comprison of 3D finite element Fig. 7. Comprison of elstic plstic stresses between 3D FM nd ) generlized-plnestrin for L/ b ) = 0, b) plne-stress for L/ b ) = 0.5 in luminum cylinder. results with the plne stress model for L / b ) = 0.5 in luminum cylinder Residul stress solutions The whole cylinder is cooled to room temperture 5 C) by imposing the cooling boundry condition in ABAQUS/Stndrd. This genertes the residul stresses in the cylinder. The finite element residul stresses in the cylinder re obtined long rdil pth t the mid-length. The results for long nd short cylinders re depicted in Fig. 8 long with the predictions of generlized plne strin model for L/ b ) = 0 nd plne stress model for L/ b ) = 0.5. It is observed tht the finite element predictions of the residul stresses re in close greement with the nlyticl models. 6. Conclusions The min objective of this work is to compre thermo-elstic plstic plne-stress nd generlized-plne-strin nlyses with 3D FM simultions. For the plne-stress nlysis, the model of Kml nd Dixit [8] is used. For the generlized-plne-strin nlysis, the model of Kml nd Dixit [9] is extended to incorporte the effect of strin hrdening. The FM results re compred with the nlyticl solutions for different length to wll thickness rtio, L / b ), of the cylinder. The comprison suggests the pplicbility of the developed nlyticl models on the bsis of L / b ) rtio of the cylinder. The generlized plne strin nlyticl model provides relistic solution for L / b ) 6. Thus, this 75

10 Acknowledgments Funding from the ngineering nd Physicl Sciences Reserch Council UK) through grnt P/K0836/ nd Deprtment of Science nd Technology Indi) through grnt DST/RC-UK/4-AM/0, project Modeling of Advnced Mterils for Simultion of Trnsformtive Mnufcturing Processes MAST) is grtefully cknowledged. The uthors re lso thnkful to the orgnizers of WCCM XII & APCOM VI 06 Congress for giving the opportunity of orl presenttion of this pper. References Fig. 8. Comprison of residul stresses between 3D FM nd ) generlized-plne-strin for L/ b ) = 0, b) plne-stress for L/ b ) = 0.5 in luminum cylinder. model is suitble for long cylinders such s gun brrels, pressure vessels nd thick pipes subjected to therml grdient. The plne stress nlyticl model provides gooesults for very short cylinders thin disks) such s fstener holes with L / b ). It is to be noted tht the CPU time ruired is much more in cse of FM nlysis compred to the solution obtined by nlyticl models. Thus, the nlyticl models hve their own importnce minly due to computtionl efficiency. The vlidity of the plne-stress nd the generlized-plne-strin ssumptions is inferred bsed on the sme mteril model nd the fixed b/ rtio. This study highlights tht for L / b ) rtio between nd 6, 3D nlysis is ruired. For other types of cylinder, the rnge of L / b ) ruiring 3D nlysis my differ. It will be interesting to crry out similr study for number of cylinder geometries, loding conditions nd mteril model formultions, nd develop generlized criterion for the vlidity of ech model. [] Hill R, Lee H, Tupper SJ. The theory of combined plstic nd elstic deformtion with prticulr reference to thick tube under internl pressure. Proc R Soc A 947;9November): [] Go XL. An exct elsto-plstic solution for n open-ended thick-wlled cylinder of strin-hrdening mteril. Int J Press Vessel Pip 99;5:9 44. [3] Durbn D. Lrge strin solution for pressurized elsto/plstic tubes. J Appl Mech 979;46:8 30. [4] Go XL. An exct elsto-plstic solution for closed-end thick-wlled cylinder of elstic liner-hrdening mteril with lrge strins. Int J Press Vessel Pip 993;56: [5] Bonn R, Hupt P. xct solutions for lrge elstoplstic deformtions of thick- -wlled tube under internl pressure. Int J Plst 995;:99 8. [6] Go XL. lsto-plstic nlysis of n internlly pressurized thick-wlled cylinder using strin grdient plsticity theory. Int J Solids Struct 003;40: [7] Drijni H, Krgrnovin MH, Nghdbdi R. Design of thick-wlled cylinders under internl pressure bsed on elsto-plstic pproch. Mter Des 009;30: [8] Blnd DR. lstoplstic thick-wlled tubes of work-hrdening mteril subject to internl nd externl pressures nd to temperture grdients. J Mech Phys Solids 956;4:09 9. [9] Wong H, Simionescu O. An nlyticl solution of thermoplstic thick-wlled tube subject to internl heting nd vrible pressure, tking into ccount corner flow nd nonzero initil stress. Int J ng Sci 996;34: [0] Orçn Y, rsln AN. Therml stresses in elstic-plstic tubes with temperture-dependent mechnicl nd therml properties. J Therm Stresses 00;4: [] Sdeghin M, Toussi H. lsto-plstic xisymmetric therml stress nlysis of functionlly grded cylindricl vessel. J Bsic Appl Sci Res 0;: [] Anni Y, Rhimi GH. Stress nlysis of thick pressure vessel composed of functionlly grded incompressible hyperelstic mterils. Int J Mech Sci 05;04: 7. [3] Orçn Y. Therml stresses in het generting elstic-plstic cylinder with free ends. Int J ng Sci 994;3: [4] Cowper GR. The elstoplstic thick-wlled sphere subjected to rdil temperture grdient. J Appl Mech 960: [5] Johnson W, Mellor PB. lstic-plstic behviour of thick-wlled spheres of non work-hrdening mteril subject to stedy-stte rdil temperture grdient. Int J Mech Sci 96;4: [6] Drijni H, Krgrnovin MH, Nghdbdi R. Design of sphericl vessels under stedy-stte therml loding using thermo-elsto plstic concept. Int J Press Vessel Pip 009;86:43 5. [7] Zre HR, Drijni H. Strengthening nd design of the liner hrdening thick- -wlled cylinders using the new method of rottionl utofrettge. Int J Mech Sci 07:4 5: 8. [8] Kml SM, Dixit US. Fesibility study of therml utofrettge process. In: Nrynn RG, Dixit US, editors. Advnces in mteril forming nd joining. New Delhi: Springer; 05. p [9] Kml SM, Dixit US. Fesibility study of therml utofrettge of thick-wlled cylinders. ASM J Press Vessel Technol 05;376): [0] Kml SM, Borsiki AC, Dixit US. xperimentl ssessment of residul stresses induced by the therml utofrettge of thick-wlled cylinders. J Strin Anl ng Des 06;5): [] Dixit PM, Dixit US. Modeling of metl forming nd mchining processes: by finite element nd soft computing methods. London: Springer; 008. [] Chkrbrty J. Theory of plsticity. 3rd ed. Burlington: Butterworth-Heinemnn; 006. [3] Nod N, Hetnrski RB, Tnigw Y. Therml stresses. nd ed. New York: Tylor nd Frncis; 003. [4] Zhu XK, Cho YJ. ffects of temperture-dependent mteril properties on welding simultion. Comput Struct 00;80: [5] ABAQUS User s Mnul, Version 6.7, Dssult Systèmes Simuli Corp., Providence, RI, USA,