PLASTIC FLOW. Sammenheng mellom spenninger og tøyninger i det plastiske området

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1 Plasticity Thory 006 Lctur 4 Plastic flow PLASTIC FLOW Sanhng llo spnningr og tøyningr i dt plastisk orådt Fundantals I dt lastisk orådt gir Hook s lov forholdt llo spnning og tøyning = ν + E = ν + E = ν + E )) )) )) På sa åt r dt nødvndig å dfinr tøyningn i dt plastisk orådt Plastisk dforasjonr r irrvrsibl vi-avhngig path dpndnt) invariant til hydrostatisk spnningr konsrvrr volu Da plastisk dforasjonr r irrvrsibl og vi-avhngig å inkrntll forulring innførs Plasticity thory Fall 006 AR 006

2 Plasticity Thory 006 Lctur 4 Plastic flow Lvy-Miss quations ) Assuptions Th lastic strains ar ngligibl: idal plastic solid Th atrial is isotropic Yilding in uniaxial tnsion 0, = = 0 =, = =, = = 0 = = = Gnralization givs th Lvy-Miss quation d d d = = = dλ Constancy of volu + isotropy d + d + d = 0 d = d = d Cobining th quations d d = = = = d d d d d = = Plastic ultiplir Plasticity thory Fall 006 Forholdt llo plastisk tøyningsinkrnt og dviatorisk spnning r konstant Lvy-Miss quations ) Th Lvy-Miss flow rul d= dλ= d λ ++ ) = d λ +) = d λ ) + d = dλ = d λ ) + d = dλ = d λ ) + Volu constancy during plastic flow rquirs d V = d + d + d = 0 It is straightforward to chck that th Lvy-Miss flow rul prsrvs bvarr) th volu during plastic dforation d V = d λ ) ) ) = 4 Plasticity thory Fall 006 AR 006

3 Plasticity Thory 006 Lctur 4 Plastic flow Effctiv strain ) 5 Th incrntal plastic work rads p dw =d +d +d Th ffctiv strain is dfind by antar at inkrntlt plastisk arbid kan uttrykks vd n ffktiv spnning og ffktiv inkrntll tøyning p dw =d +d +d =d or quivalntly d = d +d +d ) Using th Lvy-Miss flow rul and th xprssion for th von Miss ffctiv strss, w gt aftr so calculations) d = dλ Th Lvy-Miss flow in trs of th ffctiv strain incrnt d d = +) d d = +) d d = + ) Plasticity thory Fall 006 Effctiv strain ) Th following xprssions for th ffctiv plastic strain incrnt can now b drivd tdious xrcis!) d = d d ) d d ) d d ) + + d = d + d + d) Th ffctiv or accuulatd) plastic strain is dfind by = = + + d d d d ) 6 Plasticity thory Fall 006 AR 006

4 Plasticity Thory 006 Lctur 4 Plastic flow Effctiv strain ) In uniaxial tnsion, th following rlation btwn th strain incrnts xists, du to isotropy and plastic incoprssibility d = d = d and d 0 Th ffctiv plastic strain incrnt in uniaxial tnsion d = d + d + d ) = d + d = d d = In uniaxial tnsion, th ffctiv plastic strain quals th plastic strain in th dirction of th strss = = p ) 7 Plasticity thory Fall 006 Constitutiv rlation In a uniaxial strss stat, th strss-strain rlation was dfind by p = f ) but, in this cas, w hav th rlations p = & = Thus, in ultiaxial strss stats, th strss-strain rlation, or th constitutiv rlation, is dfind in trs of th ffctiv strss and strain, as = f ) 8 Plasticity thory Fall 006 AR 006

5 Plasticity Thory 006 Lctur 4 Plastic flow Th plastic paratr Graphical intrprtation of plastic paratr dλ in th Lvy-Miss flow rul = f ) d dλ= tanα α d 9 Plasticity thory Fall 006 Associatd flow rul norality ) Th von Miss yild critrion ay b writtn in th for = f,, ) = ) ) ) Y + + = whr f is th yild function, which is a function of th strss stat. Th partial drivativs of th yild function f ) )) = ) + ) + ) 4 ) = = 4 + ) f = + ) f = + ) 0 Plasticity thory Fall 006 AR 006

6 Plasticity Thory 006 Lctur 4 Plastic flow Associatd flow rul norality ) Th partial drivativs of th yild function f ar proportional to th plastic strain incrnts Lvy-Miss flow rul) f + ) d f + ) d f + ) d Th Lvy-Miss flow rul can thus b writtn as f f f d = d λ, d = d λ, d = dλ Plasticity thory Fall 006 Associatd flow rul norality ) Dnn forn av flytrgln kalls ntn assosirt flytrgl - inkrntll plastisk tøyningsvktor r proporsjonal d gradintn til flytfunksjonn, llr noralittsrgln - inkrntll plastisk tøyningsvktor r noral på flata dfinrt av konstant vrdir av flytfunksjonn i spnningsrot. Backofn, 97) Plasticity thory Fall 006 AR 006

7 Plasticity Thory 006 Lctur 4 Plastic flow Flow ruls, norality & convxity Y d p dp - Y Y - Y NB Inkrntlt plastisk arbid å vær størr llr lik 0 for plastisk dforasjonsprosssr: d p 0 Bruk av noralittsprinsippt flytflata bør vær konvks for å unngå ngativt inkrntlt plastisk arbid! Plasticity thory Fall 006 Proportional strain paths Th ratios btwn th principal strain incrnts ar constant, and intgration lads to = + + ) Th total strains ar thus givn as functions of th strsss = + ) = + ) = + ) 4 Plasticity thory Fall 006 AR 006

8 Plasticity Thory 006 Lctur 4 Plastic flow Proportional strain paths in plan strss ) Principal strains assuing ngligibl lastic strains) > 0, =β, = +β) Principal strsss > 0, =α, = 0 von Miss yild function = + = α+α =Y Th flow rul d d d = = = + ) = +α ) = +α) = = +α ) = α) d d d = = α α +α) = =α +α = α = = 0 +α ) = +α) ) ) 5 Plasticity thory Fall 006 Proportional strain paths in plan strss ) Intgration of flow rul = = α α +α) Rlations btwn strss ratio and strain ratio α β+ β=, α= α +β Effctiv plastic) strain incrnt d = d + d + d ) 4 d = +β+β ) d ) ) ) ) ) d + d + d = d +β d + +β d = d +β + +β = d +β + + β+β ) = d +β+β Intgration givs th ffctiv plastic) strain 4 = +β+β ) 6 Plasticity thory Fall 006 AR 006

9 Plasticity Thory 006 Lctur 4 Plastic flow Bibliography Backofn, W.A., 97. Dforation Procssing. Addison-Wsly, Rading, Mass. Ditr, G.E., 988. Mchanical Mtallurgy. McGraw Hill, Singapor. Khan, A.S., Huang, S., 995. Continuu Thory of Plasticity. John Wily & Sons Inc., Nw York. 7 Plasticity thory Fall 006 AR 006