Plasticity of Metals Subjected to Cyclic and Asymmetric Loads: Modeling of Uniaxial and Multiaxial Behavior

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1 Plastcty of Metals Subjecte to Cyclc an Asymmetrc Loas: Moelng of Unaxal an Multaxal Behavor Dr Kyrakos I. Kourouss Captan, Hellenc Ar Force 1/16

2 Abstract Strength analyss of materals submtte to cyclc loangs requres the knowlege of the stress state; partcularly n those cases that the materal behavor s greatly epenent to the hstory of the apple loas. Ths n turn necesstates the stuy of plastcty uner cyclc loas (cyclc plastcty), a case sgnfcantly more complcate compare to plastcty mpose by monotonc loang. The nee to escrbe the complex phenomena relate to varous engneerng applcatons has rven the evelopment of a large number of moels use for the precton of materals nelastc behavor. A new class of plastcty moels [base on the Armstrong an Freerck (AF) knematc harenng rule], capable of prectng successfully the cyclc elastoplastc response of metallc materals uner unaxal an multaxal loangs s ntrouce. The basc novelty of ths class of moels s the transformaton of one of the two constant parameters of the AF knematc harenng rule to a varable parameter (non mensonal tensor). Ths alteraton ntrouces a multplcatve scheme nterconnectng the assocate tensors. The numercal mplementaton of the basc moel, an ts assocate mofcatons, covers successfully a we range of unaxal an multaxal expermental results, ncatng ther capablty to smulate wth success the phenomena assocate wth cyclc an asymmetrc loa applcaton. /16

3 Cyclc Plastcty Phenomena (a) (a) (b) (b) Cyclc harenng: (a) Stran Controlle (b) Stress Controlle Cyclc softenng: (a) Stran Controlle (b) Stress Controlle 3/16

4 Cyclc Plastcty Phenomena (a) (b) Loang uner non zero mean stress: (a) Plastc shakeown (b) Ratchetng (a) (b) (a) Non relaxaton of mean stress (b) relaxaton of mean stress 4/16

5 Mathematcal Theory of Plastcty General Objectve: The escrpton of problems relate to tme nepenent non recoverable strans Consttutve Elastoplastcty Moels: Stran Decomposton Generalze Hooke s Law Yel Functon Flow Rule (assocatve) Harenng Rule Loang Inex Consstency Conton e p E f p qn e p : = E : σq, n 0 f f q 1 K p ˆn f : f f f σ q q σ, q σ q 0 K qˆ n n n p n n 5/16

6 Knematc Harenng Moels Armstrong & Freerck (AF) (Armstrong & Freerck, 1967) p p s a h ceqa ca n a 3 3 Mult-component Armstrong & Freerck (MAF) (Chaboche et al, 1979) a a s a c a n a 3 Mult-component Armstrong & Freerck wth Threshol Term (MAFT) (Chaboche, 1991) m1 a a s a c a n a ( 1,,..., m 1) 3 s am a c a n 1 a 3 f am f m m m m 3 a a : a m m m 6/16

7 Knematc Harenng Moels Burlet, Calletau & Dellobelle (BCD) (Burlet & Calletau, Dellobelle et al, 1995) δ parameter: constant, whle: 0 < δ < 1 BCD mofcaton wth r term (BCDr) (Dafalas & Fegenbaum, 010) The varable term r substtutes constant parameter δ a a s a c a n 1 : 3 3 a a n n r 3 a : a a s s a c a n r 1 r : 3 3 a a n n The MAFT moel can be combne wth BCD moel (-r /-δ): MAFT + BCD MAFTδ Same happens for BCDr moel: MAFT + BCDr MAFTr s a c a n 3 3 a a n n 1 : 1,,3 s a4 c4 a4n : a a n n f f 3 a a : a a4 a 4 7/16

8 Multplcatve AF Knematc Harenng Decomposton: Mult-component AF wth Multpler (MAFM) (Dafalas, Kourouss, Sars, 008) Back stress (mensonless) AFM an MAFM Moel a a 1:4 s a c a 1,,3 3 n a 3 * * s * s 4 a c4 c4 a4 4 : a a n 3 n a 3 a c a 3 n a 3 * * * s * AFM moel (AF wth Multpler): Multpler (mensonless) Plastc Stran (%) Smulates successfully the knee regon n the stress-stran curves, whch cannot be smulate by the smple AF moel Behaves smlarly to MAFT moel, whle smulatng the smooth ntal slope an smooth saturaton, wthout havng to contnuously check overshootng of the threshol value 8/16

9 MAFMδ an MAFMr Moel MAFMδ (δ term) MAFM + BCD MAFMr (r term) MAFM + BCDr s a c a n 3 3 a a n n 1 : 1,,3 * * * s s s a4 c4 c4 a4 4 : a : 3 a n 3 n 3 a a n n * * * s * a4 c4 a4 4 3 n a 3 s a c a n r r 3 3 a a n n 1 : 1,,3 * * s * s s 4 a c4 c4 a4 4 : a4 r4 4 1 r4 4 : 3 a n 3 n 3 a a n n a c a 3 n a 3 r * * * s * a : a a s 9/16

10 Cofcaton of the varous Moels Moel Descrpton Analyss AF Armstrong & Freerck moel --- MAF Mult-component AF moel AF 1 + AF + AF n MAFT Mult-component AF moel wth Threshol term MAF + AF Threhol AFM AF moel wth Multpler AF + AF Multpler MAFM Mult-component AF moel wth Multpler MAF + AFM BCD Burlet, Calletau & Dellobelle moel AF τροποποίηση δ BCDr Burlet, Calletau & Dellobelle moel wth r term BCD με r (αντί δ) MAFTδ MAFT moel wth δ term mofcaton MAFT + BCD ΜΑFMδ MAFT moel wth δ term mofcaton MAFM + BCD MAFΤr MAFT moel wth r term mofcaton MAFT + BCDr MAFMr MAFT moel wth r term mofcaton MAFM + BCDr 10/16

11 Unaxal Applcaton Stepwse cyclc harenng Ratchetng MAFT MAFM MAFM Experment: Chaboche et al, 1979 Experment-MAFT: Bar & Hassan, /16

12 Multaxal Applcaton Type Ι Loang Stran symmetrc cyclc uner constant pressure [CS106] Type ΙΙ Loang Bow-te [CS106] Type ΙΙΙ Loang Reverse Bow-te [CS1018] Type IV Loang Inclne path postve slope [CS106] Type V Loang Inclne path postve slope [CS1018] Type VI Loang Hourglass [CS1018] 1/16

13 Multaxal Applcaton Type Ι Loang Stran symmetrc cyclc uner constant pressure Comparson: MAFT vs MAFM (same performance) MAFTδ vs MAFMδ ( ) MAFTδ/ΜΑFMδ vs MAFMr ε θp (%) ε θp (%) Ν (5) (4) Ν (3) () (1) CS106 Type I Loang σ θa =,36 ks ε xc = 0,50% σ θm = 7,3 ks Πειραματικά Experment Δεδομένα MAFT MAFM MAFTδ (δ=0,18) ΜΑFMδ (δ=0,18) MAFMr CS106 Type I Loang σ θa =,36 ks ε xc = 0,5% σ θm = 7,3 ks Πειραματικά Experment Δεδομένα (1) ΜΑFMδ (δ=0,18) () MAFMδ (δ=0,5) (3) MAFMδ (δ=0,75) (4) MAFMδ (δ=0,90) (5) ΜΑFMδ (δ=0,99) 13/16

14 Multaxal Applcaton Type ΙΙΙ Loang Reverse Bow-te ε θp (%) () () () (1) σ θa =,36 ks ε xc = 0,5% (1) σ θm = 9,65 ks () σ θm = 14,54 ks 1.0 (1) (1) (1) Πειραματικά Experment Δεδομένα () Πειραματικά Experment Δεδομένα (1) MAFMδ (δ=0,18) 0.5 () MAFMδ (δ=0,18) (1) MAFMδ (δ=0,5) () ΜΑFMδ (δ=0,5) Comparson: MAFTδ vs MAFMδ (for varous δ values) ε θp (%) N () () () (1) (1) (1) σ θa =,36 ks ε xc = 0,5% (1) σ θm = 9,65 ks () σ θm = 14,54 ks (1) Πειραματικά Experment Δεδομένα () Πειραματικά Experment Δεδομένα (1) MAFΤδ (δ=0,18) Για δ=0,18: best results 0.5 () MAFΤδ (δ=0,18) (1) MAFΤδ (δ=0,5) () ΜΑFΤδ (δ=0,5) 0.0 N /16

15 Multaxal Applcaton ε θp (%) 3.0 Type VI Loang Hourglass.5.0 CS1018 Type VI Loang σ θa =,36 ks ε xc = 0,5% σ θm = 9,65 ks Πειραματικά Experment Δεδομένα MAFT MAFM MAFTδ (δ=0,18) ΜΑFMδ (δ=0,18) MAFMr ε θp (%) Ν Comparson:.5.0 () CS1018 Type VI Loang σ θa =,36 ks ε xc = 0,5% σ θm = 9,65 ks MAFT vs MAFM (same) 1.5 (1) MAFTδ vs MAFMδ (same) MAFTδ/ΜΑFMδ vs MAFMr Ν Πειραματικά Experment Δεδομένα (1) ΜΑFMδ (δ=0,18) () MAFMδ (δ=0,5) 15/16

16 Conclusons The propose moel (MAFM) s consere to be a sgnfcant mofcaton of the classcal MAF moel (atve back stress ecomposton) The MAFM moel an the mofcatons of the MAFMδ an MAFMr moels have smlar behavor performance to the MAFT moel (threshol term) an ts mofcatons MAFTδ an MAFTr moels Through the BCDr mofcaton, emboe n the moels MAFMr και ΜAFTr, the ntrnsc efcency of the BCD moel s correct, also for the cases of the MAFMδ an ΜΑFTδ moels The use of the MAFM moel rather than the MAFT moel offers computatonal smplcty snce there s no nee to contnuously check f the threshol value s exceee The unaxal an multaxal applcaton of the moels MAFM, MAFΜδ an ΜΑFMr n a large number of cases (use of publshe expermental ata an computatonal results) valate the capabltes an sutablty of the moels The multaxal smulatons may be mprove through the emboment of the rectonal stortonal harenng rule (propose by Dafalas & Fegenbaum, 007) n the MAFM moel 16/16

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