HSNV122 - Thermoplasticity and metallurgy in large deformations in simple tension

Transcription

1 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 1/11 Resonsable : Jean ANGLES Clé : V Révision : HSNV122 - Thermolasticit and metallurg in large deformations in simle tension Summarized: One treats e determination of e mechanical evolution of a clindrical bar subjected to known and uniform evolutions ermal and metallurgical (e metallurgical transformation is of bainitic te) and to a mechanical loading of tension. The behavior model is a model of lasticit in large deformations (command STAT_NON_LINE, ke word DEFORMATION: SIMO_MIEHE ) wi linear isotroic hardening and lasticit of transformation. The ield stress and e sloe of curve of tension deend on e temerature and e metallurgical comosition. The coefficient of ermal exansion deends on e metallurgical comosition. The bar is modelled b axismmetric elements. The mechanical loading alied is a following ressure. This case test is identical to e case test HSNV101 (modelization B, [V ]) in e meaning where it acts of e same material, e same loading and e same ermal and metallurgical evolutions but in a version in large deformations. Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

2 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 2/11 Resonsable : Jean ANGLES Clé : V Révision : Problem of reference 1.1 Geometr Raon : a = 0.05 m Hauteur : h = 0.2 m z P C D h r A B 1.2 Proerties of e material e material obes a constitutive law in large deformations wi linear isotroic hardening and lasticit of transformation. For each metallurgical hase, e hardening sloe is given in e logarimic strain lane - rational stress. a F = = S F S o l. l o hase E E T hase ln( l / l o ) l o and l are, resectivel, e initial leng and e current leng of e useful art of e test-tube. S o and S are, resectivel, surfaces initial and current. Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

3 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 3/11 Resonsable : Jean ANGLES Clé : V Révision : wi C 6 aust 6 o 6 E = Pa = Pa ( T T ) 10 Pa ν α α fbm aust ref fbm ref = J m C λ = W m C = 0. 3 = Pa ( T T ) 10 fbm 6 o 6 Pa o 6 aust = C h = Pa 5. ( T T ) 10 Pa o 6 C h fbm Pa T T Pa = = ( ) 10 1 K f Pa = = C b M Pa 1 T = 900 K = K = 10 C = ermal F = 2. (1- Z ) heat = conductivit ca acit E = Young's modulus = Poisson's ratio * aust = characteristic relating to e austenitic hase * fbm = characteristic relating to e hases ferritic, bainitic and martensitic = ermal coefficient of ermal exansion ref = strain of e hases ferritic, bainitic and martensitic fbm e reference temerature, austenite being regarded as not deformed wi is temerature = ield stress h = EET E ET K = coefficient relating to e lasticit of transformation F = function relating to e lasticit of transformation fbm e TRC used makes it ossible to model a metallurgical evolution of bainitic te, on all structure, of e form: Z fbm = 1.3 Boundar conditions and loadings fbm 0. si t τ1 τ1 = 60s t τ1 si τ1 t < τ2 τ2 = 112 s τ2 τ1 1. si t τ2 u Z =0 on e face AB (condition of smmetr). tension imosed (following ressure) on e face CD : Note: ( t ) o t our t τ1 = 6 t Pa our τ 1 1 o 6 = 6 10 Pa τ = 60s Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

4 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 4/11 Resonsable : Jean ANGLES Clé : V Révision : In large deformations, it is essential to use e following ressure to take account of current surface and not of initial surface (before strain). o T = T + µ t, = 5 C.s 1 on all structure. 1.4 Initial conditions T o = 900 C = T ref 2 Reference solution 2.1 Comutation of e reference solution (cf feeding-bottle [1] and [3]) For a traction test according to e direction x, e tensors of Kirchhoff and Cauch are form: τ 0 0 τ = and = wi τ = J e variation of volume J is given b e resolution of J 2 K J J ( 3 + ) 3 = where is e ermal strain. This one alies to an austenitic transformation bainitic: Note: ref ref [ fbm fbm ] = Z α ( T T ) + Z α ( T T ) + aust aust ref b The coefficient K is e bulk modulus (not to be confused wi e coefficients K hase relating to e model of lasticit of transformation) In lastic load, for a linear isotroic R hardening, such as: R = ( Z h + Z h ) aust aust b fbm e cumulated lastic strain is wor = J Z h + Z h aust aust b fbm wi = Z + Z aust aust b fbm Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

5 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 5/11 Resonsable : Jean ANGLES Clé : V Révision : e tensor gradients of e transformation F and F e strain tensor lastics G are form: F = J= det F = FF F = J / F F 0 0 F F = F = = - 1/ 3 F J F - 1/ 3 = J 0 F 0 et det 1 / F = F 0 0 F G = F F F G G G et et det G The law of evolution of e lastic strain G P is written: 2 = 1 G G / G = 2 4τ K ( 1 Z ) Z b b b 1 2 = ( G ) For 0s t 60s, ere A. Zb = 0 It is not ere lasticit of transformation. One obtains en: G = e 2 For 60s t 176s, one A. = constante to integrate e law of evolution of e lastic strain, it should be suosed at e stress of Kirchhoff varies ver little, i.e. e variation of volume J is ver small. Under is assumtion, one obtains G e e 2 ( ) K b Z b Z b = 2 4τ / 2 1/ 2 e comonent F of e gradient of e transformation is given b e resolution of: F 3 τ G F = 0 µ ( G ) / Lastl, e field of dislacement u (in e initial configuration) is form u = uxx + uy + uz Z. The comonents are given b: u = ( F 1) X x ( / 1) ( / 1 ) u = J F Y u = J F Z z Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

6 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 6/11 Resonsable : Jean ANGLES Clé : V Révision : Notice In e case test HSNV101 (modelization B), e coefficients of e material were selected of such wa not to have classical lasticit = 0 during e metallurgical transformation which takes lace between times 60 and 122s. Indeed if one writes e criterion of load-discharge in is time interval, one obtains f = wi = 360 MPa whom cancels oneself onl for onl one value of e cumulated lastic strain. For e constitutive law written in large deformations, e criterion of load-discharge is written between ese two times f = J( t ) wi = 360 MPa In is case, as long as e variable J remains lower an e value obtained at time t=60s, one will have = 0. However e value of J is function onl of e value of e ermal strain (e stress is constant and e coefficient K is indeendent of e metallurgical hases and e temerature). In is time interval, e ermal strain is given b e following equation: = t t One traces below e ermal strain as well as e variation of volume J, solution of e equation of e 3rd degree, according to time. Title: e Creator: cano avec agraf + Ilog Views CreationDate: 08/11/ :42:57 Thermal strain according to time Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

7 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 7/11 Resonsable : Jean ANGLES Clé : V Révision : Title: J Creator: cano avec agraf + Ilog Views CreationDate: 08/11/ :50:01 Variation of volume J according to time One notes at e variable J decreases and increases same wa as e ermal strain. In is case, to know time from which e variable J is higher an e value obtained at time 60s, it is enough to know e time for which e ermal strain is identical to at obtained at time t=60s. One finds b e resolution of e equation above t=84.46s. 2.3 Uncertaint on e solution e solution is analtical. Two mistakes are made on is solution. The first door on e comutation of e bainitic roortion of hase created. The comutation metallurgical recondition does not restore exactl e equation of [ 1.2] giving Z fbm according to time, is is wh e results of reference resented below are calculated wi e bainitic roortion of hase calculated b Code_Aster. The second error is e assumtion made on e stress of Kirchhoff which is not constant on e time interval understood enters 60 and 176s. This will imact e comutation of dislacement u x and e lastic strain G P. 2.4 Results of reference One will adot like results of reference dislacement in e direction of e loading of tension, e stress of Cauch, e Boolean indicator of lasticit and e cumulated lastic strain. Various times of comutations are t=47, 48,60,83,84,85 and 176s. For e comutation of dislacement, e initial leng of e bar in e direction of loading is of 0.2m. Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

8 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 8/11 Resonsable : Jean ANGLES Clé : V Révision : In all e cases, one has 3K=500000MPa (bulk modulus) = MPa At time t=47s, one has Z b = 0 T = 665 C, = 282 MPa = J = τ = = MPa = 0 χ = 0 MPa G = 1 F = u = At time t=48s, one has Z b = 0 T = 660 C, = 288 MPa = J = τ = = 280. MPa = χ = 1 MPa G = F = u = At time t=60s, one has Z b = 0 T = 600 C, = 360 MPa = J = τ = = 250. MPa = χ = 1 G = F = u = At time t=83s, one has Z b = T = 485 C, = 360 MPa 2 MPa 4 m 4 m m = J = τ = MPa -2 = MPa = χ = 0-2 m G = F = u = At time t=84 S, one has Z b = T = 480 C, = 360 MPa = J = τ = MPa -2 = MPa = χ = 1-2 G = F = u = At time t=85s, one has Z b = T = 475 C, = 360 MPa = J = τ = MPa -2 = MPa = χ = 1-2 G = F = u = At time t=176s, one has Z b = 1 T 2 = 20 C, = 360 MPa = J = τ = = 90. MPa = χ = 1 G = F = u = MPa 2 m Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

9 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 9/11 Resonsable : Jean ANGLES Clé : V Révision : bibliograhical References One will be able to refer to: 1) V. CANO, E. LORENTZ: Introduction into e Code_Aster of an elastolastic model of behavior in large deformations wi isotroic hardening internal Note EDF DER HI - 74/98/006/0 2) A.M. DONORE, F. WAECKEL: Influence structure transformations in e elastolastic constitutive laws Notes HI-74/93/024 3). WAECKEL F, V. CANO: Constitutive law large deformations élasto (visco) lastic wi metallurgical transformations [R ] Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

10 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 10/11 Resonsable : Jean ANGLES Clé : V Révision : Modelization A 3.1 Characteristic of e modelization C D N13 N11 N12 N9 N10 N7 N6 N8 N1 N3 N4 N2 N5 A B A= N4 B=N5 C= N13 D= N12. Charge: e nombre total of increments is of 102 (4 increments of 0 wi 46s, 2 increments of 46 wi 48s, 6 increments of 48 wi 60s, 26 of 60 wi 112s, 4 of 112 wi 116s and 60 increments until 176s ). Convergence is carried out if residue (RESI_GLOB_RELA) is lower or equal to Characteristics of e mesh Man nodes: 13 Number of meshes and tes: 2 meshes QUAD8, 6 meshes SEG3 3.3 Values tested Identification Reference t=47 Dislacement DY N m t=47 Variable VARI M1, PG1 0. t=47 VARI M1, PG1 0 t=47 Stress SIGYY M1, PG Pa t=48 Dislacement DY N m t=48 Variable VARI M1, PG t=48 VARI M1, PG1 1 t=48 Stress SIGYY M1, PG Pa t=60 Variable DY N13 Dislacement t=60 m VARI M1, PG t=60 VARI M1, PG1 1 t=60 Stress SIGYY M1, PG Pa Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://

11 Titre : HSNV122 - Thermo-lasticité et métallurgie en gran[...] Date : 27/02/2013 Page : 11/11 Resonsable : Jean ANGLES Clé : V Révision : t=83 Variable DY N13 Dislacement t=83 m VARI M1, PG t=83 VARI M1, PG1 0 t=83 Stress SIGYY (M1, PG1) Pa t=84 Variable DY N13 Dislacement t=84 m VARI M1, PG t=84 VARI M1, PG1 1 t=84 Stress SIGYY M1, PG Pa t=85 Variable DY N13 Dislacement t=85 m VARI M1, PG t=85 VARI M1, PG1 1 t=85 Stress SIGYY (M1, PG1) Pa t=176 Variable DY N13 Dislacement t=176 m VARI M1, PG t=176 VARI M1, PG1 1 t=176 Stress SIGYY M1, PG Pa 4 Summar of e results Them results found wi Code_Aster are ver satisfactor wi ercentages of error lower an 0.9%, knowing at e analtical solution of reference makes e dead end on certain asects which into account recisel e solution of Code_Aster takes. This can exlain e differences observed. Warning : The translation rocess used on is website is a "Machine Translation". It ma be imrecise and inaccurate in whole or in art and is rovided as a convenience. Licensed under e terms of e GNU FDL (htt://