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1 Local resolution of constitutive laws Code_Aster, Salome-Meca course material GNU FDL licence (htt://

2 GENERAL CONCEPTS 2 - Code_Aster and Salome-Meca course material GNU FDL Licence

3 General concets Defining solid Ω which is in equilibrium with external forces ( u) : ( u). d f ( u) : u. d Unknowns (3D): - Dislacement field (3 comonents) - Stress field (6 comonents) Equations (3D): - (weak) equilibrium (3 equations) Six equations missing! => behaviour law required to close the system g g( u) : u. d 3 - Code_Aster and Salome-Meca course material GNU FDL Licence

4 General concets The behaviour law: Identify from exerimental: from try to comlete structure (reresentativity) Using formalism (general rooves for convergence) Develo in non-linear framework Test (verification and validation) 4 - Code_Aster and Salome-Meca course material GNU FDL Licence

5 General concets Examle: Behaviour law for metal from 1D tensile-test (lasticity) 5 - Code_Aster and Salome-Meca course material GNU FDL Licence

6 General concets: examle 1/ Elaborate a reresentative try F Loading monotone t We can observe : An elasticity domain with a Yield Stress An irreversible strain P A hardening caracterized by P Y Y Plasticity, a henomenon that is indeendent of velocity 6 - Code_Aster and Salome-Meca course material GNU FDL Licence

General concets : examle 2/ Identify to extend to the REV (Reresentative Elementary Volume) First art: the elastic domain until Y The elasticity domain is defined by : f ( ) 0 Exerimentaly

7 General concets : examle 2/ Identify to extend to the REV (Reresentative Elementary Volume) First art: the elastic domain until Y The elasticity domain is defined by : f ( ) 0 Exerimentaly : f is a function of more articulary D 1 dev() J1 dev( ). dev( ) with 2 f ( D ) f ( J, J 1 ) J2 tracdev( ). dev( ). dev( ) Code_Aster and Salome-Meca course material GNU FDL Licence

8 General concets : examle 2/ Identify to extend to the REV (Reresentative Elementary Volume) Second art: the lastic domain from Y The artition of strains: e For elastic case : e E:. tr( ) 2. e e 8 - Code_Aster and Salome-Meca course material GNU FDL Licence

9 General concets : examle A general theoretical framework: mechanics and thermodynamics First law of thermodynamics: E int Ecin Pext Qext t t With PPV, we can formulate a variational internal energy: e : r div( q) t The second law of thermodynamics: the Inegality of Clausius-Duhem D d q : st. T dt T 0 With the Helmholtz s free energy 9 - Code_Aster and Salome-Meca course material GNU FDL Licence

10 General concets : examle The method of local state The thermodynamic state, at the oint and the instant considered, is entirely defined at this instant by the state variables (observable ( T, ) and internal V, V ) ). ( 1 k d dt e e T T V k V k D int d e d : d dt d dv e : s. k V k if V k T 0 and if V k 0, T The first state law : e The second state law : s T 10 - Code_Aster and Salome-Meca course material GNU FDL Licence

11 General concets : examle State Variables observable T e internal Associated thermodynamic forces s State laws s T e Vk Ak A k V k Evolution of internal state variables => irreversible dissiation formalism 11 - Code_Aster and Salome-Meca course material GNU FDL Licence

General concets : examle A formalism for lasticity: the maximum lastic work (Hill 1951) In short, the rincile ostulate two imortant ideas : - the yield surface must be convex function, - the

12 General concets : examle A formalism for lasticity: the maximum lastic work (Hill 1951) In short, the rincile ostulate two imortant ideas : - the yield surface must be convex function, - the lastic strain rate is normal to the yield surface Evolution law (or law of normality) f Outward normal to the boundary of the domain n f 12 - Code_Aster and Salome-Meca course material GNU FDL Licence

13 General concets : examle A formalism for lasticity: Intensity of the flow The lasticity multilier is determined by the consistency relation df 0 f A k dak 0 Consistency relation f Code_Aster and Salome-Meca course material GNU FDL Licence

14 General concets : examle Summary, a lasticity theory: Defining hardening Defining yield surface Defining flow direction (normal = associative law) Défining flow intensity (lastic multilicator) 14 - Code_Aster and Salome-Meca course material GNU FDL Licence

15 General concets : isotroic hardening 3 D f (, ) R( ) 2 An isotroic extension of the elasticity domain is taking into account: Dilatation of the elasticity domain Evolution of criterion is governed by a single scalar (internal state variable: cumulated lastic strain) y 15 - Code_Aster and Salome-Meca course material GNU FDL Licence

16 General concets : kinematic hardening f (, ) D X y An translation of the elasticity domain is taking into account: Translation of the elasticity domain Evolution of criterion is governed by a tensor (internal state variable: centre of the elasticity domain) 16 - Code_Aster and Salome-Meca course material GNU FDL Licence

17 17 - Code_Aster and Salome-Meca course material GNU FDL Licence General concets : final system to solve Partition of strains Elastic strains Plasticity criterion Flow law (normality) Material arameters e e e tr. 2 ) (. y D R f ) ( 2 3 ), ( y D X f ), ( D D ), ( 0 if 0 ), ( 0 if f f 2 3 X X D D ( ) R C C X

18 SOLVING BEHAVIOUR LAWS 18 - Code_Aster and Salome-Meca course material GNU FDL Licence

19 Solving behaviour law Algorithm: 1. Define functional 2. Solve functional Behaviour law is a functional for stress from strains, external state variables (temerature ) and internal state variables F, T, V k Newton s method: need jacobian too!

20 Solving behaviour law Load increment: R( 0 u ) 1 F i Fi 1 n-1 Calculation tangent matrix K( : ) u i u n i u n-1 i n1 1 n1 K ( u ) R( u ) i i Calculation : n i and n i Residu calculation Test of convergence 20 - Code_Aster and Salome-Meca course material GNU FDL Licence

21 Solving behaviour law From ODE equations => time discretization Imlicit choice: stability The choice of the time ste deends on the radial nature of the roblem Unknown variables at time ste: incremental scheme for stress,v k Scheme for internal state variables F, V k V Incremental choice for stress udate: only for small strains! F V and F are non-linear functionals to solve F V k 21 - Code_Aster and Salome-Meca course material GNU FDL Licence

22 Solving behaviour law: examle of algorithm Test : 3 2 D D R( ) y D D 2 3Ktr Oui Non D R'( ) 3 2 n D D : D D 2 R( - +) R( - ) D 3Ktr( ) Code_Aster and Salome-Meca course material GNU FDL Licence

23 Solving behaviour law F V and F (seudo)-time integration: imlicit or exlicit (code_aster: mainly imlicit) Deending of Non-linear solving: Newton s method, line-search, Warning! Hyothesis to solve non-linear equation! Imlicit algorithm => unconditional stability BUT when solve ODE using RADIAL hyothesis of loads Parameters for non-linear solving of behaviours laws: In the COMPORTEMENT keyword Sometimes, you can choose local non-linear algorithm to solve 23 - Code_Aster and Salome-Meca course material GNU FDL Licence

24 Integration of constitutive laws: sum u General non-linear algorithm: In ractice, only a few laws in code_aster Solving the NL local system of n equations Exlicit method (Runge-Kutta) or imlicit method (Newton) Secific non-linear algorithms: For some laws (in fact, most of the laws in code_aster!) The system is reduced to one single scalar equation Solved by various methods (secant, Newton, Dekker, Brent) Analytical solution for some laws (ex: Von Mises isotroic hardening and / or linear kinematic) 24 - Code_Aster and Salome-Meca course material GNU FDL Licence

25 BEHAVIOUR LAWS IN CODE_ASTER 25 - Code_Aster and Salome-Meca course material GNU FDL Licence

26 Constitutive laws available More than 160 laws in the 13 stable version Various fields of alications Metals, olycrystalline metals Concrete Soils Various henomena Irradiation Damage or cracking Metallurgical hases Documentation Synthesis of non-linear constitutive laws: U DEFI_MATERIAU syntax: U Code_Aster and Salome-Meca course material GNU FDL Licence

27 Constitutive laws available 2D and 3D continuum media Non linear elasticity Von Mises isotroic Pseudo-hardening ELAS_VMIS_LINE ELAS_VMIS_TRAC ELAS_HYPER Incremental elasto-lasticity Von Mises isotroic hardening, kinematic linear, mixed VMIS_ISOT_TRAC VMIS_ISOT_PUIS VMIS_ISOT_LINE VMIS_CINE_LINE VMIS_ECMI_TRAC VMIS_ECMI_LINE 27 - Code_Aster and Salome-Meca course material GNU FDL Licence

28 Constitutive laws available 2D and 3D continuum media (cont.) Other elastolastic models (metals) VMIS_CIN1_CHAB VMIS_CIN2_CHAB VMIS_CIN2_MEMO Polycrystalline metals POLY_CFC MONOCRISTAL POLYCRISTAL Elasto-visco-lasticity (metals) LEMAITRE, LEMA_SEUIL VISC_CIN1_CHAB VISC_CIN2_CHAB VISC_ISOT_LINE VISC_ISOT_TRAC VISC_TAHERI VISCOCHAB Limit loads NORTON_HOFF Elasto-visco-lasticity under irradiation LMARC LEMAITRE_IRRA GATT_MONNERIE VISC_IRRA_LOG GRAN_IRRA_LOG IRRAD3M 28 - Code_Aster and Salome-Meca course material GNU FDL Licence

29 Constitutive laws available 2D and 3D continuum media (cont.) Damage or cracking of metals ENDO_FRAGILE VENDOCHAB ROUSSELIER ROUSS_PR ROUSS_VISC RUPT_FRAG BARENBLATT Metallurgical hases (elasto-visco-lastic) for steel or zirconium META_X_Y_Z X = P (lasticity) or V (viscosity) Y = IL (linear isotroic) or INL (nonlinear isotroic) or CL (linear kinematic) Z = RE (restoration) and/or PT (transformation lasticity) Concrete BETON_DOUBLE_DP GRANGER_FP GRANGER_FP_V GRANGER_FP_INDT BAZANT_FP ENDO_ISOT_BETON ENDO_ORTH_BETON MAZARS JOINT_BA CORR_ACIER KIT_DDI BETON_REGLE_PR BETON_UMLV_FP BETON_BURGER_FP BETON_RAG 29 - Code_Aster and Salome-Meca course material GNU FDL Licence

30 Constitutive laws available 2D and 3D continuum media (cont.) Soils and geomaterials DRUCK_PRAGER(N_A) CAM_CLAY, BARCELONE CJS, HUJEUX LAIGLE, LETK HOEK_BROWN KIT_HM, KIT_HHM, KIT_THH, KIT_THM, KIT_THHM Plates, shells and ies (local behaviour = lane stress) All 3D constitutive laws (thanks to the method if C_PLAN is not suorted: ALGO_C_PLAN = 'DEBORST') Bars, multi-fiber beams, grids All the laws of 1D behaviour (thanks to the DeBorst method if 1D is not suorted: ALGO_1D = 'DEBORST') Discrete elements, shear connections, reinforcements 30 - Code_Aster and Salome-Meca course material GNU FDL Licence

31 SYNTAX IN CODE_ASTER 31 - Code_Aster and Salome-Meca course material GNU FDL Licence

32 Syntax for the constitutive laws integration Choice of arameters for the integration: under the factor key word COMPORTEMENT General algorithm: Resolution of the local NL system of n equations ALGO_INTE = Exlicit resolution Imlicit resolution by a local Newton, with the ossibility of LInear REsearch for certain laws RUNGE_KUTTA NEWTON NEWTON_RELI VISCOCHAB, VENDOCHAB, POLYCRISTAL, MONOCRISTAL VMIS_POU_FLEJOU, VMIS_POU_LINE VISCOCHAB, LMARC, MONOCRISTAL, IRRAD3M, CJS, HUJEUX VISCOCHAB, LMARC, MONOCRISTAL, IRRAD3M 32 - Code_Aster and Salome-Meca course material GNU FDL Licence

33 Syntax for the constitutive laws integration General algorithm: Convergence Residue to achieve: RESI_INTE_RELA (10-6 ) Maximum number of iterations: ITER_INTE_MAXI (20) Tis For behaviour which are "difficult" to integrate, increase ITER_INTE_MAXI ssnd105b where ssnv172a where ssnl106i where ITER_INTE_MAXI = 250 for VISCOCHAB ITER_INTE_MAXI = 100 for MONOCRISTAL ITER_INTE_MAXI = 500 for VMIS_POU_LINE For certain behaviours, it is better to integrate finely the behaviour (ex: Hujeux) RESI_INTE_RELA = Code_Aster and Salome-Meca course material GNU FDL Licence

34 Syntax for the constitutive laws integration Secific algorithms: For some laws (in fact, most of the laws in code_aster!) The system is reduced to one single scalar equation : ( ) 0 Solved by various methods: ALGO_INTE = SECANTE, DEKKER, NEWTON_1D, BRENT Convergence: RESI_INTE_RELA (10-6 ), ITER_INTE_MAXI (20) Analytical resolution VMIS_ISOT_LINE, VMIS_ISOT_TRAC, VMIS_ISOT_PUIS, CZM_*, ENDO_SCALAIRE, No additional keyword is required! (excet for lane stresses) Ex: hsnv125a: VMIS_ISOT_LINE in 3D and ITER_INTE_MAXI = Code_Aster and Salome-Meca course material GNU FDL Licence

35 INCOMPRESSIBILITY 35 - Code_Aster and Salome-Meca course material GNU FDL Licence

36 Quasi-incomressibility For secial choice of Poisson ratio Examle: hyerelasticity (for elastomer) υ 0.5 In case of high level of lasticity (generalized lasticity) Plastic flow at constant volume imlies incomressibility condition Numerical consequences Too rigid behaviour Possibility of oscillations of the stress (fluctuation on the tensor trace) 36 - Code_Aster and Salome-Meca course material GNU FDL Licence

37 Quasi-incomressibility Notched secimen: yy Scour at the rib foot of a valve: Tr( )/ eq 37 - Code_Aster and Salome-Meca course material GNU FDL Licence

38 Quasi-incomressibility Two solutions: Sub-integration Quasi-incomressible model Sub-integration Integrals evaluated numerically (Gauss): Choice of sub-integrated finite elements in AFFE_MODELE: 3D_SI, AXIS_SI, D_PLAN_SI, C_PLAN_SI e ng f x d In 2D: QUAD4 1 Gauss oint instead of 4 i1 In 2D: QUAD8 4 Gauss oints instead of 9 In 3D: HEXA20 8 Gauss oints instead of 27 f i i Linear cell with sub-integration: hourglass control required 38 - Code_Aster and Salome-Meca course material GNU FDL Licence

39 Quasi-incomressibility Quasi-incomressible formulations Mixed formulation with 2 or 3 fields: Fields: dislacement, volumetric strain and the associated Lagrange multilier (which would corresond to the ressure in the incomressible case). Two fields for quadratic cells: 3D_INCO_UP, D_PLAN_INCO_UP, AXIS_INCO_UP Two fields for linear cells (OSGS stabilization or bubble function): 3D_INCO_UPO, D_PLAN_INCO_UPO, AXIS_INCO_UPO Three fields only for quadratic cells: 3D_INCO_UPG, D_PLAN_INCO_UPG, AXIS_INCO_UPG R : Finite elements dealing with the quasi-incomressibility Remarks: Formulation very effective, but a little more "exensive" Led to non-ositive matrices solver: MUMPS has to be referred 39 - Code_Aster and Salome-Meca course material GNU FDL Licence

40 KINEMATIC HYPOTHESIS 40 - Code_Aster and Salome-Meca course material GNU FDL Licence

41 Models of small deformations DEFORMATION = 'PETIT' HPP (assumtion of small erturbations, french abbreviation) everything is small! Lagrangian descrition = Eulerian descrition DEFORMATION = 'GROT_GDEP' Large dislacement & rotation Green-Lagrange strains Piola-Kirchhoff stresses in initial configuration ure hyer-elastic media (linear or nonlinear elasticity? rubber!) h: all other behaviours + large dislacements and rotations DEFORMATION = 'PETIT_REAC' Aroximation of large strains Small strains with geometry udate at each iteration: order 2 terms are neglected Total deformation = sum of linearized strain increments based on different configurations. Physical sense? Stresses: simle time derivation => not objective i.e. not rigid rotation invariance of the structure => PETIT_REAC is not suitable for large rotations Aroximation valid only for very small time increments, quasi-radial loading, elastic deformation small comared to lastic deformation Absence of geometric contribution to the tangent matrix 41 - Code_Aster and Salome-Meca course material GNU FDL Licence

42 Models of small deformations DEFORMATION = 'SIMO_MIEHE' Comrehensive aroach of large deformations Elastic deformations are measured in the current configuration (deformed) Plastic deformations are measured in the initial configuration Model incrementally objective: exact solution in the resence of large rotations. Very robust, quadratic convergence. Requires secially adated behaviour laws: elasto-(visco)-lasticity isotroic hardening von Mises, Rousselier. DEFORMATION = GDEF_LOG' Another aroach of large deformations: Logarithmic deformations (Miehe & Ael) Used with all HPP laws (kinematic, anisotroy, C_PLAN...) Costs more CPU time than SIMO_MIEHE 42 - Code_Aster and Salome-Meca course material GNU FDL Licence

43 Conclusion Documentation Utilisation U : synthesis of non-linear behaviour U : syntax of the command DEFI_MATERIAU Reference R : integration of isotroic hardening or kinematic linear laws R : taking into account the lane stresses R5.03.XX: integration of other behaviours R : imlicit and exlicit integration of nonlinear laws R : Hyothesis of lane stresses in non-linear behaviours R : Finite elements dealing with the quasi-incomressibility R : Elasto(visco)lastic modelling with isotroic hardening in large strains (SIMO_MIEHE) R : Behaviour law in large rotations and small deformations (GROT_GDEP) R : Models for large deformations GDEF_LOG 43 - Code_Aster and Salome-Meca course material GNU FDL Licence

44 End of resentation Is something missing or unclear in this document? Or feeling hay to have read such a clear tutorial? Please, we welcome any feedbacks about Code_Aster training materials. Do not hesitate to share with us your comments on the Code_Aster forum dedicated thread Code_Aster and Salome-Meca course material GNU FDL Licence

Continuation methods for non-linear analysis

Continuation methods for non-linear analysis FR : Méthodes de pilotage du chargement Code_Aster, Salome-Meca course material GNU FDL licence (http://www.gnu.org/copyleft/fdl.html) Outline Definition of