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Local resolution of constitutive laws

Local resolution of constitutive laws. Outline. General concepts The tensile test General framework of thermodynamics Definition of the domain of elasticity, general laws Two examples: isotropic hardening and kinematic hardening Constitutive laws in Code_Aster

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Local resolution of constitutive laws

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  1. Local resolution of constitutive laws

  2. Outline • General concepts • The tensile test • General framework of thermodynamics • Definition of the domain of elasticity, general laws • Two examples: isotropic hardening and kinematic hardening • Constitutive laws in Code_Aster • What is exactly computed at local point (Gauss point)? • Numerical considerations about the integration • Analytical integration of the isotropic hardening • Analytical integration of the linear kinematic hardening • Integration in general • Practical considerations • Definition of the constitutive law • Setting material parameters • Setting parameters for the integration • Advanced concepts • Linear search • Quasi-incompressibility • Small / Large deformations 2 - Code_Aster and Salome-Meca course material GNU FDL Licence

  3. Outline • General concepts • The tensile test • General framework of thermodynamics • Definition of the domain of elasticity, general laws • Two examples: isotropic hardening and kinematic hardening • Constitutive laws in Code_Aster • What is exactly computed at local point (Gauss point)? • Numerical considerations about the integration • Analytical integration of the isotropic hardening • Analytical integration of the linear kinematic hardening • Integration in general • Practical considerations • Definition of the constitutive law • Setting material parameters • Setting parameters for the integration • Advanced concepts • Linear search • Quasi-incompressibility • Small / Large deformations 3 - Code_Aster and Salome-Meca course material GNU FDL Licence

  4. General concepts • Wecan observe : • An elasticitydomainwith a Yield Stress , • An irreversiblestrain , • A hardeningcaracterized by , • Plasticity, a phenomenon that is independent of velocity. 4 - Code_Aster and Salome-Meca course material GNU FDL Licence

  5. General concepts Elastic domain The elasticity domain is defined by : Experimentaly : • f is a function of • more particulary : with 5 - Code_Aster and Salome-Meca course material GNU FDL Licence

  6. General concepts Total Strain Partition of the deformation : In elastic case : 6 - Code_Aster and Salome-Meca course material GNU FDL Licence

  7. General concepts Framework: mechanics and thermodynamics (Eckart, Biot 1950) First law of thermodynamics law: With PPV, we can formulate a variationnal internal energy With second law of thermodynamics and considering a Helmholtz’s free energy 7 - Code_Aster and Salome-Meca course material GNU FDL Licence

  8. General concepts The method of local state : The thermodynamic state, at the point and the instant considered, is entirely defined at this instant by the state variables (observable and internal ). The first state law : The second state law : 8 - Code_Aster and Salome-Meca course material GNU FDL Licence

  9. General concepts Can we describe an irreversible evolution using only state laws???? 9 - Code_Aster and Salome-Meca course material GNU FDL Licence

  10. General concepts Irreversible dissipation • Definition of a (pseudo-)potential of dissipation • Then the supplementary laws are resulting, describing irreversible processes 10 - Code_Aster and Salome-Meca course material GNU FDL Licence

  11. General concepts Theory of plasticity 11 - Code_Aster and Salome-Meca course material GNU FDL Licence

  12. General concepts • Definition of the elasticity domain in 3D • to define an yield surface in the stress space: Definitionf is a negative function in the elastic domain and null during the plastic transformation and she can't evoluate differently. In resume : 12 - Code_Aster and Salome-Meca course material GNU FDL Licence

  13. General concepts • Evolution law • Maximum plastic work (Hill 1951) In short, the principle postulate two important ideas : - the yield surface (f) must be convex function, - the plastic strain rate is normal to the yield surface. • Direction of the plastic flow • Evolution law (or law of normality) • outward normal to the boundary of the domain : Using Maximum plastic work => D>0 13 - Code_Aster and Salome-Meca course material GNU FDL Licence

  14. General concepts • Evolution law • Maximum plastic work (Hill 1951) In short, the principle postulate two important ideas : - the yield surface (f) must be convex function, - the plastic strain rate is normal to the yield surface. • Evolution law (or law of normality) • Intensity of the flow The plasticity multiplier is determined by the consistency relation By example, for an isotropic hardening: We must verify : 14 - Code_Aster and Salome-Meca course material GNU FDL Licence

  15. Example of isotropic hardening Isotropic hardening laws • An isotropic extension of the elasticity domain is taking into account • dilatation of the elasticity domain • The evolution of the criterion is governed by a single scalar • Internal variable: cumulated plastic strain • Associated variables: thermodynamic force • Plasticity criterion: 15 - Code_Aster and Salome-Meca course material GNU FDL Licence

  16. Example of isotropic hardening Nonlinear system (3D constitutive laws, small perturbations, isothermal) with derived from the stress-strain curve 16 - Code_Aster and Salome-Meca course material GNU FDL Licence

  17. Example of isotropic hardening • derived from the stress-strain curve • Uniaxial tensile test: 17 - Code_Aster and Salome-Meca course material GNU FDL Licence

  18. Example of kinematic hardening • The centre of the elasticity domain evolves with plastic strain • Translation of the elasticity domain • Introduction of a tensor variable : centre of the elasticity domain • Plasticity criterion: 18 - Code_Aster and Salome-Meca course material GNU FDL Licence

  19. Example of kinematic hardening with (C is the coefficient of hardening of the material, derived from the curve of tension-compression) 19 - Code_Aster and Salome-Meca course material GNU FDL Licence

  20. Example of kinematic hardening • Uniaxial tensile test: • C : coefficient of kinematic hardening is eliminated from the first two equations And finally, 20 - Code_Aster and Salome-Meca course material GNU FDL Licence

  21. Outline • General concepts • The tensile test • General framework of thermodynamics • Definition of the domain of elasticity, general laws • Two examples: isotropic hardening and kinematic hardening • Constitutive laws in Code_Aster • What is exactly computed at local point (Gauss point)? • Numerical considerations about the integration • Analytical integration of the isotropic hardening • Analytical integration of the linear kinematic hardening • Integration in general • Practical considerations • Definition of the constitutive law • Setting material parameters • Setting parameters for the integration • Advanced concepts • Linear search • Plane stresses • Quasi-incompressibility • Small / Large deformations 21 - Code_Aster and Salome-Meca course material GNU FDL Licence

  22. What is being done locally? Initialisations and - Compute Load increment: Calculation tangent matrix : Yes : equilibrium verified Calculation : No n=n+1 Residu calculation: Test of convergence 22 - Code_Aster and Salome-Meca course material GNU FDL Licence

  23. What is being done locally? Incremental discretization with equilibrium iterations Incremental discretization without equilibrium iterations 23 - Code_Aster and Salome-Meca course material GNU FDL Licence

  24. What is being done locally? • A) Calculation of the tangent matrix for prediction RIGI_MECA_TANG • At iteration 0 (prediction), the tangent matrix at the previous equilibrium time step • For each element, we want the elementary tangent matrix • For each Gauss point , the tangent operator for prediction is computed from constraints and internal variables at the previous equilibrium time step: • Updating REAC_INCR= M in file command with M=N*q where i: time step n: Newton iteration 24 - Code_Aster and Salome-Meca course material GNU FDL Licence

  25. What is being done locally? • B) Calculation of the consistent tangent matrix at each iteration of Newton FULL_MECA • At iteration n, one must solve the linear system: • Requires the knowledge of the tangent matrix • For each finite element, one wants the elementary tangent stiffness matrix • Updating REAC_ITER= M in command file with M=N*q where i: time step n: Newton iteration 25 - Code_Aster and Salome-Meca course material GNU FDL Licence

  26. Integration of the isotropic hardening 3D constitutive laws (small perturbations, isothermal) 26 - Code_Aster and Salome-Meca course material GNU FDL Licence

  27. Integration of the isotropic hardening • Time discretization • Implicit choice: stability • The choice of the time step depends on the radial nature of the problem • Input: • solutions at • : estimation at iteration • Unknown variables at time step 27 - Code_Aster and Salome-Meca course material GNU FDL Licence

  28. Integration of the isotropic hardening R(p- +Dp) R(p-) Test : Oui Non 28 - Code_Aster and Salome-Meca course material GNU FDL Licence 28 - Code_Aster and Salome-Meca course material GNU FDL Licence

  29. Integration of the isotropic hardening • Calculation of the tangent operator for prediction • Calculation of the tangent operator from the stresses and internal variables at the previous equilibrium time step • Analytical solution • Calculation the tangent operator at Newton iteration n • Calculation of the tangent operator from the stress and internal variables at the previous equilibrium time step and from the increment of deformation • Analytical solution 29 - Code_Aster and Salome-Meca course material GNU FDL Licence

  30. Integration of the kinematic hardening • Time discretization • Implicit choice: stability • The choice of the time step depends on the radial nature of the problem • Input: • solutions at • : estimation at iteration • Unknown variables at time step 30 - Code_Aster and Salome-Meca course material GNU FDL Licence

  31. Integration of the kinematic hardening 31 - Code_Aster and Salome-Meca course material GNU FDL Licence

  32. Integration of the kinematic hardening Test : Oui Non 32 - Code_Aster and Salome-Meca course material GNU FDL Licence

  33. General integration of constitutive laws • General case: Test : Oui Non • Solving the local system of n NL equations • Explicit method (Runge-Kutta) or implicit method (Newton) With n is usually quite small (10-50) 33 - Code_Aster and Salome-Meca course material GNU FDL Licence

  34. Integration of constitutive laws: sum up • General case • In practice, only a few laws in Code_Aster • Solving the NL local system of n equations • Explicit method (Runge-Kutta) or implicit method (Newton) • Particular cases • 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 isotropic hardening and / or linear kinematic) 34 - Code_Aster and Salome-Meca course material GNU FDL Licence

  35. How to define a constitutive law Constitutive law Represented mechanisms State laws Dissipation laws Name of the constitutive law Numerical representation (in STAT_NON_LINE: COMP_INCR or COMP_ELAS) (in DEFI_MATERIAU) Tests Material parameters Experimental curves, handbooks … 35 - Code_Aster and Salome-Meca course material GNU FDL Licence

  36. Constitutive laws available • More than 160 laws in the 12 stable version • Various fields of applications • Metals, polycrystalline metals • Concrete • Soils • Various phenomena • Irradiation • Damage or cracking • Metallurgical phases • Documentation • Synthesis of non-linear constitutive laws: U4.51.11 • DEFI_MATERIAU syntax: U4.43.01 36 - Code_Aster and Salome-Meca course material GNU FDL Licence

  37. Constitutive laws available • 2D and 3D continuum media • Non linear elasticity • Von Mises isotropic Pseudo-hardening • ELAS_VMIS_LINE • ELAS_VMIS_TRAC • ELAS_HYPER • Incremental elasto-plasticity • Von Mises isotropic hardening, kinematic linear, mixed • VMIS_ISOT_TRAC • VMIS_ISOT_PUIS • VMIS_ISOT_LINE • VMIS_CINE_LINE • VMIS_ECMI_TRAC • VMIS_ECMI_LINE 37 - Code_Aster and Salome-Meca course material GNU FDL Licence

  38. Constitutive laws available 2D and 3D continuum media (cont.) • Other elastoplastic models (metals) • VMIS_CIN1_CHAB • VMIS_CIN2_CHAB • VMIS_CIN2_MEMO • Elasto-visco-plasticity (metals) • LEMAITRE, LEMA_SEUIL • VISC_CIN1_CHAB • VISC_CIN2_CHAB • VISC_ISOT_LINE • VISC_ISOT_TRAC • VISC_TAHERI • VISCOCHAB • Limit loads • NORTON_HOFF • Polycrystalline metals • POLY_CFC • MONOCRISTAL • POLYCRISTAL • Elasto-visco-plasticity under irradiation • LMARC • LEMAITRE_IRRA • GATT_MONNERIE • VISC_IRRA_LOG • GRAN_IRRA_LOG • IRRAD3M 38 - Code_Aster and Salome-Meca course material GNU FDL Licence

  39. 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 phases (elasto-visco-plastic) for steel or zirconium • META_X_Y_Z • X = P (plasticity) or V (viscosity) • Y = IL (linear isotropic) or INL (nonlinear isotropic) or CL (linear kinematic) • Z = RE (restoration) and/or PT (transformation plasticity) • 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 39 - Code_Aster and Salome-Meca course material GNU FDL Licence

  40. 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 pipes (local behaviour = plane stress) • All 3D constitutive laws (thanks to the method if C_PLAN is not supported: ALGO_C_PLAN = 'DEBORST') • Bars, multi-fiber beams, grids • All the laws of 1D behaviour (thanks to the DeBorst method if 1D is not supported: ALGO_1D = 'DEBORST') • Discrete elements, shear connections, reinforcements 40 - Code_Aster and Salome-Meca course material GNU FDL Licence

  41. Data catalogue for materials • “material” entries • reference material (AFNOR name) • vessel steel 16MND5 • Inconel 600 tubes (GV), ... • Two possible accesses • document from www.code-aster.org (documentation / Materials) • Automated access from Code_Aster • Quality Assurance • Independent validation by the MMC department • 55 material entries for the stable 12 version • Only needs to be enriched Note that some values can vary greatly depending on the nature of the product (grade) INCLUDE_MATERIAU (NOM_AFNOR='A42', TYPE_MODELE='REF', VARIANTE='A', TYPE_VALE='NOMI', NOM_MATER='MAT', EXTRACTION=(_F(COMPOR='ELAS', TEMP_EVAL=20.), _F(COMPOR='THER', TEMP_EVAL=20.)) ) 41 - Code_Aster and Salome-Meca course material GNU FDL Licence

  42. Data catalogue for materials 42 - Code_Aster and Salome-Meca course material GNU FDL Licence

  43. Fitting of material parameters N experimental curves Master file MACR_RECAL or ADAO (command file from the study) Fitting loops P parameters N computed curves Slave file • The constitutive law is known • Search for the values ​​of the parameters of the law • To do this: • one has at his disposal several experimental data from tests • one will seek the parameters that best simulate these tests (optimization methods) See the test case SSNA110 43 - Code_Aster and Salome-Meca course material GNU FDL Licence

  44. Fitting of material parameters Iteration 0 44 - Code_Aster and Salome-Meca course material GNU FDL Licence

  45. Fitting of material parameters Iteration 3 45 - Code_Aster and Salome-Meca course material GNU FDL Licence

  46. Syntax for the constitutive laws integration • Choice of parameters for the integration: • under the factor key word COMP_INCR • General case: • Resolution of the local NL system of n equations • ALGO_INTE = 46 - Code_Aster and Salome-Meca course material GNU FDL Licence

  47. Syntax for the constitutive laws integration • General case: • Convergence • Residue to achieve: RESI_INTE_RELA (10-6) • Maximum number of iterations: ITER_INTE_MAXI (20) • Tips • For behaviour which are "difficult" to integrate, increase ITER_INTE_MAXI • ssnd105b where ITER_INTE_MAXI = 250 for VISCOCHAB • ssnv172a where ITER_INTE_MAXI = 100 for MONOCRISTAL • ssnl106i where ITER_INTE_MAXI = 500 for VMIS_POU_LINE • For certain behaviours, it is better to integrate finely the behaviour (ex: Hujeux) RESI_INTE_RELA =10-8 47 - Code_Aster and Salome-Meca course material GNU FDL Licence

  48. Syntax for the constitutive laws integration • Particular cases: • 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: • 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 ! (except for plane stresses) • Ex: hsnv125a: VMIS_ISOT_LINE in 3D and ITER_INTE_MAXI = 100 48 - Code_Aster and Salome-Meca course material GNU FDL Licence

  49. Outline • General concepts • The tensile test • General framework of thermodynamics • Definition of the domain of elasticity, general laws • Two examples: isotropic hardening and kinematic hardening • Constitutive laws in Code_Aster • What is exactly computed at local point (Gauss point)? • Numerical considerations about the integration • Analytical integration of the isotropic hardening • Analytical integration of the linear kinematic hardening • Integration in general • Practical considerations • Definition of the constitutive law • Setting material parameters • Setting parameters for the integration • Advanced concepts • Linear search • Quasi-incompressibility • Small / Large deformations 49 - Code_Aster and Salome-Meca course material GNU FDL Licence

  50. Linear search for the local Newton • Purpose: To improve the robustness of the local Newton • Principles: • We introduce a functional related to the system to be solved • We will seek to "minimize this functional" • The Newton direction is a descent direction (locally) for the function • Choice of a progress step that reduces  along the descent direction • Algorithm: • System • Requires to compute the progress step • as , the best is • Requires introducing • Syntax: ALGO_INTE = ‘NEWTON_RELI’ 50 - Code_Aster and Salome-Meca course material GNU FDL Licence

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