1. COMING FROM? IMDEA Materials Institute (GETAFE) Polytechnic University of Madrid Vicente Herrera Solaz 1 Javier Segurado 1,2 Javier Llorca 1,2 1 Politechnic University of Madrid 2 Imdea Materials Institute Vicente Herrera Solaz 1 Javier Segurado 1,2 Javier Llorca 1,2 1 Politechnic University of Madrid 2 Imdea Materials Institute

2. An inverse optimization strategy to determine single crystal mechanics behavior from polycrystal tests: application to Mg alloys WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

3. 2. Crystal Plasticity Model 3. Optimization Strategy 4. Results 5. Conclusions WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

4. News alloys and different manufacturing systems are influence alloyed elements The influence of the alloyed elements WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

5. Macroscopic Properties Macroscopic Properties (E, s y..)  Mechanical Tests Microscopic Properties Microscopic Properties (grains)  Hard estimation  nº slip and twinning def systems  Micromechanical Tests  Lower scale Models (MD, DD)  Inverse analysis of mechanical tests with FE models WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

6. Objetives Objetives:  Develop a CP model for HCP materials + twinning  Apply CP in a Polycrystalline homogenization Model  Implement an optimization technique  Inverse analysis WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

7. Multiplicative decomposition of the deformation gradient is considered Lp The velocity Gradient Lp contains three terms: Composite material model: parent and twin phases Composite material model: parent and twin phases WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

8. With: Three slip deformation modes (basal, prismatic and pyramidal [c+a]) and tensile twinning (TW) have been considered. WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

9. UMATABAQUSimplicit scheme The crystal plasticity model has been programmed using a subroutine (UMAT) in ABAQUS and was resolved on an implicit scheme. Shear rate Twinning rate WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

10. behaviorpolycristal: The behavior of the polycristal: HomogenizationFE  Numerical Homogenization: Calculation by FE of a boundary problem in a RVE of the microstructure. Voxels model with 2 3 element/crystal Dream 3D model with Realistic microstructure (grain size and shapes) ≈ 200 elements/crystal Voxels model with 1 element/crystal tensioncompressionperiodic boundary Uniaxial tension and compression are simulated under periodic boundary conditions statistically representative The grain orientations are generated by Montecarlo to be statistically representative of ODF RVEs Different RVEs can be used: WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

11. Experimental curves Micromechanical properties (known) Numerical curves Comparison Micromechanical properties (????) Experimental curves Numerical curves Inverse analysis Comparison Micromechanical properties fit Validation numerical model WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

12. Inverse analysis Trial-error Optimization algorithm (Levenberg-Marquardt) Optimization algorithm (Levenberg-Marquardt) Subjective  Time Subjective  Time Objective, Automatic  Time Micromechanical properties (????) Experimental curves Numerical curves Inverse analysis Comparison Micromechanical properties fit WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

13. Micromechanical properties (????) Experimental curves Numerical curves Inverse analysis Comparison Micromechanical properties fit Inverse analysis Optimization algorithm (Levenberg-Marquardt) Optimization algorithm (Levenberg-Marquardt) Objective, Automatic  Time Objective, Automatic  Time IMPLEMENTATIONIMPLEMENTATION WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

14. n Experimental data: pair of n points (xi, yi) defining an experimental curve y(x) n Numerical data: pair of n points (xi, yi*) defining a numerical curve, where: yi*=f(xi, β )=f( β ) and β m a set of m parameters on wich our numerical model depends increases d If we do small increases d β parameters in the β parameters, the response (modified numerical curve) can be written as: O( β ): Objective function: O( β ): WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

15. δ minimum equations The perturbance of parameters δ which results in a minimum of the objective function is obtained with the following linear system of equations β The new set of β parameters will be: minimization iterative goalreached The minimization process is iterative, each iteration k is based on the results of the k-1. The loop iteration ends when a goal is reached or it is impossible to minimize the error. The initial set of parameters is arbitrary algorithmpython The optimization algorithm has been programmed in python J Jacobian Matrix, Where J is the Jacobian Matrix, obtained here numerically KEYPOINTS The procedure is applied hierarchically: From simplistic RVEs to realistic ones → Time saving Experimental data used have to be representative: Number of curves, load direction → To avoid multiple solutions The values obtained have to be critically assessed: Predictions of independent load cases → Validation WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

16. Fitting done on several Mg alloys: AZ31, MN10 and MN11 Validation Initial and Final textures Temperature influence on MN11 WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

27. VALIDATION Prediction AZ31 Fitting 1 curves error= 31 MPa/pt To accept the results obtained you need to check the predictive ability of the model in other load cases which are not included in the iterative process. The independence and representativeness of the initial curves used for the adjustment will influence in the quality of these predictions WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

28. VALIDATION Prediction AZ31 Fitting 2 curves error= 25 MPa/pt To accept the results obtained you need to check the predictive ability of the model in other load cases which are not included in the iterative process. The independence and representativeness of the initial curves used for the adjustment will influence in the quality of these predictions WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

29. AZ31 Fitting 3 curves Prediction VALIDATION error= 11 MPa/pt To accept the results obtained you need to check the predictive ability of the model in other load cases which are not included in the iterative process. The independence and representativeness of the initial curves used for the adjustment will influence in the quality of these predictions WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

30. Prediction MN10 Fitting 3 curves VALIDATION error= 9.5 MPa/pt To accept the results obtained you need to check the predictive ability of the model in other load cases which are not included in the iterative process. The independence and representativeness of the initial curves used for the adjustment will influence in the quality of these predictions WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

31. Prediction MN11 Fitting 3 curves VALIDATION error= 11.3 MPa/pt To accept the results obtained you need to check the predictive ability of the model in other load cases which are not included in the iterative process. The independence and representativeness of the initial curves used for the adjustment will influence in the quality of these predictions WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

32. AZ31MN10 MN11 Experimental Numerical INITIAL TEXTURES WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

33. AZ31MN10 MN11 Experimental Numerical FINAL TEXTURES WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

34. Curves Fit Polar effect (↑Tª) TEMPERATURE INFLUENCE on MN11 WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

35. The Polar effect could be attributed to the twinning mechanism but it doesn't appears at high Tª then… non-Schmidt stresses The Inclusion of the non-Schmidt stresses on Pyramidal c+a is the only way to explain it (by modifying Schmidt law) In other HCP materials (Ti), Pyramidal c+a has this role, but never on Mg. At high Tª, pyramidal c+a has a great activity due to its low CRSS TEMPERATURE INFLUENCE on MN11 WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)

36. CPFE A CPFE model has been developed for Magnesium. An optimization algorithm Inverse analysis An optimization algorithm has been implemented  Inverse analysis. Precise fit Numerical results  Precise fit  Experimental curves representativepredictive capacity Experimental curves input (representative)  predictive capacity effect of alloyed elements and Tª on the micromechanical parameters Three Mg alloys were analyzed  effect of alloyed elements and Tª on the micromechanical parameters Future work: Future work:  Optimization: Textureobjective function  Optimization: Texture inclusion as objective function  Others representations of microstructures  Inclusion of grain boundary effects  Inclusion of grain boundary effects  Crack propagation, fatigue crack initiation, grain boundary sliding WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS PARIS (2-3 October)