CONSTITUTIVE MODELLING AND PARAMETERIZATION OF ALUMINA POWDER Scot Swan 24 November 2015 Trento, Italy 2 Advanced Ceramic Production 3 Image from made-in-china.com Advanced Ceramic Production 4 5 Image from vesuvius.com Material Model Development Used for powder compaction Developed for very large deformations Transition between disparate behaviors Flexible yield surface formulation Is convex and smooth at all points Can mimic many well-known plasticity models Other notable aspects: Non-associativity Cap evolution/elastoplastic coupling Arbitrary hardening/softening with elective yield surface geometry change Tridentum Material Model 6 Transition Function 7 Elastic-Plastic Coupling 8 Bigoni-Piccolroaz Yield Surface 9 BP Yield function Bigoni-Piccolroaz Yield Surface 10 BP Yield surface for alumina powder Alumina YS parameterization: Stupkiewicz S, et al. Elastoplastic coupling to model cold ceramic powder compaction. J Eur Ceram Soc (2013) Meridional Profiles 11 Octahedral Profiles Type TXC SHR TXE0 12 Implicit Yield Function Pros: No false elastic domains Smooth/Derivatives defined everywhere Cons: Non-uniqueness based on arbitrary reference point Image taken from S. Stupkiewicz, Reference Point Dependence 14 Cutting-Plane Return, Version 1 15 Cutting-Plane Return, Version 2 16 Return Algorithm of Choice Multi-stage Return Algorithm Implicit return algorithm Independent of yield function only requires derivatives on yield surface. Use any (possibly incorrect) helper return algorithm 17 A multi-stage return algorithm for solving the classical damage component of constitutive models for rocks, ceramics, and other rock-like media, R.M. Brannon, S. Leelavanichkul, 2009 Cooper-Eaton Compaction Law 18 Compaction Behavior of Several Ceramic Powders, A.R. Cooper JR. and L.E. Eaton, Journal of the American Ceramic Society, 1962 Cooper-Eaton Compaction Law 19 Cooper and Eaton: Current Model: Cooper-Eaton Compaction Law 20 Cooper-Eaton Compaction Law 21 uniaxial strain Non-associativity 22 z Parameterization is Easy! 23 Tridentum Based on the Bigoni-Piccolroaz model described by Stupkiewicz* 28 Parameters Nonlinear elastic powder phase (4 parameters) Linear elastic green body solid phase (2 parameters) Yield surface, powder and solid phases (2x6 parameters) Cap evolution (4 parameters) Elastic-plastic coupling/transition (5 parameters) Nonassociativity (1 parameter) 24 *S. Stupkiewicz et. al., Elastoplastic coupling to model cold ceramic powder compaction, 2013 Simple Green Body Production 25 Disk forming Some benefits of disk-forming experiments are: Simple Quick Cheap 26 What can we learn from one? Continuous loading behavior Axial and lateral plastic strains for a given axial stress Also density and porosity Constrained modulus for a given axial stress Magnitude of frictional forces (loosely related to Poissons ratio) 27 slope=M Axial plastic strain Ejection Force: 60N (85KPa axial stress, 250KPa traction) What can we learn from many? Continuous plastic strain evolution Continuous constrained modulus evolution Approximation of the evolution of Poissons ratio 28 What do I do with this information? The Material Model Laboratory (matmodlab) is a material point simulator developed as a tool for developing and analyzing material models for use in larger finite element codes. Open source (MIT license) available on Github https://github.com/tjfulle/matmodlab Written in python available through the Python Package Index $ pip install matmodlab Serial optimization of parameters Native support for Abaqus UMATs Speed 29 Example input file 30 Input syntax is for matmodlab 3.0.6 Example Output 31 Output formats: text, g-zipped text, XLS, XLSX, JSON, python pickle Time series viewer (tsviewer) 32 https://github.com/sswan/tsviewer Optimization method After applying engineering judgement, it was decided to optimize 19 of the 28 possible parameters using the downhill simplex optimization algorithm using matmodlab While it is possible to simultaneously optimize multiple different experiment types, we will limit the present scope to disk forming. Comparison of three different possible methods: For one representative experiment, drive one simulation with experimental input strain and optimize the loading and unloading axial stress response For one representative experiment, drive one simulation with experimental input strain and optimize the loading axial stress response and the axial plastic strain response For seven experiments, drive seven simulations with experimental input strain and optimize the loading and unloading axial stress responses from each 33 Parameterization With four parameters I can fit an elephant, and with five I can make him wiggle his trunk John von Neumann 34 Drawing an elephant with four complex parameters by Jurgen Mayer, Khaled Khairy, and Jonathon Howard, Am. J. Phys. 78, 648 (2010) Convergence Rate 35 1 test, stress (load and unload) 36 1 test, stress (load) and plastic strain 37 7 tests, stress (load and unload) 38 Stress load/plastic strain optimization 39 Plastic Strains 40 Conclusions Know your experimental error! Where? How much? When choosing an objective function, the information density of the experimental data that is used is more important than the total amount of data Using matmodlab can give large speedups in evaluation time and significantly decrease complexity for experiments that can be approximated as homogeneous deformations matmodlab also simplifies and accelerates model development Cheap, abundant computational power can help minimize the disparity between complex constitutive models and the sparse experimental data available to parameterize them i.e. parameterization is made easier 41 Acknowledgement The authors gratefully acknowledge financial support from the European Unions Seventh Framework Programme FP7/ / under REA grant agreement number PITN-GA CERMAT2. 42 Simulation - Prediction 43 Conclusion New material model: Elastoplastic coupling Cap evolution BP yield surface Abaqus UMAT Short-term goals Include friction and green body ejection in compaction simulation Include triaxial compression and extension data in parameterizaion Long-term goals Inclusion of aleatory uncertainty in material parameters to better predict final green body heterogeneity Acknowledgement: The authors gratefully acknowledge financial support from theEuropean Unions Seventh Framework Programme FP7/ / under REA grant agreement number PITN-GA CERMAT2. 44 Simulation Axial Stress 45 Simulation volumetric plastic strain 46 Simulation von Mises Stress 47 Material Model Laboratory (MML) Python code-base Built-in path visualizer Open-source Independently prescribe Strain Stress Electric field Temperature Deformation gradient Parameter optimization Large benchmark suite Easily interfaces with Abaqus UMATs Some of the included material models: Linear Elasticity Linear Drucker Prager Mooney-Rivlin Anisotropic hyperelasticity Viscoelastity 48 Cooper-Eaton Hardening Law 49 Cooper-Eaton Hardening Law 50 Cap Evolution and Elastoplastic Coupling 51