NSF Grant Number: DMI PI: Prof. Nicholas Zabaras Institution: Cornell University Title: Development of a robust computational design simulator for industrial deformation processes Research Objectives: To develop a mathematically and computationally rigorous gradient-based optimization methodology for virtual materials process design that is based on quantified product quality and accounts for process targets and constraints including economic aspects. Significant Results: Development of a general purpose continuum sensitivity method for the design of multi-stage industrial deformation processes Applications to process design for microstructure control using various constitutive models Deformation process design for porous materials Approach: Selection of the sequence of processes and initial process parameter designs using knowledge based expert systems, microstructure evolution paths and/or ideal forming techniques Selection of the design variables (e.g. die parameterization) Interactive optimization environment Continuum multi-stage process sensitivity analysis consistent with the direct process model Assessment of automatic process optimization Reliability of the optimal design to physical and computational model errors Broader Impact: Successful development will lead to a virtual process laboratory that will assist industry in reducing lead time for process development, in trimming the cost of an extensive experimental trial-and-error process development effort, in developing processes for tailored material properties and in increasing volume/time yield. Materials Process Design and Control Laboratory, Cornell University Convection/ Radiation Conduction Rigid Die Forging rate Unfilled die cavity Flash Damage/ microstructure Single stage process preform design Multi stage process design Preforming stage Finishing stage

Deformation Process Design for Tailored Material Properties Difficult Insertion of new materials and processes into production Numerical Simulation Trial-and-error and with no design information Conventional Design Tools Material Modeling Incremental improvements in specific areas Development of designer knowledge base Time consuming and costly Computational Material Process Design Simulator Sensitivity Information points to most influential parameters so as to optimally design the process Virtual Material Process Laboratory Reliability Based Design for material/tool variability & uncertainties in mathematical & physical models Data Mining of Designer Knowledge for rapid solution to complex problems and to further drive use of knowledge Materials Process Design control of microstructure using various length and time scale computational tools Accelerated Insertion of new materials and processes Innovative Processes for traditional materials

Reliability Based Design Sensitivity Information Designer Knowledge Materials Process Design Virtual Materials Process Laboratory Selection of a virtual direct process model Selection of the sequence of processes (stages) and initial process parameter designs Selection of the design variables like die and preform parameterization Continuum multistage process sensitivity analysis consistent with the direct process model Optimization algorithms Interactive Optimization Environment Virtual Deformation Process Design Simulator

Equilibrium equation Design derivative of equilibrium equation Material constitutive laws Design derivative of the material constitutive laws Design derivative of assumed kinematics Assumed kinematics Incremental sensitivity constitutive sub-problem Time & space discretized modified weak form Time & space discretized weak form Sensitivity weak form Contact & friction constraints Regularized design derivative of contact & frictional constraints Incremental sensitivity contact sub-problem Conservation of energy Design derivative of energy equation Incremental thermal sensitivity sub-problem Schematic of the Continuum Sensitivity Method (CSM) Consider continuum problem Design differentiate Discretize

  Lagrangian analysis   Multiplicative decomposition of F   Rate-dependent material models   State variable based microstructure evolution   Thermal effects & mechanical dissipation Reference configuration r n Inadmissible region Current configuration Admissible region Contact/friction model B o B F e F p F  F Initial configuration temperature:  o void fraction: f o grain size: L o Deformed configuration temperature:  void fraction: f grain size: L Intermediate thermal configuration temperature:  void fraction: f o grain size: L o Constitutive model   Damage evolution   Dynamic recrystallization Stress free (relaxed) configuration temperature:  void fraction: f grain size: L Schematic of the constitutive and contact sub-problems   Augmented Lagrangian approach to contact and friction

Description of parameter sensitivities: Take F R = I with the design velocity gradient L 0 = 0. Main features:   Gateaux differential referred to the fixed configuration Y   Rigorous definition of sensitivity   Driving force for the sensitivity problem is L R =F R F R -1 o Shape and Parameter Continuum Sensitivity Analysis

Consider the non-differentiability of contact and friction conditions Sensitivity deformation is a linear problem Iterations are avoided within a single time increment Additional augmentations are avoided by using large penalties in the sensitivity contact problem y = y + y υ r υ + υ o r + r o x + x o X y = y ( ξ ) Die o o y + [y] x = x ( X, t, β p ) ~ x = x ( X, t, β p + Δ β p ) ~ B0B0 B΄B΄ B x ParameterSensitivityAnalysis υ r υ r y, ξ ξ y o + x = x ( X, t, β s ) B0B0 B’0B’0 BRBR X + X X o x = x ( X + X, t, β s + Δ β s ) ~ o X = X (Y ; β s + Δ β s ) ~ Y X = X (Y ; β s ) ~ ~ x + x B΄B΄ o B y = y ( ξ ) Die y = y ( ξ ) x ShapeSensitivityAnalysis REGULARIZATION Contact and friction sensitivity assumptions REMARKS Continuum Sensitivity Contact sub-problem

Continuum Sensitivity problem for a Multi-Stage Deformation Process Sequential transfer of sensitivities from one stage to the next Design objective Knowledge-based methods Shape sensitivity analysis Die and process parameter sensitivity analysis Selection of stages Design of preforms Design of dies Generic Forming Stage

  Models describing grain refinement and growth through state variable which defines the average grain size (L).   Evolution of the deformation resistance involves hardening as well as recovery due to recrystallization.   Sensitivity constitutive problem - relate the stress sensitivity and the sensitivity of the deformation gradient (Ref. [2]). PHYSICS INVOLVED Equations representing the direct dynamic recrystallization processes Grain size evolution Evolution of the deformation resistance Detailed description of the constitutive problem for a material undergoing dynamic recrystallization Corresponding sensitivity problem evaluate Using along with and Critical strain for the onset of recrystallization Evolution of equivalent plastic strain

Preforming stage Finishing stage Design the preforming die for a fixed volume of the workpiece such that the variation in mean grain sizes in the finished product is minimal. 0.2%C Steel workpiece Initial temperature 1213 K Axisymmetric problem Standard ambient conditions 2 pre-defined stages - preforming & finishing DESIGN OBJECTIVE DEFORMATION PROCESS SYSTEM Product using guess die shape Product using optimal die shape Control of microstructure through a two stage die design process for 0.2%C Steel

Objective: Design the extrusion die for a fixed reduction so that the deviation in grain size is minimized Material: 0.2%C steel, friction coefficient of Using a guess die shape Using the optimum die shape Extrusion die design for grain size control

  Existence of a potential providing the equivalent stress in terms of the Cauchy stress and void fraction.   Evolution of the damage parameter (void fraction) through conservation of mass.   Normality rule – evolution of F.   Sensitivity constitutive problem - relate the stress sensitivity and the sensitivity of the deformation gradient (Ref. [1]). p PHYSICS INVOLVED Chevron damage Detailed description of the constitutive problem for a material with damage evolution Equations representing the damage evolution processes Void fraction evolution Evolution of the deformation resistance Potential relating the stress and void fraction – Gurson model Corresponding sensitivity problem to evaluate Use Ram speed during extrusion (mm/s) Cylinder Extrusion

Objective: Minimize the flash and the deviation between the die and the workpiece for a preforming shape and volume design Material: T351Al, 300K, 5% initial void fraction, varying elastic properties (using Budiansky method), co-efficient of friction between die & workpiece = 0.1 Product using guess preform Product using optimal preform Distribution of the void fraction in product Variation of preform shape with optimization iterations Iteration number Non-dimensional objective Preform design for porous material

Optimal extrusion process design Initial extrusion process design Objective: Design the extrusion die for a fixed reduction of the workpiece s.t. chevron defects are avoided. Design the extrusion die for a fixed reduction of the workpiece s.t. chevron defects are avoided. Initial design has chevron defects, characterized here by the void fraction being Initial design has chevron defects, characterized here by the void fraction being > 1%. > 1% Iteration index Nondimensionalized Objective function r - axis Final Initial z - axis Isothermal frictionless, material with ductile damage area reduction 10.7% 1% initial void fraction Power law model Extrusion die design for control of Chevron cracking

Minimize the flash and the deviation between the die and the workpiece through a preforming shape design Unfilledcavity Flash The same material in a conventionaldesign The same material with an optimum design Noflash Fully filled cavity Process design for the manufacture of an engine disk – two possible approaches Minimize the gap between the finishing die and the workpiece in a  two stage forging, with given finishing die;  unknown die but prescribed stroke in the preforming stage. Initial design Unfilledcavity Optimal design Iteration Number Objective Function (x1.0E-05) Al 1100-O Initially at 673K; Preform and die parameteriza -tions

Current research efforts in multi-length scale analysis Control of microstructure during deformation: A multi-length scale approach Extend towards relevant, realistic optimization problems.   Introduce a framework for the sensitivity analysis.   Develop models for the evolution of microstructure for a material under prescribed deformation.   Example highlighting the direct problem – FCC crystal Fundamental region Orientation distribution function Design for an idealistic crystal with planar microstructure Desired ODF ODF obtained through design Exact solution is obtained. More results/schemes in forthcoming publications.

[1] Ganapathysubramanian, S. and N. Zabaras, “Computational design of deformation processes for materials with ductile damage”, Comp. Methods Appl. Mech. Engrg., Vol. 192, Issues 1-2, [2] Shankar, G. and Zabaras, N., “Deformation process design for control of microstructure in the presence of dynamic recrystallization and grain growth mechanisms”, Intl. J. Solids Struct., submitted. [3] Shankar, G. and Zabaras, N., “Polycrystalline materials: Is the control of properties at the micro level feasible?”, in preparation. Further developments for single-stage designs – 3D geometries   Regularized contact/ friction sensitivity modeling   Simultaneous thermal & mechanical design   Sensitivity analysis for multi-body deformations Design across length scales   Coupled length scale analysis with control of texture Robust design algorithms   Can we design a process with desired robustness limits in the objective?   Work includes a spectral method for the design of thermal systems – V. A. Badri Narayanan and N. Zabaras, Ref. [4]. Design for the control of microstructure   Variational principles towards analysis and control of grain size and orientation ACKNOWLEDGEMENTS The work presented here was funded by NSF grant DMI with additional support from AFOSR, AFRL and ALCOA. [4] V. A. Badri Narayanan and N. Zabaras, “Uncertainty propagation in analysis and inverse- design of heat conduction processes using a spectral finite element approach”, Intl. J. Numer. Methods Engrg., submitted. REFERENCES Forthcoming research efforts