Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 101 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: Babak Kouchmeshky and Nicholas Zabaras Advances on multi-scale design of deformation processes for the control of material properties

Materials Process Design and Control Laboratory Forging Goals : Goals : 1.Minimal material wastage due to flash 2.Filling up the die cavity 3.Minimal variation of macro- scale properties Material : Cu Why multi scale? Why multi scale? –The evolution of the material properties at the macro scale has a strong correlation with the underlying microstructure. Problem at hand

Materials Process Design and Control Laboratory Multi-scale polycrystal plasticity Meso Macro formulation for macro scale Update macro displacements Texture evolution update Polycrystal averaging for macro-quantities Evolution of plastic strain rate on slip systems Macro-deformation gradient microscale stress Macro-deformation gradient Micro

Multi-length scale design environment Coupled micro-macro direct model Selection of the design variables (e.g. variables (e.g. preform parameterization, process parameters process parameters Coupled micro-macro sensitivity model Design based on: Polycrystal plasticity, evolution of texture, multi-length scale analysis Deformation problem: -Updated Lagrangian framework -Connection to the micro scale through Taylor hypothesis - Microstructure represented as orientation distribution function (ODF) in Rodrigues space. Materials Process Design and Control Laboratory

Crystallographic slip, twinning and re-orientation of crystals are assumed to be the primary mechanisms of plastic deformation Evolution of various material configurations for a single crystal as needed in the integration of the constitutive problem. Evolution of plastic deformation gradient The elastic deformation gradient is given by (Ashby; Kocks; Anand) B0B0 mm nn nn mm mm nn ^ nn mm nn ^ mm _ _ BnBn BnBn B n+1 _ _ FnFn FnFn FnFn F n+1 p p e F trial e e FrFr FcFc Intermediate configuration Deformed configuration Intermediate configuration Reference configuration INCREMENTAL KINEMATICS Materials Process Design and Control Laboratory

THE DIRECT CONTACT PROBLEM r n Inadmissible region Reference configuration Current configuration Admissible region Impenetrability Constraints Augmented Lagrangian approach to enforce impenetrability

Materials Process Design and Control Laboratory Constitutive theory D = Macroscopic stretch = Schmid tensor = Lattice spin W = Macroscopic spin = Lattice spin vector Reorientation velocity Symmetric and spin components Velocity gradient Divergence of reorientation velocity Polycrystal plasticity Initial configuration B o B F * F p F Deformed configuration Stress free (relaxed) configuration n0n0 s0s0 n0n0 s0s0 n s (2) Ability to capture material properties in terms of the crystal properties (1) State evolves for each crystal

Polycrystal average of orientation dependent property Continuous representation of texture Materials Process Design and Control Laboratory REORIENTATION & TEXTURING

Materials Process Design and Control Laboratory Orientation distributions Any macroscale property can be expressed as an expectation value if the corresponding single crystal property χ (r,t) is known. Determines the volume fraction of crystals within a region R' of the fundamental region R Probability of finding a crystal orientation within a region R' of the fundamental region Characterizes texture evolution ORIENTATION DISTRIBUTION FUNCTION – A(r,t) ODF EVOLUTION EQUATION – LAGRANGIAN DESCRIPTION

Materials Process Design and Control Laboratory A sample direct problem

Materials Process Design and Control Laboratory Convergence studies for the direct problem

Materials Process Design and Control Laboratory Connection between two process This phase usually consists of an unloading process where one of the dies is removed from contact with the workpiece. The unloading process is modeled as a non-linear (finite deformation) boundary value problem.

Materials Process Design and Control Laboratory Connection between two process The workpiece material also undergoes a recovery process, whereby the equivalent stress evolves in the absence of an applied stress. Equivalent stress without the unloading step Equivalent stress with the unloading step

Materials Process Design and Control Laboratory The load reduction rate does not have any effect in this problem (Since a rate independent constitutive model is used). The effect of unloading the workpiece can be seen in the stress response. The effect of elasticity is most noticeable in the initial portion of the stress-strain curve where the elasticity is predominant. At large strains where all crystals deform plastically the stress level is controlled by the flow stress of the material Connection between two process

Materials Process Design and Control Laboratory The steep reduction of load at the unloading stage suggests that the elastic part of strain is negligent in front of the plastic part. This is due to the fact that the elastic constants of the material are at least one order of magnitude larger than the strength of slip systems.

Materials Process Design and Control Laboratory After unloading Connection between two process Before unloading

Materials Process Design and Control Laboratory The effect of sequence of processes on the macro-scale properties

Materials Process Design and Control Laboratory SCHEMATIC OF THE CONTINUUM SENSITIVITY METHOD Advantage : Fast Multi-scale optimization Requires 1 Non-linear and n Linear multi-scale problems for each step of the optimization algorithm. ( n: number of design parameters) Equilibrium equation Design derivative of equilibrium equation Material constitutive laws Design derivative of the material constitutive laws Incremental sensitivity constitutive sub-problem Time and space discretized weak form Sensitivity weak form A continuum sensitivity approach using extended Taylor hypothesis is used to obtain the sensitivity of macro-scale properties on perturbation of parameters on multi-scales.

Materials Process Design and Control Laboratory DEFINITION OF PARAMETER SENSITIVITY state variable sensitivity contour w.r.t. parameter change X = X (Y;  s ) o F R + F R Y X X+X o x n +x n o xnxn o F n + F n FRFR FnFn BRBR BoBo I+L o x+x oo F r + F r x B x n + x n = x (Y, t n ;  s +   s ) o ~ Q n + Q n = Q (Y, t n ;  s +   s ) o ~ x = x (x n, t ;  s ) ^ B n x n = x (X, t n ;  s ) ~ Q n = Q (X, t n ;  s ) ~ I+L n Main Features Main Features Mathematically rigorous definition of sensitivity fieldsMathematically rigorous definition of sensitivity fields Gateaux differentials (directional derivatives) referred to fixed Y in the configuration B RGateaux differentials (directional derivatives) referred to fixed Y in the configuration B R o X + X= X (Y;  s +   s ) o ~ ~ FrFr x + x = x (x+x n, t ;  s +   s ) ^ oo

Materials Process Design and Control Laboratory SENSITIVITY KINEMATIC PROBLEM Continuum problem Differentiate Discretize Design sensitivity of equilibrium equation Calculate such that x = x (x r, t, β, ∆β ) o o Variational form - F r and x o o o λ and x o Constitutive problem Regularized contact problem Kinematic problem Sensitivity of ODF evolution P r and F, o  o o

Materials Process Design and Control Laboratory Verification of sensitivities of displacement

Materials Process Design and Control Laboratory Verification of sensitivities of macro-scale properties

Materials Process Design and Control Laboratory Initial guess : Final values : Design (example1)

Materials Process Design and Control Laboratory Result for the design example (1)

Materials Process Design and Control Laboratory Result for the design example (1)

Materials Process Design and Control Laboratory Initial guess :Final values : Design (example2)

Materials Process Design and Control Laboratory Result for the design example (2)

Materials Process Design and Control Laboratory Result for the design example (2)

Materials Process Design and Control Laboratory Conclusion A multi scale design methodology is applied for the problem of forging FCC copper -A methodology is presented to design for process parameters of a multi-stage process. -A multi-scale model has been used to obtain the distribution of macro-scale properties at the end of a multi- stage process. -The goal has been to fill the die cavity and minimize the variation of macro-scale properties -A continuum sensitivity approach using extended Taylor hypothesis is used to obtain the sensitivity of macro-scale properties on perturbation of parameters on multi-scales.