DESIGN OF MICROSTRUCTURE-SENSITIVE PROPERTIES IN ELASTO-VISCOPLASTIC POLYCRYSTALS USING MULTISCALE HOMOGENIZATION TECHNIQUES V. Sundararaghavan, S. Sankaran and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/ Materials Process Design and Control Laboratory

RESEARCH SPONSORS U.S. Army research office (ARO) Mechanical Behavior of Materials Program U.S. Air Force Partners Materials Process Design Branch, AFRL Computational Mathematics Program, AFOSR NATIONAL SCIENCE FOUNDATION (NSF) Design and Integration Engineering Program CORNELL THEORY CENTER people Materials Process Design and Control Laboratory

Meso-scale Nano Micro-scale MULTISCALE MODELING grain/crystal Metallic materials are composed of a variety of features at different length scales Meso-scale Twins Continuum scale Inter-granular slip Nano Micro-scale atoms Materials Process Design and Control Laboratory

Meso-scale Homogenization Mechanics of slip Nano Micro-scale MD MULTISCALE MODELING Material property evolution is dictated by different physical phenomena at each scale. grain/crystal Meso-scale Twins Continuum scale Inter-granular slip Homogenization Mechanics of slip Nano Micro-scale MD atoms Materials Process Design and Control Laboratory

CONTROL PROPERTY EVOLUTION THROUGH PROCESS DESIGN g: orientation of crystal Microstructures are complex and the response depends on crystal orientations, higher order correlations of orientations, grain boundary and defect sensitive properties. Control these features through careful design of deformation processes f(g) One point statistic: Texture two point statistics f(g,g’|r) Process? Strain rate? Property Time Desired response Final response Materials Process Design and Control Laboratory

MULTI-LENGTH SCALE CONTROL Process: Cold working Meso-scale representation Forging Evolving microstructure Properties Intermediate step OBJECTIVES VARIABLES Material usage Design properties Identification of stages Desired shape Control process parameters Number of stages Microstructure Preform shape Desired properties Forging rates Materials Process Design and Control Laboratory

Sample reference frame PHYSICAL APPROACH TO PLASTICITY Crystal/lattice reference frame e1 ^ e2 Sample reference frame e’1 e’2 crystal e’3 e3 Crystallographic orientation Rotation relating sample and crystal axis Properties governed by orientation

CONVENTIONAL MULTISCALING SCHEMES APPROXIMATION 1: All grains will take the same deformation – TAYLOR Relaxed constraints model: takes grain shapes into account for relaxing certain stress components APPROXIMATION 2: All grains have the same stresses – SACHS ASSUMPTION APPROXIMATION 3: Assume each grain is surrounded by an equivalent medium: Identify an interaction law between a grain and its surroundings – Self consistent scheme Satisfies compatibility, Equilibrium across GBs fails How does macro loading affect the microstructure Strong kinematic constraint: gives stiff response (upper bound) Taylor assumption Gives softest response (lower bound) Failure to predict evolution of texture within grains Failure to predict GB misorientation development Sachs assumption Materials Process Design and Control Laboratory

Homogenization scheme How does macro loading affect the microstructure Microstructure is a representation of a material point at a smaller scale Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure. (Hill, R., 1972) Materials Process Design and Control Laboratory

HOMOGENIZATION OF DEFORMATION GRADIENT Macro-deformation can be defined by the deformation at the boundaries of the microstructure (Hill, Proc. Roy. Soc. London A, 1972) Decompose deformation gradient in the microstructure as a sum of macro deformation gradient and a micro-fluctuation field (Miehe, CMAME 1999). X x Macro Meso x = FX y = FY + w N n Mapping implies that Use BC: = 0 on the boundary Note = 0 on the volume is the Taylor assumption, which is the upper bound Sundararaghavan and Zabaras, IJP 2006. Materials Process Design and Control Laboratory

Virtual work considerations How to calculate homogenized stresses? Hill Mandel condition: The variation of the internal work performed by homogenized stresses on arbitrary virtual displacements of the microstructure is required to be equal to the work performed by external loads on the microstructure. Apply BC Homogenized stresses Must be valid for arbitrary variations of dF Materials Process Design and Control Laboratory

Equilibrium state of the microstructure An equilibrium state of the micro-structure is assumed This assumes stress field variation is quasi-static and inertia forces are instead included in the equations of motion of the homogenized continuum. Thermal effects linking assumption Equate macro and micro temperatures Macro dissipation = average micro dissipation Assumed (Nemat-Nasser, 1999) Is assumed and used to calculate the averaged Cauchy stress Materials Process Design and Control Laboratory

Single crystal constitutive laws Reference configuration Crystallographic slip and re-orientation of crystals are assumed to be the primary mechanisms of plastic deformation Fn+1 na Fn Bn Bn+1 ma B0 _ ^ na Fr ma _ ^ _ ma na e Fn p Fn Bn e Ftrial Deformed configuration Evolution of plastic deformation gradient na ma Intermediate configuration e Fn+1 p Fn+1 _ Fc Bn+1 na The elastic deformation gradient is given by ma Intermediate configuration Incorporates thermal effects on shearing rates and slip system hardening (Ashby; Kocks; Anand) Evolution of various material configurations for a single crystal as needed in the integration of the constitutive problem. Materials Process Design and Control Laboratory

Constitutive integration scheme Constitutive law for stress Evolution of slip system resistances Shearing rate Coupled system of equations for slip system resistances and stresses at each time step is solved using Newton-Raphson algorithm with quadratic line search Athermal resistance (e.g. strong precipitates) Thermal resistance (e.g. Peierls stress, forest dislocations) If resolved shear stress does not exceed the athermal resistance , otherwise Materials Process Design and Control Laboratory

Consistent tangent moduli at meso-scale The consistent tangent moduli required for non-linear solution of the microstructure equilibrium problem is calculated using a implicit solution scheme by direct differentiation of crystal constitutive equations. Definition of stresses Variation in Cauchy stress Implicit solution scheme dT = dEetrial Materials Process Design and Control Laboratory

Implementation Macro Meso Micro Macro-deformation information Homogenized (macro) properties Boundary value problem for microstructure Solve for deformation field meso deformation gradient Mesoscale stress, consistent tangent Integration of constitutive equations Continuum slip theory Consistent tangent formulation (meso) Materials Process Design and Control Laboratory

Pure shear of an idealized aggregate Materials Process Design and Control Laboratory

Comparison of texture from Taylor and Homogenization approach Materials Process Design and Control Laboratory

Plane strain compression of idealized aggregate Materials Process Design and Control Laboratory

Three-dimensional shear of an idealized aggregate Initial texture Final texture Experiment (Carreker and Hibbard, 1957) Homogenization with Taylor-calibrated parameters from (Balasubramanian and Anand 2002) Materials Process Design and Control Laboratory

Homogenization of real microstructures X Y Z (a) (b) 3D microstructure from Monte Carlo Potts simulation 24 x 24 x 24 Pixel based grid 1000 mins on 60 X64 Intel processors with a clock speed of 3.6 GHz using PetSc KSP solvers on the Cornell theory center’s supercomputing facility Materials Process Design and Control Laboratory

Design of microstructure-sensitive properties Process? Strain rate? Property Time Desired response Final response Design Problems: Microstructure selection: How do we find the best features (e.g. grain sizes, texture) of the material microstructure for a given application? Process sequence selection: How do we identify the sequences of processes to reach the final product so that properties are optimized? Process parameter selection: What are the process parameters (e.g. forging rates) required to obtain a desired property response? Materials Process Design and Control Laboratory

PROCESS DESIGN FOR STRESS RESPONSE AT A MATERIAL POINT Given an initial microstructure Problem 1) Selection of optimal strain rates to achieve a desired property response during processing? Problem 2) A more relevant problem: What should be the straining rates during processing so that a desired response can be obtained after processing? Sensitivity of a homogenized property dv z: Deviation from desired property x: Strain rate of stage 1 y: Strain rate of stage 2 starting point Steepest descent: Need to evaluate gradients of objective function (deviation from desired property) with respect to strain rates. Materials Process Design and Control Laboratory

CONTINUUM SENSITIVITY METHOD FOR MICROSTRUCTURE DESIGN Microstructure homogenization Sundararaghavan and Zabaras, IJP 2006. Discretize infinite dimensional design space into a finite dimensional space Differentiate the continuum governing equations with respect to the design variables Discretize the equations using finite elements Solve and compute the gradients Gradient optimization Bn+1 Linking and homogenization Bo X B’n+1 Sensitvity linking and perturbed homogenization Perturbed homogenization COMPUTE GRADIENTS Materials Process Design and Control Laboratory

Design variables and objectives Definition of homogenized velocity gradient Decomposition of homogenized velocity gradient into basic 2D modes – Plane Strain Compression, Shear and Rotation Design objective – to minimize mean square error from discretized desired property (W) Design variables Materials Process Design and Control Laboratory

Multi-scale sensitivity analysis perturbed homogenized deformation gradient Sensitivity linking assumption: The sensitivity of the averaged deformation gradient at a material point is taken to be the same as the sensitivity of the deformation gradient on the boundary of the underlying microstructure, in the reference frame. Sensitivity equilibrium equation (Total Lagrangian) perturbed macro deformation gradient Sensitivity of (macro) properties Solve for sensitivity of microstructure deformation field Perturbed meso deformation gradient Perturbed Mesoscale stress, consistent tangent Integration of sensitivity constitutive equations Materials Process Design and Control Laboratory

Sensitivity equations for the crystal constitutive problem Sensitivity hardening law perturbed macro deformation gradient Sensitivity of (macro) properties Sensitivity flow rule Sensitivity constitutive law for stress Solve for sensitivity of microstructure deformation field Perturbed meso deformation gradient Perturbed Mesoscale stress, consistent tangent Integration of sensitivity constitutive equations From this derive sensitivity of PK 1 stress Materials Process Design and Control Laboratory

Initial microstructures for the examples <111> <110> Contains 151 and 162 grains, respectively, generated using a standard Voronoi construction Meshed with around 4000 quadrilateral elements. Mesh conforms to grain boundaries. An initial random ODF is assigned to the microstructures as shown in the pole figures Materials Process Design and Control Laboratory

Problem 1: Design of process modes for a desired response Equivalent stress (MPa) Equivalent stress (MPa) Initial response Intermediate Final response Desired response Time (sec) Time (sec) Final microstructure of the design solution (b) Cost function Change in Neo-Eulerian angle (deg) Iterations Misorientation map (c) (d) Materials Process Design and Control Laboratory

Design of multi-stage processes Modeling unloading Unloading process is modeled as a non-linear (finite deformation) elasto-static boundary value problem. Assumptions during unloading: No evolution of state variable during unloading Unloading is fast enough to prevent crystal reorientation during unloading The bottom edge of the microstructure is held fixed in the normal direction during unloading Stage 1: Plane strain compression Unloading Stage – 2 shear Materials Process Design and Control Laboratory

Iterations of the design problem Design for response in the second stage after unloading . 2 4 6 8 5 1 3 Equivalent strain Equivalent stress (MPa) Stage 1: Shear Stage 2: Compression Iterations of the design problem What should be the strain rate used in the first stage be for getting desired microstructure-response in the second stage after unloading? Materials Process Design and Control Laboratory

Design for response in the second stage after unloading b c d At the end of stage 1 e After unloading f During stage 2 Materials Process Design and Control Laboratory

Conclusions and Future work A multi-scale homogenization approach was derived and employed for modeling elasto-viscoplastic behavior and texture evolution in a polycrystal subject to finite strains. The model was validated with ODF-Taylor, aggregate-Taylor and experimental results with respect to the equivalent stress–strain curves and texture development. A continuum sensitivity analysis of homogenization was developed to identify process parameters that lead to desired property evolution. Design extensions Address process sequence selection and initial feature selection to obtain a desired response after loading (- Statistical learning problems) Model thermal processing stages, designing thermal stages Inclusion of grain boundary accommodation and failure effects Materials Process Design and Control Laboratory

URL: http://mpdc.mae.cornell.edu/ INFORMATION RELEVANT PUBLICATIONS S. Ganapathysubramanian and N. Zabaras, "Modeling the thermoelastic-viscoplastic response of polycrystals using a continuum representation over the orientation space", International Journal of Plasticity, Vol. 21/1 pp. 119-144, 2005 V. Sundararaghavan and N. Zabaras, "Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization", International Journal of Plasticity, in press CONTACT INFORMATION Prof. Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/ Materials Process Design and Control Laboratory