1. 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
URL:
Materials Process Design and Control Laboratory

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

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

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

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

20. 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

21. Homogenization of real microstructuresX 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

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

33. 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

34. 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 , 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
URL:
Materials Process Design and Control Laboratory