Multiscale Modeling Using HomogenizationPI: Prof. Nicholas Zabaras Participating students: Veera Sundararaghavan, Megan Thompson

Material Process Design and Control Laboratory

How loading affects the microstructure FEM and Taylor texture predictions

Homogenization of a 2D polycrystal

Implementation

Homogenization of a 3D polycrystal

Why multiscale? • Material properties are dictated by the micro-

structure• Microstructures are complex and the response

depends on loading history, topology of grains, crystal orientations, higher order correlations of orientations, and grain boundary (defect sensitive) properties.

• A few relevant questions arise:• How do we find the best features (listed above)

for the material microstructure for a given application?

• How do we design sequences of processes to reach the final product so that properties are optimized?

Design for desired materials response

a) Microstructure is a representation of a material point at a smaller scale

b) Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure. (Hill, R., 1972)

a) Idealized 2D polycrystal with 400 grains and one finite element per grain

b) Equivalent stress field after deformation in pure shear mode with a strain rate of 6.667e-4 s-1

c) Comparison of the equivalent stress-strain curve predicted through homogenization with Taylor simulation

d) The initial texture of the polycrystalse) Texture prediction using finite element homogenizationf) Texture prediction using the Taylor model

Update macroscopic displacementsLargedef formulation for macroscaleUpdate macroscopic displacements

Boundary Value Problem for microstructureSolve for deformation field

Consistent tangent formulation (macro)

Integration of constitutive equationsContinuum slip theory

Consistent tangent formulation (meso)

Macrodeformation gradient

Homogenized stressConsistent tangent

Mesodeformation gradient

Mesoscale stressConsistent tangent

a) Microstructure obtained from an MC growth simulation

b) Equivalent stress after simple shear

c) Equivalent stress after plane strain compression

a) The final ODF obtained after simple shearb) (top) The initial random texture of the material and (bottom) The

final texture of the materialc) Equivalent stress field after deformation in pure shear moded) Comparison of the equivalent stress-strain curve predicted through

homogenization with experimental results from Carreker and Hibbard (1957)

a) pure shear and b) plane strain compression

a) Desired response of the material given by a smooth cubic interpolation of four desired coordinates

b) Change in the microstructure response over various iterations of t he optimization problem

c) Final microstructure at time t = 11 s of the design solution with misorientation distribution over grains

d) Change in objective function over various design iterations of gradient minimization algorithm

a) Desired response in the second stage and response obtained at various design iterations

b) Microstructure response in the first deformation stage at various design iterations

c) Change in objective function over various design iterations of gradient minimization algorithm

d) Equivalent stress distribution (at final design solution) at the end of first deformation stage (time t = 1 s)

e) Residual equivalent stress distribution after unloading at the end of the first stage

f) Equivalent stress distribution at the microstructure at time t = 0.45 s of the second stage (plane strain compression)