Multiscale Modeling Using Homogenization

1. Multiscale Modeling Using Homogenization PI: 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 • Microstructure obtained from an MC growth simulation • Equivalent stress after simple shear • Equivalent stress after plane strain compression • 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) a) pure shear and b) plane strain compression Design for desired materials response Homogenization of a 2D polycrystal Homogenization of a 3D polycrystal • Desired response of the material given by a smooth cubic interpolation of four desired coordinates • Change in the microstructure response over various iterations of t he optimization problem • Final microstructure at time t = 11 s of the design solution with misorientation distribution over grains • Change in objective function over various design iterations of gradient minimization algorithm • Idealized 2D polycrystal with 400 grains and one finite element per grain • Equivalent stress field after deformation in pure shear mode with a strain rate of 6.667e-4 s-1 • Comparison of the equivalent stress-strain curve predicted through homogenization with Taylor simulation • The initial texture of the polycrystals • Texture prediction using finite element homogenization • Texture prediction using the Taylor model • The final ODF obtained after simple shear • (top) The initial random texture of the material and (bottom) The final texture of the material • Equivalent stress field after deformation in pure shear mode • Comparison of the equivalent stress-strain curve predicted through homogenization with experimental results from Carreker and Hibbard (1957) 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? Implementation Update macroscopic displacements Largedef formulation for macroscale Update macroscopic displacements Macrodeformation Homogenized stress gradient Consistent tangent • Desired response in the second stage and response obtained at various design iterations • Microstructure response in the first deformation stage at various design iterations • Change in objective function over various design iterations of gradient minimization algorithm • Equivalent stress distribution (at final design solution) at the end of first deformation stage (time t = 1 s) • Residual equivalent stress distribution after unloading at the end of the first stage • Equivalent stress distribution at the microstructure at time t = 0.45 s of the second stage (plane strain compression) Boundary Value Problem for microstructure Solve for deformation field Consistent tangent formulation (macro) Mesoscale stress Mesodeformation Consistent tangent gradient Integration of constitutive equations Continuum slip theory Consistent tangent formulation (meso)