Scale-bridging models for micrstructure evolution in processing of copper

Schematic illustration of the vertical scale-bridging approach which is based on a meshless max-ent approximation on the macroscale, a recrystallization Taylor model on the mesoscale, and a crystal plasticity model on the subgranular microscale.

Severe plastic deformation (SPD), occurring ubiquitously across metal forming processes, has been utilized to significantly improve bulk material properties such as the strength of metals. The latter is achieved by ultra-fine grain refinement at the polycrystalline mesoscale via the application of large plastic strains on the macroscale. We here present a multiscale framework that aims at efficiently modeling SPD processes while effectively capturing the underlying physics across all relevant scales. At the level of the macroscale boundary value problem, an enhanced maximum-entropy (max-ent) meshless method is employed. Compared to finite elements and other meshless techniques, this method offers a stabilized finite-strain updated-Lagrangian setting for improved robustness with respect to mesh distortion arising from large plastic strains. At each material point on the macroscale, we describe the polycrystalline material response via a Taylor model at the mesoscale, which captures discontinuous dynamic recrystallization through the nucleation and growth/shrinkage of grains. Each grain, in turn, is modeled by a finite-strain crystal plasticity model at the microscale. We focus on equal-channel angular extrusion (ECAE) of polycrystalline pure copper as an application, in which severe strains are generated by extruding the specimen around a 90°-corner. Our framework describes not only the evolution of strain and stress distributions during the process but also grain refinement and texture evolution, while offering a computationally feasible treatment of the macroscale mechanical boundary value problem. Though we here focus on ECAE of copper, the numerical setup is sufficiently general for other applications including SPD and thermo-mechanical processes (e.g., rolling, high-pressure torsion, etc.) as well as other materials systems.

Ref: S. Kumar, A. D. Tutcuoglu, Y. Hollenweger, D. M. Kochmann, A meshless multiscale approach to modeling severe plastic deformation of metals: Application to ECAE of pure copper, Comp. Mater. Sci., 173 (2020), 109329.

Finding energy‐minimizing microstructures associated with nonconvex energetics

Microstructural patterns emerge ubiquitously during phase transformations, deformation twinning, or crystal plasticity. Challenges are the prediction of such microstructural patterns and the resulting effective material behavior. Mathematically, the experimentally observed patterns are energy‐minimizing sequences produced by an underlying non‐(quasi)convex strain energy. Therefore, identifying the microstructure and effective response is linked to finding the quasiconvex, relaxed energy. Due to its nonlocal nature, quasiconvexification has traditionally been limited to (semi‐)analytical techniques or has been dealt with by numerical techniques such as the finite element method (FEM). Both have been restricted to primarily simple material models. We here contrast three numerical techniques—FEM, a Fourier‐based spectral formulation, and a meshless maximum‐entropy (max‐ent) method. We demonstrate their performance by minimizing the energy of a representative volume element for both hyperelasticity and finite‐strain phase transformations. Unlike FEM, which fails to converge in most scenarios, the Fourier‐based spectral formulation (FFT) scheme captures microstructures of intriguingly high resolution, whereas max‐ent is superior at approximating the relaxed energy. None of the methods are capable of accurately predicting both microstructures and relaxed energy; yet, both FFT and max‐ent show significant advantages over FEM. Numerical errors are explained by the energy associated with microstructural interfaces in the numerical techniques compared here.

Ref: S. Kumar, A. Vidyasagar, D. M. Kochmann, An assessment of numerical techniques to find energy‐minimizing microstructures associated with nonconvex potentials, Int. J. Numer. Meth. Engng., 121 (2020), 1595-1628 (cover article).

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