A novel level-set finite element formulation for grain growth with heterogeneous grain boundary energies

Abstract

Grain growth in polycrystals is one of the principal mechanisms that take place during heat treatment of metallic components. This work treats an aspect of the anisotropic grain growth problem. By applying the first principles of thermodynamics and mechanics, an expression for the velocity field of a migrating grain boundary with an inclination dependent energy density is expressed. This result is used to generate the first, to the authors' knowledge, analytical solution (for both the form and kinetics) to an anisotropic boundary configuration. This new benchmark is simulated in order to explore the convergence properties of the proposed level-set finite element numerical model in an anisotropic setting. Convergence of the method being determined, another configuration, using a more general grain boundary energy density, is investigated in order to show the added value of the new formulation.

Full Field and Mean Field Modeling of Grain Growth in a Multiphase Material Under Dry Conditions: Application to Peridotites

“…This formulation ensures that triple junctions respect Young's equilibrium. It has been shown by Fausty et al () that only considering the first and third terms of equation (i.e., classical strong formulation) with a heterogeneous field leads to triple junctions equilibrated at 120°. Taking into account the second term of equation permits to respect Young's law at triple junctions according to the different interfacial energies in place but also to respect the local values in the boundaries kinetics.…”

“…At each resolution time step, the DSP volume gained or lost is then redistributed throughout the microstructure (see Appendix ). To summarize, in order to model the grain growth within a forsterite (Mg‐rich end‐member of olivine) + enstatite (Mg‐rich end‐member of pyroxene) system, we proceed as follows: The heterogeneous fields and are defined at the different types of interfaces (Fo/Fo, En/En, Fo/En boundaries, see Table and section ). These fields are extended and regularized in order to make them differentiable by using the same method as Fausty et al (). The pre‐Laplacian term of equation is calculated using these two heterogeneous fields. The preconvective term of equation is calculated by using the heterogeneous field and a homogeneous field at a value equal to the one used for the of grain boundaries. The transport of the LS functions is obtained by solving equation through a FE framework (see ; Fausty et al, ) for details on FE integration procedure). The DSP volume gained or lost is then redistributed throughout the microstructure during a last transport step of the LS functions in order to ensure the volume conservation of each phase (see Appendix ). …”

“…• The heterogeneous fields and M are defined at the different types of interfaces (Fo/Fo, En/En, Fo/En boundaries, see Table 1 and section 3.1). • These fields are extended and regularized in order to make them differentiable by using the same method as Fausty et al (2018). • The pre-Laplacian term of equation (6) is calculated using these two heterogeneous fields.…”

“…• The preconvective term of equation (6) is calculated by using the heterogeneous field and a homogeneous M field at a value equal to the one used for the of grain boundaries. • The transport of the LS functions is obtained by solving equation (6) through a FE framework (see ;Fausty et al, 2018) for details on FE integration procedure). • The DSP volume gained or lost is then redistributed throughout the microstructure during a last transport step of the LS functions in order to ensure the volume conservation of each phase (see Appendix A).…”

“…The full field level-set (LS) approach has demonstrated its capability to model several microstructural evolutions in metallic materials (Bernacki et al, 2009;Maire et al, 2016; and was recently used to model isotropic grain growth in pure olivine aggregates (Furstoss et al, 2018). In this paper, we use recent (Fausty et al, 2018) and well-established (Agnoli et al, 2012; techniques of the LS framework to take into account the presence of second phases (SP), in order to simulate, using full field modeling, the grain size evolution of real mantle rocks under dry conditions. We then compare the full field results obtained with a mean field model based on the work of Bercovici and Ricard (2012), which describes the grain size evolution as a function of the different SP fractions and is calibrated for the temperatures of the upper mantle.…”