Microstructure evolution accounting for crystal plasticity in the context of the multiphase-field method

Computational Mechanics https://doi.org/10.1007/s00466-023-02423-7 ORIGINAL PAPER Microstructure evolution accounting for crystal plasticity in the context of the multiphase-field method Thea Kannenberg1,2 · Lukas Schöller1,2 · Andreas Prahs1 · Daniel Schneider2,3 · Britta Nestler1,2,3 Received: 29 August 2023 / Accepted: 13 November 2023 © The Author(s) 2023 Abstract The role of grain boundaries (GBs) and especially the migration of GBs is of utmost importance in regard of the overall mechanical behavior of polycrystals. By implementing a crystal plasticity (CP) theory in a multiphase-field method, where GBs are considered as diffuse interfaces of finite thickness, numerically costly tracking of migrating GBs, present during phase transformation processes, can be avoided. In this work, the implementation of the constitutive material behavior within the diffuse interface region, considers phase-specific plastic fields and the jump condition approach accounting for CP. Moreover, a coupling is considered in which the phase-field evolution and the balance of linear momentum are solved in each time step. The application of the model is extended to evolving phases and moving interfaces and approaches to strain inheritance are proposed. The impact of driving forces on the phase-field evolution arising from plastic deformation is discussed. To this end, the shape evolution of an inclusion is investigated. The resulting equilibrium shapes depend on the anisotropic plastic deformation and are illustrated and examined. Subsequently, evolving phases are studied in the context of static recrystallization (SRX). The GB migration involved in the growth of nuclei, which are placed in a previously deformed grain structure, is investigated. For this purpose, three approaches to strain inheritance are compared and, subsequently, different grain structures and distributions of nuclei are considered. It is shown, how the revisited method contributes to a simulation of SRX. Keywords Crystal plasticity theory · Multiphase-field method · Grain boundary migration · Strain inheritance B Thea Kannenberg Lukas Schöller Andreas Prahs Daniel Schneider Britta Nestler 1 Institute for Applied Materials - Microstructure Modelling and Simulation (IAM-MMS), Karlsruhe Institute of Technology (KIT), Straße am Forum 7, 76131 Karlsruhe, Germany 2 Institute of Digital Materials Science (IDM), Karlsruhe University of Applied Sciences, Moltkestraße 30, 76133 Karlsruhe, Germany 3 Institute of Nanotechnology - Microstructure Simulations (INT-MSS), Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany 1 Introduction Motivation The investigation of microstructural mechanisms in materials science and associated phenomena is fundamental for developing and improving materials regarding their effective properties such as strength and ductility. Furthermore, the prediction of the material behavior under certain conditions is essential for various applications. Microstructural evolution involves a large diversity of often complex processes. In order to manipulate microstructural evolution and thereby tailor the effective material properties, a thorough understanding of the phenomena occurring on a microscopic scale is essential. Modeling and simulation of material behavior, help to gain insights into microstructural transformation processes. In contrast to experiments, specific phenomena can be modeled and investigated, separately. Regarding the simulation 123 Computational Mechanics of microstructure evolution during recrystallization the modeling of migrating grain boundaries (GBs) is required. In this context, plastic deformation energy provides a driving force on GBs, amongst other sources of energy, cf., e.g., [1]. The plastic behavior of polycrystalline materials can be directly related to the evolution of lattice defects such as dislocations and experimental investigations are commonly based on the exploitation of dislocation density measurements. In the present work, the plastic deformation is considered in terms of classical crystal plasticity (CP) theory, cf., e.g., [2]. In this context, the plastic deformation is described by so-called plastic slips acting on specific slip systems characteristic for each crystal symmetry. Thus, the CP constitutes a phenomenological model that takes into account the underlying microstructure such as the characteristic slip systems. Based on Orowan’s law, cf., e.g., [3, Eq. (6.31b)], the plastic slips can be related to the dislocation densities in an approximative sense and the considered model is applied to describe the microstructural evolution during load as well as morphological changes of the microstructure after loading. Multiphase-field method The tracking of GBs in the sharp interface context is numerically challenging and costly. It can be circumvented by the use of a multiphase-field method (MPFM). Instead of modeling GBs as material singular surfaces, cf., e.g., [4], as commonly done in classical continuum mechanics, GBs are modeled as diffuse interfaces within the MPFM, cf., e.g., [5–9]. The position of interfaces is implicitly given by the contour of the field of order parameters, thus, no explicit tracking is needed, cf., e.g., Chen [9]. The MPFM is widely established for simulating microstructural evolution. Applications include solidification [10, 11], solid-solid phase transformations, cf., e.g., [12, 13], and more recently crack propagation [14–16]. In the work at hand, the MPFM is based on the work by Steinbach et al. [17], Steinbach and Pezzolla [18] and Nestler et al. [19]. Crystal plasticity in the context of the multiphase-field method To implement CP in an MPFM, an approach is required to account for the mechanical fields in the diffuse interface. In literature, three approaches are discussed: the interpolation approach, the homogenization approach, cf., e.g., [20, 21], and, more recently, the jump condition approach, cf., e.g., [22–25]. Regarding the interpolation approach, all material points follow the same set of mechanical constitutive equations. The material parameters, however, may differ and are interpolated within the diffuse interface, cf., e.g., [20]. More possibilities are offered by homogenization approaches, where different constitutive equations may apply in different phases. The behavior of the diffuse interface is significantly influenced by the chosen homogenization scheme, cf., e.g., [20]. The inheritance of plastic strains 123 during phase transitions is an important topic of an ongoing discussion. Within interpolation and homogenization approaches the transformation behavior is significantly different, as discussed, e.g., by Ammar et al. [26]. The jump condition approach can be seen as an addition to the homogenization approach. Schneider et al. [24] introduced a method for the calculation of stresses in a multiphase system, which satisfies the mechan (...truncated)