A multiphase-field approach to small strain crystal plasticity accounting for balance equations on singular surfaces

Abstract

An implementation of the crystal plasticity theory in the context of the multiphase-field method provides a numerically efficient tracking of evolving grain boundaries, modeled as diffuse interfaces. In literature, several approaches exist for the implementation of the plastic material behavior within the diffuse interface, based on interpolation, homogenization, or the mechanical jump conditions. Among these, only the jump condition approach exhibits an intrinsic relationship to the sharp interface (SI) theory. Therefore, in the work at hand, the implementation of the crystal plasticity theory within the jump condition approach, referred to as phase-specific plastic fields approach (PSPFA), is discussed in detail. The PSPFA is compared to the interpolation approach, referred to as common plastic fields approach (CPFA), using three-dimensional benchmark simulations of a bicrystal set-up. The comparison reveals that the PSPFA and SI coincide convincingly regarding the accumulated plastic slip and the Mises stress. In contrast, a significant deviation of CPFA and SI is observed both quantitatively and qualitatively, not only within the diffuse interface region, but throughout the complete simulation domain. A variation of the interface width illustrates that this observation can be transferred to the normal components of the total strain, even for smaller interface widths. Consequently, a quantitative estimate of the plastic material behavior, which is crucial for the prediction of the dynamic behavior of grain boundaries, is only provided by the PSPFA. The application of the crystal plasticity in the context of PSPFA to more complex microstructures is illustrated with respect to a periodic honeycomb-structure and an octotuple.

Investigation of microstructure evolution accounting for crystal plasticity in the multiphase‐field method

“…In the work at hand, the implementation of CP within the diffuse interface is based on the jump condition approach, compare, for example, Refs. [9][10][11], as introduced and described in detail by Prahs et al [12]. Thereby, the balance of linear momentum at a singular surface and the Hadamard jump condition are satisfied at each point within the diffuse interface.…”

“…The balance of linear momentum is formulated in the terms of the interpolated stress tensor σ. It can be derived by minimizing the free energy functional  , compare Equation (4), with respect to the displacement 𝒖 reading div( σ) = 𝟎, with σ = ∑ 𝑁 * 𝛼=1 𝜙 𝛼 𝝈 𝛼 , compare, for example, Prahs et al [12]. Within the phase-specific plastic field approach, the plastic energy density fp is given by the interpolation of the phase-specific plastic fields 𝑓 𝛼 p , reading fp = ∑ 𝑁 * 𝛼=1 𝜙 𝛼 𝑓 𝛼 p , compare, for example, Prahs et al [12].…”

“…As pointed out above, each phase can be associated with specific material properties and specific grain orientations. Therefore, a phase-specific plastic energy density can be derived from 𝜏 𝛼 C = 𝜕𝑓 𝛼 p ∕𝜕𝛾 𝛼 ac + 𝜏 𝛼 0 , compare, for example, Prahs et al [12], reading…”

“…which applies for the bulk and the diffuse interfaces, compare, for example, Prahs et al [12]. The condition 𝑓 𝛼 p (𝛾 𝛼 ac = 0) = 0 is met, by introducing an integration constant 𝐶 𝛼 = −(𝜏 𝛼 ∞ − 𝜏 𝛼 0 ) 2 ∕Θ 𝛼 0 .…”

“…as derived in detail by Prahs et al [12]. Herein, the jump of a quantity 𝜓 across a diffuse interface between two phases 𝛼 and 𝛽 is expressed by )) ,…”