Phase field microelasticity accommodating large deformation and modeling of voids evolution under creep

Abstract

The phase-field microelasticity theory of Khachaturyan has been widely applied in materials science that is based on the small strain assumption. Here we develop an incremental realization of inelastic deformation (IRID) algorithm to deal with finite/large deformation with yet small elastic strain. A large deformation process is decomposed into a sequence of small deformation processes with intervals. At each interval the grids are stretched to accommodate the average inelastic deformation and advection/rotation operations are performed to account for the heterogeneous inelastic deformation. It is shown that Kachaturyan’s Fourier-transform based solutions can be adapted to the stretched grids. The IRID algorithm is compared against literature results regarding growth of a single void under remote stress of different triaxialities. Based on the IRID algorithm a phase-field model is developed that incorporates material microstructure, plasticity, multicomponent diffusion, and surface and grain boundary diffusion. The void growth kinetics and morphology evolution under coupling of diffusion and plasticity under creep are studied and the results show that coupling of the two mechanisms can significantly accelerate growth and coalescence of multiple voids at grain boundaries.

Introduction

Finite or large deformation is common during both the manufacturing/processing and service of metallic materials. At the microstructure length scale often when the macroscopic deformation is yet small there can be severe local deformation and distortion. Typical examples include creep damage and fracture of metals at high temperatures [1], [2], [3], and high temperature oxidation and corrosion of metals owing to the volume expansion associated with the oxidation reactions [4].

Creep rupture is one of the leading life-limiting factors for structural alloy components applied at high temperatures [5], [6], and for polycrystals it typically starts from growth and coalescence of voids at grain boundaries [7], [8], [9], [10]. Void growth and coalescence are intrinsic multiscale problems. At smaller length scales, the surface tension determines the shape of voids but it gradually loses its dominance when voids grow up. In the meanwhile, the growth kinetics of voids is subject to diffusional processes and dislocation creep. Typically the former is important for lower stresses and low damage-stage and the latter for higher stresses and high-damage stages [1]. To model and simulate this multiscale process has been a challenge. The single void growth under diffusional flow or dislocation creep in the bulk or at grain boundaries has been extensively discussed via analytical models [9], [11], [12], [13], [14]. In the literature, numerical creep damage models typically focus on one of the growth mechanisms and apply to either large or small length scale, although theoretical models indicate that there are often coupling, such as between dislocation creep and grain boundary diffusion [9]. Traditionally, void growth by dislocation creep (plasticity) is modeled by finite element methods [15], [16]. Rather recently, void growth models dominated by plasticity was modeled by Lebensohn et al. [17] by using an efficient Fast-Fourier-Transform (FFT) based algorithm on a representative volume element (RVE).

Also based on the FFT algorithm, the phase-field microelasticity theory of Khachaturyan has been widely applied in materials science in modeling phase-transformation and micromechanical problems in solids [18], [19], [20], [21], [22]. Nevertheless, Khachaturyan’s close-form solution is derived under small strain assumption, which appears to be a barrier for simulation of large deformation processes. Very recently, Zhao et al. formulated a phase-field model based on finite-strain theory [23], which employs the FFT-based finite-strain elasticity solver of Eisenlohr et al. [24] instead of the small-strain based solution method of Khachaturyan. Shanthraj et al. also developed a finite-strain phase-field model adapted to the finite-element framework [25]. Most of the existing finite-strain phase-field models are based on Lagrangian description and address displacive phase transformations without considering diffusional processes [23], [25], [26], [27]. While Lagrangian algorithms have the advantage of being able to apply the history dependent constitutive relations of materials, purely Lagrangian description is known to be weak in tracking severe distortion and topological changes that requires frequent remeshing schemes [28]. On the other hand, Eulerian description that is traditionally applied in fluid mechanics, has been introduced in phase-field modeling by Borukhovich et al. [29], [30]. In Borukhovich’s models, the large deformation is decomposed into a series of small deformation. An advection/rotation operation is defined to map the phase-field state variables and material properties to the new configuration according to the deformation of the previous step. A major critique on refs. [29], [30] is that the constitutive relation is of hypoelasticity type that lacks thermodynamic consistency [23]. In addition, a rate form of constitutive equation requires the stress to be updated by the advection and rotation operations in every time step in the simulation. It is also a concern whether the consistency of stress (the basic balance equation of elasticity) can be maintained after a large amount of mapping operations in phase-field modeling due to error accumulation such as in creep simulations.

In this work, we propose an algorithm to deal with large deformation with phase-field modeling, so-called incremental realization of inelastic deformation (IRID). It is assumed that the elastic strain is maintained small throughout the process, which is true for conventional metals and ceramics. In the IRID algorithm, the large inelastic deformation is decomposed into a sequence of small deformation steps, in analogy to the idea of Borukhovich et al. [29], [30]. Consequently, the efficient FFT-based solution method of Khachaturyan et al. [18], [20] can still be applied. In the IRID the major differences from Borukhovich’s approach are two folds: (a) only inelastic deformation rather than the total deformation is mapped to the new configuration; and (b) the hypoelasticity type constitutive models are abandoned. Motivated by the adaptive remeshing in Arbitrary-Lagrangian–Eulerian finite element method [28], [31], the regular FFT grid is adapted by uniform stretching which can accommodate large macroscopic deformation while maintaining the periodicity of the RVE. It is shown that Khachaturyan’s solution is valid under the stretched grid. With integration of the IRID algorithm, a general phase-field framework is developed that can include voids, precipitates, and grain structure in a multicomponent alloy system. The growth of a spherical void under remote stress of different triaxialities is simulated and compared with literature data. Finally, multiple voids growth and coalescence under coupled diffusion and plasticity is investigated.