Phase field microelasticity accommodating large deformation and modeling of voids evolution under creep
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.
Section snippets
Method
The finite strain elastoplastic theory starts from the assumption of the existence of an intermediate configuration [32] so that the total deformation gradient can be taken as a multiplicative decomposition of the form . That is, the intermediate configuration is characterized by the deformation gradient which is caused by plastic or any inelastic strain, and the elastic deformation () is assumed to be based on that stress-free configuration. The assumption of the stress-free
Simulations and validations
Here we simulate a series of cases regarding void growth under creep. We first study a single spherical void under remote triaxial stress to validate the model. The matrix assumes linear viscous creep. This work has been studied by Budiansky et al. [12] who provide analytical asymptotic solutions for various situations. Subsequently, five-power-law creep is assumed for the matrix, and the simulation results is compared to Lebensohn’s dilatational-plasticity-FFT [17] and Garajeu’s finite element
Discussion
Influence of the incompatibility of inelastic displacement in the intermediate configuration
Our simulation results have shown that the IRID algorithm is valid and can quantitatively simulate void growth under creep. However, one may still question the justification of neglecting the incompatibility of inelastic deformation in the previous step at each IRID interval. In fact, whether the intermediate configuration is physical has been an issue of debate [34]. It is commented by Simo and Hughes
Conclusion
An IRID (Incremental Realization of Inelastic Deformation) algorithm is developed to deal with large deformation with Euler description in phase-field modeling. In this approach, the inelastic deformation is implemented by advection (and rotating when necessary) of the materials according to the inelastic displacement at finite number of intervals. During each interval, small strain assumption is still valid and the efficient FFT-based microelasticity solution methods of Khachaturyan et al. can
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This technical effort was performed in support of the Cross-Cutting Technologies Program of National Energy Technology Laboratory (NETL), United States under the RSS contract 89243318CFE000003. The research was executed through NETL Research & Innovation Center’s project coordinated by David E. Alman and was specifically under the advanced alloy development FWP led by Drs. David Alman and Omer Dogan and in part through the XMAT project. Discussions with Prof. Yunzhi Wang, Prof. Ingo Steinbach,
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