Elastoplastic phase-field simulation of self- and plastic accommodations in Cubictetragonal martensitic transformation

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Abstract

An elastoplastic phase-field (PF) model of martensitic transformation is proposed to study the effects of self- and plastic accommodations on the evolution of a stress-accommodating martensitic structure and elastic strain energy minimization. Cubictetragonal martensitic transformation in elastic and elastoplastic materials is simulated using our elastoplastic PF model. The PF model can predict the self- and plastic accommodation behaviors during the martensitic transformation, which relax the transformation-induced stress field. Therefore, it can be shown that the elastic strain energy associated with the transformation is largely reduced by both accommodations. Our PF simulation also indicates that plastic deformation in the matrix arrests the growth of the tetragonal martensite phase, as suggested by some theoretical and experimental studies. In the evolution of the martensitic microstructure with both self- and plastic accommodations, we can find that the delay of the formation of the self-accommodating variant takes place, because the transformation-induced stress field is considerably relaxed by plastic deformation before self-accommodation. Furthermore, we can confirm that plastic accommodation largely reduces the elastic strain energy during the formation of the tetragonal phase that is affected by both self- and plastic accommodations, such as lath martensite formation, by the proposed PF model.

Introduction

The martensitic microstructure possesses desirable mechanical properties and, therefore, is used as an important constituent phase of high-strength steel. Thus, the metallurgy and crystallography of martensite and the mechanical properties of martensitic steel have been intensively studied experimentally and theoretically [1], [2]. It is well known that the morphology of the martensite phase in steel strongly affects the mechanical properties of steel. In a ferrous alloy, various morphologies of the martensite phase, such as thin plate, lenticular and lath martensite, are observed depending on the chemical composition of the alloy and transformation temperature [3], [4]. These different martensite morphologies are characterized by the minimization process of the elastic strain energy that arises during martensitic transformation. The elastic strain energy is lowered by either the formation of a multi variant domain or plastic deformation in the matrix and the martensite phase, which are called self-accommodation and plastic accommodation, respectively [5], [6].

Recently, a phase-field (PF) method has been extensively studied as a powerful tool for predicting the microstructural evolution and has been applied to the martensitic transformation [7], [8], [9], [10], [11], [12], [13], [14]. In particular, Khachaturyan and co-workers researchers have proposed time-dependent Ginzburg-Landau (TDGL) or phase-field models of the martensitic transformation on the basis of micromechanics theory or inclusion theory [15], [16], [17], [18], [19]. These models are now known as the phase-field microelasticity (PFM) models of martensitic transformation. These PFM models have been used to successfully simulate evolution process of twinned martensite, and some new aspects of the martensitic transformation have been found. These micromechanics approaches to the martensitic transformation are essentially equivalent to the phenomenological crystallographic theory of martensitic transformation, such as the Wechsler–Lieberman–Read (WLR) theory and the Bowles–Mackenzie (BM) theory, in terms of a solution of the minimization of elastic strain energy [16], [20], [21], [22], [23]. Therefore, the PFM model of martensitic transformation can predict the morphology and crystallographic feature of the martensitic microstructure. And, it is suggested that the results obtained by the PFM model are identical to that of the WLR and BM theories [16]. Although the formation of self-accommodated twinned martensite has been simulated using the PFM models, no PF model of martensitic microstructure associated with both self- and plastic accommodations has been established. Thus, in order to simulate the formation of the lenticular and lath martensites taking into consideration the self- and plastic accommodations, it is quite essential to develop a PF model describing multi variant formation and plastic deformation during the martensitic transformation.

In this study, we construct a new elastoplastic PF model of the martensitic transformation using the PFM theory and the TDGL description for plastic deformation. With our PF model, we investigate the contributions of self- and plastic accommodations to the evolution of the martensitic microstructure and the minimization process of the elastic strain energy.

In Section 2, the elastoplastic PF model of martensitic transformation is formulated. The main idea of the PF model is that the total eigen strain is defined as a sum of transformation-induced eigen strain and plastic strain, and the evolution of the plastic strain is described by the TDGL equation related to the shear strain energy proposed by Guo et al. [24]. After the computational model is presented in Section 3, the two-dimensional simulation of cubictetragonal martensitic transformation in elastic and elastoplastic materials is performed with our PF model in Section 4. According to Wang and Khachaturyan [16], the two-dimensional simulation of martensitic transformation is inadequate because the volumetric contribution of the elastic strain energy is eliminated. However, since the main purpose of this study is to propose the elastoplastic PF model, this paper is aimed at investigating the typical features of plastic deformation behavior (the plastic accommodation) during the martensitic transformation and their effects on transformation kinetics in two-dimensional space.

Section snippets

Phase-field model

The elastoplastic PF model of the martensitic transformation is constructed by combining the PFM model of the martensitic transformation [15], [16], [17], [18] with the elastoplastic PF model in order to describe plastic strain [24].

To describe the evolution of the microstructure and elastoplastic deformation using the PF theory, the total free energy of the system, G, is defined by the following Ginzburg-Landau-type Gibbs free energy functional,G=V(gch+ggr+gel)dV,where gch,ggr and gel denote

Numerical simulation

The cubictetragonal martensitic transformation with both self- and plastic accommodations is simulated in two dimensions. The governing equations, Eqs. (17) and (18), are solved by the finite difference method. We use a square computational domain with 256×256 meshes and periodic boundary conditions. The mesh size is Δx=2 nm and, therefore, the size of the system is 512nm×512nm.

Because we perform a two-dimensional simulation assuming the plane strain condition, the governing equation (Eq. (18))

Results and discussion

To gain a basic understanding of the effects of plastic accommodation on the evolution of the tetragonal phase and elastic strain energy minimization, we first simulate the growth of an isolated plate of the tetragonal variant 1 in elastic and elastoplastic materials. In this case, the formation of the opposite variant (variant 2) by self-accommodation is not considered. Then, the cubictetragonal transformation that generates two variants in elastic and elastoplastic materials is simulated to

Conclusions

The elastoplastic PF model of martensitic transformation was developed on the basis of the PFM theory and the TDGL equation that describes the evolution of plastic strain. The PF model allows us to simulate the martensitic transformation accompanied by self- and plastic accommodations and to investigate the contributions of both accommodation processes to the evolution of the martensitic microstructure and the minimization of the elastic strain energy. In this study, to study the effects of

Acknowledgements

We gratefully acknowledge the financial support of the Japan Society for the Promotion of Science for Young Scientists. This work was supported financially by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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