Interplay of martensitic phase transformation and plastic slip in polycrystals

Abstract

we propose a modelling framework to explore the interplay between martensitic phase transformations and plastic slip in polycrystalline materials, with an eye towards computational efficiency. The resulting framework uses a convexified potential for the internal energy density to capture the stored energy associated with transformation at the meso-scale, and introduces kinetic potentials to govern the evolution of transformation and plastic slip.The framework is novel in the way it treats plasticity on par with transformation.We implement the framework in the setting of anti-plane shear, using a staggered implicit/explict update: we first use a Fast-Fourier Transform (FFT) solver based on an Augmented Lagrangian formulation to implicitly solve for the full-field displacements of a simulated polycrystal, then explicitly update the volume fraction of martensite and plastic slip using their respective stick-slip type kinetic laws. We observe that, even in this simple setting with an idealized material comprising four martensitic variants and four slip systems, the model recovers a rich variety of SMA type behaviors. We use this model to gain insight into the isothermal behavior of stress-stabilized martensite, looking at the viii effects of the relative plastic yield strength, the memory of deformation history under non-proportional loading, and several others.We extend the framework to the generalized 3-D setting, for which the convexified potential is a lower bound on the actual internal energy, and show that the fully implicit discrete time formulation of the framework is governed by a variational principle for mechanical equilibrium. We further propose an extension of the method to finite deformations via an exponential mapping. We implement the generalized framework using an existing Optimal Transport Mesh-free (OTM) solver. We then model the α-γ and α-ε transformations in pure iron, with an initial attempt in the latter to account for twinning in the parent phase. We demonstrate the scalability of the framework to large scale computing by simulating Taylor impact experiments, observing nearly linear (ideal) speed-up through 256 MPI tasks. Finally, we present preliminary results of a simulated Split-Hopkinson Pressure Bar (SHPB) experiment using the α-ε model. ix

A micromechanics-inspired constitutive model for shape-memory alloys that accounts for initiation and saturation of phase transformation

“…This finding is also supported by the mesoscale observations of Daly et al (2008) that deviations in linearity of the stress-strain curve occurs well before the formation of macroscopic transformed regions. Bhattacharya and Schlömerkemper (2010) have recently proved in an idealized setting with uniform modulus that the transformation begins in isolated grains, while Richards et al have made the same observation using a full-field micromechanical model (Richards et al, 2013). Finally, Sittner and Novak (2000) have noted that the constant stress Sachs or Ruess bound accurately describes the initiation of the transformation in polycrystals.…”

“…As the best oriented grains begin to transform, whole grains cannot due to the constraints of the neighbors and the transformation proceeds in isolated regions (Richards et al, 2013). Thus, the internal stress distribution amongst grains becomes extremely heterogeneous as stress is redistributed from ideally oriented grains where transformation has nucleated to slightly misoriented grains, which subsequently nucleate (Richards et al, 2013;Paranjape and Anderson, 2014).…”

“…Furthermore, the quantified characteristics of superelastic responses vary significantly with material, processing, and loading mode. Non-proportional loading provides further complexities (Thamburaja and Anand, 2002;McNaney et al, 2003;Richards et al, 2013). The models we previously mentioned each focus on certain aspects of these phenomena with varying levels of complexity and fidelity, but an engineering demand for a simple, unified framework that describes all of these observations still exists.…”

“…All of these mechanisms in combination give rise to the observed hardening BC. As a result, the transformation of the polycrystal is quite heterogeneous (Brinson et al, 2004;Daly et al, 2008;Richards et al, 2013;Stebner et al, 2015;Paranjape and Anderson, 2014), as is the state of internal stress . As transformation and reorientation proceed, the poorly oriented grains exhibit smaller transformation strain in the direction of loading, and thus they begin to saturate and lock together eventually leading to a network of fully transformed grains surrounded by partially transformed and untransformed grains at C (Brinson et al, 2004).…”

Generalized grain cluster method for multiscale response of multiphase materials

“…The need for understanding the evolution of microstructural characteristics with deformation has stimulated the development of advanced micromechanical models that accurately describe the underlying physical phenomena, e.g., recrystallization [3,17], martensitic phase transitions [20,23,35], phase separation and coarsening by diffusion [5], twinning and detwinning [12,39], dislocation interactions [7,9], and cracking and damage growth [2,4,8,29]. In order to apply state-of-the-art micromechanical models for the analysis of large-scale engineering problems, efficient and generic multiscale methods need to be developed for keeping the computational times within manageable bounds.…”