Stochastic modeling of twin nucleation in polycrystals: An application in hexagonal close-packed metals

https://doi.org/10.1016/j.ijplas.2013.11.005Get rights and content

Highlights

  • Twin nucleation is modeled as a stochastic process.

  • Variable multiscale microstructure effects captured via probability distributions.

  • Model predicts distribution of critical nucleation strengths in hcp metals.

  • Model implemented in a visco-plastic self consistent framework.

  • Model correctly predicts twin variant selection and temperature dependence on twinning in zirconium.

Abstract

Twinning in hexagonal close-packed (hcp) metals is a multi-scale process that depends on the microstructural and mechanical response details at the polycrystalline aggregate, grain, micro, and atomic scales. Twinning can generally be regarded as a two-step process, a nucleation event followed by propagation and growth. This articles presents a stochastic model for the nucleation of deformation twins in hcp polycrystals. Twin nucleation is modeled through its dependence on lower length scale material details, such as the defect configurations at potential nucleation sites within grain boundaries, and mechanical details such as highly localized stress concentrations at the microscale in a probabilistic manner. These two aspects, the material and mechanical, must align for a successful nucleation event. The nucleation process is cast as a survival model parameterized by the local stress at the grain boundary. The model gives an explicit form for the probability distribution for the critical stress values required for twin nucleation. The model is implemented into a viscoplastic self-consistent (VPSC) crystal plasticity framework in order to test its predictive capability against previously reported statistical characterization in deformed zirconium at multiple temperatures. For implementation in VPSC, the stress concentrations are sampled from a distribution calibrated to full-field crystal plasticity simulations and a three-dimensional model of grain neighbors and distribution of grain boundary areas are implemented.

Introduction

Deformation twinning in hexagonal close-packed (hcp) metals, such as Zr, Mg, Be, Hf, Ti, etc., plays an important role in determining the plastic properties and overall stress/strain response. The growth of twins during deformation is one of the primary mechanisms for accommodating plasticity along the c-axis of the individual crystallites and has a dramatic effect on the hardening behavior and texture evolution of the polycrystal as a whole, particularly at room temperature and below. The incorporation of twinning into continuum scale deformation models is difficult in that, from the perspective of the simulation length scale, twinning is a highly variable event. Recent statistical analysis of 101¯2 twinning in hexagonal metals by Capolungo et al., 2009, Beyerlein et al., 2010 highlight this variability and demonstrate some of the basic features which a physics based twinning model must incorporate. These authors found that for Zr, under the same conditions explored in this study, only about 60% of the observed twins were the ones with the highest macroscopic Schmid factor (40% in textured Mg), while the remaining twins were of less favorable variants. Additionally it was observed that, while 90% of the grains most favorably oriented for twinning contained at least one twin, the remainder did not nucleate. On the other hand, it was also observed that a small, but non-negligible, fraction of grains poorly oriented for twinning (highest Schmid factor in grain 0.2) did in fact twin. An interesting feature of the quoted statistical analysis is that while the nucleated twin variants deviated from expectation, the subsequent growth of the twins largely followed intuition, in that the twins with higher macroscopic Schmid factors grew to be thicker than twins with lower Schmid factors. This suggests that twin nucleation and the subsequent twin growth are distinct processes that occur at fundamentally different length and time scales.

The nucleation event itself is dependent on the myriad details of the multi-scale material structure (i.e. grain lattice orientation, grain boundary character, defect structures in and near the grain boundary, etc.), as well as the highly local stress state and deformation rate present at the defect structures and internal material interfaces. As a result, the twin density, volume fraction, and morphology can vary widely in grains of the same crystallographic orientation. As such, nucleation is a truly multi-scale event, and from the perspective of the continuum scale, without the benefit of knowledge of the details at the lower length scales, twinning appears to have a large random or stochastic component.

Ongoing research on the possible twin nucleation mechanisms, based on energetic and kinetic considerations, coupled with qualitative or descriptive evidence from optical and electron backscatter diffraction (EBSD) microscopy suggests that twin nucleation occurs at grain boundaries (Beyerlein et al., 2011, Khosravani et al., 2013). Grain boundaries, or other interfaces, are the most likely sources for high densities of partial dislocations, which current nucleation theories require (Wang et al., 2010). Grain boundaries and other defects are the primary locations for stress concentrations in polycrystals, which can supply the energy necessary to overcome activation barriers for nucleation to take place. Additionally, high energy interfaces such as grain boundaries (relative to the bulk crystal lattice) are more likely to be able to accommodate the complex rearrangement necessary to support a stable twin nucleus.

The explicit insertion of twin nucleation criteria in polycrystalline mechanical models is problematic in that the actual nucleation event can be considered as a statistically rare microstructure event. In general the term statistically rare microstructure event refers to phenomena where the volume of occurrence is infinitesimal compared to the material volume, or that the probability of observation is virtually zero. The case of a single twin nucleation event in hcp metals can be considered as a statistically rare event in that the nascent twin nucleus forms on a small grain boundary domain which has an area that is vanishing when compared to the entire grain boundary surface in a volume consisting of hundreds or thousands of grains. When considering the very large number of grain boundary segments and strain increments, statistically rare twin nucleation events will still occur thousands of times in the material volume during deformation and, given sufficient time for growth, twinned domains will make up a significant volume fraction of the material.

Crystal plasticity models that incorporate twinning generally do not include a physically based mechanism to account for the observed variability in twin nucleation. As a consequence constitutive models for hcp metals often rely on completely empirical rules to determine when a twin forms, which variant is selected, the number of twins per grain etc. Most models depend on a simple deterministic critical resolved shear stress (CRSS) based law for describing nucleation and subsequent propagation of twins (Salem et al., 2005, Proust et al., 2007, Proust et al., 2009, Abdolvand et al., 2011). The two main problems with CRSS-based approaches are: (1) that the CRSS is typically applied at a length scale (micron to millimeter) several magnitudes larger than the defect scale interactions that trigger twin nucleation and (2) the actual stress to cause twinning is going to be strong function of the myriad details at the atom/defect length scales.

In this work, we describe an approach to include twin nucleation into constitutive laws for application into crystal plasticity models. This approach is grounded in two underlying assumptions: (1) Twin nucleation, propagation, and growth, are distinct physical processes that are driven by forces acting at different length scales. (2) Twin nucleation is a statistically rare event, which is largely controlled by local atomistic configurations and highly-localized stress concentrations at the grain boundaries. While the model does not explicitly take an atomistic description of the grain boundary into account, the distribution of local structure is implicitly captured in that the critical nucleation strength of a twin is not constant but will adopt a distribution that is an implicit function of the atomistic details of the boundary. The distribution of critical nucleation strengths will be referred to as the material component of twin nucleation and the dependence on local stresses at the grain boundary will be referred to as the mechanical component.To this end we present a stochastic model for twin nucleation, the form of which is largely influenced by the ongoing work in studying grain boundary and interfacial behavior at the atomistic scale. In this model nucleation is not deterministic but instead will occur with a probability that is governed by both material and mechanical factors.

The constitutive model was implemented into a visco-plastic self consistent (VPSC) crystal plasticity simulation code (Lebensohn and Tomé, 1993, Lebensohn et al., 2007). As VPSC is a homogenized model, the local stresses at grain boundaries are not explicitly available. To overcome this, full field crystal plasticity simulations based on a fast Fourier transform algorithm (Moulinec and Suquet, 1998, Lebensohn, 2001, Lebensohn et al., 2012) were carried out to characterize the distribution for grain boundary stress concentrations (see Section 5). Given an appropriate probability distribution for the stress concentrations, the mechanical component to twin nucleation can also be cast in a probabilistic framework. The net effect is the inclusion of multi-scale effects into a homogenized meso-scale simulation with minimal computation requirements.

While initially inspired by the prior stochastic treatment of nucleation by Beyerlein and Tomé, 2010, Beyerlein et al., 2011, this model is quite distinct both in philosophy and implementation. In particular the model: (1) incorporates a statistical representation of the tensorial stress fluctuations at grain boundaries which maintains a geometric consistency between the stress projections on the various twin systems, (2) separates the probability of twin nucleation at a boundary from the subsequent propagation of twinning, (3) is parameterized by stress and (4) predicts the distribution of critical twin nucleation strengths and explicitly describes the dependence of the twinning rate on stress. Several novel features have been implemented within VPSC to support the twin nucleation framework. Specifically: a statistical model for the number of 3D neighbors for a given grain which accounts for the grain size and shape distribution of the sample. The grain boundary surface surrounding the grain is also explicitly described as a series of facets, where the area of a specific facet depends on the number of neighbors, the grain size and the size of the neighboring grains.

Section snippets

Deformation twinning in hexagonal close-packed metals

For hcp metals, the dominant deformation mechanisms are (1) dislocation slip along the close-packed 112¯0 directions, either on the (0001) basal plane or on the 101¯0 family of prism planes, (2) c+a pyramidal slip on the 101¯1 planes, (3) 101¯2 or 112¯1 tensile twinning or (4) 101¯1 or 112¯2 compressive twinning. Deformation parallel to the c-axis can be accommodated by c+a slip or by twinning, however pyramidal slip is energetically unfavorable due to large Burgers vectors. By

Modeling the material aspects of twin nucleation

We will construct our model of twin nucleation from grain boundaries based on MD evidence showing the feasibility of dislocation reactions at grain boundaries leading to formation of twin seeds via local atomic shuffling followed by seed coalescence into stable twin nuclei (Wang et al., 2009). The model will start from the following core assumptions: (i) Twin nucleation occurs when some number of grain boundary dislocations (GBDs) undergo stress-driven transformations which then coalesce into a

Modeling the mechanical aspects of twin nucleation

In this section we will focus on the mechanical aspects, in particular the variance of the stress state within a grain and the multi-scale stress concentrations present at the grain boundary. In the same manner that the unknown defect content of the grain boundary introduced randomness into the critical stress required for nucleation, here the unknown local neighborhood will introduce a randomness into the stress concentration. From Eq. (7) we see that the Hazard Rate is a direct function of

Implementation

To validate the proposed nucleation theory through comparison with experimental mechanical testing and statistical characterization (EBSD), we have implemented the nucleation criteria into a multi-scale hcp constitutive model originally developed by Beyerlein and Tomé (2008) and further extended by Capolungo et al., 2009, Capolungo et al., 2009. The hcp constitutive model is exercised within the well known visco-plastic self-consistent (VPSC) framework, which relates the effective or

Results

Validation of the simulation results obtained by implementing the twin nucleation model presented above into VPSC is performed by comparison with two types of experimental information. The first is the effective polycrystalline response and sample average microstructural evolution such as the crystallographic texture evolution and evolution of twin volume fraction. Additional experimental validation is based on comparison of the model to detailed grain level twin statistics from EBSD (Capolungo

Discussion

The theory developed here treats twin nucleation in a probabilistic manner and subsequent growth in a deterministic manner. Nucleation and growth are fundamentally different processes and are controlled by stress states at different length scales. Very short range localized fluctuations are the driving force behind nucleation, while long range intergranular stresses drive the growth. The accuracy with which the model predicts both the number (nucleation) and volume fraction (propagation) of

Conclusion

In this article a new stochastic theory of twin nucleation in hcp polycrystalline metals is formulated. Conceptually the model can be broken into two aspects the material and the mechanical. The material aspect defines a probability of twinning given a stress that captures the unknown nature of the grain boundary defect structures and mechanisms necessary to nucleate a twin. The mechanical aspect deals with unknown stress concentrations at lower length scales. The theory is implemented into a

Acknowledgments

The authors acknowledge full support from the U.S. Department of Energy, Office of Science, Basic Energy Science, Project FWP 06SCPE401. Los Alamos National Laboratory is operated by LANS, LLC, for the National Nuclear Security Administration of the U.S. DOE under contract DE-AC52–06NA25396.

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