Elsevier

International Journal of Plasticity

Volume 72, September 2015, Pages 151-167
International Journal of Plasticity

A microstructure based analytical model for tensile twinning in a rod textured Mg alloy

https://doi.org/10.1016/j.ijplas.2015.05.003Get rights and content

Abstract

A model for tensile twinning during the compression of rod textured magnesium is developed based on the idea that these twins nucleate at grain boundaries and that when the twin number density per grain is low these twins readily give rise to the formation of other ‘interaction’ twins in adjacent grains. Experimental observations of twin aspect ratios measured at a single grain size and twin number densities measured over four grain sizes were used to determine model material parameters. Using these, the model provides reasonable predictions for the observed magnitudes and trends for the following observations:

  • 1)

    Effect of grain size and stress on twin volume fraction, fractional twin length and the fraction of twin contact.

  • 2)

    Effect of grain size on the yield stress.

  • 3)

    Effect of grain size on the general shape of the stress-strain curve at low strains.

A parametric study shows the model to be quite robust but that it is particularly sensitive to the value of the exponent assumed for the twin nucleation rate law. It is seen that preventing the formation of interaction twins provides an important avenue for hardening and that the flow stress is also particularly sensitive to the relaxation of the twin back stresses. The model shows the importance of taking microstructure into account when modelling twinning.

Introduction

Deformation twinning exerts a controlling influence on the stress-strain behaviour of wrought magnesium (e.g. (Clausen et al., 2008, Gharghouri et al., 1999, Kelley and Hosford, 1968, Muránsky et al., 2010b, Wang et al., 2010a)). To understand the strength of this material requires an understanding of twinning. For instance, the rod textured extrusions considered in the present study (i.e. material produced by uniaxial extrusion, with an axisymmetric fibre texture comprising the c-axis perpendicular to the rod axis) are often found to display axial compressive yield strengths approximately half those seen in tension (Reed-Hill, 1973). This arises because the sharp texture favours {101¯2} ‘tensile’ twinning in compression, and the critical stress for twinning is so much lower (by at least ∼ 2 times (Agnew et al., 2006, Agnew et al., 2001, Wang et al., 2010a)) than that required to activate prismatic or pyramidal slip, with which it competes. In tension, the polarity of twinning precludes it from being activated and yielding occurs at the onset of prismatic, and possibly pyramidal, slip (Agnew et al., 2006, Muránsky et al., 2008).

The dominance of twinning in compression is striking and, along with the interest in the structural properties of this material, this has resulted in it becoming somewhat of a standard hcp material for academic studies of twinning. The effect of twinning has been incorporated into various crystal plasticity codes (Beyerlein et al., 2011, Beyerlein and Tome, 2010, Capolungo et al., 2009, Clausen et al., 2008, Fernández et al., 2013, Gu et al., 2014, Li et al., 2014, Liu and Wei, 2014, Niezgoda et al., 2014, Oppedal et al., 2011, Oppedal et al., 2013, Proust et al., 2009, Wang et al., 2010a, Wang et al., 2013a, Wang et al., 2013b). These capture the strong asymmetry but typically do so without explicitly considering twin nucleation and growth, which operate under different stresses (Christian and Mahajan, 1995). Niezgoda et al. (Niezgoda et al., 2014) extended the earlier models (Beyerlein et al., 2011, Beyerlein and Tome, 2010) within which the stochastic nature of twinning is considered whereby twin nucleation is seen to depend both upon mechanical fluctuations of local stresses and microstructural fluctuations of dislocation and interfacial configurations. A focus of these studies has been to predict the variation in twinning events seen over the grains within a polycrystal both in terms of the grain size and orientation. By assuming that twin nucleation occurs only at grain boundaries, good agreement was obtained with experimental measurements of the twin number density per grain, which is seen to increase as the grain size increases (Barnett, 2008, Beyerlein et al., 2010a, Beyerlein et al., 2010b). This is consistent with the idea that twin nucleation requires the defect configuration available at grain boundaries (Wang et al., 2010b).

The fact that twins nucleate at grain boundaries obviously introduces a strong size effect (Barnett, 2008) (Fig. 1a). However, grain boundary nucleation also means that events in neighbouring grains that cause sharp local stress concentrations at the boundary can become significant. Consequently, twinning events in neighbouring grains are often correlated (Barnett et al., 2012b, Beyerlein et al., 2010a, Wang et al., 2010c) (see Fig. 1b). This can be manifested as ‘butterfly twinning’ where two twins form apparently simultaneously either side of a boundary (Wang et al., 2010c) or as ‘catalytic’ twinning whereby twin events cascade from one grain to the next over the sample (Barnett et al., 2012b), see also (Baird et al., 2012, Min et al., 2014). Such propagation of twinning is seen to be prolific in the present material, particularly at fine grain sizes (<∼15 μm) (Barnett et al., 2012b) and under these conditions it is also common to see a Lüders-like yield elongation (i.e. a region of low, negligible or even slightly negative work hardening following plastic yielding – see Fig. 1a and also the recent study (Wu et al., 2015).

The present work develops a simple analytical twin nucleation and growth model to capture these effects for {101¯2} tensile twinning in extruded magnesium. The main aim is to gain confidence in the essential elements of the phenomena and to gain insight into strengthening effects, which will guide alloy development.

Statistical treatments (Beyerlein and Tome, 2010, Niezgoda et al., 2014) and the rather limited experimental data available (e.g. (Barnett et al., 2014, Ghaderi and Barnett, 2011)) have suggested that the following twin nucleation rate law provides a reasonable approximation in the absence of saturation:Ngb=1a0(ττ0)nwhere Ngb is the number of twins formed over unit grain boundary surface area, τ is the resolved shear stress, n is a rate exponent (≥1) and a0 and τ0 are reference values. Thus, τ0 is the resolved stress required to form one twin within area a0. This value is an averaged value. Locally, there is an important effect of boundary character on twin formation (Beyerlein et al., 2011, Niezgoda et al., 2013, Wang et al., 2010b, Wang et al., 2014), which would need to be taken into account in a full field model, for example. In the present case a mean field approach is adopted, so Equation (1) should be adequate, although it may introduce some sensitivity to significant changes in texture, or more strictly, grain boundary character distribution. Interestingly, Equation (1) bears resemblance to the visco-plastic rate sensitive slip law but note that here the twin nucleation is not a time derivative. The expression can be understood, in part, to be a reflection of the low stress end of the distribution of ‘strengths’ of the various nucleation sites present in the material. Some boundaries, for instance, will be more likely to produce twin nuclei from incoming dislocations than others. There are also likely to be different defect configurations that can decompose into a twinning nucleus. Phenomenologically, this is somewhat similar to the Weibull distribution of fracture strengths, an analogy that has been drawn by a number of workers (Meyers et al., 2001, Niezgoda et al., 2014). The idea is that increased stresses are required to raise the twin density because the weaker sites are used up at lower stress levels.

This rationale for Equation (1) reflects ‘microstructural’ effects (Beyerlein et al., 2011, Beyerlein and Tome, 2010, Niezgoda et al., 2014) but there are also stress variations that occur over the sample. These arise from slip induced plastic anisotropy and also to other twinning events. For instance, twinning relaxes local stresses, so subsequent nearby twinning events on the same system in the same grain are likely to require additional stresses to rebuild the field. Beyerlein and co-workers have adopted various approaches to capture aspects of these stress fluctuations (Beyerlein et al., 2010a, Beyerlein and Tome, 2010, Niezgoda et al., 2014); including full field modelling (Niezgoda et al., 2014). In the present case, they are lumped into the parameters of Equation (1), apart from the stresses at the tips of blocked twins, which we believe in magnesium to be the dominating source of stress fluctuation at low strains. So, we define τ in Equation (1) as a mean resolved shear stress over the sample. Further below we introduce a means of taking into account the effect of blocked twins, which are frequently seen to contact other twins, indicating co-operative formation (Barnett et al., 2012b, Beyerlein et al., 2010a, Fernández et al., 2013), as is also the case in zinc (Ecob and Ralph, 1983a, Ecob and Ralph, 1983b).

Following nucleation, twins expand in diameter rapidly and, in the present material, are often found to propagate over the area available on the habit plane of their parent grain (Barnett et al., 2013a). Thus, to a first approximation, their initial ‘propagation’ can be reduced to a matter of geometry. Subsequent twin ‘growth’ by thickening can be understood in terms of the resolved shear stress (Ghaderi and Barnett, 2011, Ghaderi et al., 2013); statistical studies show reasonable correlation between twin thickness and aspect ratio (twin thickness/twin length) and the Schmid factor (Barnett et al., 2013a, Beyerlein et al., 2010a). Although considerable fluctuations can be found in individual cases, as one would expect from the stress state required for plasticity and from the stress variations mentioned above (e.g. (El Kadiri et al., 2013)). In the present work, we will assume a near linear relation between twin aspect ratio and the mean resolved shear stress (Barnett et al., 2013b).

At low strains, twins in magnesium are frequently seen to adopt elliptical shapes on the metallographic surface (Ghaderi and Barnett, 2011). With increasing strain more complex shapes can appear and twin ‘blunting’ is commonly seen where twins intersect at grain boundaries (e.g. (Beyerlein et al., 2010a, Ma et al., 2012)). For the present case the grains are equiaxed and we assume the twins can be described by an equivalent average oblate spheroid with thickness t and diameter l (aspect ratio q = t/l). The average twin volume is thus assumed to be adequately described by the following function of the average effective diameter and average aspect ratio: v=πtl2/6=πl3q/6. The twin volume fraction, fv, in terms of the number of twins per unit volume, Nv, can then be given as:fv=π6l3qNv

The average contribution the twinning shear, γ (=0.13 for {101¯2} twinning in magnesium), makes to the axial strain can be written in terms of the mean scalar Schmid factor for twinning, m, as (Partridge, 1967):ε=fvmγ

If we can determine the stress dependencies of q and Nv, combining Equations (2), (3)) will provide a stress-strain relation. This is the basis for the present model. The key challenge is the derivation of an expression for Nv that considers the interaction between twinning events in one grain and those in its neighbours.

The remainder of the article is constructed as follows: model development is described first, then determination of material constants is undertaken and this is followed by model validation and a brief parametric exploration.

Section snippets

Model development

In the following section the model is developed starting with a description of an idealized microstructure. Then the nucleation and growth laws will be presented followed by a summary of the main equations.

Determination of material parameters n, c and a0τ0n

As noted above, the Lüders like yield elongation seen in fine grained rod textured magnesium (Fig. 1) corresponds to frequent observances of interconnected twins in the microstructure. Our interpretation, as presented previously (Barnett et al., 2012b), is that the formation of {101¯2} interaction twins permits twins to propagate from grain to grain in a front that moves over the sample at a more or less constant level of stress. We assume that at this stress the probability of a twin producing

Model validation

In the following section, we examine the ability of the model to explain trends and estimate the magnitudes of the following fractional measures determined from the microstructure: {101¯2} twin volume fraction, {101¯2} twin fractional length and fraction of {101¯2} twins in contact. These measures are implicit in the present model and the relevant data are obtained from references (Barnett et al., 2012b, Ghaderi and Barnett, 2011). Following this, the model is used to estimate the effects of

Discussion

The present model is based on the idea that {101¯2} twins nucleate at grain boundaries and that when the twin number density per grain is low these twins give rise to the formation other ‘interaction’ twins. The rate of formation of interaction twins drops as the twin number density rises, due to twin impingement within the parent grains. When the grain size is fine, higher stresses are required to provide each grain with twins, because of the lower surface area of smaller grains. This provides

Conclusions

A model for {101¯2} twinning in rod textured (extruded) magnesium was developed based on the idea that twins nucleate at grain boundaries and that when the twin number density per grain is low these twins give rise to the formation other ‘interaction’ twins in neighboring grains. The rate of formation of interaction twins drops as the twin number density rises, due to twin impingement within the parent grains.

Experimental observations of twin aspect ratios measured at a single grain size and

Acknowledgements

The first author is grateful for discussions with Alireza Ghaderi, Sean Agnew, Bevis Hutchinson and Laurent Capoloungo. Thanks to the Deakin University Academic Studies Program and to the “Chercheur d’Excellence” scheme of the Lorraine Champagne Region for support of the first author's sabbatical in France, which permitted this work to be carried out. This work was supported by the ARC Future Fellowship Scheme and the French State program “Investment in the future” operated by the National

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