Strain localisation patterns under equal-channel angular pressing

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Abstract

Instabilities of plastic flow in the form of localised shear bands were experimentally observed to result from equal-channel angular pressing (ECAP) of magnesium alloy AZ31. The appearance of shear bands and their spacing were dependent on velocity of the pressing and applied back-pressure. A generic gradient plasticity theory involving second-order strain gradient terms in a constitutive model was applied to the case of AZ31 deformed by ECAP. Linear stability analysis was applied to the set of equations describing the deformation behaviour in the process zone idealised as a planar shear zone. A full analytical solution providing a dispersion relation between the rate of growth of a perturbation and the wave number was obtained. It was shown that the pattern of incipient localised shear bands exhibits a spectrum of characteristic lengths corresponding to admissible wave numbers. The interval of the spectrum of wave numbers of viable, i.e. growing, perturbations predicted by linear stability analysis was shown to be in good agreement with the experimentally observed spectrum. The effect of back-pressure applied during ECAP was also considered. The predicted displacement of the shear band spectrum towards lower wave numbers, shown to be a result of the decreased shear strain rate in the shear zone, was consistent with the experimentally observed increase of the band spacing with increased back-pressure. A good predictive capability of the general modelling frame used in conjunction with linear stability analysis was thus demonstrated in the instance of the particular alloy system and the specific processing conditions considered.

Introduction

Plastic strain localisation in metals in the form of shear bands has been repeatedly observed in several conventional experiments, including uniaxial tensile (Tvergaard et al., 1981) and compressive (daSilva and Ramesh, 1997) loading as well as torsion (Cizek, 2002) and shear (Gasperini et al., 2001; Wright and Ockendon, 1996) testing. The phenomenon of shear banding becomes a particularly important issue in severe plastic deformation processing, such as equal-channel angular pressing (ECAP) (Valiev et al., 2006) of low-ductility materials (Semiatin and DeLo, 2000), as in these processes crack initiation and failure of the billet occur along the shear bands formed. ECAP processing has recently emerged as an important technique for grain refinement by severe plastic deformation, see recent reviews by Valiev and Langdon (2006) and Beyerlein and Tóth (2008). The process can be viewed as basically simple shear limited to a zone near the plane where the entrance and the exit channels of a die intersect (commonly at 90°). The benefits of this processing technique are the extreme grain refinement (often down to submicron scale), which leads to a significant improvement of the mechanical properties of metallic materials. Generally, a fairly uniform microstructure is obtained throughout the bulk of the work piece deformed by ECAP (Baik et al., 2003). However, in some cases strain localisation into deformation bands, which are detrimental to the material's performance and can even cause failure, may occur (Semiatin and DeLo, 2000). The formation of shear bands is normally attributed to the intrinsic characteristics of the material flow, notably strain softening and/or low strain rate sensitivity of the flow stress (DeLo and Semiatin, 1999), texture effects, dynamic recrystallization or adiabatic heating (Rittel et al., 2008). The propensity of a material for strain localisation under isothermal ECAP was quantified in terms of a criterion for the onset of strain localisation by Semiatin and Jonas (1984). However, the spatial characteristics of the strain localisation pattern cannot be predicted on this basis (Semiatin and DeLo, 2000). Patterning in the form of localised shear bands was recently modelled by finite element (FE) simulations of the ECAP process (Figueiredo et al., 2006), but the results are of limited validity as a particular form of strain softening was used. An analytical approach to the strain localisation under ECAP conditions is therefore desirable.

Among the various theoretical approaches to the analysis of plastic flow instabilities (Boudeau and Gelin, 2000), the perturbation technique proposed by Molinari (1997) has emerged as a potent tool for predicting the spatial strain localisation patterns. This technique is used in the present paper to explain our experimental observations of strain localisation patterns in ECAP processed magnesium alloy AZ31. A particular feature of our experimental technique is the use of a back-pressure that imposes a high hydrostatic pressure on the ECAP billet (Lapovok, 2005). Experiments reported below have demonstrated that not only does the distance between shear bands formed during ECAP depend on the intrinsic constitutive properties of the material, but it is also sensitive to the hydrostatic pressure applied. We use a constitutive description that includes second-order gradient terms (Estrin et al., 2008; Estrin and Mühlhaus, 1996) in conjunction with the perturbation approach (Molinari, 1997) to account for the spatial patterns formed in the ECAP processing of alloy AZ31 and demonstrate a good predictive capability of the theoretical approach used.

Section snippets

Stress–strain curves for magnesium alloy AZ31

The AZ31 material was received in the form of continuously cast billet. Before testing, samples were homogenised at 420 °C for 4–5 h followed by water quenching.

Uniaxial compression tests were conducted on cylindrical samples with a diameter of 10 mm and a length of 15 mm at 250 °C (which corresponds to a homologous temperature of about 0.65) at the strain rates in the range of 0.03–3 s−1. The compression rig consisted of two heated platens incorporated in thermally insulated compression dies. The

ECAP of magnesium alloy AZ31

One pass of ECAP of homogenised cylindrical as-cast billets (35 mm in length and 10 mm in diameter) with the initial grain size in the range from 200 μm to 1.2 mm was performed in a 90° circular cross-section die 10 mm in diameter at 250 °C with different ram velocities: 0.5, 5.0 and 15 mm/s. Two levels of back-pressure were applied during ECAP, namely 130 and 260 MPa. ECAP tests without back-pressure were also conducted. After pressing, the samples were cut longitudinally, polished and etched

Linear perturbation analysis of shear band formation

In an idealised picture of plastic flow in a 90° ECAP die, shear deformation is localised in a narrow layer (Segal, 2002, Segal, 2003). We adopt this picture and denote the thickness of this layer by 2h. The analysis, by Molinari (1997), of shear banding in a layer of thickness 2h that experiences a simple shear in direction x by applying velocities ±V* to the boundaries of the layer, Fig. 5, can thus be adopted. It is interesting to note a similarity between ECAP and machining: in both

Stability analysis applied to ECAP processing of AZ31

The linear stability analysis carried out in the preceding section was applied to the present experiments. During the strain hardening stage (see Fig. 1), the material behaviour can be approximated by the same type of equation as Eq. (4), yet with a positive exponent for the strain-dependent term, which corresponds to a negative p value. However, for negative p, the η value is always negative, and hence a uniform regime is stable. Instability can only occur for positive p values; this is why

Discussion

Previous plastic instability analyses normally predicted a well-defined maximum in the growth rate of perturbation, η, as a function of the wave number ξ (Molinari, 1997). The preferred length scale of the emerging strain localisation pattern was thus ‘selected’ by the system naturally. By contrast, in the present modelling, the peak is quite flat—although it does have a maximum, cf. Fig. 6, Fig. 8. The location of the maximum ξηmax can be expressed analytically using Eq. (21)ξηmax=pγ˙(2ρ-(mτ/γ˙

Conclusions

In this paper, both a theoretical analysis and an experimental study were carried out to examine strain localisation patterns in ECAP of a technologically important magnesium alloy AZ31. Dynamic recrystallization occurring in this material was identified as a crucial factor that promotes strain softening, which is at the core of the concomitant strain localisation. The process shows an important difference with strain localisation under plane strain compression in that localised deformation

Acknowledgement

The authors gratefully acknowledge help with the ECAP processing of AZ31 samples by Monash University student Stuart Rundell.

References (32)

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