Elsevier

Acta Materialia

Volume 46, Issue 15, 18 September 1998, Pages 5509-5522
Acta Materialia

A dislocation-based model for all hardening stages in large strain deformation

https://doi.org/10.1016/S1359-6454(98)00196-7Get rights and content

Abstract

A new model is presented to describe the hardening behaviour of cell-forming crystalline materials at large strains. Following previous approaches, the model considers a cellular dislocation structure consisting of two phases: the cell walls and the cell interiors. The dislocation density evolution in the two phases is considered in conjunction with a mechanical analysis for the cell structure in torsional deformation in which the cell walls are lying at 45° with respect to the macroscopic shear plane and are strongly elongated in the direction perpendicular to the applied shear direction. Guided by recent results on the volume fraction of cell walls [Müller, Zehetbauer, Borbély and Ungár, Z. Metallk. 1995, 86, 827], the cell-wall volume fraction is considered to decrease as a function of strain. Within a single formulation, all stages of large strain behaviour are correctly reproduced in an application for copper torsion. Moreover, strain rate and temperature effects are accounted for correctly and the predicted dislocation densities are in accord with experimental measurements. It is suggested that the factor responsible for the occurrence of hardening Stages IV and V is a continuous decrease of the volume fraction of the cell walls at large strains. A significant effect of the deformation texture variation on strain hardening is also discussed.

Introduction

In the seventies and eighties, modelling of strain hardening of metals was concentrating on the so-called Stage III hardening. Dislocation-based models proposed1, 2, 3, 4, 5, 6, 7 were able to describe a gradual decrease of the strain hardening coefficient θ characteristic of this hardening stage. In many cases the decrease of θ was shown to be linear in the flow stress1, 2. While this type of strain hardening behaviour could be easily explained in terms of a constitutive model using a single internal variable related to the total dislocation density1, 2, 3, 4, two-internal-variable models5, 6, 7 had to be invoked to account for deviations from linearity which were also observed. It is now generally accepted that, regardless of the particular detail of the underlying dislocation mechanism, the decrease of the strain hardening coefficient with stress is a result of dynamic recovery processes.

The Stage III theories mentioned predict a continual decrease of θ down to zero signifying saturation of stress at large strain. Of course, this saturation cannot be reached in a tensile test as necking intervenes when θ drops to a level of the applied stress σ. An interesting observation is, however, that even under deformation conditions preventing plastic instabilities, such as torsion testing, the gradual decrease of the strain hardening coefficient is interrupted by the onset of a new strain hardening stage, referred to as Stage IV8, 9, 10. This stage is characterized by a nearly constant hardening rate which depends slightly on temperature and strain rate. It was found that the value of this Stage IV hardening rate, θIV, scales with the “would-be” saturation shear stress and ranges between 0.05 and 0.10 of the latter (cf. Ref.[11]). A plateau on the θ vs stress diagram associated with Stage IV is followed by a fall-off signifying yet another stage of hardening, referred to as Stage V, which eventually leads to stress saturation.

The discovery of late stages of strain hardening (Stage IV and beyond) prompted significant activity on the theory side, and the eighties and nineties have seen publication of a number of models for these stages5, 7, 12, 13, 14, 15. These models are summarized in recent reviews16, 17, so that a detailed description of them is deemed unnecessary here. The majority of Stage IV models go back to the notion[18] that a material can be treated as a two-phase composite consisting of cell walls of high dislocation density and cell interiors relatively poor in dislocations. The stress supported by this structure is considered to be given by the stresses in both “phases” weighted according to a rule of mixtures. The stresses, in turn, are related to the dislocation densities in the two “phases” whose different rates of evolution with strain give rise to two separate stages of hardening (Stage III and Stage IV).

In a variant of the two-phase model[14], Stage IV hardening was associated with an athermal resistance to dislocation glide due to dislocation debris in the cell interiors. In the approach of Argon and Haasen[13], the cell interiors are basically void of dislocations. Yet, they are responsible for the hardening in Stage IV, because of the long-range stresses that arise from the increasing misorientations between cell walls and cell interiors. These authors present a comprehensive elasticity theory analysis for the stress state deriving from the misfit dislocations. Then they obtain the resultant resolved shear stress τr from the rule of mixtures for the stress τrw in the walls and τrc in the cell interiors asτr=fτrw+(1−f)τrcwhere f is the volume fraction of the cell walls. The authors relate τrw directly to the dislocation density ρw in the walls using the formulaτrw=αGbρwwhere α is a constant (typically around 0.25), G is the shear modulus, and b is the magnitude of the dislocation Burgers vector. With respect to τrc, Argon and Haasen calculate it as an internal stress stemming from the misorientation angle between the cell walls and cell interiors. Although Argon and Haasen obtained interesting results concerning Stage IV, it is important to note that their analysis is valid for a very particular configuration of the cell walls, namely when they are all parallel (see Figs 6, 7 and 9 and Appendix II in Ref.[13]). However, a real cell structure is at least two-dimensional, containing walls also in perpendicular positions, normally in equal proportions. Such a situation will be addressed in the present paper, albeit, with a completely different interpretation of the quantities τrc and τrw in Eq. (1).

While the existing models of late stages of strain hardening give a picture semi-quantitatively consistent with the experimental results, no particular model appears to describe the entirety of experimental facts related to these stages quantitatively. In our opinion, this is due to the fact that no attempt has been made so far to combine in a single model a dislocation theory-based approach with a sound solid-mechanics treatment taking into account the real topology of the dislocation structure that makes our two-phase “composite” more akin to a cellular solid, with a topologically continuous skeleton enclosing the soft cell interior phase, rather than to a soft matrix reinforced with isolated strong particles.

In the present paper, such an attempt at combining a physically based constitutive model for the constituent phases of the “composite” with a sound mechanics framework is made. In the model considered we return to the concept of the glide resistance in the cell interior being determined by the dislocation density therein, as suggested in the early models5, 7, 14, 18. An important new aspect, which turns out to be crucial in modelling Stage IV behaviour, is the variation of the cell-wall volume fraction during the deformation process observed by Müller et al.19, 20 and Zehetbauer and Ungár[21]. The problem is greatly simplified by a particular cellular arrangement that develops under torsion deformation conditions. Combined with a relatively simple kinetics of the dislocation density evolution considered, the resulting model is very simple in structure. It provides a unified description of Stages III, IV and V within a single formalism. Model predictions are in good quantitative agreement with experiment, no observed experimental fact being left unaccounted.

The paper is organized in the following way. In Section 2, a mechanistic model that takes the topology of the dislocation structure into account is presented. A constitutive description of the mechanical behaviour of the cell wall and cell interior phases based on the dislocation density evolution is given in Section 3. A comparison of model predictions with data in the literature for pure copper is given in Section 4. Finally, the main findings of this work are summarized in the Conclusions (Section 5).

Section snippets

The mechanics of cellular structure in torsion

A specimen deforming in torsion is considered. In this case the material deforms uniformly at the macroscopic scale. Effects of elasticity will be neglected so that the model to be developed is of a purely viscoplastic type. A dislocation structure that develops under torsion deformation can be described as cellular, with cell walls containing a high dislocation density separating cell interiors where the dislocation density is significantly lower. The mechanisms leading to the development of

Kinetics of plastic flow and dislocation density evolution

It was suggested in the composite model of Mughrabi[18] that there are internal stresses within the cell walls and the cell interiors after unloading. These stresses are also present when there is plastic flow. In fact, it is due to the internal stresses that the two phases deform plastically in a coherent way. For each of the two phases, we now specify constitutive relations based on dislocation theory considerations at the slip system level. As dislocation glide is rate and temperature

Application of the model: torsion deformation of copper

Before presenting the calculated hardening curves it is important to discuss the way they are going to be plotted.

Conclusions

In this paper a new model was presented to describe the strain hardening behaviour of dislocation cell-forming crystalline materials at large strains. The model is based on the dislocation densities in the two phases treated as internal variables. Coupled evolution equations for the two dislocation densities were considered. The mechanics of a cell structure consisting of strongly elongated cells oriented at 45° to the applied shear, which is characteristic of the case of torsional deformation,

Acknowledgements

The authors acknowledge many fruitful discussions on the experimental aspects of large strain hardening and cell formation with T. Ungár (Eötvös University, Budapest) and M. Zehetbauer (University of Vienna). Three authors (AM, LST and YB) gratefully acknowledge the Gledden Senior Visiting Fellowships from the University of Western Australia.

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