Gradient plasticity constitutive model reflecting the ultrafine micro-structure scale: the case of severely deformed copper

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Abstract

A further development of the mechanism-based strain gradient plasticity model well established in literature is reported. The major new element is the inclusion of the cell size effect in dislocation cell forming materials. It is based on a ‘phase mixture’ approach in which the dislocation cell interiors and dislocation cell walls are treated as separate ‘phases’. The model was applied to indentation testing of copper severely pre-strained by equal channel angular pressing. The deformation behaviour and the intrinsic length scale parameter of the gradient plasticity model were related to the micro-structural characteristics, notably the dislocation cell size, resulting from the deformation history of the material.

Introduction

The quest for miniaturisation of various metallic parts and components, such as electrical connectors, micro-screws or fasteners used in electronic and telecommunication products as well as prosthetics, has grown substantially over the last decade. A recent trend is to manufacture such parts by ‘micro-forming’, which is defined as the production of metallic parts by forming with dimensions in the submillimeter range (Geiger et al., 1997).

The production of parts by micro-forming cannot be simply scaled down from macro-processing using similarity principles and dimensional analysis. Scaling down the dimensions of a part while decreasing the grain size proportionally results in a change of the material behaviour at the micro-level. The well-known Hall–Petch relation that expresses the yield stress as a function of the grain size may break down when the grain size is reduced to the nanometer range (Morris, 1998) and, therefore, a constitutive model describing the behaviour of materials with the grain size in submillimeter range is required.

A second problem of miniaturisation is connected with modelling of the micro-forming processes because classical plasticity was formulated under the assumption that the dimensions of the deformation zone are much larger than the characteristic length scale of the micro-structure, e.g. the grain size. In the classical formulation, the equations of the plasticity boundary problem do not contain a dependence on strain gradients, which become significant when the dimensions of the deformed body and the tool are decreased, even if the grain size is reduced simultaneously and proportionally with the dimensions of the workpiece and the tool.

Two types of strain gradient plasticity theories (Fleck and Hutchinson, 1997, Gao et al., 1999, Yuan and Chen, 2001) can be distinguished. The first approach (Fleck and Hutchinson, 1997, Gao et al., 1999) assumes that the strain tensor εij and the strain gradient tensor ηijk produce a stress response in the form of a symmetric Cauchy stress tensor σij and a symmetric higher-order (moment) stress tensor τijk. Strain gradients, scaled by the ‘internal material length parameter’, l, were introduced in the constitutive description either in the Fleck–Hutchinson form (Fleck and Hutchinson, 1997):σ=σreff(εe),where the equivalent strain, εe, is a function of the components of the strain tensor εij, the strain gradient tensor ηijk, and three components of the parameter l, or in the Gao–Huang form (Gao et al., 1999):σ=σreff2(εe)+lηe.Here, ηe is the equivalent strain gradient and f(εe) is a classical function representing the strain dependence of stress in absence of a strain gradient. In both equations, σref denotes a reference stress.

The second approach to the formulation of strain gradient plasticity theory (Yuan and Chen, 2001) is based on the assumption that the actual flow stress of the material depends on the equivalent plastic strain gradient in the following way:σ=σreff2(εe)+lεe.

The main difference between this approach and that due to Fleck and Hutchinson is that this model does not contain higher-order strains or moment stresses. A conventional gradient plasticity model not involving higher-order gradients was also very recently proposed by Huang et al. (2004).

The definition and interpretation of the internal material length parameter in the aforementioned theories (Fleck and Hutchinson, 1997, Gao et al., 1999, Yuan and Chen, 2001) are somewhat different, l being obtained from micro-tension, micro-torsion or micro-indentation tests. In Huang et al. (2004) size effects were considered for micro-bend, micro-torsion and void growth. Alternatively, this length scale parameter can be obtained using a dislocation model, cf. the so-called ‘mechanism-based model’ (Nix and Gao, 1998). The latter results from the assumption that plastic deformation during micro-indentation into a {111} oriented plane of a Cu single crystal is accommodated by geometrically necessary dislocations, further assuming that this approach also holds for cold worked polycrystalline Cu. The expression for the (scalar) length scale parameter, l, following from the mechanism-based model (Nix and Gao, 1998) reads as follows:l=18α2μσref2b,where α is an empirical constant relating stress to the square root of the dislocation density (0.1α0.5), μ is the shear modulus and b is the magnitude of the Burgers vector.

The experimental data on hardness versus indentation depth support the mechanism-based model (Nix and Gao, 1998). It should be noted, however, that the indentation tests are typically restricted to indentation depths ranging from 0.15 to 2.0μm. The size of the projected area under the Berkovich indenter used for these experiments varies typically between 24.5 and 98μm2. In an indentation test on a nanocrystalline or ultrafine grained material many grains will thus be involved within this depth range, and it is to be expected that the indentation response will be strongly dependent on the grain size. Therefore, a generalisation of the model based on the strain accommodation mechanism in a single crystal to an ultrafine grained polycrystal requires additional considerations with regard to the grain size. As recently demonstrated by Chen et al. (2003), the hardness response of coarse-grained and nanocrystalline Cu in nanoindentation experiments with a Berkovich indenter differs significantly in the indentation range from 4 to 80 nm. This can be explained if the strain accommodation mechanisms are considered to be different for different grain size ranges (Conrad, 2003).

The dislocation glide mechanism that controls plastic flow gives rise to dislocation cell structure formation. The cell size formed under severe plastic deformation turns out to be in the order of several hundreds of nanometers (but not to fall below 100 nm) (Zhu et al., 2004, Dalla Torre et al., 2004). As the length scale of a gradient plasticity model for cell forming materials may be expected to be related to the cell size, an extension of the mechanism-based model (Nix and Gao, 1998) to include cell size effects is required.

Such an extension is discussed in the present paper. The mechanism-based strain gradient plasticity model is modified accordingly, and this makes it possible to identify the structural length scale parameter relevant to indentation testing in terms of the characteristic size of the dislocation cell structure formed in the material over its deformation history. To illustrate model applications and also to identify the parameters involved, the constitutive behaviour of ultra-fine grained copper produced by up to 16 passes (Route Bc) of Equal Channel Angular Pressing (ECAP), cf. (Zhu et al., 2004, Dalla Torre et al., 2004), is considered. In this process, the dislocation cell/grain size drops with an increasing number of passes from 10μm for homogenised copper to 170nm. Nano-indentation tests performed on samples processed by ECAP as well as transmission electron microscopy (TEM) analysis of the dislocation cell size and the cell wall width were used to verify the constitutive model proposed. The tensile yield strength and the ultimate tensile strength were derived using the model, along with estimates of the appropriate Taylor factor and the dislocation density.

Section snippets

Theoretical framework

We consider a scalar constitutive formulation in terms of the von Mises equivalent stress, σ, and strain, ε. The equivalent stress will be taken in the generally accepted form:σ=Mαμbρt,where ρt is the total dislocation density and M is the Taylor factor. The constant α can be taken as 0.25. Following the approach due to Ashby (1970), the total dislocation density can be represented as the sum of the geometrically necessary dislocations (of density ρg) and statistically stored dislocations (of

Prediction of tensile characteristics

An assumption of small strain was made to enable analytical integration of differential Eqs. (10) and (11). An estimate showed that the prediction of ultimate tensile stress is sufficiently precise if the strain is restricted to about 10%. The tensile tests on ECAP processed Cu samples showed that the maximum uniform strain, εu, i.e. the strain at which the flow curves exhibit a maximum, was well within this strain range. That was not the case for annealed copper, where the uniform strain

Conclusion

An analytical model describing the deformation behaviour of copper was derived. It is based on a ‘phase mixture’ of the dislocation cell interiors and cell walls and in addition includes a gradient term. The classical mechanism-based strain gradient plasticity model due to Nix and Gao was modified to include the dislocation cell size effect. An interpretation of the intrinsic length scale parameter that is related to the deformation history of the material in terms of the dislocation cell size

Acknowledgements

This research was supported by a Monash Research Fund—New Research Areas grant, the Victorian Centre for Advanced Materials Manufacturing and DFG (Grant ES 74/12-1).

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