Strain gradient plasticity modelling of high-pressure torsion

Abstract

Gradient plasticity modelling combining a micro-structure-related constitutive description of the local material behaviour with a particular gradient plasticity frame is presented. The constitutive formulation is based on a phase mixture model in which the dislocation cell walls and the cell interiors are considered as separate ‘phases’, the respective dislocation densities entering as internal variables. Two distinct physical mechanisms, which give rise to gradient plasticity, are considered. The first one is associated with the occurrence of geometrically necessary dislocations leading to first-order strain gradients; the second one is associated with the reaction stresses due to plastic strain incompatibilities between neighbouring grains, which lead to second-order strain gradients. These two separate variants of gradient plasticity were applied to the case of high-pressure torsion: a process known to result in a fairly uniform, ultrafine grained structure of metals. It is shown that the two complementary variants of gradient plasticity can both account for the experimental results, thus resolving a controversial issue of the occurrence of a uniform micro-structure as a result of an inherently non-uniform process.

Introduction

Severe plastic deformation (SPD) processing has emerged as one of the most promising techniques for producing ultrafine grained materials. Various SPD methods, such as high-pressure torsion (HPT), equal-channel angular pressing, twist extrusion, etc., have been established to achieve high strain and the attendant grain refinement (Valiev et al., 2006; Segal, 1995; Valiev and Langdon, 2006). Owing to strain non-uniformity inherent in SPD, strain gradients become a significant feature of these processes. The prevailing view in the solid mechanics community is that the influence of strain gradient effects needs to be included in the formulation of the constitutive behaviour of materials at micro-scale (Aifantis, 1984; Fleck and Hutchinson, 1993, Fleck and Hutchinson, 1997). A number of gradient plasticity models accounting for these effects have been proposed (Aifantis, 1987; Gao et al., 1999; Huang et al., 2000, Huang et al., 2004; Fleck and Hutchinson, 2001; Aifantis, 2003; Gurtin and Anand, 2005).

One popular approach, referred to as the mechanism-based gradient plasticity theory (Gao et al., 1999), is based on Ashby's concept of geometrically necessary dislocations (GND) (Ashby, 1970) used in conjunction with Taylor's theory of the flow stress (Taylor, 1938). This theory relates the flow stress, σ, to the total dislocation density, ρ_T:

$$\sigma =M\alpha \mathit{Gb}\sqrt{{\rho }_{\mathrm{T}}}\text{,}$$

where M is the Taylor factor, α the numerical constant, G the shear modulus and b the magnitude of the dislocation Burgers vector. Following Ashby's approach, the total dislocation density is represented as a sum of the density of GND, ρ_G, which are required for strain compatibility across the material, and that of statistically stored dislocations, ρ_S, accumulated by dislocation trapping at obstacles in a more or less random way.

The presence of GND leads to additional storage of defects, and it was suggested by Ashby (1970) that ρ_G is related to the absolute value of the equivalent plastic strain gradient

$$\eta =||\nabla {\epsilon }^{\mathrm{p}}||$$

through the equation

$${\rho }_{\mathrm{G}}=M\frac{\eta }{b}\text{,}$$

while the density of statistically stored dislocations, ρ_S, is related to the flow stress in the case of uniform deformation. It is given by the Taylor relation as used in the conventional plasticity theory:

$${\rho }_{\mathrm{S}}={\left(\frac{{\sigma }_{\mathrm{Y}}f({\epsilon }^{\mathrm{p}})}{M\alpha \mathit{Gb}}\right)}^2\text{.}$$

Here, the flow stress in the absence of gradient effects has been written as σ_Yf(ε^p), where σ_Y is the yield stress, ε^p is the plastic strain and the function f(ε^p) represents the strain hardening. This function can be most conveniently identified through an inherently uniform test, such as uniaxial tensile test in the pre-necking range.

Substitution of Eqs. (3), (4) in Eq. (1) leads to a modified, gradient dependent stress–strain relation:

$$\sigma =\sqrt{({\sigma }_{\mathrm{Y}}f({\epsilon }^{\mathrm{p}}){)}^2+M^3{\alpha }^2{G}^{2}b\eta }={\sigma }_{\mathrm{Y}}\sqrt{f^2({\epsilon }^{\mathrm{p}})+l\eta }\text{,}$$

where $l=M^3{\alpha }^2(G/{\sigma }_{\mathrm{Y}}{)}^{2}b$ is a parameter with the dimension of length. Relating this quantity to an intrinsic material length scale is a serious challenge to gradient plasticity modelling.

The length scale parameter l plays a significant role in gradient plasticity. It was observed in numerous experiments (Huang et al., 2000; Stolken and Evans, 1998; Shrotriya et al., 2003; Fleck et al., 1994; Ma and Clarke, 1995) that down-scaling of the specimen dimensions leads to deviations in the mechanical response of materials once at least one of the dimensions becomes comparable with intrinsic material length scale parameter. For specimen dimensions in the micro- or nanometre range the strength of metallic materials was found to be different from that predicted by classical theories. For example, micro-bending (Stolken and Evans, 1998; Shrotriya et al., 2003) and micro-torsion (Fleck et al., 1994) tests demonstrated that strength increases with decreasing specimen size. Furthermore, nano-indentation tests (Stelmashenko et al., 1993; DeGuzman et al., 1993; Ma and Clarke, 1995) revealed an increase of hardness with decreasing size of the indentor. It can be conjectured that such size effects occur when one of the specimen dimensions becomes comparable with the intrinsic material length scale. Gurtin and Anand (2005) distinguish between ‘energetic’ and ‘dissipative’ length scales associated with a back-stress and a dissipative hardening, respectively.

The significance of the intrinsic length scale calls for a physically sound definition of l, rather than determining it by fitting the experimental results of the above-mentioned tests to gradient plasticity models, such as those based on Eq. (5). A physical interpretation of the length scale parameter has been discussed by several researchers. Nix and Gao (1998) related l to L²/b, where L is the average spacing between dislocations. In a recent study, Al-Rub and Voyiadjis, 2004, Al-Rub and Voyiadjis, 2006 suggested that l may assume different values during the deformation history, depending on various quantities including the grain size, the characteristic dimension of the specimen and the hardening exponent. In our previous study (Lapovok et al., 2005), a physical definition of the length scale parameter was proposed which relates l to the dislocation cell size, d, resulting from the pre-straining history:

$$l=\frac{M}{\mathit{bK}}{d}^2\text{,}$$

where $K$ is a function of plastic strain.

Other approaches (Fleck and Hutchinson, 1997; Gao et al. 1999) make use of high-order continuum theory and assume that the strain tensor ε_ij and the strain gradient tensor η_ijk produce a stress response in the form of a symmetric Cauchy stress tensor σ_ij and a symmetric higher-order (moment) stress tensor τ_ijk. The quantity η entering Eq. (5) then assumes the following form:

$$\eta =\sqrt{c_1{\eta }_{\mathit{iik}}{\eta }_{\mathit{jjk}}+c_2{\eta }_{\mathit{ijk}}{\eta }_{\mathit{ijk}}+c_3{\eta }_{\mathit{ijk}}{\eta }_{\mathit{kij}}}\text{,}$$

where the three constants c₁, c₂ and c₃ are related to intrinsic material length scale parameters.

This theory requires high-order boundary condition and is therefore difficult to implement in existing FEM codes. An approach that would not involve higher-order strain or moment stress tensors would have the advantage of being suitable for use with standard finite elements methods. As indicated by Niordson and Hutchinson (2003), questions remain as to whether ‘lower-order’ gradient plasticity theories (in terms of their definition) can be justified from the physical viewpoint. However, as demonstrated by Huang et al. (2004), the effects associated with higher-order gradient plasticity are ‘only significant within a thin boundary layer of the solid’.

A phenomenological approach suggested by Aifantis (1984) introduces a strain gradient term of second order in the constitutive relation between the equivalent stress, σ, and the equivalent plastic strain, ε^p, in the following manner:

$$\sigma ={\sigma }_{\mathrm{Y}}f({\epsilon }^{\mathrm{p}})-c{\nabla }^2{\epsilon }^{\mathrm{p}}\text{,}$$

where c is a coefficient which contains a material length scale. This model was later modified and generalized (Aifantis, 1987; Zbib and Aifantis 1988; Mühlhaus and Aifantis, 1991; Aifantis, 2003) to include both first- and second-order gradient terms:

$$\sigma ={\sigma }_{\mathrm{Y}}f({\epsilon }^{\mathrm{p}})+c_1{\Vert \nabla {\epsilon }^{\mathrm{p}}\Vert }^q+c_2{\nabla }^2{\epsilon }^{\mathrm{p}}\text{,}$$

where c₁ and c₂ can be seen as containing hidden length scale parameters, which can be generally strain dependent and the exponent q is considered as a constant.

A further development of the model was suggested by Estrin and Mühlhaus (1996) who considered the function f to include micro-structure related internal variables Z₁, Z₂, … and to be plastic strain rate dependent. The model was represented by the following equation:

$$\sigma ={\sigma }_{\mathrm{Y}}f({\dot{\epsilon }}^{\mathrm{p}},Z_1,Z_2,\dots )+\tilde{c}{\nabla }^2{\epsilon }^{\mathrm{p}}\text{.}$$

In their analysis, Estrin and Mühlhaus (1996) considered elastic reaction stresses between adjoining discrete material elements, which arise from plastic strain incompatibility between the elements in the case of uniform strain distribution. The nearest neighbour approximation was used. In the limit case when the material elements are much smaller than the characteristic distances over which the plastic strain varies, these reaction stresses reduce to the second-order gradient term represented by the last term in Eq. (10). Higher-order terms, notably the fourth-order gradient, can also be included to ascertain the well-posedness of the problem (Estrin and Mühlhaus, 1996). Without that, the model considered is to be classified as ‘low-order strain gradient plasticity’ one in the sense defined by Niordson and Hutchinson (2003). If the ‘material elements’ are identified with the individual grains in a polycrystalline aggregate, the coefficient $\tilde{c}$ assumes the form $\tilde{c}=s{d}^2$, where d is the average grain size and s is a ‘spring constant’, which can be identified with Young's modulus.

The internal variables entering the first term on the right-hand side of Eq. (10) are related to dislocation densities described by evolution equations (Toth et al., 2002; Baik et al., 2003; Estrin, 1996; Roters et al., 2000). The constitutive models based on the dislocation density evolution have the benefit of providing an adequate description of strain hardening behaviour in terms of micro-structure development. They have been successfully applied to the simulation of strength and plasticity of a range of materials (Toth et al., 2002; Estrin, 1996; Roters et al., 2000). The model in the Toth et al. (1998) form is particularly suited for SPD, as was demonstrated for the case of equal channel angular pressing (Baik et al., 2003). The two internal variables of the latter model are associated with the dislocation densities in the dislocation cell walls and dislocation cell interiors. The model was already adopted in a gradient model (Lapovok et al., 2005), which involves the first derivative in strain, to account for a size effect, and will also be used in the present work in conjunction with the second derivative model (Estrin and Mühlhaus, 1996), cf. Eq. (10).

In the present study, the first-order gradient model (Lapovok et al., 2005) and the second-order one (Estrin and Mühlhaus, 1996), both of which are interpretable in terms of the micro-structural characteristics of a material will be applied for simulation of a highly non-uniform deformation process of HPT.