Generalized Aifantis strain gradient plasticity model with internal length scale dependence on grain size, sample size and strain

1. Introduction

Strain gradient plasticity (SGP) theories [1-17] have been successfully employed to explain the size effect of metals under nonuniform deformation since the advent of Aifantis’ SGP theory in the 1980s [18,19], such as wire tension/torsion [20,21] and deformation of composites [22,23]. Such a capability is achieved by introducing strain gradients to account for nonlocal deformation and internal length scale (ILS) to consider the region where strain gradients are significant. However, ILSs are usually introduced into strain gradient theories for dimensional consistency. Although efforts have been made to address this issue [6,7,9,24-31], their physical meaning and relation with the material microstructure remain open issues [32]. Moreover, the value of ILS varies from one SGP model to another and changes from one experiment to another [32]. For instance, Fleck et al. [33] predicted the ILS to be approximately l = 2.6 μm-5.1 μm from the size-dependent torsion tests of copper wires; Li et al. [34] reported that the ILS takes different values by using strain gradient theory and modified couple stress theory; Liu et al. [28] confirmed that the ILS in Nix and Gao’s SGP theory changes from several millimeters to tens of millimeters, while it is several micrometers in the models of Fleck and Aifantis. Thus, it is controversial whether the ILS is an externally introduced phenomenological parameter or an intrinsic material parameter.

Based on the physical background of size effects, the ILS should depend on the microstructure determining the heterogeneous deformation field. With the evolution of the material microstructure, the ILS will also change. However, most researchers have treated ILS as a constant in the strain gradient modeling of experiments [33,35-38]. Aifantis and his coworkers [39] first employed a nonconstant ILS in their SGP modeling of thin wires under torsion. Voyiadjis and Abu Al-Rub [7,25] also found that a constant internal length could not predict some experiments successfully. In the past few years, they made much effort to correlate ILS with microstructures [9,25,32,40]. However, this issue has not been well addressed, which stimulates this study to establish such connections using systematic torsion experiments considering both grain (intrinsic) and sample size (extrinsic) effects.

In the current work, the interaction between dislocations is considered to control the plastic flow of metals. Then, the ILS in the classical SGP theory is reformulated with a power-law relationship derived from the Taylor hardening rule [27], and the model is further modified to consider the intrinsic size effect. Comparisons with experimental results show that the initial ILS depends on both the geometrical size (sample size) and the microstructure (grain size). Furthermore, the generalized Aifantis’ SGP model predicted larger ILSs than the generalized Fleck-Hutchinson model, and possible reasons are discussed in detail.

2. Aifantis’ SGP theory

In Aifantis’ SGP model [39], the relationship between effective stress $\overline{\sigma }\text{=}\sqrt{\text{3}/\text{2}{{\sigma }^{\prime }}_{ij}{{\sigma }^{\prime }}_{ij}}$ and effective strain $\overline{\epsilon }\text{=}\sqrt{2/3{{\epsilon }^{\prime }}_{ij}{{\epsilon }^{\prime }}_{ij}}$ is

where ${{\sigma }^{\prime }}_{ij}$ and ${{\epsilon }^{\prime }}_{ij}$ denote the deviatoric stresses and deviatoric strains, respectively. $\kappa \left(\overline{\epsilon }\right)$, $c_1\left(\overline{\epsilon }\right)$, and $c_2\left(\overline{\epsilon }\right)$ are functions of effective strain. $\kappa \left(\overline{\epsilon }\right)$ describes the contribution of statistically stored dislocations (SSDs) to local hardening, while the second and third terms describe the nonlocal hardening coming from geometrically necessary dislocations (GNDs) in materials under nonuniform deformation. A power-law relation is utilized to depict the SSD hardening,

where ${\kappa }_0$ and n denote the hardening modulus and exponent, respectively.

The coefficients of the gradient terms are considered to depend on the effective strain as

where c₀ corresponds to the gradient coefficient at a strain level of 1. Then, Eq. (1) takes the form of

The combined contributions of local and nonlocal hardening are modeled through the superposition of corresponding flow stresses, while an alternative choice would be the superposition of effective strains [41].

For torsion problems, the effective stress and strain appear to be $\overline{\sigma }\text{=}\sqrt{3}\tau$ and $\overline{\epsilon }\text{=}\gamma /\sqrt{3}$, where $\tau$ denotes the shear stress in the cross-section of the wire, and $\gamma$ represents the shear strain. According to Eq. (4), the constitutive equation for torsion is formulated as

In a cylindrical coordinate system, where r, θ, and z represent the radial, circumferential, and axial directions, $\gamma$ is linearly proportional to the twist per unit length of the wire $\phi$ as

The Laplacian of $\gamma$ is calculated as

Combining Eqs. (5)-(7) gives the shear stress at a twit of $\phi$ as

Using the static equivalence principle and integrating $\tau$ over the cross-section of a cylinder, we have the torque

Then, the relation between the normalized torque $Q/a^3$ and surface shear strain ${\gamma }_{\text{s}}=\phi a$ is given as

Tsagrakis et al. [39] defined the internal length as $l_0=\sqrt{c_0/{\kappa }_0}$; then, Eq. (10) is alternatively written as

It represents the constitutive response of thin wire considering nonuniform deformation along the radial direction. If the strain gradient is neglectable, i.e., $l_0/a\to 0$, Εq. (11) degenerates into the classical form

where ${\overline{\kappa }}_0={\kappa }_0/{\left(\sqrt{3}\right)}^{n+1}$, ${\tau }_{\text{s}}={\overline{\kappa }}_0{\gamma }_{\text{s}}^n$. This relation indicates that the normalized torque is proportional to the surface shear stress. It is also noted that using Eq. (12), the plots of surface stress versus strain for wires with various geometrical sizes are supposed to coincide into a single curve.

Tsagrakis et al. [39] applied the above model to study the torsion of thin wires [33], and the comparison between modeling results and experimental data was perfect (see Fig. 1 in Ref. [39]). The model parameters are summarized in Table 1. All curves share the same set of parameters except for the wire with D = 2a = 20 μm. Herein, Aifantis’ SGP model is further applied to study the recent experimental results of Liu et al. [42]. To determine the strain hardening exponent n in the constitutive model, the log-log fit to the curves of $Q/a^3$ versus ${\gamma }_{\text{s}}$ is performed. Then, other undetermined parameters are fitted and summarized in Table 1. The theoretical modeling results coincide well with the experimental measurements, as given in Fig. 1.Comparison of the modeling results using Aifantis’ theory with experimental results [42].Model parameters for the experimental results of Fleck [33] and Liu et al. [42] using the original Aifantis SGP model, i.e., Eq. (11)

Apart from the sample size, the grain size is also supposed to affect the deformation behavior of the material at the microscale. Here, taking the grain size effect into account, the parameter ${\kappa }_0$ in the SGP model is supposed to depend on grain size through the Hall-Petch relation [43,44], i.e.,

where d denotes the grain size. Then, combining Eq. (13) with Eq. (10), we have a generalized torsion response that depends on both grain size and sample size as

It is noted that the gradient parameter c₀ is supposed to be a constant, without grain size dependence.

Using this modified model to investigate the experiments of Gan et al. [45], in which both the grain (intrinsic) and sample (extrinsic) size effects were studied, we obtained the model parameters listed in Table 2. Here, the ILS l₀ is grain size-dependent, i.e.,Model parameters for torsion experiments of Gan et al. [45] using Aifantis’ SGP model with grain size dependence, as expressed by Eq. (14)

Table 2 indicates a grain size- and sample size-dependent ILS. However, different ILSs should be adopted for different grain sizes. A common assumption is that the ILS is linked to the mean spacing among dislocations [25,40] and should be less affected by grain size at a high strain level (e.g., $\overline{\epsilon }=1,{\gamma }_s=\sqrt{3}$) [46,47]. Thus, a constant value should be pursued. Then, Eq. (14) is modified to be

leading to a grain size-dependent c₀ in the Hall-Petch form of

where ${\widehat{c}}_0={\widehat{\kappa }}_0{l}_0{}^2$ and $k_{\text{c}}=k_{\text{HP}}{l}_0{}^2$. The comparisons of theoretical results utilizing Eq. (16) with experimental results [23] are given in Fig. 2. Table 3 lists the corresponding parameters.The comparison between the modeling results by the grain size-dependent Aifantis’ SGP theory and the experimental measurements of Gan et al. [45].Parameters for the torsion experiments of Gan et al. [45] using the grain size-dependent Aifantis SGP model, as expressed in Eq. (16)

3. Generalized Aifantis’ SGP model

As the ILS has been defined as $l_0=\sqrt{c_0/{\kappa }_0}$, the evolution of the gradient parameter $c_2(\overline{\epsilon })$ is rewritten as

If we further assume a strain dependence of ILS in the form of

then Eq. (4) changes to be

In Eq. (19), l₀ is considered the ILS at a strain level of 1, which is termed the reference ILS in the current work. It is clear that this strain-dependent ILS in Eq. (19) takes a similar form to the relation of $l=l_0{\epsilon }^{-n}$, which was derived based on the Taylor hardening law in our previous work [27], and they will be the same for n = 1/3. In this work, the influence of flow stress $\kappa \left(\overline{\epsilon }\right)$ on the ILS is also considered to achieve a more general framework; then, Eq. (20) can be rewritten as

with $l\left(\overline{\epsilon }\right)=\sqrt{c_2\left(\overline{\epsilon }\right)/\kappa \left(\overline{\epsilon }\right)}$. To maintain consistency of the strain-dependent ILS in this work with previous work [27], we further adopt the above-stated power-law relation $l=l_0{\epsilon }^{-n}$ to reflect the strain dependence of the internal length:

For wire torsion, $\overline{\sigma }=\sqrt{3}\tau$, $\overline{\epsilon }\text{=}\gamma /\sqrt{3}$, $\gamma =\phi r$, and ${\nabla }^2\gamma =\phi /r$. Similar to Eq. (8), the shear stress can be written as

Combining Eq. (23) with Eq. (9) gives

We then have

Comparisons between the modeling results using the generalized SGP model and experimental results [20] are shown in Fig. 3; the model parameters are given in Table 4. The values of l₀ and n are almost the same as those in the original SGP model. Furthermore, experimental results show that the copper wire with a larger diameter maintains a larger hardening exponent (confirmed by Table 4), indicating a larger initial ILS for larger samples based on the relation of $l=l_0{\epsilon }^{-n}$. This conclusion coincides with the results of Begley and Hutchinson [48], Abu Al-Rub and Voyiadjis [25], as well as the recent discovery of Khorshidi [49] using a modified couple stress theory. In the work of Abu Al-Rub and Voyiadjis [25], the plastic strain and strain hardening exponent is involved in formulating the ILS, behind which the sample size was implied.Comparisons between the modeling results by the generalized Aifantis’ SGP theory and the experimental measurements of Liu et al. [42].Model parameters for torsion experiments of Liu et al. [42] using the generalized Aifantis’ SGP model, as expressed in Eq. (25)

The physical mechanism for the sample size-dependent ILS can be interpreted in two aspects. First, in Aifantis’ SGP model, gradient strengthening is associated with ${\nabla }^2\overline{\epsilon }$, which coincides with the physical model derived by Groma et al. [50]. During their derivation, an ILS, i.e., the range of dislocation correlations that is proportional to the dislocation spacing, emerges. In other words, the ILS controlling the gradient hardening in Aifantis’ theory is a quantity proportional to the dislocation spacing. For copper wire with a smaller diameter, the strain gradient, and thus the GND density, is higher, which results in a higher total dislocation density and a smaller dislocation spacing. Therefore, the ILS is larger for wires with larger diameters. Second, for the copper wires studied in this work, their stress-strain relation can be described by a power-law formula; see, for example, Eqs. (11) and (16) and Figs. 1 and 2. By means of the Considère criterion ${\sigma }_{\text{e}}=\partial {\sigma }_{\text{e}}/\partial {\epsilon }_{\text{e}}$ ($Q/a^3=\partial \left(Q/a^3\right)/\partial \left(\phi a\right)$), the uniform elongation of the copper wire is calculated to be ${\epsilon }_{\text{e}}=n$. Based on the experimental results of Liu et al. [42], copper wires with larger diameters have larger uniform elongation ${\epsilon }_{\text{e}}$ and thus larger n. Moreover, the total dislocation density ρ in copper wires can be calculated by the sum of GND density ${\rho }_{\text{GND}}$ and SSD density ${\rho }_{\text{SSD}}$, i.e., $\rho ={\rho }_{\text{SSD}}+{\rho }_{\text{GND}}$. For wires with smaller diameters, the strain gradient is more significant, and ${\rho }_{\text{GND}}$ and ρ are higher, which will promote the stress concentration and premature failure of wires; thus, the uniform elongation and hardening exponent are smaller for wires with smaller diameters. Therefore, the sample size dominates the hardening exponent of the thin copper wire.

Combining the above-stated two aspects, it can be concluded that a larger sample size leads to a lower GND density and total dislocation density, so a higher hardening exponent and a larger dislocation spacing are expected, resulting in a larger ILS. This mechanism of sample size-dependent ILS is well described by the evolution law of $l=l_0{\epsilon }^{-n}$ through the hardening exponent in this work.

If the grain size effect is accounted for in this generalized model, i.e., ${\kappa }_0={\widehat{\kappa }}_0+k_{\text{HP}}{d}^{-1/2}$, Eq. (25) will be modified as

This generalized Aifantis model considering the grain size effect is used to model both the grain size and strain gradient-dependent torsion response. Figure 4 shows the comparisons between the modeling results and experimental measurements for polycrystalline wires with sample sizes of 20 μm and 50 μm. The model parameters are given in Table 5, which shows that the reference ILSs are almost the same as those obtained by the original Aifantis model.Comparisons between the generalized Aifantis’ SGP modeling results and the experimental results of Gan et al. [45].Model parameters for the torsion experiments of Gan et al. [45] using the generalized Aifantis’ SGP model with grain size dependence, as expressed by Eq. (27)

Furthermore, the sample with a larger grain size has a larger strain hardening exponent, which is consistent with the evolution law of polycrystalline ductility, where grain refinement will reduce the ductility. According to the relation of $l=l_0{\epsilon }^{-n}$, the initial ILS for copper with a larger grain size is larger since copper with a larger grain size maintains a larger n. This conclusion is physically reasonable since the initial mean free path of dislocations is usually larger in larger grains. The underlying mechanism for the grain size-dependent ILS can also be interpreted through the influence of grain size on the dislocation mean free path. With the increase in grain size, the dislocation density decreases, resulting in a larger dislocation spacing and larger ILS. Moreover, the lower dislocation density leads to a larger hardening exponent based on the above discussions. This mechanism of grain size-dependent ILS can also be well described by the evolution law of $l=l_0{\epsilon }^{-n}$ through the hardening exponent.

Table 6 summarizes the ILSs obtained by the original and generalized SGP models, as well as their values obtained in our previous work by a generalized Fleck-Hutchinson’s model [27], which also employed a strain level-dependent ILS. It is found that the original and generalized SGP models give almost the same ILSs, which are larger than those obtained by the modified Fleck-Hutchinson’s model. The possible reasons are discussed below. A conceptual difference between Aifantis’ SGP model [19,39] and Fleck-Hutchinson’s model [33] is that Aifantis’ model used a superposition of stresses through a classical term and a gradient term, while Fleck-Hutchinson’s model employed a superposition of a classical effective strain term and a strain gradient term. However, the substantial difference between the two models is that Fleck-Hutchinson introduced the (first-order) strain gradient, which represents the effect of GNDs, while in Aifantis’ model, the second-order strain gradient, namely, the gradient of GNDs, is employed. This leads to the diversity of the two models not only in formulation but also in physical meaning. In Aifantis’ SGP model, the gradient term (stress) $c_2\left(\overline{\epsilon }\right){\nabla }^2\overline{\epsilon }$ is similar to that derived by Groma et al. [50] and Evers et al. [51], in which it was considered to be the long-range internal stress resulting from the nonuniform spatial distribution of GNDs, i.e., back stress. In Fleck-Hutchinson’s model, the gradient term leads to forest dislocation (Taylor) hardening. Therefore, Fleck-Hutchinson’s theory introduces the strain gradient, while Aifantis’ theory involves the Laplacian of strain, probably having a weaker effect than that of the first-order derivative. Therefore, a larger internal length is needed for the SGP model to match the stress level in the experiments.Obtained ILSs using various models in the torsion of thin copper wires

4. Conclusions

Based on the Taylor hardening law, Aifantis’ SGP model was modified to involve a power-law dependence of the ILS on the strain level. It was further generalized to consider the grain size effect by means of the Hall-Petch relation. The modification enables the model to describe each set of torsion measurements using one reference ILS successfully. According to the Taylor hardening law, the ILS is a nonconstant parameter but depends on the mean dislocation spacing, thus evolving during deformation. Furthermore, the initial ILS is geometry (sample size) and microstructure (grain size) dependent, and its evolution during deformation depends on the dislocation mean spacing and the strain hardening exponent.

Executive Editor: Huiling Duan