Thermodynamic dislocation theory: Size effect in torsion

https://doi.org/10.1016/j.ijplas.2018.11.009Get rights and content

Highlights

  • The thermodynamic dislocation theory for polycrystalline copper wires undergoing torsion is proposed.

  • The torque-twist curves are simulated that agree with the experimental ones by Liu et al. [2012].

  • The size effect is caused by the nucleation and pile-up of excess dislocations.

Abstract

The thermodynamic dislocation theory developed for non-uniform plastic deformations is used for the analysis of twisted copper wires. With a small set of physical parameters, assumed to be independent of strain rate and temperature, we can simulate the torque-twist curves that match the experimental ones of Liu et al. [2012]. It is shown that the size effect results from the nucleation and pile-up of excess dislocations.

Introduction

The theory of dislocation mediated plasticity, originally proposed by Langer et al. (2010) (called LBL-theory for short) and further developed in (Langer, 2015, 2016, 2017; Le et al., 2017, 2018a), deals with the uniform plastic deformations of crystals driven by a constant strain rate. During these uniform plastic deformations the crystal may only have redundant dislocations with vanishing resultant Burgers vector. As shown in (Le, 2018; Le and Piao, 2018; Le and Tran, 2018), the extension of LBL-theory to more general thermodynamic dislocation theory (TDT) for non-uniform plastic deformations should account for excess dislocations due to the incompatibility of the plastic distortion (Bilby, 1955; Kröner, 1955; Nye, 1953). There are various examples of non-uniform plastic deformations in material science and engineering, the most typical of which being those in twisted wires (Fleck et al., 1994; Horstemeyer et al., 2002; Liu et al., 2012, 2013a) and bent beams (Demir et al., 2010; Stölken and Evans, 1998; Wang et al., 2003). Phenomenological approaches dealing with non-uniform plastic deformations include the strain gradient plasticity (Al-Rub and Voyiadjis, 2004; Fleck et al., 1994; Fleck and Hutchinson, 2001; Huang et al., 2004) which can capture the size effect. Its main weakness is that the chosen gradient of the plastic strain rate is not related to the dislocation density, so that the proposed constitutive equations are not based on the physics of dislocations. It is therefore impossible to derive the theory from the first principle calculation. The alternative continuum dislocation theory (CDT) accounting for excess dislocations, proposed for instance in (Berdichevsky, 2006; Kaluza and Le, 2011; Le and Stumpf, 1996; Le and Piao, 2016; Liu and Dunstan, 2017; Liu et al., 2018), is more predictive as kinematic hardening and size effect are captured by the first principle calculation of energy of dislocated crystals. However, since redundant dislocations and the disorder temperature are completely ignored, the above CDT cannot describe the isotropic hardening and the sensitivity of the torque-twist curves to temperature and strain rate. Also worth mentioning is the continuum dislocation dynamics (CDD), developed for example in (Hochrainer et al., 2014; Leung et al., 2015), that can be used to predict the size effect. The effective disorder temperature is also missing in CDD, so that the proposed evolution equations for the dislocation densities cannot be derived from the non-equilibrium thermodynamics of driven systems.

The purpose of this paper is to explore use of TDT for non-uniform plastic deformations (Le, 2018; Le and Piao, 2018; Le and Tran, 2018) in modeling twisted copper wires. Our challenge is to simulate the torque-twist curves that show the hardening behavior and the size effect. We also want to compare these torque-twist curves with those obtained by a simple extension of the LBL theory (Le et al., 2018b) in which the excess dislocations are ignored, by the micro-torsion tests reported in (Liu et al., 2012) and by an empirical formula proposed therein. To make this comparison possible we will need to identify from the experimental data obtained in (Liu et al., 2012) a list of material parameters for twisted copper wires. For this purpose, we will use the large scale least-squares analysis described in (Le et al., 2017, 2018a; Le and Tran, 2017). The comparison shows that: (i) the LBL-theory cannot predict the torque-twist curves and the size effect for wires in the micrometer range, (ii) the empirical formula proposed in (Liu et al., 2012) is not based on the solution of the coupled boundary-value problem of dislocated crystal in equilibrium and therefore does not allow to find both the stress distribution and the dislocation densities inside the bar, (iii) the TDT provides an accurate prediction of the torque-twist curves and the size effect as well as enables one to find the distributions of stress, strain, and dislocation densities. Regarding the distribution of the excess dislocation density: It is still early to judge its validity by comparing it with the available results in the literature, neither with the experimental data obtained by the EBSD measurement (Ziemann et al., 2015), nor with the discrete dislocation dynamics simulations (Senger et al., 2011) (see the discussion in Section 4 and in the conclusion).

The paper is organized as follows. After this short Introduction we present in Section 2 the governing equations of TDT. In Section 3 we discretize the obtained system of differential-algebraic equations and develop the numerical method for its solution. The parameter identification based on the large scale least squares analysis and the results of the numerical simulations are presented in Section 4. We conclude in Section 5 with some remarks about the excess dislocations.

Section snippets

Thermodynamic dislocation theory

Suppose a thin polycrystalline copper wire with a circular cross section A, of radius R and length L, is subjected to torsion (see the wire with its cross-section in Fig. 1). For this particular geometry of the wire and under the condition RL it is natural to assume that the circumferential displacement is uφ=ωrz, with ω being the twist angle per unit length, while the displacement uz does not depend on φ. Thus, the total shear strain of the wire γ=2εφz=ωr and the shear strain rate γ˙=ω˙r turn

Method of solution

For the purpose of numerical integration of system (3), (4), (6), and (7) let us introduce the following variables and quantitiesr˜=r/R,τ˜=τ/μ,τ˜Y=τY/μ,τ˜B=τB/μ,ω˜=Rω,χ˜=χeD,η=bR,ρ˜=a2ρ.

The variable r˜ changes from 0 to 1. The dimensionless quantity ω˜ has the meaning of the maximum shear strain achieved at the outer radius. The calculation of the rescaled torque T˜=T/R3 as function of ω˜=ωR is convenient for the later comparison with the experimental data from (Liu et al., 2012). Then we

Parameter identification and numerical simulations

The experimental data of Liu et al. (2012) include four torque-twist curves for polycrystalline copper wires with different radii R=9 micron, R=15 micron, R=21 micron, and R=52.5 micron. It was mentioned in (Liu et al., 2012) that all wires were annealed for 2.4 h in a vacuum furnace with argon shielding at 410 C to ensure that each sample had the same mean grain size. Torsion tests were performed at room temperature, and for all tests the twist rate Lω˙ was π/30 per second (6/s). We show

Conclusions

The results obtained show the principal applicability of TDT to torsion tests. We found that the behavior of the torque-twist curves is controlled not only by the microstructure of material (the grain size, the initial dislocation density, and the initial disorder temperature), which affects isotropic hardening, but above all the sample size, which affects the nucleation and pile-up of excess dislocations and kinematic hardening. For wires of micron sizes under torsion, the back stress

Acknowledgments

Y. Piao acknowledges financial support from the Chinese Government Scholarship Program. K.C. Le is grateful to J.S. Langer for helpful discussions and to D. Liu for informing us about the details of experimental setting in (Liu et al., 2012).

References (40)

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