Characterizing Torsional Properties of Microwires Using an Automated Torsion Balance

Abstract

In order to characterize the torsional behavior of microwires, an automated torsion tester is established based on the principle of torsion balance. The main challenges in developing a torsion tester at small scales are addressed. An in-situ torsional vibration method for precisely calibrating the torque meter is developed. The torsion tester permits the measurement of torque to nN m, as a function of surface shear strain to a sensitivity of sub-microstrain. Using this technique, we performed (monotonic and/or cyclic) torsion tests on polycrystalline copper and gold wires. It is found that (i) a size effect appears in both the initial yielding and the plastic flow of torsional response; (ii) a reverse plasticity occurs upon unloading in cyclic torsion response; and (iii) the Hall-Petch effect and the strain gradient effect are synergistic. We also performed cyclic torsion tests on human hairs and spider silk which are natural protein fibers with a different morphological structure to metallic wires. It is shown that the single hair exhibits torsional recovery, and that the spider silk displays torsionally superelastic behavior whereby it is able to withstand great shear strain.

Appendices

Appendix 1

Strain gradient plasticity applied to wire torsion

Phenomenological strain gradient theories of isotropic plasticity represent a relatively simple approach to predict size effects in wire torsion, in which the spatial gradients of plastic strain associated with the GNDs lead to enhanced hardening [4, 6]. Here, the torsion data in Fig. 7 are analyzed by means of the strain gradient plasticity theory proposed by Fleck and Willis [52, 53]. To simplify the analysis, the wires are assumed to be isotropic, incompressible and rigid-plastic. In this case, the plastic behavior can be characterized by the plastic strain rate ${\dot{\varepsilon}}_{ij}^{\mathrm{pl}}$ and its spatial gradient ${\dot{\varepsilon}}_{ij,k}^{\mathrm{pl}}$. We restrict attention to the simplest model that contains only one material length scale l. Following Fleck and Hutchinson [6], we express the generalized overall effective plastic strain rate ${\dot{E}}_P$ as

$$ {\dot{E}}_{\mathrm{P}}={\left[{\dot{\varepsilon}}_{\mathrm{P}}^{\mu }+{\left(l{\dot{\varepsilon}}_{\mathrm{P}}^{\ast}\right)}^{\mu}\right]}^{{\scriptscriptstyle \frac{1}{\mu }}} $$

(A1)

Here, ${\dot{\varepsilon}}_{\mathrm{P}}=\sqrt{{\scriptscriptstyle \frac{2}{3}}{\dot{\varepsilon}}_{ij}^{\mathrm{pl}}{\dot{\varepsilon}}_{ij}^{\mathrm{pl}}}$ is the standard Von Mises plastic strain rate used to consider the plastic dissipation results from the motion of SSDs, while ${\dot{\varepsilon}}_{\mathrm{P}}^{\ast }=\sqrt{{\scriptscriptstyle \frac{2}{3}}{\dot{\varepsilon}}_{ij,k}^{\mathrm{pl}}{\dot{\varepsilon}}_{ij,k}^{\mathrm{pl}}}$ is the rate of plastic strain gradient for considering the contribution of GNDs [6, 54]. The exponent μ is an additional material parameter dependent on these two contributions. A Cartesian reference frame (x ₁, x ₂, x ₃) and a cylindrical polar coordinate system (r, θ, x ₃) are introduced, as shown in the inset of Fig. 7. The generalized effective plastic strain rate is then given by

$$ {\dot{E}}_{\mathrm{P}}=\frac{\dot{\kappa}}{\sqrt{3}}{\left[{r}^{\mu }+{\left(\sqrt{2}l\right)}^{\mu}\right]}^{{\scriptscriptstyle \frac{1}{\mu }}} $$

(A2)

where $\dot{\kappa}$ is the twist rate per unit length. The plastic response considering the strain gradient effect for a rigid-hardening solid can be identified as

$$ {\sum}_f={\sigma}_{ref}+{\sigma}_0{E}_P^N $$

(A3)

where ${E}_P={\displaystyle \int {\dot{E}}_P}\mathrm{d}t$ is the accumulated effective plastic strain. If the strain gradient effect is absent, Equation (A3) is degenerated to a conventional power-law relation, and σ _ref is the initial yield stress, σ ₀ is a measure of the hardening modulus, N is the hardening exponent. These parameters can be obtained by fitting the tensile data, as listed in Table 2. In wire torsion, the torque Q is work-conjugate to κ, so that the external work rate is $Q\dot{\kappa}$. And the torque is given by

$$ Q=\frac{1}{\dot{\kappa}}{\displaystyle {\int}_V{\sum}_f{\dot{E}}_P}\mathrm{d}V=\frac{2\pi }{\dot{\kappa}}{\displaystyle {\int}_0^a{\sum}_f{\dot{E}}_Pr\ \mathrm{d}r} $$

(A4)

Table 2 Parameters of tensile stress-strain curves

Substituting Equation (A2) and (A3) into Equation (A4) with μ = 2, one can obtain the normalized torque

$$ \frac{Q}{a^3}=\frac{6\pi {\sigma}_0{\left(\kappa a\right)}^N}{N+3}\left\{{\left[{\scriptscriptstyle \frac{1}{3}}+{\scriptscriptstyle \frac{2}{3}}{\left(l/a\right)}^2\right]}^{\frac{N+3}{2}}-{\left[\sqrt{{\scriptscriptstyle \frac{2}{3}}}\left(l/a\right)\right]}^{N+3}\right\}+\frac{2\sqrt{3}\pi {\sigma}_{ref}}{9}\left\{{\left[1+2{\left(l/a\right)}^2\right]}^{\frac{3}{2}}-2\sqrt{2}{\left(l/a\right)}^3\right\} $$

(A5)

If the torsional response of 2a = 20μm is used as a calibration curve, we obtain l = 0.92μm. The theoretical prediction agrees well with the experimental result, as shown in Fig. 7.

Appendix 2

Estimation of yielding torque

According to Von Mises yield criterion, the conventional yielding torque is estimated as [4]

$$ {Q}_Y=\frac{\pi {a}^3{\sigma}_Y}{2\sqrt{3}} $$

(B1)

where σ _Y is the yield strength in tension, a the wire radius. Following Liu et al. [4], we assume σ _Y  = 88.0 ± 13.0 MPa, then the yielding torque for copper wires with diameters ranging from 10 to 100 μm is in the range 9.98nN • m ~ 9.98 × 10³nN • m.