Solid Cylinder Torsion for Large Shear Deformation and Failure of Engineering Materials

Abstract

Background

Using a thin-walled tube torsion test to characterize a material’s shear response is a well-known technique; however, the thin walled specimen tends to buckle before reaching large shear deformation and failure. An alternative technique is the surface stress method (Nadai 1950; Wu et al. J Test Eval 20:396–402, 1992), which derives a shear stress-strain curve from the torque-angular displacement relationship of a solid cylindrical bar. The solid bar torsion test uniquely stabilizes the deformation which allows us to control and explore very large shear deformation up to failure. However, this method has rarely been considered in the literature, possibly due to the complexity of the analysis and experimental issues such as twist measurement and specimen uniformity.

Objective

In this investigation, we develop a method to measure the large angular displacement in the solid bar torsion experiments to study the large shear deformation of two common engineering materials, Al6061-T6 and SS304L, which have distinctive hardening behaviors.

Methods

Modern stereo-DIC methods were applied to make deformation measurements. The large angular displacement of the specimen posed challenges for the DIC analysis. An analysis method using multiple reference configurations and transformation of deformation gradient is developed to make the large shear deformation measurement successful.

Results

We successfully applied the solid bar torsion experiment and the new analysis method to measure the large shear deformation of Al6061-T6 and SS304L till specimen failure. The engineering shear strains at failure are on the order of 2–3 for Al6061-T6 and 3–4 for SS304L. Shear stress-strain curves of Al6061-T6 and SS304L are also obtained.

Conclusions

Solid bar torsion experiments coupled with 3D-DIC technique and the new analysis method of deformation gradient transformation enable measurement of very large shear deformation up to specimen failure.

Appendix I

Consider the deformation described by Eq. (1), the deformation gradient is decomposed into a product of the right stretch tensor U and the rotation tensor R as F = RU, where

$$ R=\left[-\begin{array}{cc} co\mathrm{s}\varTheta \kern0.5em sin\varTheta & 0\\ {}\begin{array}{cc}\begin{array}{c} sin\varTheta \\ {}0\end{array}& \begin{array}{c} co s\varTheta \\ {}0\end{array}\end{array}& \begin{array}{c}0\\ {}1\end{array}\end{array}\right], $$

(11)

$$ U=\left[\begin{array}{ccc} pcos\varTheta & psin\theta & 0\\ {} psin\varTheta & ksin\theta + qcos\theta & 0\\ {}0& 0& s\end{array}\right], $$

(12)

and

$$ k=\left(p+q\right) tan\theta . $$

(13)

The deformation and rotation angle θ are illustrated in Fig. 1. Biot strain $\overline{U}$ is defined from the right stretch tensor

$$ \overline{U}=\left[\begin{array}{ccc}{e}_{x-B}& {e}_{xy-B}& 0\\ {}{e}_{xy-B}& {e}_{y-B}& 0\\ {}0& 0& {e}_{z-B}\end{array}\right]=U-I, $$

(14)

which is usually one of output options from DIC analysis packages [14, 17]. The relationship among the components of F and $\overline{U}$ are obtained in the following:

$$ {\displaystyle \begin{array}{c} tan\theta =\frac{e_{xy-B}}{e_{x-B}+1}\\ {}p={e}_{xy-\mathrm{B}}/ sin\theta =\kern0.5em \left({e}_{x-B}+1\right)/ cos\theta \\ {}q=\left(1+{e}_{y-B}-{e}_{xy-B} tan\theta \right) cos\theta \end{array}} $$

(15)