Measurement and analysis of differential work hardening behavior of pure titanium sheet using spline function

Abstract

Biaxial stress tests and in-plane tension/compression tests of pure titanium sheet (JIS #1) have been carried out in order to elucidate its anisotropic plastic deformation behavior under linear stress paths. Contours of plastic work and the directions of plastic strain rates at different levels of plastic work have been precisely measured in the stress space. The measured work contours bulged significantly toward the equibiaxial direction and showed strong asymmetry, and moreover, changed its shapes significantly with increasing plastic work (differential work hardening). Using the data of the measured work contours, the applicability of selected anisotropic yield functions, i.e., Hill’s quadratic, the Yld2000-2d and Cazacu’s yield functions, to the accurate prediction of the plastic deformation behavior of the pure titanium has been discussed. It was found that these yield functions were not able to reproduce the measured data. A new method for analyzing the differential work hardening behavior of the pure titanium sheet has been developed. This method uses the spline function of Bezier curves which approximates a work contour, inspired by the methodology proposed by Vegter and Boogaard (Int. J. Plasticity 22 (2006) 557-580). The procedure for determining the spline function is described in detail. The calculated results have been in good agreement with the differential work hardening behavior of the pure titanium sheet.

Appendix: derivation of Eq. 2

When the coordinates of stress points making up a work contour in stress space are given in polar coordinates $({\sigma_x} = l\cos \varphi$, ${\sigma_y} = l\sin \varphi )$, a slope of a tangent to the work contour is given by

$$ \frac{{d{\sigma_y}}}{{d{\sigma_x}}} = \frac{{l\prime\sin \varphi + l\cos \varphi }}{{l\prime\cos \varphi + l\sin \varphi }}, $$

(A.1)

where l is expressed as a function of φ. $l\prime \equiv dl/d\varphi$.

From the associated flow rule, a slope of a tangent to the work contour is determined as

$$ \frac{{d{\sigma_y}}}{{d{\sigma_x}}} = \tan (\beta + \frac{\pi }{2}), $$

(A.2)

using the direction of plastic strain rate β. From Eqs. A.1 and A.2,

$$ \begin{gathered} \frac{{l\prime}}{l} = \frac{{\tan (\beta + {{\pi } \left/ {2} \right.})\sin \varphi + \cos \varphi }}{{\tan (\beta + {{\pi } \left/ {2} \right.})\cos \varphi - \sin \varphi }} \hfill \\= \frac{1}{{\tan (\beta - \varphi + {{\pi } \left/ {2} \right.})}}. \hfill \\\end{gathered} $$

(A.3)

Indefinite integral of Eq. A.3 with respect to φ leads to

$$ \ln l + C = \int {\frac{1}{{\tan (\beta - \varphi + {{\pi } \left/ {2} \right.})}}} d\varphi $$

(A.4)

Expressing β and φ using the cubic Bezier curve parameter, t, l is given as

$$ \begin{array}{*{20}{c}} {\ln l + C = \int {\frac{{\varphi \prime}}{{\tan (\beta - \varphi + {{\pi } \left/ {2} \right.})}}} dt} \\{l(t) = {l_0}\exp \left( {\int_0^t {\frac{{\varphi \prime}}{{\tan (\beta - \varphi + {{\pi } \left/ {2} \right.})}}dt} } \right),} \\\end{array} $$

(A.5)

where φ' ≡ dφ/dt.