Stretchability of Commercial Purity Titanium Sheet

Abstract

The article presents the results of tests to assess stretchability of commercial purity titanium sheet. The plastic strain ratio (r value) indicates strong normal anisotropy and weak planar anisotropy in the plastic properties. The forming limit diagram (FLD), determined by hemispherical punch stretching, exhibits limiting strains that are large at the smallest strain ratios, low at plane strain, and minimum at a strain ratio close to equibiaxial stretching. Hill’s criterion for localized necking, in conjunction with a best fit constitutive equation from the tensile test and the quadratic Hill’s yield surface, predicts the negative minor strain region of the FLD well. However, fracture as the mechanism limiting the deformation cannot be ruled out, especially as the negative minor strain region extends to the positive side with a minimum in the positive side. The increase in limit strains in the positive minor strain region beyond the minimum and the plane strain condition occurring in full dome stretching are explained as “anomalies” due to planar anisotropy. Electron backscatter diffraction (EBSD) studies show a strong basal texture that intensifies when deformed under plane strain and equibiaxial conditions. Under uniaxial tension, the basal plane texture changed to $\left\{ {0001} \right\}\left\langle {11\bar{2}0} \right\rangle$ texture. Twinning is also found to be more under uniaxial tension. Trends in the mechanical properties correlate well with the EBSD results.

Appendix

Hill’s quadratic yield function for the plane stress condition $g\left( {\sigma_{1} ,\sigma_{2 } } \right)$, for planar isotropic material is[3,4,30]

$$ g\left( {\sigma_{1} ,\sigma_{2 } } \right) = \bar{\sigma }^{2} = \frac{3}{2}\left( {\frac{1 + r}{2 + r}} \right)\left[ {1 - \left( {\frac{2r}{r + 1}} \right)\alpha + \alpha^{2} } \right]\sigma_{1}^{2} $$

(A1)

where $\sigma_{1} ,\sigma_{2 }$ are the principal stresses, $\bar{\sigma }$ is the equivalent stress, r is plastic strain ratio and the $\alpha = \sigma_{2} /\sigma_{1}$ is the stress ratio. The corresponding equivalent strain ${\text{d}}\bar{\varepsilon}$ is[3,4]

$$ {\text{d}}\bar{\varepsilon } = \sqrt {\frac{2}{3}} \sqrt {\frac{{\left( {2 + r} \right)\left( {1 + r} \right)}}{1 + 2r}} \left[ {1 + \left( {\frac{2r}{r + 1}} \right)\rho + \rho^{2} } \right]^{{\frac{1}{2}}} {\text{d}}\varepsilon_{1} $$

(A2)

where ${\text{d}}\varepsilon_{1}$, ${\text{d}}\varepsilon_{2}$ are the principal incremental strains and the $\rho = {\text{d}}\varepsilon_{2} /{\text{d}}\varepsilon_{1}$ is the strain ratio. From the flow rule it can be shown that, $\alpha$ and $\rho$ are related by the relation[3,4]

$$ \alpha = \frac{{\left( {1 + r} \right)\rho + r}}{1 + r + r\rho }. $$

(A3)

Hill’s condition for the formation of a localized neck for negative minor strains is[24]

$$ \frac{1}{{\bar{\sigma }}}\frac{{{\text{d}}\bar{\sigma }}}{{{\text{d}}\bar{\varepsilon }}} \le \frac{{\partial g/\partial \sigma_{1} + \partial g/\partial \sigma_{2} }}{{{\text{d}}g/{\text{d}}\bar{\sigma }}}. $$

(A4)

From Eq. [A1] we obtain

$$ \frac{\partial g}{{\partial \sigma_{1} }} = \frac{3}{2} \frac{{\left( {r + 1} \right)}}{{\left( {r + 2} \right)}}\left[ {2 - \left( {\frac{2r}{1 + r}} \right)\alpha } \right]\sigma_{1} $$

(A5)

$$ \frac{\partial g}{{\partial \sigma_{2} }} = \frac{3}{2} \frac{{\left( {r + 1} \right)}}{{\left( {r + 2} \right)}}\left[ {2\alpha - \frac{2r}{1 + r}} \right]\sigma_{1} $$

(A6)

and

$$ \frac{\partial g}{{\partial \bar{\sigma }}} = 2\bar{\sigma } $$

(A7)

Substituting Eqs. [A5], [A6] and [A7] in the right hand side of Eq. [A4] we obtain

$$ \frac{{{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial \sigma_{1} }}} \right. \kern-0pt} {\partial \sigma_{1} }} + {{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial \sigma_{2} }}} \right. \kern-0pt} {\partial \sigma_{2} }}}}{{{{{\text{d}}g} \mathord{\left/ {\vphantom {{{\text{d}}g} {{\text{d}}\bar{\sigma }}}} \right. \kern-0pt} {{\text{d}}\bar{\sigma }}}}} = \frac{{\sqrt 3 \left( {1 + \alpha } \right)}}{{\sqrt {2\left( {1 + r} \right)\left( {2 + r} \right)\left[ {1 - \left( {{{2r} \mathord{\left/ {\vphantom {{2r} {1 + r}}} \right. \kern-0pt} {1 + r}}} \right)\alpha + \alpha^{2} } \right]} }} $$

(A8)

The Swift constitutive equation in terms of equivalent stress as a function of equivalent strain is[22]

$$ \bar{\sigma } = K_{\text{S}}^{\prime } (\varepsilon_{\text{s}}^{\prime } + {\bar{{\varepsilon }}})^{{n_{\text{s}}^{\prime } }} $$

(A9)

where $K_{\text{S}}^{'}$, $\varepsilon_{\text{s}}^{'}$ and $n_{\text{s}}^{\prime }$ are constants. Differentiating the above we obtain

$$ \frac{{{\text{d}}\bar{\sigma }}}{{{\text{d}}\bar{\varepsilon }}} = K_{\text{S}}^{\prime } n_{\text{s}}^{\prime } (\varepsilon_{\text{s}}^{\prime } + {\bar{{\varepsilon }}})^{{\left( {n_{\text{s}}^{\prime } - 1} \right)}} . $$

(A10)

Substituting Eqs. [A9] and [A10] in the left hand side of Eq. [A4] we have

$$ \frac{1}{{\bar{\sigma }}}\frac{{{\text{d}}\bar{\sigma }}}{{{\text{d}}\bar{\varepsilon }}} = \frac{{n_{\text{s}}^{\prime } }}{{\varepsilon_{\text{s}}^{\prime } + {\bar{{\varepsilon }}}}}. $$

(A11)

From Eqs. [A2], [A8] and [A11] we obtain the critical equivalent strain for the onset of localized necking ${\bar{{\varepsilon }}}^{*}$to be

$$ {\bar{{\varepsilon }}}^{*} = \frac{{f\left( \rho \right)n_{\text{s}}^{\prime } }}{1 + \rho } - \varepsilon_{\text{s}}^{\prime } $$

(A12)

where

$$ f\left( \rho \right) = \sqrt {\frac{2}{3}} \sqrt {\frac{{\left( {2 + r} \right)\left( {1 + r} \right)}}{1 + 2r}} \left[ {1 + \left( {\frac{2r}{r + 1}} \right)\rho + \rho^{2} } \right]^{{\frac{1}{2}}} . $$

(A13)

When deformation is along linear paths the limiting major strain $\varepsilon_{1}^{*}$, determined by substituting Eq. [A2] in Eq. [A12] and replacing the principal strain in place of the incremental strain, is:

$$ \varepsilon_{1}^{*} = \frac{{n_{\text{s}}^{\prime } }}{1 + \rho } - \frac{{\varepsilon_{\text{s}}^{\prime } }}{f\left( \rho \right)} . $$

(A14)

Since volume is conserved during plastic deformation $\varepsilon_{3} = - \left( {1 + \rho } \right)\varepsilon_{3}$, the thickness strain at the onset of localized necking $\varepsilon_{3}^{*}$obtained from Eq. [A11] is

$$ \varepsilon_{3}^{*} = n_{\text{s}}^{\prime } - \frac{{\left( {1 + \rho } \right)\varepsilon_{\text{s}}^{\prime } }}{f\left( \rho \right)} . $$

(A15)