Micromechanics-based strain energy study of $\{\textbf{1}\,\textbf{0}\,\bar{\textbf{1}}\,\textbf{2}\}$ twin-band pattern in a three-point bend Mg alloy

Abstract

$\{\textbf{1}\,\textbf{0}\,\bar{\textbf{1}}\,\textbf{2}\}$ twinning in magnesium AZ31 alloy activates profusely under c-axis tension (tension twinning) or d-axis compression (extension twinning). In a three-point bending condition, a strongly basal textured Mg AZ31 alloy offers a d-axis compression along $\langle \textbf{1}\,\textbf{0}\,\bar{\textbf{1}}\,\textbf{0} \rangle$ direction ensuing in a pattern of localized twinning events. A micromechanics-based analytical solution is used to calculate strain energy and energies involved for various patterns of localized twins. The results reveal a minimum energy path of twinning evolution, predicting first twin propagation to the neutral axis, nucleation of a second twin at a certain distance from the first one, with the microstructure finally evolving to a favorable twinning pattern to accommodate the total bending strain. The results show the effect of multivariant twinning on the overall accommodation of bending strain. The twinning behaviors observed from this analysis compare well with the experimental results.

Graphical abstract

Appendix

The analytical solution to the elliptical integral expressions $I_i$, and $I_{ij}$ used to obtain the elastic fields:

$$\begin{aligned}{} & {} \begin{aligned} I_1 =&I_2 = -\frac{\left( \pi a_1^2 a_3\right) \left( 2 \sqrt{a_1^2-a_3^2} \lambda -\pi \sqrt{\frac{1}{a_1^2-a_3^2}} \sqrt{a_1^2-a_3^2} \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }+2 a_1^2 \sqrt{a_3^2+\lambda } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) +2 \lambda \sqrt{a_3^2+\lambda } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) +2 \sqrt{a_1^2-a_3^2} a_3^2\right) }{\left( a_1^2-a_3^2\right) {}^{3/2} \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }} \end{aligned} \end{aligned}$$

(9)

$$\begin{aligned}{} & {} I_3 = \frac{2 \pi a_1^2 a_3 \left( \sqrt{a_1^2-a_3^2} \left( 2 \lambda -\pi \sqrt{\frac{1}{a_1^2-a_3^2}} \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }\right) +2 a_1^2 \left( \sqrt{a_3^2+\lambda } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) +\sqrt{a_1^2-a_3^2}\right) +2 \lambda \sqrt{a_3^2+\lambda } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) \right) }{\left( a_1^2-a_3^2\right) {}^{3/2} \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }} \end{aligned}$$

(10)

$$\begin{aligned}{} & {} \begin{aligned} I_{11}&= I_{22} = I_{12} = I_{21} = \frac{\pi a_1^2 a_3}{4 \left( a_1^2-a_3^2\right) {}^{5/2} \left( a_1^2+\lambda \right) {}^3 \sqrt{a_3^2+\lambda }}\\&\left( a_1^2 \left( \sqrt{a_1^2-a_3^2} \sqrt{a_3^2+\lambda } \left( 9 \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \lambda ^2-10 \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }\right) -12 \lambda \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) \right) \right. \\&+3 \lambda \left( \sqrt{a_1^2-a_3^2} \sqrt{a_3^2+\lambda } \left( \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \lambda ^2-2 \sqrt{\left( a_1^2+\lambda \right) ^2 \left( a_3^2+\lambda \right) }\right) -2 \lambda \sqrt{\left( a_1^2+\lambda \right) ^2 \left( a_3^2+\lambda \right) } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) \right) \\&+3 \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \sqrt{a_1^2-a_3^2} a_1^6 \sqrt{a_3^2+\lambda }+4 \sqrt{a_1^2-a_3^2} a_3^2 \sqrt{a_3^2+\lambda } \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }\\&\left. +3 a_1^4 \left( 3 \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \sqrt{a_1^2-a_3^2} \lambda \sqrt{a_3^2+\lambda }-2 \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) \right) \right) \end{aligned} \end{aligned}$$

(11)

$$\begin{aligned}{} & {} \begin{aligned} I_{13}&= I_{23} = I_{31} = I_{32} = \frac{\pi a_1^2 a_3}{\left( a_1^2-a_3^2\right) {}^{5/2} \left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) {}^{3/2}} \\&\left( a_3^2 \left( \sqrt{a_1^2-a_3^2} \sqrt{a_3^2+\lambda } \left( 2 \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }-3 \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \lambda ^2\right) +6 \lambda \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) \right) \right. \\&-3 \lambda \left( \sqrt{a_1^2-a_3^2} \sqrt{a_3^2+\lambda } \left( \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \lambda ^2-2 \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }\right) -2 \lambda \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) \right) \\&-2 a_1^2 (\sqrt{a_1^2-a_3^2} \sqrt{a_3^2+\lambda } \left( 3 \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \lambda ^2-2 \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }\right) \\&+3 a_3^2 \left( \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \sqrt{a_1^2-a_3^2} \lambda \sqrt{a_3^2+\lambda }-\sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) \right) \\&\left. -3 \lambda \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) )-3 \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \sqrt{a_1^2-a_3^2} a_1^4 \left( a_3^2+\lambda \right) {}^{3/2}\right) \end{aligned} \end{aligned}$$

(12)

$$\begin{aligned}{} & {} \begin{aligned} I_{33} =&\frac{2 \pi a_1^2 a_3}{3 \left( a_1^2-a_3^2\right) {}^{5/2} \left( \left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) \right) {}^{3/2}} \\&\left( a_1^2+\lambda \right) {}^2\left( -2 a_1^2 \left( a_3^2 \left( 3 \sqrt{a_3^2+\lambda } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) +4 \sqrt{a_1^2-a_3^2}\right) +\lambda \left( 3 \sqrt{a_3^2+\lambda } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) +2 \sqrt{a_1^2-a_3^2}\right) \right) \right. \\&+a_3^2 \left( \sqrt{a_1^2-a_3^2} \left( 3 \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }-8 \lambda \right) -6 \lambda \sqrt{a_3^2+\lambda } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) \right) \\&\left. +3 \lambda \left( \sqrt{a_1^2-a_3^2} \left( \pi \sqrt{\frac{1}{a_1^2-a_3^2}} \sqrt{\left( a_1^2+\lambda \right) {}^2 \left( a_3^2+\lambda \right) }-2 \lambda \right) -2 \lambda \sqrt{a_3^2+\lambda } \tan ^{-1}\left( \frac{\sqrt{a_3^2+\lambda }}{\sqrt{a_1^2-a_3^2}}\right) \right) +2 \sqrt{a_1^2-a_3^2} a_1^4\right) \end{aligned} \end{aligned}$$

(13)