Size Effects on the Mechanical Responses and Deformation Mechanisms of AZ31 Mg Foils

Abstract

An integrated experimental and microstructure-based simulation research was carried out to study the effect of thickness and grain size on the mechanical response and deformation mechanism of AZ31 Mg foils. Equal channel angle pressing (ECAP) and subsequent annealing were applied to fabricate the billets with tailored microstructures. Ex situ micro-tensile tests of the foils were conducted to explore the meso-scale size effect. The experimental results show that the flow stress, ductility, and microstructure evolution of the foils are significantly affected by both grain size and thickness. With the increase of grain number (λ) in thickness, the flow stress curve changes from convex-up to a typical sigmoidal shape, and the extension twinning is remarkably suppressed. Full-field crystal plasticity simulations successfully captured the micromechanical interaction between dislocation slip and twinning. Specifically, the decrease of λ enhances the dominance of extension twinning on the mechanical response and ductility of the foils and further intensifies the interaction of deformation twinning on slip resistance. As a measurement for describing the combined effect of grain size and geometrical size, λ is a critical factor affecting the interaction and competition between dislocation slip and deformation twinning.

Appendix: Finite Strain Crystal Plasticity Theory

This work adopts a finite strain rate-dependent plasticity flow law conjugated with a modified twinning model from Salem et al.[43] to describe the plasticity response of HCP crystals. The plastic velocity gradient containing the contribution of both slip and twinning is expressed as:

$$ {\mathbf{L}}^{{\text{p}}} = \left( {1.0 - f_{{{\text{tw}}}} } \right)\sum \dot{\gamma }_{{{\text{sp}}}}^{\alpha } {\mathbf{S}}_{{{\text{sp}}}}^{\alpha } + f_{{{\text{tw}}}} \sum \dot{\gamma }_{{{\text{tw}}}}^{\beta } {\mathbf{S}}_{{{\text{tw}}}}^{\beta } + f_{{{\text{tw}}}} \sum \dot{\gamma }_{{{\text{sp}}}}^{\alpha } {\hat{\mathbf{S}}}_{{{\text{sp}}}}^{\alpha }, $$

(A1)

where ${\mathbf{S}}_{{{\text{sp}}}}^{\alpha },\;{\hat{\mathbf{S}}}_{{{\text{sp}}}}^{\alpha }$ and ${\mathbf{S}}_{{{\text{tw}}}}^{\beta }$ are the Schmid tensors of the potential slip (the untwined matrix and the twined region) and twinning systems, $\dot{\gamma }_{{{\text{sp}}}}^{\alpha }$ and $\dot{\gamma }_{{{\text{tw}}}}^{\beta }$ denote the plastic shearing rates associated with slip and twinning on the αth slip system and the βth twinning system, and $f_{{{\text{tw}}}}$ is the accumulated volume fraction of the twinned areas.

The plastic shearing rates of both slip and twinning are describe by a power-law constitutive relation. They are written as:

$$ \dot{\gamma }_{{{\text{sp}}}}^{\alpha } = \dot{\gamma }_{{{\text{sp}}}}^{0} \left| {\frac{{\tau_{{{\text{sp}}}}^{\alpha } }}{{g_{{{\text{sp}}}}^{\alpha } }}} \right|^{{1/m_{{{\text{sp}}}} }} {\text{sign}}\left( {\tau_{{{\text{sp}}}}^{\alpha } } \right),{\text{ and }}\dot{\gamma }_{{{\text{tw}}}}^{\beta } = \left\langle \dot{\gamma }_{{{\text{tw}}}}^{0} \frac{{\tau_{{{\text{tw}}}}^{\beta } }}{{g_{{{\text{tw}}}}^{\beta } }}\right\rangle^{{1/m_{{{\text{tw}}}} }} $$

(A2)

where $\dot{\gamma }_{{{\text{sp}}}}^{0}$ and $\dot{\gamma }_{{{\text{tw}}}}^{0}$ are the reference shearing rates, $m_{{{\text{sp}}}}$ and $m_{{{\text{tw}}}}$ the rate sensitivity parameters, $g_{{{\text{sp}}}}^{\alpha }$ and $g_{{{\text{tw}}}}^{\beta }$ the slip and twinning resistances, respectively. The evolutions of $g_{{{\text{sp}}}}^{\alpha }$ and $g_{{{\text{tw}}}}^{\beta }$ are described as follows[43]:

$$ \left\{ {\begin{array}{*{20}c} {\dot{g}_{{{\text{sp}}}}^{\alpha } = h_{{{\text{sp}}}}^{0} \left( {1 + c_{1} f_{{{\text{tw}}}} } \right)\mathop \sum \limits_{\xi } \left( {1 - \frac{{g_{{{\text{sp}}}}^{\xi } }}{{g_{{{\text{sp}}}}^{\infty } }}} \right)^{n} \dot{\gamma }_{{{\text{sp}}}}^{\xi } } \\ {\dot{g}_{{{\text{tw}}}}^{\beta } = h_{{{\text{tw}}}}^{0} \left( {f_{{{\text{tw}}}} } \right)^{{c_{2} }} \mathop \sum \limits_{\zeta } \dot{\gamma }_{{{\text{tw}}}}^{\zeta } + h_{{{\text{tw}} - {\text{sp}}}}^{0} \left( {\mathop \sum \limits_{\alpha } \gamma_{{{\text{sp}}}}^{\alpha } } \right)^{{c_{3} }} \mathop \sum \limits_{\xi } \dot{\gamma }_{{{\text{sp}}}}^{\xi } } \\ \end{array} } \right.{ ,} $$

(A3)

where $h_{{{\text{sp}}}}^{0}$ represents the initial hardening modulus of slip systems, and $g_{{{\text{sp}}}}^{\infty }$ denotes the saturated slip resistance in the absence of twinning. $h_{{{\text{tw}}}}^{0}$ and $h_{{{\text{tw}} - {\text{sp}}}}^{0}$ denote the initial twin hardening moduli due to twin-twin interaction and slip-twin interaction, respectively. The parameters $n$, $c_{1}$, $c_{2}$, $c_{3}$, and $s_{{{\text{pr}}}}$ are estimated constants. In particular, $c_{1}$ describes the effect of the Basinski-type transmutation of slip resistance. More details on the physical basis of these parameters can be found in the work of Salem et al.[43] It is supposed that there is a linear relationship between the twining shear rate and the rate of twin volume fraction, i.e., $\dot{\gamma }_{{{\text{tw}}}}^{\beta } = \dot{f}_{{{\text{tw}}}}^{\beta } \gamma_{{{\text{tw}}}}$, where $\gamma_{{{\text{tw}}}}$ is the twinning shear which can be expressed in terms of the axial ratio c/a. For Mg crystals, c/a = 1.624, and $\gamma_{{{\text{tw}}}} = 0.13$ for the $\left\{ {10\overline{1}2} \right\}\langle 10\overline{11} \rangle$ extension twinning.[45] In the end, the total volume fraction of twins $f_{{{\text{tw}}}} = \mathop \sum_{\beta } f_{{{\text{tw}}}}^{\beta }$ with $f_{{{\text{tw}}}}^{\beta } \ge 0$ and $f_{{{\text{tw}}}} \le 1$.

Mg alloys have multiple families of potential slip systems and twinning systems to accommodate deformation. In this work, we consider three $\left\{ {0001} \right\}\langle 11\overline{2}0 \rangle$ basal $a$ slip systems, three $\left\{ {10\overline{1}0} \right\}\langle 11\overline{2}0 \rangle$ prismatic $a$ slip systems, and six $\left\{ {11\overline{2}3} \right\}\langle \overline{1}011 \rangle$ pyramidal $c + a$ slip systems, as well as six $\left\{ {10\overline{1}2} \right\}\langle 10\overline{11} \rangle$ extension twinning systems. At room temperature, the basal slip systems are most easily activated comparing with other non-basal slip systems because their initial critical resolved shear stress (CRSS) are much smaller than that of non-basal slip systems.[52,53] As the flow stress curves of the foils exhibits a complex dependence of grain size and thickness, developing a non-local CP model[30] currently is not the aim of this work. The CRSSs and hardening parameters of these deformation modes for the material are referenced to the well-calibrated parameters of Agnew et al.,[46] in which the studied material is also the extruded AZ31 alloy. The specific material parameters are listed in Table A1.

Table A1 Material Parameters of the CP Model for the AZ31 Alloy

In Table A1, the rate sensitivity coefficient $m$ was routinely chosen to be a small value for all the deformation modes in view of the studied room temperature deformation; all parameters of slip systems were fitted by correlating the evolution law of $\dot{g}_{{{\text{sp}}}}^{\alpha }$ (Eq. A3 with $c_{1} = 0$) to the Voce-type hardening law with calibrated material parameters from the literature[46]; the initial resistance ($g^{0} = 30{\text{MPa}}$) of extension twinning was also obtained from Reference 46; $h_{{{\text{tw}}}}^{0}$ and $h_{{{\text{tw}} - {\text{sp}}}}^{0}$ are the estimated coefficients that describe the slightly hardening of twinning resistance. Other parameters related to the twinning process and the ratio of $h_{{{\text{tw}}}}^{0}$ to $h_{{{\text{tw}} - {\text{sp}}}}^{0}$ were referenced to the suggestion of Salem et al.,[43] and they are $c_{1} = 0.1$, $c_{2} = c_{3} = 10$, and $s_{{{\text{pr}}}} = 10$. In the end, the elastic properties of the Mg single crystal were referred to Reference 54, i.e., $C_{11} = 59.3{\text{GPa}}$, $C_{33} = 61.5{\text{GPa}}$, $C_{44} = 16.4{\text{GPa}}$, $C_{12} = 25.7{\text{GPa}}$, and $C_{13} = 21.4{\text{GPa}}$.