Cyclic Bending Behaviors of Extruded AZ31 Magnesium Alloy Beams

Abstract

Bending behavior holds significant importance, as it plays an unavoidable role in a variety of forming processes. Therefore, gaining insight into the cyclic bending mechanical behavior of magnesium (Mg) alloys is crucial for designing, fabricating, and ensuring the performance of end products. This study conducts cyclic bending simulations of an extruded AZ31 Mg alloy beam using crystal-plasticity-based methods. The study encompasses the macroscopic aspects of stress distribution and strain distribution, along with a detailed discussion of microscopic factors such as the relative activities of deformation mechanisms and twin volume fraction during cyclic bending. Different from conventional alloys with high crystal symmetry, the apparent the neutral layer shifting phenomena is observed in the extruded AZ31. The simulated results reveal that this distinct behavior is ascribed to the alternating twinning and detwinning mechanisms.

1 Introduction

The global interest in magnesium (Mg) alloys has surged owing to their lightweight characteristics, high specific strength, and exceptional recyclability [1,2,3,4]. The distinctive hexagonal close-packed (HCP) structure, the unidirectional deformation twinning mechanism, and the developed texture all play a role in the pronounced anisotropic behavior exhibited by wrought Mg alloys [5,6,7,8]. In order to accommodate arbitrary deformation in magnesium (Mg) alloys, a minimum of five independent deformation systems are necessary. Experimentally observed major deformation mechanisms in Mg alloys include basal <a> slip ({$0001\}\langle 11\overline{2 }0\rangle$) [9], prismatic <a> slip $\left( {\left\{ {10\overline{1} 0} \right\}\left\langle {11\overline{2} 0} \right\rangle } \right)$ [10], pyramidal <a> slip $\left( {\left\{ {10\overline{1} 1} \right\}\left\langle {11\overline{2} 0} \right\rangle } \right)$ [11], pyramidal <c+a> slip $\left( {\left\{ {11\overline{2} 2} \right\}\left\langle {11\overline{2} 3} \right\rangle } \right)$ [12] and extension twin $\left( {\left\{ {10\overline{1} 2} \right\}\left\langle {10\overline{1} 1} \right\rangle } \right)$ [13]. The cyclic deformation behaviors of wrought Mg alloys are examined under relatively straightforward paths, including tension–compression cycles and simple shear [14,15,16,17]. Extensive documentation supports the crucial roles of twinning and detwinning in the cyclic plasticity of both pure Mg and Mg alloys [18,19,20]. In common manufacturing processes or in-service conditions, bending and cyclic bending behaviors are frequently encountered as instances of inhomogeneous cases [21,22,23]. In the pursuit of improved processing technology and elevated product quality, understanding and quantifying the cyclic bending behaviors in Mg alloys are of paramount importance.

The investigation presented in this work focuses on the cyclic bending behaviors of Mg alloys featuring different curvature amplitudes. The study utilizes the EVPSC-BEND, which is designed for bending loads based on the elasto-viscoplastic self-consistent model [24, 25]. The moment–curvature loading curves, relative activity and twin volume fraction (TVF) during cyclic loading, and the distribution of stress and strain components at certain loading processes are discussed.

2 Simulation Methods

A brief overview of the EVPSC-BEND approach for simulating pure bending is provided here, with a more comprehensive description available elsewhere [24, 25]. When the beam is in a state of pure bending, a local coordinate $(x)$ is established along the thickness of the beam, as illustrated in Fig. 1. The beam is discretized into numerous layers along the thickness direction. The brown dashed line denotes the neutral layer and serves as the coordinate origin. The intrados and extrados coordinates are designated as $a$ and $b$, respectively. The curvature of the neutral layer is $1/\rho$.

Fig. 1

Numerous discretized layers beam under pure bending

The stress increments ($\Delta {\Sigma }_{r}^{i}$, $\Delta {\Sigma }_{\theta }^{i}$) in response to bending ($\Delta \rho$) within the beam can be expressed as follows:

$$\begin{array}{c}\left[\begin{array}{ccc}\begin{array}{cc}\frac{t{\overline{M} }_{\theta r}^{e1}}{\rho +{x}_{1}}+{\overline{M} }_{\theta \theta }^{ei}& 0\end{array}& \cdots & 0\\ \begin{array}{cc}\frac{t{\overline{M} }_{\theta r}^{e2}}{\rho +{x}_{2}}& \frac{t{\overline{M} }_{\theta r}^{e2}}{\rho +{x}_{2}}+{\overline{M} }_{\theta \theta }^{e2}\end{array}& \cdots & 0\\ \begin{array}{cc}\begin{array}{c}\vdots \\ \frac{t{\overline{M} }_{\theta r}^{en}}{\rho +{x}_{n}}\end{array}& \begin{array}{c}\vdots \\ \frac{t{\overline{M} }_{\theta r}^{en}}{\rho +{x}_{n}}\end{array}\end{array}& \begin{array}{c}\ddots \\ \cdots \end{array}& \begin{array}{c}\vdots \\ \frac{t{\overline{M} }_{\theta r}^{en}}{\rho +{x}_{n}}+{\overline{M} }_{\theta \theta }^{en}\end{array}\end{array}\right]\left[\begin{array}{c}\begin{array}{c}\Delta {\Sigma }_{\theta }^{1}\\\Delta {\Sigma }_{\theta }^{2}\end{array}\\ \begin{array}{c}\vdots \\\Delta {\Sigma }_{\theta }^{n}\end{array}\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}\left(\frac{t{\overline{M} }_{\theta r}^{e1}}{{\left(\rho +{x}_{1}\right)}^{2}}{\Sigma }_{\theta }^{1}-\frac{{x}_{1}}{{\rho }^{2}}\right)\Delta \rho -\Delta {E}_{\theta }^{p1}\\ \left(\frac{t{\overline{M} }_{\theta r}^{e2}}{{\left(\rho +{x}_{2}\right)}^{2}}\sum_{k=1}^{2}{\Sigma }_{\theta }^{k}-\frac{{x}_{2}}{{\rho }^{2}}\right)\Delta \rho -\Delta {E}_{\theta }^{p2}\end{array}\\ \begin{array}{c}\vdots \\ \left(\frac{t{\overline{M} }_{\theta r}^{en}}{{\left(\rho +{x}_{n}\right)}^{2}}\sum_{k=1}^{n}{\Sigma }_{\theta }^{n}-\frac{{x}_{n}}{{\rho }^{2}}\right)\Delta \rho -\Delta {E}_{\theta }^{pn}\end{array}\end{array}\right]\end{array}$$

(1)

Here, $\Delta {E}_{\theta }^{pi}$ represents the increment of plastic strain of the ith layer, while ${\overline{M} }_{\theta r}^{ei}$ and ${\overline{M} }_{\theta \theta }^{ei}$ denote the components of elastic compliance. In the EVPSC-TDT model, every layer is treated as a homogeneous effective medium (HEM) consisting of numerous grains. The relationship between the strain rate ${\dot{E}}_{ij}$ and the stress of the HEM (${\Sigma }_{ij}$) is given by:

(2)

where ${\overline{M} }_{ijkl}^{e}$ is the elastic compliance, and ${\dot{E}}_{ij}^{e}$ and ${\dot{E}}_{ij}^{p}$ are the elastic and plastic parts of the strain rate, respectively.

The strain rate of the single crystal ${\dot{\varepsilon }}_{ij}$ comprises both elastic and plastic components: ${\dot{\varepsilon }}_{ij}^{e}$ and ${\dot{\varepsilon }}_{ij}^{p}$. The plastic deformation of Mg alloys arises from both slip and twinning:

$$\begin{array}{c}{\dot{\varepsilon }}_{ij}={\dot{\varepsilon }}_{ij}^{e}+{\dot{\varepsilon }}_{ij}^{p}={M}_{ijkl}^{e}{\dot{\sigma }}_{kl}+\sum_{\alpha }{\dot{\gamma }}^{\alpha }{P}_{ij}^{\alpha }\#\end{array}$$

(3)

Here, ${M}_{ijkl}^{e}$ represents the elastic compliance, ${\dot{\gamma }}^{\alpha }$ denotes the shear rat, and ${P}_{ij}^{\alpha }$ is the Schmid tensor for a slip/twinning system $\alpha$. For both slip and twinning systems, the resolved shear stress (RSS) serves as the driving force to induce the shear rate ${\dot{\gamma }}^{\alpha }$. And the RSS on a system $\alpha$ is calculated by the Cauchy stress ${\sigma }_{ij}$: ${\tau }^{\alpha }={\sigma }_{ij}{P}_{ij}^{\alpha }$.

The shear rate for a slip system $\alpha$ can be defined by

$$\begin{array}{c}{\dot{\gamma }}^{\alpha }={\dot{\gamma }}_{0}{\left|\frac{{\tau }^{\alpha }-{\tau }_{b}^{\alpha }}{{\tau }_{cr}^{\alpha }}\right|}^\frac{1}{m}sgn\left({\tau }^{\alpha }-{\tau }_{b}^{\alpha }\right)\#\end{array}$$

(4)

incorporating the reference shear rate ${\dot{\gamma }}_{0}$, the critical resolved shear stress (CRSS) ${\tau }_{cr}^{\alpha }$, the back stress ${\tau }_{b}^{\alpha }$, and the strain rate sensitivity $m$.

For a twin system $\alpha$, the shear rate can be expressed similarly:

$$ \begin{array}{*{20}c} {\dot{\gamma }_{T}^{\alpha } = \left\{ {\begin{array}{*{20}c} {\dot{\gamma }_{0} \left| {\frac{{\tau^{\alpha } - \tau_{b}^{\alpha } }}{{\tau_{cr}^{\alpha } }}} \right|^{1/m} } & {\tau^{\alpha } - \tau_{b}^{\alpha } {\text{suit}}\,{\text{the}}\,{\text{situation}}} \\ 0 & {{\text{else}}} \\ \end{array} } \right.;\dot{f}_{T}^{\alpha } = \frac{{\left| {\dot{\gamma }_{T}^{\alpha } } \right|}}{{\gamma^{tw} }}} \\ \end{array} $$

(5)

where ${\dot{f}}_{T}^{\alpha }$ represents the rate of TVF associated with a certain twinning operation, ${\gamma }^{tw}$ is the characteristic twinning shear. The reduction of matrix (MR) and propagation of twin (TP) lead to an increase in TVF. Conversely, the propagation of matrix (MP) and reduction of twin (TR) result in a decrease in TVF, indicating the occurrence of de-twinning.

The evolution of the TVF of the ${\alpha }^{th}$ twinning system, ${f}^{\alpha }$ can be expressed as:

$$\begin{array}{c}{\dot{f}}^{\alpha }={f}^{M}({\dot{f}}_{MR}^{\alpha }+{\dot{f}}_{MP}^{\alpha })+{f}^{\alpha }({\dot{f}}_{TP}^{\alpha }+{\dot{f}}_{TR}^{\alpha })\#\end{array}$$

(6)

where ${f}^{M}=1-{f}^{tw}=1-{\sum }_{\alpha }{f}^{\alpha }$ is the volume fraction of the matrix.

If the TVF, represented as ${f}^{tw}$, attains a threshold value ${V}^{th}$, twinning within a grain is halted. This threshold is determined by the accumulated twin fraction ${V}^{{\text{acc}}}$ and the effective twinned fraction ${V}^{{\text{eff}}}$,

$$\begin{array}{c}{V}^{th}={\text{min}}\left(1.0, {A}_{1}+{A}_{2}\cdot \frac{{V}^{{\text{eff}}}}{{V}^{{\text{acc}}}}\right)\#\end{array}$$

(7)

where ${A}_{1}$ and ${A}_{2}$ are two governing parameters.

For a slip or twinning system $\alpha$, the change of the CRSS ${\tau }_{cr}^{\alpha }$ can be expressed as:

$$\begin{array}{c}{\dot{\tau }}_{cr}^{\alpha }=\frac{d{\widehat{\tau }}^{\alpha }}{d\Gamma }\sum_{\beta }{h}^{\alpha \beta }\left|{\dot{\gamma }}^{\beta }\right|\#\end{array}$$

(8)

in this equation, Γ stands for the accumulated shear strain of all the deformation systems in the grain, expressed as $\Gamma ={\sum }_{\alpha }\int {\dot{\gamma }}^{\alpha }dt$. The coefficients ${h}^{\alpha \beta }$ are latent hardening coupling parameters that empirically address obstacles on system $\alpha$ associated with system $\beta$. The threshold stress ${\widehat{\tau }}^{\alpha }$ is defined by an extended Voce law:

$$\begin{array}{c}{\widehat{\tau }}^{\alpha }={\tau }_{0}^{\alpha }+\left({\tau }_{1}^{\alpha }+{h}_{1}^{\alpha }\Gamma \right)\left(1-{\text{exp}}\left(-\frac{{h}_{0}^{\alpha }\Gamma }{{\tau }_{1}^{\alpha }}\right)\right)\#\end{array}$$

(9)

here, ${\tau }_{0}$, ${h}_{0}$, ${h}_{1,}$ and ${\tau }_{0}+{\tau }_{1}$ stand for the initial Critical Resolved Shear Stress (CRSS), the initial hardening rate, the asymptotic hardening rate, and the back-extrapolated CRSS, respectively.

The Bauschinger effect, linked to back stress, is induced by the generation of geometrically necessary dislocations. In the EVPSC-TDT model, to approximate the development of a permanent forest dislocation structure and the structure of reversible dislocations, the back stress (${\tau }_{b}^{\alpha }$) is introduced. The rate of back stress (${\dot{\tau }}_{b}^{\alpha }$) is expressed as follows:

$$\begin{array}{c}{\dot{\tau }}_{b}^{\alpha }={\xi }^{\alpha }sgn\left({\dot{\gamma }}^{\alpha }\right)\sum_{\beta }{h}^{\alpha \beta }\left|{\dot{\gamma }}^{\beta }\right|-{\eta }^{\alpha }{\tau }_{b}^{\alpha }\sum_{\beta }{h}^{\alpha \beta }\left|{\dot{\gamma }}^{\beta }\right|\#\end{array}$$

(10)

the effects of linear kinematic hardening and dynamic recovery are represented by the first and second terms in the above equation, respectively. ${\xi }^{\alpha }$, ${\eta }^{\alpha }$ are two governing material coefficients.

The bending moment on the beam is given by the equation:

$$\begin{array}{c}B={\int }_{a}^{b}{xw\Sigma }_{\theta }\left(x\right)dx=t\sum_{k=1}^{n}w{{x}_{k}\Sigma }_{\theta }^{k}\#\end{array}$$

(11)

where $w$ denotes the beam width, $t$ is the layer thickness, and $n$ represents the number of layers. As demonstrated in our prior research, opting for 10 discretized layers has demonstrated satisfactory outcomes and will be applied in the current study.

3 Results and Discussion

Figure 2a illustrates the coordinate system utilized in the cyclic bending simulation, while the initial texture, represented by the $\{0001\}$ and $\left\{ {10\overline{1} 0} \right\}$ pole figures (Fig. 2b), has been previously investigated in related research [24, 25]. The parameters associated with slip systems, including basal $\langle a\rangle \left( {\left\{ {0001} \right\}\left\langle {11\overline{2} 0} \right\rangle } \right)$, prismatic $\langle a\rangle \left( {\left\{ {10\overline{1} 0} \right\}\left\langle {11\overline{2} 0} \right\rangle } \right)$ and pyramidal $\langle c + a\rangle \left( {\left\{ {\overline{1} \overline{1} 22} \right\}\left\langle {\overline{1} \overline{1} 23} \right\rangle } \right)$, and extension twin system $\left\{ {10\overline{1} 2} \right\}\left\langle {10\overline{1} 1} \right\rangle$, in the EVPSC-BEND model, were determined by fitting the stress–strain curves obtained from tensile and compressive experiments along the longitude direction (LD) of the sample. The value of these parameters is listed in Table 1. The simulated results, as depicted in Fig. 2c, exhibit good agreement with the corresponding experimental results.

Fig. 2

a The layout of the pure-bending simulation. b The initial texture of the simulation. c Comparison of stress–strain curves for uniaxial tension and uniaxial compression along the LD direction between experimental and simulation results. d The relative activity of tension. e The relative activity of compression

Table 1 List of the parameters involved in the EVPSC-BEND model. ${h}^{\alpha \beta }$ denotes the latent hardening parameters associated with slip systems corresponding to the extensive twin systems

The simulated moment–curvature curve has been compared with the experimental result, as shown in Fig. 3 (The corresponding four-point bending test has been described in detail in the previous work [24]). The comparison demonstrates a good agreement, confirming the predictability of the EVPSC-BEND model. Based on the credibility of this model, this study further conducted simulations of cyclic bending. As illustrated in Fig. 4, the cyclic Moment–curvature curves depict the curvature amplitudes of Cycle A, B, and C as 0.003, 0.006, and 0.010, respectively. The marker points indicate three instances when the loading reached its maximum amplitude. The asymmetry of the yielding moment becomes increasingly evident as the amplitude increases from Cycle A to Cycle C. The relative activity and evolution of twin volume fraction (TVF) during cyclic bending are presented in Fig. 5. The extension twin remains at a high level throughout the entire cycle, while the apparent TVF fluctuation illustrates the involvement of the detwinning mechanism in the deformation. Basal slip's role diminishes as the curvature amplitude increases from Cycle A to Cycle C. In each monotonous bending process ($O-{P}_{1}, {P}_{1}-{P}_{2}, {P}_{2}-{P}_{3}$), prismatic slip exhibits a nearly monotonic increase, except for a short but significant drop after ${P}_{1}$ and ${P}_{2}$. The amount of pyramidal slip remains low. The mechanical behavior of magnesium alloys is closely related to the deformation systems they activate, particularly the twinning system. The results of activity analysis indicate that the twinning-detwinning behavior is significant throughout the entire cyclic bending process. The influence of twinning during the cyclic process is worthy of careful discussion.

Fig. 3

The simulated and experimental moment–curvature curves

Fig. 4

The cyclic loading curves of three different amplitude cycle

Fig. 5

The relative activity and twin volume fraction under cyclic bending in a Cycle A, b Cycle B, and c Cycle C

The stress distribution at ${P}_{1}$, ${P}_{2}$, and ${P}_{3}$ during loading cycles A–C is presented in Fig. 6, while Fig. 7 displays the corresponding strain distribution at the same points. The stress distribution at ${P}_{1}$–${P}_{3}$ in loading Cycle A–C follows a pattern of alternation between tensile and compressive zones. For a circumferential force balance of zero, it is imperative that the shaded areas in the tensile and compressive regions are equal, prompting the need for the neutral layer shifting. The strain component ${E}_{\theta }$ displays a linear distribution along the thickness, whereas the other two components, ${E}_{r}$ and ${E}_{z}$, exhibit significant non-linearity in their distribution. The stress distribution at ${P}_{1}$ reveals that twinning, by influencing the yield and hardening of different layers, creates curves on the bending cross-section resembling uniaxial tension and uniaxial compression. This characteristic distribution is also evident at ${P}_{2}$ and ${P}_{3}$. The distribution in ${P}_{2}$ is reversed relative to ${P}_{1}$ and ${P}_{3}$. Additionally, there is a clear hardening effect from P1 to P3, which weakens the asymmetricity between the tensile and compressive zone.

Fig. 6

The stress distribution at different loading stages. a–c The distribution at ${{\varvec{P}}}_{1-3}$ of Cycle A, respectively; d–f The distribution at ${{\varvec{P}}}_{1-3}$ of Cycle B, respectively; g–i The distribution at ${{\varvec{P}}}_{1-3}$ of Cycle C, respectively

Fig. 7

The strain distribution at different loading stages. a–c The distribution at ${{\varvec{P}}}_{1-3}$ of Cycle A, respectively; d–f The distribution at ${{\varvec{P}}}_{1-3}$ of Cycle B, respectively; g–i The distribution at ${{\varvec{P}}}_{1-3}$ of Cycle C, respectively

In Fig. 8, the distribution of TVF at ${P}_{1}$, ${P}_{2}$, and ${P}_{3}$ during loading cycles A–C is presented. The polar nature of the extension twin and the initial texture lead to the apparent occurrence of twinning and detwinning effects. The wider compressive zone at ${P}_{2}$ (as shown in Fig. 6) results in a higher TVF value at ${P}_{2}$ than at ${P}_{1}$. The TVF distribution at ${P}_{3}$ is similar to that at ${P}_{2}$, but with a higher value. During loading from ${P}_{1}$ to ${P}_{2}$, detwinning occurs in the upper half, and the twin does not completely disappear. From ${P}_{2}$ to ${P}_{3}$, a similar loading condition causes twinning to become more profound again, resulting in a higher TVF at the end of the cycle.

Fig. 8

The TVF distribution at different loading stages: a Cycle A; b Cycle B; c Cycle C

4 Conclusion

This work investigated the cyclic bending behavior of Mg alloys with curvature amplitudes of 0.003, 0.006, and 0.010 using the EVPSC-BEND model. The relative activity and twin volume fraction (TVF) during cyclic loading indicate the twinning-detwinning mechanism is actively involved throughout the entire cyclic bending process. The results of stress distribution also demonstrate that the twinning mechanism induces different yield stresses in different layers. The stress distribution results demonstrate that the twinning mechanism induces different yield stresses in different layers. Combined with the distribution of twin volume fraction (TVF), it can be observed that the action of twinning and detwinning reproduces similar stress distributions during different stages of the cyclic process. Our future research will involve conducting cyclic four-point bending experiments to further validate the predictive capability of the model. Additionally, we will perform microscopic characterization, focusing on the changes in texture and twin volume fraction during the cyclic bending process, to further analyze the role of twinning and detwinning mechanism.