Experimental observations on the nonproportional multiaxial ratchetting of cast AZ91 magnesium alloy at room temperature

1.   Introduction

Magnesium (Mg) alloys, which are lightweight materials for structural applications, have attracted considerable attention in the transportation, aerospace, and biological fields because of their high specific strength, large specific modulus, good vibration dissipation performance, and low cost [1–5]. However, these structures experience unavoidable cyclic loading in practical applications. Therefore, studying the cyclic deformation of Mg alloys, especially ratchetting, is essential to promote their application in engineering. The activities of nonbasal plane slipping systems (including prismatic and pyramidal) in Mg alloys with a hexagonal close-packed crystal structure are low at room temperature (RT) because of their high critical resolved shear stresses [6–8]. However, the von Mises compatibility condition of plastic deformation [9] for polycrystalline metallic materials indicates that five independent slip systems should be activated to deform metals uniformly. Therefore, the plastic deformation of Mg alloys requires the contribution of twinning deformation, which is also a major mechanism of plastic deformation in Mg alloys [10].

Mg alloys are commonly categorized as either cast or wrought (such as hot-rolled and extruded) on the basis of their different manufacturing processes. The intense basal texture and polarity of twinning deformations [11–12] result in the macroscopic tension–compression asymmetry of wrought Mg alloys at RT [13–16]. The cast Mg alloys have been widely used in industrial production due to their economic advantages, such as short processing time, low assembly expenses, and ability to produce complex components, although their strengths are lower than those of the wrought ones. The cast Mg alloys have no intense basal texture, resulting in generally random grain orientations that lead to tension–compression symmetry at RT [17]. Furthermore, at the unloading stage, cast Mg alloys exhibit pseudo-elasticity due to the participation of twinning/detwinning deformations [18], indicating that the secant modulus at the unloading stage is less than the original elastic modulus, and the recovery strain after unloading is considerably larger than the expected in consideration of linear elastic behaviors. This phenomenon is not observed in other polycrystalline metals with body-centered cubic [19] and face-centered cubic [20–21] crystal structures that also exhibit tension–compression symmetry.

Recent experimental results [17,22–24] regarding the uniaxial cyclic plastic deformation of cast Mg alloys have demonstrated that symmetrical stress–strain curves are acquired in the strain-controlled symmetrical cyclic tests of cast Mg alloys. Li et al. [25] observed that cyclic hardening occurred in cast AZ91 Mg alloys. Further findings from Liu et al. [26] revealed that the degree of cyclic hardening increased with increasing loading strain amplitude. Ratchetting is expected in the stress-controlled cyclic tests of metal alloys under nonzero mean stress and stress levels exceeding the yield strength of the tested materials [19–21,27–28]. The ratchetting observed in Mg alloys shows unique evolutionary features because of the participation of multiple plastic deformation mechanisms. For example, Kang et al. [13] discussed the uniaxial ratchetting of extruded AZ31 Mg alloy at RT by exploring the participation of three plastic deformation mechanisms: dislocation gliding, twinning, and detwinning individually or alternately. The results [13] indicated that in the cyclic tension–compression tests of extruded AZ31 Mg alloy under certain loading conditions, sigmoidal stress–strain curves and points representing peak and valley strains continuously shifted in opposite directions individually. Furthermore, Lin et al. [29] reported that ratchetting strain and its evolution rate increased with increases in the applied mean stress and stress amplitude, revealing that loading levels have a clear effect on ratchetting. Currently, research on the cyclic deformation of cast Mg alloys under stress-controlled loading conditions is limited. For example, Cáceres et al. [18] investigated the pseudo-elasticity of cast AZ91 Mg alloy during cyclic loading–unloading and provided an explanation for their underlying deformation mechanisms on the basis of optical microscope (OM) observations, which identified a correlation between the pseudo-elasticity of cast Mg alloys and the mechanisms associated with twinning/detwinning. In addition, Lei et al. [17] conducted uniaxial asymmetric stress-controlled experiments on the cast AZ91 alloy at different stress levels at RT, and then explained the observed evolutionary features of cyclic deformation on the basis of electron backscattered diffraction and OM observations. Microscopic observation [17] verified that during the cyclic deformation of cast Mg alloys, twinning/detwinning is activated alongside dislocation gliding. In addition, the presence of residual twins is identified to contribute to the cyclic hardening behavior observed in cast Mg alloys. Components made from cast Mg alloys inevitably experience multiaxial stress states during operational service, thus necessitating experimental investigations into the multiaxial cyclic deformation, including the ratchetting. Nevertheless, no study has been conducted on the ratchetting of cast Mg alloys under multiaxial loading conditions.

Existing studies on the cyclic deformation of Mg alloys under uniaxial and multiaxial loading conditions have demonstrated that the interaction of multiple plastic deformation mechanisms makes the multiaxial cyclic plasticity of Mg alloys more intricate than uniaxial cyclic plasticity. For example, some scholars [30–34] conducted practical studies on the torsional cyclic deformation of Mg alloys and found that a part of extension twinning would be activated under a torsional loading level high enough. Zhang et al. [34] revealed that under cyclic torsion, extruded Mg alloys exhibited symmetrical shear stress–shear strain hysteresis loops due to the combined influence of multiple mechanisms. Furthermore, the stress–strain response of the alloys is highly dependent on shear stress amplitude. Albinmousa et al. [35–36] compared the axial and torsional stress–strain hysteresis loops of extruded Mg alloy under the strain-controlled multiaxial loading conditions, including the nonproportional and proportional loading paths, and revealed that the torsional stress–strain curves exhibited a symmetrical shape in the nonproportional loading tests, while displaying a sigmoidal pattern in the proportional loading tests, analogous to the axial stress–strain curves. Li et al. [37] investigated the impact of loading paths on the multiaxial ratchetting of extruded AZ31 Mg alloy through a sequence of multiaxial cyclic tests at RT and found that cyclic tests with circular and rhombic loading paths exhibited higher axial ratchetting strains than those with proportional 45° linear and uniaxial loading paths. Gryguć et al. [38] observed that the effect of nonproportionality induced by multiaxial loading paths was only evident in the shear response of forged AZ80 Mg alloys, whereas axial stress–strain behavior remained unaffected. However, studies on the multiaxial cyclic deformation of cast Mg alloys are rare, particularly on the multiaxial ratchetting.

Therefore, in this work, a sequence of cyclic deformation experiments with various loading paths and stress levels was performed on the cast AZ91 Mg alloy to elucidate the evolutionary features and physical mechanisms of the multiaxial ratchetting. Consequently, this work will enrich the database for the multiaxial ratchetting of cast Mg alloys and will be helpful in developing corresponding constitutive models in future works.

2.   Materials and procedures

The experimental material selected for this study was cast AZ91 Mg alloy, which consisted of 8.33% aluminum, 0.34% zinc, and 0.17% manganese by mass, with the remainder being Mg. The specimens were thin-walled tubes, as shown in Fig. 1(a), wherein a gauge length of 30 mm was set with a wall thickness of 1.5 mm and an inner diameter of 15 mm. The specimens were annealed before the experiments to relieve the residual stress resulting from manufacturing. The initial pole figure of the cast Mg alloy was characterized by X-ray diffraction, as depicted in Fig. 1(b). It is evident that the alloy exhibits no pronounced texture with the maximum density of 1.5, displaying a random grain orientation.

Fig. 1.  Initial macroscopic and microscopic figures of the cast AZ91 Mg alloy specimen: (a) shape and size of the multiaxial specimen (unit: mm) and (b) pole figures.

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The experiments were conducted with the MTS809 material testing machine. The multiaxial extensometer with the measuring ranges of axial strain −10% to 10% and torsional angle −5° to 5° was used to measure the axial strain and shear strain. The loading mode controlled by stress with a triangular loading waveform and rate of 50 MPa/s was adopted in the experiments. The loading paths employed in this study included a uniaxial path, a pure torsional path, an elliptical path, and a circular path and its inscribed paths, as depicted in Fig. 2. In comparison with correspondent uniaxial ratchetting, nonzero mean stress was exclusively applied in the axial direction, while the mean shear stress was held at zero for all multiaxial loading paths, as depicted in Fig. 2 (from Fig. 2(c) to (g)). It should be noted that some experiments on the prescribed loading cases with various loading paths were repeated to eliminate the dispersion of experimental results.

Fig. 2.  Loading paths: (a) uniaxial, (b) pure torsional, (c) elliptical, (d) circular, (e) rhombic, (f) square, and (g) 45° linear paths.

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The axial and equivalent shear stresses are represented as σ and $\sqrt{\text{3}}\tau$, while $\varepsilon$ and $\textit{γ}/\sqrt{\text{3}}$ denote the axial and equivalent shear strains, respectively. For describing axial and torsional ratchetting in the multiaxial cyclic tests controlled by stress, the axial ratchetting strain $\varepsilon_{\text{r}}$ and torsional ratchetting strain ${\textit{γ}}_{\text{er}}$ are quantified as follows:

where $\varepsilon_{\text{p}}$, $\varepsilon_{\text{v}}$, ${\textit{γ}}_{\text{p}}$, and ${\textit{γ}}_{\text{v}}$ are the peak and valley axial and shear strains in each cycle.

3.   Results and analysis

3.1   Uniaxial and pure torsional ratchetting

In comparison to the multiaxial ratchetting of cast AZ91 Mg alloy, correspondent uniaxial and pure torsional ratchetting tests were performed by using the thin-walled tubular specimens depicted in Fig. 1, respectively. The uniaxial ratchetting experiments were performed under two uniaxial loading conditions with high mean stress (i.e., the mean stress ${\textit{σ}}_{\text{m}}=60\; \text{MPa}$ and the stress amplitude ${\textit{σ}}_{\text{a}}=60\; \text{MPa}$) and low mean stress (i.e., ${\textit{σ}}_{\text{m}}=20\; \text{MPa}$ and ${\textit{σ}}_{\mathrm{a}}= 100\; \text{MPa}$), individually. Fig. 3 displays the obtained results.

Fig. 3.  Responses of the cast AZ91 Mg alloy with uniaxial loading path: (a) axial stress–strain at (60 ± 60) MPa, (b) corresponding evolution of ε_p, ε_v, and ε_r at (60 ± 60) MPa, (c) axial stress–strain at (20 ± 100) MPa, and (d) corresponding evolution of ε_p, ε_v, and ε_r at (20 ± 100) MPa.

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Similar to the uniaxial ratchetting obtained by Lei et al. [17] by using the solid dumbbell specimens, Fig. 3(a) and (b) shows that the alloy only yields plastically in the tensile direction because the minimum stress with the relatively high mean stress is below the compressive yield strength of the cast alloy. The stress–strain hysteresis loops exhibit a convex and symmetrical shape. The $\varepsilon_{\text{p}}$ and $\varepsilon_{\text{v}}$ evolve both along the tensile direction with an occurrence of obvious ratchetting, and the $\varepsilon_{\text{v}}$ exhibits a greater velocity than $\varepsilon_{\text{p}}$ at the early cyclic loading stage because of the interplay among multiple plastic deformation mechanisms of the cast alloy involved in the cyclic deformation process [18]. However, Fig. 3(c) and (d) shows that: similar to that obtained by Lei et al. [17], the cast Mg alloy yields plastically in the directions of both tension and compression during the cyclic loading with a low ${\textit{σ}}_{\text{m}}$ and a high ${\textit{σ}}_{\text{a}}$, which means that both the dislocation gliding and twinning/detwinning play important roles in the cyclic plastic deformation. Consistent with the findings obtained by Cáceres et al. [18] through the in-situ OM observations, the twinning occurs during loading and the detwinning occurs during unloading. With the increase of cyclic numbers, the numerous twins induced in the tensile process cannot be adequately detwinned during unloading. The rapid accumulation of residual twins during cyclic loading impedes subsequent dislocation gliding and twinning deformation, thus, the cast AZ91 alloy manifests a cyclic hardening characterized by the progressive narrowing of hysteresis loops [22,39]. The evolutions of $\varepsilon_{\text{p}}$ and $\varepsilon_{\text{v}}$ with the number of cycles exhibit opposite trends due to the various plastic deformation mechanisms involved, wherein $\varepsilon_{\text{p}}$ slowly moves toward the compressive direction, whereas $\varepsilon_{\text{v}}$ moves toward the tensile direction at a higher rate. Consequently, the $\varepsilon_{\text{r}}$ evolves in the mean stress direction (namely, the tensile direction under this loading condition) at a decreasing rate. In general, it is evident from the comparison between the uniaxial ratchetting results acquired by using the thin-walled tubular specimens and solid dumbbell specimens that: the evolution rule of uniaxial ratchetting obtained by the specimens with different shapes are the same, but the specific values of ratchetting strain are distinguished from each other. So, the uniaxial ratchetting results acquired by using the thin-walled tubular specimens described in this subsection are compared with the axial ratchetting of the alloy obtained with the multiaxial loading paths in the next subsection.

Subsequently, the pure torsional ratchetting tests of the cast AZ91 Mg alloy were performed by setting the equivalent shear stress amplitude ${\textit{τ}}_{\text{ea}}$ as 60 MPa and the equivalent shear mean stresses ${\textit{τ}}_{\text{em}}$ as 0, 20, and 40 MPa. The experimental results, as depicted in Fig. 4, reveal that: 1) The cast Mg alloy exhibits a narrow and symmetrical equivalent shear stress–strain hysteresis loop, and no significant evolution of the hysteresis loop is observed with the increase in the cyclic number if the ${\textit{τ}}_{\text{em}}$ is 0 MPa. 2) The stress–strain curves in the pure torsional cyclic tests with nonzero mean shear stress are convex and symmetrical and are analogous to the results acquired in the uniaxial cyclic tests illustrated in Fig. 3(a). This result can be attributed to a greater number of twins activated by the higher shear mean stress (if the applied shear stress amplitude remains constant), which enhances the ${\textit{γ}}_{\text{er}}$ and makes the pseudo-elastic behavior more prominent during cyclic deformation. 3) The points representing the ${\textit{γ}}_{\text{p}}/\sqrt{\text{3}}$ and ${\textit{γ}}_{\text{v}}/\sqrt{\text{3}}$ move toward the same direction, while the evolution rate of ${\textit{γ}}_{\text{v}}$ is faster than that of ${\textit{γ}}_{\text{p}}$ due to the interplay of multiple deformation mechanisms under specific loading conditions with a high shear mean stress ${\textit{τ}}_{\text{em}}$, as illustrated in Fig. 4(c). This phenomenon is also observed in the uniaxial cyclic test with high ${\textit{σ}}_{\text{m}}$ presented in Fig. 3(b).

Fig. 4.  Responses of the cast AZ91 Mg alloy with pure torsional loading path: (a) torsional stress–strain with τ_em of 0 MPa, (b) torsional stress–strain with τ_em of 40 MPa, (c) evolution of γ_p$/ \sqrt{\text{3}}$ and γ_v$/\sqrt{\text{3}}$, and (d) evolution of γ_er.

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3.2   Multiaxial ratchetting

3.2.1   Circular loading path

To examine the multiaxial ratchetting of cast AZ91 Mg alloy, the cyclic tests with the circular path (shown as Fig. 2(d)) and different stress levels are presented in this subsection. Figs. 5 and 6 illustrate the strain responses of the alloy and evolution curves of $\varepsilon _{\text{p}}$, $\varepsilon _{\text{v}}$, and $\varepsilon _{\text{r}}$ with the cyclic number, where the ${\textit{σ}}_{\text{m}}$ was 60 MPa and the ${\textit{σ}}_{\text{a}}$ was 60 MPa in the axial direction, and the ${\textit{τ}}_{\text{ea}}$ was set as 60 MPa and ${\textit{τ}}_{\text{em}}$ was applied as 0 MPa in the torsional direction. Fig. 6 also presents the uniaxial ratchetting results discussed in Section 3.1 for comparison.

Fig. 5.  Responses of the cast AZ91 Mg alloy with circular path: (a) shear–axial strain, (b) axial stress–strain, and (c) torsional stress–strain.

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Fig. 6.  Corresponding evolution of (a) ε_p and ε_v and (b) ε_r of the cast AZ91 Mg alloy with circular path.

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Figs. 5 and 6 show the following: 1) The axial stress–strain curves display symmetrical and convex contours. With the increase in the number of loading cycles, the axial hysteresis loop gradually moves to the right (as depicted in Fig. 5(b)), reflecting an obvious axial ratchetting. The axial ratchetting strain increases, while its rate decreases with the increase in loading cycles (as depicted in Fig. 6(b)). Meanwhile, the hysteresis loops gradually narrow, and the corresponding axial strain amplitude gradually decreases with the accumulation of loading cycles (as depicted in Fig. 5(a)). The decrease in the axial strain amplitude indicates that the cast Mg alloy exhibits a certain degree of cyclic hardening. 2) The equivalent shear stress–strain curves show a symmetric elliptical shape in the torsional direction. The evolutions of ${\textit{γ}}_{\text{p}}/\sqrt{\text{3}}$ and ${\textit{γ}}_{\text{v}}/\sqrt{\text{3}}$ exhibit opposite trends with increasing loading cycles. Specifically, the point representing the ${\textit{γ}}_{\text{p}}/\sqrt{\text{3}}$ evolves toward the negative direction, whereas that representing the ${\textit{γ}}_{\text{v}}/\sqrt{\text{3}}$ evolves toward the positive direction, as depicted in Fig. 5(c). Furthermore, the shear strain amplitude also decreases gradually with increasing the loading cycles, indicating a certain level of cyclic hardening in the torsional direction. 3) Due to the application of nonzero mean stress solely in the axial direction, the multiaxial ratchetting of cast Mg alloy occurs only along the axial direction. The ratchetting strain in the torsional direction is close to zero and can be neglected. 4) As depicted in Fig. 6, in comparison to its uniaxial ratchetting, the cast alloy demonstrates more pronounced multiaxial ratchetting, and the axial ratchetting strain and its rates under the nonproportional multiaxial cyclic loading with a circular path are higher. This result can be attributed to the continuous change in the directions of principal stresses during multiaxial cyclic deformation, which activates a greater amount of dislocation gliding and twinning/detwinning in the grains with various orientations [10].

For the comparison of the multiaxial ratchetting of the extruded Mg alloys with that of the cast Mg alloys, Fig. 7 presents the experimental multiaxial ratchetting results of extruded AZ31 Mg alloy acquired by Li et al. [37] with a similar circular path. Because of the intense basal texture in the extruded alloy and the random grain orientations in the cast alloy, it is manifested that a large difference exists between the multiaxial ratchetting results of the two alloys, as depicted in Figs. 5 and 7, that is: 1) The extruded Mg alloy yields only in the tensile direction at the prescribed stress level, and no obvious plastic deformation occurs in the compressive direction. The axial plastic deformation is dominantly governed by dislocation gliding under this loading condition. Therefore, the stress–strain curves in the axial direction with the circular path are almost linear, and no noticeable hysteresis loop occurs. However, in the torsional direction, the activation of a large number basal slipping and twinning results in an obvious hysteresis loop. In addition, with increasing loading cycles, obvious ratchetting appears not only in the axial direction but also in the torsional direction. The ${\textit{γ}}_{\text{p}}/\sqrt{\text{3}}$ and ${\textit{γ}}_{\text{v}}/\sqrt{\text{3}}$ evolve along the same direction at similar rates, except for that in the first two cycles. 2) Although the cast Mg alloy is subjected to a cyclic tensile–tensile loading in the axial direction, its deformation mechanism is not only the dislocation gliding but also the twinning/detwinning [17]. Therefore, its axial stress–strain hysteresis loop shows a symmetric convex contour with an occurrence of obvious ratchetting. However, the ${\textit{γ}}_{\text{p}}$ and ${\textit{γ}}_{\text{v}}$ in the torsional direction evolve in the opposite direction without an occurrence of obvious ratchetting.

Fig. 7.  Responses of the extruded AZ31 Mg alloy with circular path: (a) shear–axial strain, (b) axial stress–strain, and (c) torsional stress–strain [37]. Reprinted from Mater. Sci. Eng. A, Vol. 671, H. Li, G.Z. Kang, Y.J. Liu, and H. Jiang, Non-proportionally multiaxial cyclic deformation of AZ31 magnesium alloy: Experimental observations, 70-81, Copyright 2016, with permission from Elsevier.

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(1) Effect of stress amplitude on ratchetting.

Although several scholars [36,40] have performed experimental studies on the multiaxial cyclic plasticity of wrought Mg alloys with different loading levels, these studies have mostly focused on the influence of strain levels on multiaxial fatigue life. Few studies have been conducted on the effect of stress levels on the multiaxial ratchetting of extruded Mg alloys. Specifically, there is no research on the multiaxial ratchetting of cast Mg alloys. The effect of ${\textit{σ}}_{\text{a}}$ on the multiaxial ratchetting of the cast Mg alloy with a circular path is first discussed here to examine its dependence on the stress level. Therefore, in this subsection, the ${\textit{σ}}_{\text{a}}$ and ${\textit{τ}}_{\text{ea}}$ were jointly specified as 50, 60, and 70 MPa with the ${\textit{σ}}_{\text{m}}$ of 60 MPa and ${\textit{τ}}_{\text{em}}$ of 0 MPa. Fig. 8 depicts the evolution curves of $\varepsilon _{\text{p}}$, $\varepsilon_{\text{v}}$, and $\varepsilon_{\text{r}}$ derived from the multiaxial cyclic tests on the cast Mg alloy with the circular path at different stress amplitudes. Although the cast AZ91 alloy only yields plastically in the tensile direction under the loading condition with a high ${\textit{σ}}_{\text{m}}$ and low ${\textit{σ}}_{\text{a}}$, the twinning/detwinning is also involved during the cyclic tests owing to its random grain orientation, different from the extruded Mg alloy [13]. With different stress amplitudes, the cyclic axial stress–strain curves of the cast AZ91 Mg alloy all exhibit a symmetrical and convex shape, as depicted in Fig. 5(b). Fig. 8 illustrates the following: 1) When the multiaxial loading path and mean stress are identical, the alloy presents the ratchetting apparently in the tensile direction under the multiaxial cyclic loading conditions with different stress amplitudes. Both the $\varepsilon _{\text{p}}$ and $\varepsilon _{\text{v}}$ move toward the tensile direction with increasing the loading cycles and exhibit quasi-shakedown after certain loading cycles. 2) The higher the stress amplitude, the more significant the axial ratchetting is. The evolution rates of $\varepsilon _{\text{p}}$ and $\varepsilon _{\text{v}}$ accelerate with increasing the stress amplitude. These findings align with those obtained by Lei et al. [17] in the uniaxial loading experiments.

Fig. 8.  Corresponding evolution of (a) ε_p and ε_v and (b) ε_r of the cast AZ91 Mg alloy with circular path and various stress amplitudes.

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(2) Effect of axial mean stress.

To examine the effect of various axial mean stresses on the multiaxial ratchetting of the cast Mg alloy with a circular path, the tests in this subsection were conducted under a symmetric cyclic loading condition controlled by stress in the torsional direction with the ${\textit{σ}}_{\text{a}}$ and ${\textit{τ}}_{\text{ea}}$ both set to 60 MPa and the ${\textit{σ}}_{\text{m}}$ set to 50, 60, and 70 MPa, respectively. Fig. 9 presents the experimental axial ratchetting of the cast AZ91 Mg alloy with a circular path and various ${\textit{σ}}_{\text{m}}$. As depicted in Fig. 9, it is evident that: 1) With the same loading path and stress amplitude, the significant axial ratchetting is observed in the multiaxial cyclic tests with various ${\textit{σ}}_{\text{m}}$, and the quasi-shakedown of ratchetting eventually occurs after certain loading cycles. The $\varepsilon_{\text{p}}$, $\varepsilon_{\text{v}}$, and $\varepsilon_{\text{r}}$ progressively increase in the same tensile direction with increasing the cyclic number. 2) The axial ratchetting with a circular path gets promoted with the increase in axial mean stress, which is consistent with the uniaxial ratchetting observed by Lei et al. [17]. This is because a higher ${\textit{σ}}_{\text{m}}$ means a higher peak stress if the stress amplitude remains the same, which leads to more activated dislocation gliding and twinning, thus resulting in more significant axial ratchetting.

Fig. 9.  Corresponding evolution of (a) ε_p and ε_v and (b) ε_r of the cast AZ91 Mg alloy with circular path and various axial mean stresses.

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3.2.2   Effect of loading paths on ratchetting

(1) Circular path and its inscribed paths.

To further discuss the effect of multiaxial loading paths with various shapes on the multiaxial ratchetting of cast AZ91 Mg alloy, the circular path and its inscribed rhombic, square, and proportional 45° linear paths (depicted in Fig. 2(e), (f), and (g)) were used in the tests discussed here. The ${\textit{σ}}_{\text{m}}$ was specified as 60 MPa, and symmetric cyclic loading was established in the torsional direction. For the circular and its inscribed rhombic paths, the ${\textit{σ}}_{\text{a}}$ and ${\tau }_{\mathrm{e}\mathrm{a}}$ were both specified as 60 MPa, while for the inscribed square and 45° linear paths, the ${\textit{σ}}_{\text{a}}$ and ${\tau }_{\mathrm{e}\mathrm{a}}$ were both specified as 42.43 MPa. Figs. 10 and 11 provide the strain response curves of the alloy derived from the tests with the rhombic and square paths (the experimental results with the circular path are presented in Fig. 5). Furthermore, Fig. 12 demonstrates the evolution curves of $\varepsilon_{\text{r}}$ with loading cycles for the cast AZ91 Mg alloy and extruded AZ31 Mg alloy derived from the cyclic tests controlled by stress with various loading paths.

Fig. 10.  Responses of the cast AZ91 Mg alloy with rhombic path: (a) shear–axial strain, (b) axial stress–strain, and (c) torsional stress–strain.

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Fig. 11.  Responses of the cast AZ91 Mg alloy with square path: (a) shear–axial strain, (b) axial stress–strain, and (c) torsional stress–strain.

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Fig. 12.  Corresponding evolution of $\varepsilon_{\text{r}}$ with cyclic number for Mg alloys with various loading paths: (a) cast AZ91 Mg alloy and (b) extruded AZ31 Mg alloy [37]. (b) Reprinted from Mater. Sci. Eng. A, Vol. 671, H. Li, G.Z. Kang, Y.J. Liu, and H. Jiang, Non-proportionally multiaxial cyclic deformation of AZ31 magnesium alloy: Experimental observations, 70-81, Copyright 2016, with permission from Elsevier.

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The following conclusions are obtained on the basis of the experimental results shown in Fig. 10 to Fig. 12: 1) With the aforementioned loading paths, the cast AZ91 Mg alloy shows notable axial ratchetting but negligible torsional ratchetting due to the symmetric stress-controlled shear loading condition. 2) In comparison to the uniaxial path, the $\varepsilon_{\text{r}}$ with the nonproportional loading paths is higher, demonstrating a nonproportional additional softening in the cast Mg alloy. 3) The ratchetting of the cast Mg alloy is greatly affected by the shape of the loading path. Among the results acquired with the circular, rhombic, and square paths, the most significant axial ratchetting is observed in the square path as it involves both peak and valley stress holdings in the axial and torsional directions. Furthermore, considering the nonproportional loading paths without any peak and valley stress holdings, more pronounced ratchetting in the axial direction with the circular path is observed in comparison to that with its inscribed rhombic path. The axial ratchetting obtained with both the circular and rhombic paths is higher than that obtained with the uniaxial and 45° linear paths. This is because with the circular path, the phase angle $\textit{φ}$ between the axial and torsional loads is $\text{π}/\text{2}$. The twinning density in the Mg alloy displays a minimum at $\textit{φ}$ $=\text{π}/\text{2}$ [41], which reduces the hindrance on the subsequent plastic deformation. Therefore, the development of axial ratchetting is faster with the circular path. This observation is consistent with the ratchetting of extruded Mg alloy with various loading paths [37] (as depicted in Fig. 12(b)). However, different from the increase in the $\varepsilon_{\text{r}}$ at a constant rate observed in the extruded Mg alloy with the nonproportional loading paths, the axial ratchetting strain rate of cast Mg alloy tends to approach zero after certain loading cycles, leading to a quasi-shakedown of ratchetting, whether with the uniaxial or multiaxial loading paths.

(2) Elliptical loading path.

It is noted that high axial mean stresses are employed in the aforementioned multiaxial loading cases, which results in the axial valley stress lowering the compressive yield strength of the cast Mg alloy. Thus, for exploring the multiaxial ratchetting of the cast Mg alloy with plastic yields in the directions of tension and compression simultaneously, a cyclic test was conducted with the elliptical path shown in Fig. 2(c), where the ${\textit{σ}}_{\text{m}}$ was set as 20 MPa and the ${\textit{σ}}_{\text{a}}$ was set as 100 MPa, while the ${\textit{τ}}_{\text{em}}$ was set as 0 MPa in the torsional direction with the ${\textit{τ}}_{\text{ea}}$ of 50 MPa. The strain responses and evolution of $\varepsilon_{\text{r}}$ with the cyclic number are depicted in Figs. 13 and 14, respectively. The experimental results lead to the following conclusions: 1) As loading cycles increase, the $\varepsilon_{\text{p}}$ moves toward compression, whereas the $\varepsilon_{\text{v}}$ moves toward tension. Furthermore, the strain amplitude steadily decreases, but the $\varepsilon_{\text{r}}$ continuously grows. Compared with the uniaxial ratchetting, the cast AZ91 Mg alloy with the elliptical path exhibits a higher evolution rate of $\varepsilon_{\text{v}}$, resulting in a more pronounced axial ratchetting than the uniaxial one. 2) The torsional stress–strain curves show a symmetric shape. The evolution trends of ${\textit{γ}}_{\text{p}}$ and ${\textit{γ}}_{\text{v}}$ with the increase of cyclic number are consistent with those in the axial direction. Furthermore, the equivalent shear strain amplitude progressively decreases during the cyclic loading, in accordance with the evolutionary trend observed in the axial direction.

Fig. 13.  Responses of the cast AZ91 Mg alloy with elliptical path: (a) shear–axial strain, (b) axial stress–strain, and (c) torsional stress–strain.

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Fig. 14.  Corresponding evolution of (a) ε_p and ε_v and (b) ε_r of the cast AZ91 Mg alloy with elliptical path.

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In summary, the cast AZ91 Mg alloy presents obvious ratchetting in the axial direction under the multiaxial cyclic loading conditions with a nonzero axial mean stress at RT. The peak and valley strains show the same evolutionary trends generally, except for the case with an elliptical path and a low ${\textit{σ}}_{\text{m}}$. Furthermore, $\varepsilon_{\text{r}}$ is more pronounced in the nonproportional multiaxial loading cases than the uniaxial ratchetting. The shape of loading path significantly influences the multiaxial ratchetting. In contrast to the extruded Mg alloys, the axial ratchetting of the cast AZ91 Mg alloy generally reaches to a quasi-shakedown after certain loading cycles. These findings are valuable for developing a corresponding cyclic plasticity model for the multiaxial ratchetting of cast Mg alloys. However, there are still some limitations in this work, that is: 1) This work only focuses on the macroscopic experimental results, lacking the microstructural characterization to explain the corresponding physical nature of multiaxial ratchetting reasonably. 2) Although previous studies have illustrated that there is an influence of temperature on the cyclic deformation in Mg alloys [42–43], all the experiments here are performed at RT without considering the temperature dependence of the multiaxial ratchetting of cast Mg alloys. These limitations will be investigated and improved in the subsequent study.

4.   Conclusions

The conclusions from the multiaxial ratchetting tests of the cast AZ91 Mg alloy at RT are summarized as follows:

(1) Stress–strain curves exhibit symmetry in the uniaxial and pure torsional ratchetting tests. Nevertheless, the evolutions of peak and valley strains follow different trends because of the involvement of twinning/detwinning and the interplay of multiple plastic deformation mechanisms. These trends also exhibit dependence on changes in stress levels.

(2) Noticeable axial ratchetting is observed in the multiaxial stress-controlled cyclic tests on the cast AZ91 Mg alloy. The nonproportional additional softening is also demonstrated clearly in the axial direction.

(3) The multiaxial ratchetting of the cast Mg alloy is highly influenced by the applied stress amplitude and axial mean stress. As the stress amplitude increases, the axial ratchetting strain and its rate also increase. Furthermore, consistent with the uniaxial ratchetting observed in a previous study [17], axial ratchetting is promoted by an increase in axial mean stress.

(4) The multiaxial ratchetting of the cast AZ91 Mg alloy shows a strong correlation with the shape of loading path. Among various loading paths, nonproportional multiaxial loading paths result in more significant axial ratchetting than uniaxial and 45° linear paths. In contrast to that of the extruded Mg alloy, the axial ratchetting strain rate of the cast Mg alloy gradually decreases with increasing cyclic number, and the quasi-shakedown of ratchetting eventually occurs.