Anelastic and Microplastic Damping of an Mg–Zn–Y–Al Alloy

Abstract

In this study, the Mg₉₇Zn₁Y₂–Al alloy was selected as a high damping material, anelastic and microplastic damping were analyzed by studying strain amplitude-dependent damping curves of the alloy. The C₁ and C₂ values of the Granato–Lücke (G–L) model, the theoretical basis of anelastic damping, can be calculated for each alloy, then, the corresponding numbers of strong and weak pinners are deduced. However, since the occurrence of microplastic damping in a high strain amplitude cannot be explained by the G–L model, a new microplastic damping theory was introduced, by comparing the activation volume of the material dislocation slip, the microplastic damping capacity of the Mg₉₇Zn₁Y₂–xwt%Al (x = 0.3, 1, 3) alloys at the microplastic stage are compared. The elastic and plastic deformation of the materials could be sensitively reflected by studying the damping behavior of these two stages.

1 Introduction

With the rapid development of industrialization, the harm caused by noise and vibration to both machinery and human body has become a serious issue that cannot be ignored [1]. The application of damping materials is one of the methods to reduce such vibration and noise [2]. As early as in the 1950 s, Zener [3] published the renowned work about the elasticity and anelasticity of metals and people began to explore high-damping materials based on the anelastic theory. The study of the damping mechanism of metals in various states is crucial for developing high-performance metal-based materials.

In our previous study [4], the Mg₉₇Zn₁Y₂–xwt%Al (x = 0.3, 1, 3) alloy has been defined as a promising high-strength and high-damping magnesium alloy due to the presence of a long-period stacking ordered (LPSO) phase, which can balance its damping and mechanical properties. Therefore, in this work, the same alloy was selected to investigate the damping-strain amplitude dependence of high-damping Mg alloys in a wide range of strain amplitudes. This study could not only clarify the damping capacity of this material but also provide its damping response under different deformation states, through which its elastic and plastic deformation could be sensitively reflected.

According to the Granato–Lücke (G–L) model, the damping-strain amplitude curve of a material is usually divided into two segments [5, 6], the strain amplitude-independent segment and the strain amplitude-dependent one. However, some studies suggest that [7, 8] when the strain amplitude exceeds a certain value, this curve can no longer be explained by the classical G–L model. In this study, an Mg–Zn–Y–Al alloy was used as an example to explain the anelastic and microplastic damping in detail, and a model for describing the microplastic damping of magnesium alloys was introduced.

2 Experimental

Mg₉₇Zn₁Y₂–xwt%Al alloys were prepared by conventional casting method. Pure Mg (99.9% purity), pure Zn (99.9% purity), Mg–25wt%Y binary alloy, and high-purity Al flakes were selected as the raw materials. The traditional casting process was used to smelt in XMT well resistance furnace under the condition of 720 °C. The molten metal was pouring into the metal mold for air cooling. The microstructure of the specimens was observed by an optical microscope (COOLPIX-4500) and a scanning electron microscope (SEM) with an energy dispersive spectroscopy (EDS) analyzer (JSM 6701F). The damping performance was tested by dynamic mechanical thermal analysis (DMA Q800). The temperature of the strain amplitude dependent damping test is at room temperature, and the vibration frequency is 1 Hz. In order to facilitate observation, the strain amplitude is increased as much as possible in the damping test, the strain amplitude interval is 1 × 10⁻³ to 4 × 10⁻³, which is covered over anelastic and microplastic zone. The test-pieces for damping were machined with dimension of 50 mm × 5 mm × 1 mm on electric spark cutter and properly installed in the DMA testing head, was constrained at each end by a clamping bar arrangement with one end fixed to a rigid frame and the other end driven by an electromagnetic vibrator via a composite drive shaft. The resulting sinusoidal force and deflection data were recorded and analyzed by modulus. The internal friction value of material in the damping-strain spectrum was evaluated by loss tangent (tanδ), which can be expressed by Q⁻¹.

3 Results and Discussion

3.1 Microstructure

Figure 1a shows the optical microscopy images of the Mg₉₇Zn₁Y₂ alloy, which is mainly revealed a coarse dendrite structure and the uneven distribution of a secondary phase, while the morphologies resulting from the Al addition are displayed in Fig. 1b–d. When the Al content added was 0.3 wt%, the morphology of the secondary phase did not change significantly; however, the increasing of its content hindered the dendrite structure growth, refined the dendrite size to a certain extent, and promoted the generation and even distribution of the secondary phase [4].

Fig. 1

Microstructure of cast Mg₉₇Zn₁Y₂–Al_x alloy under optical microscope. a x = 0; b x = 0.3; c x = 1; d x = 3

Figure 2 and Table 1 show the SEM images and EDS results of the Mg₉₇Zn₁Y₂–xwt%Al alloy. The 001 point was located on the dendrite structure where the Zn/Y ratio was close to 1:1, possibly corresponding to an LPSO phase. For the Al content of 1.0 wt%, the alloy components at the 002 point were similar to those at the 001 one, while the 003 point stood out, according to the Al content, the phase at this point was probably an Al-containing phase. As shown in Fig. 2b, this phase exhibited a network-like distribution, which normally hinders grain growth during crystallization. According to the Mg–Al phase diagram [9], during the crystallization, the eutectic reaction L → α-Mg + β-Mg₁₇Al₁₂ occurs at the eutectic temperature (437 °C); hence, the above-mentioned Al-containing phase at the 003 point could be Mg₁₇Al₁₂. The diffraction peaks were observed in the XRD analysis (Fig. 3) suggested that the Mg₉₇Zn₁Y₂–xwt%Al alloy consists of α-Mg, Mg₁₂ZnY phase (LPSO), and Mg₁₇Al₁₂ phase. Thus, the dendrite structures observed at the 001 and 002 points belonged both to the Mg₁₂ZnY phase, while the network structure at the 003 point probably corresponded to the Mg₁₇Al₁₂ one.

Fig. 2

Microstructure of as-cast Mg₉₇Zn₁Y₂–Al_x under scanning electron microscope (SEM). a x = 0; b x = 1

Table 1 The energy spectrum analysis of alloy components

Fig. 3

X-ray diffraction (XRD) analysis of Mg₉₇Zn₁Y₂–Al_x alloy

3.2 Damping Analysis of Anelasticity

Magnesium presents the highest damping properties among light metals, which is typical dislocation damping mechanism material [10,11,12,13]. Dislocation damping is mainly caused by the dislocation movement among defects under stress and its theoretical model has been proposed by Granato and Lücke [14]. According to this G–L model, the damping of materials can be divided into two stages, observed at two different strain amplitude ranges: (1) at a low strain amplitude range, the dislocation lines move around weak pinners and the damping, denoted by $Q_{0}^{ - 1}$ at this stage, is independent or weakly dependent of the strain amplitude; (2) at a strain amplitude greater than the critical strain amplitude, the dislocations slip between strong pinners and the damping value, now denoted by $Q_{h}^{ - 1}$, increases with the strain amplitude. Therefore, the damping value can be expressed as follows [15, 16]:

$$Q^{ - 1} = Q_{0}^{ - 1} + Q_{h}^{ - 1}$$

(1)

$Q_{0}^{ - 1}$ and $Q_{h}^{ - 1}$ are obtained by

$$Q_{0}^{ - 1} = \frac{{\rho BL_{c}^{4} \omega }}{{36Gb^{2} }}$$

(2)

and

$$Q_{h}^{ - 1} = \frac{{C_{1} }}{\varepsilon }{ \exp }\left( { - \frac{{C_{2} }}{\varepsilon }} \right)$$

(3)

where $C_{1} = \frac{{\rho F_{B} L_{N}^{3} }}{{6bEL_{C}^{2} }},C_{2} = \frac{{F_{B} }}{{bEL_{C} }}$, ρ is the movable dislocation density, b is the Berkeley vector of a dislocation, ω is the corner frequency, B is a constant, it is a movable dislocation density, L_C and L_N are the distances between adjacent weak and strong pinners, respectively, F_B is the point force between defects and dislocations, E is the elastic modulus, and G is the shear modulus. Equation (3) can be also expressed as

$$\ln (Q_{h}^{ - 1} \cdot \varepsilon ) = \ln C_{1} - C_{2} /\varepsilon$$

(4)

The $\ln (Q_{h}^{ - 1} \cdot \varepsilon ) - 1/\varepsilon$ curve shown in Fig. 4a was plotted according to Eq. (4); it is called G–L plot and is often used as an applicability criterion for the G–L model.

Fig. 4

The G–L curve of Mg₉₇Zn₁Y₂–Al_x alloy. a The range of the whole strain amplitude; b ε_cr1 < ε < ε_cr2

Figure 4 shows the G–L curve of the Mg₉₇Zn₁Y₂–xwt%Al alloy. In the first stage (ε < ε_cr1), the damping value changed slightly; according to the G–L model, since the dislocation line was pinned by the weak pinners and could only perform reciprocating motion between them, it was less affected by the strain amplitude. In the second strain range (ε_cr1 < ε < ε_cr2, Fig. 4b), the G–L curve was clearly a straight line and the damping value at this stage could be explained by the dislocation damping mechanism and yield to the G–L dislocation model. In this range, with the increase of the strain amplitude, an increasing number of dislocation lines that break away from the weak pinners and bowed out between with the strong pinners. This suggests that the damping capacities strongly depend on the strain amplitude. When the strain amplitude exceeded the second critical strain amplitude (ε_cr2), the G–L curve deviated from the straight line and the dislocation lines started break away from the strong pinners; this is considered the beginning of the microplastic deformation of the material.

According to Eq. (4), the line slope is –C₂, the intersection of the y-axis is lnC₁, the specific numerical results of C₁ and C₂ were calculated, as listed in Table 2. A smaller L_C value implies a larger number of weak pinners in the dislocation and the C₂ value is proportional to the latter; therefore, C₂ is inversely proportional to L_C. The C₂ value increases with the Al content, meaning that also the number of weak pinners increases with it. Aluminum has a maximum solubility of 12.5% in Mg [17]; with the Al addition, more solute atoms are solid-solutioned into the Mg matrix, resulting in more weak pinners on the dislocation line, which consecutively requires a greater strain to be removed from them. This conclusion can be supported by Table 2 that shows the increase of ε_cr1 with the Al addition.

Table 2 C₁, C₂ value and critical strain amplitude of Mg₉₇Zn₁Y₂–Al_x alloy

According to Table 2, C₁ is proportional to the Al content and $\rho L_{N}^{3} /L_{C}^{2}$. Based on the alloy microstructure shown in Fig. 1, as the Al content increases, more secondary phase (Mg₁₇Al₁₂), which corresponds to the strong pinners on the dislocation line, is formed in the matrix. Therefore, as the Al content increases, the number of strong pinners increases, while the L_N value decreases. Figure 1 also shows that an increase in the Al content further refined the microstructure and generated more interfaces. These factors improve the mechanical properties of the material [4], hence, its microplastic deformation requires a greater strain amplitude, as supported by the results in Table 2. Trojanov et al. [18] reported that, due to the different thermal expansion coefficients between matrix and particles, the dislocation density at their interface is much larger than that in the matrix. Therefore, the formation of the secondary phase also increases the dislocation density of the material. Comprehensive consideration of the above analysis, the increase in C₁ value is in line with previous theories.

3.3 Damping in Microplastic Deformation

When the strain amplitude was greater than ε_cr2, the G–L curve clearly deviated from the straight line. Since this aspect cannot be explained by the G–L theory, a more appropriate theory for the microplastic deformation is required. Dislocation lines sweep a certain area and consume energy; when they get far from all the weak pinning points, the elastic energy consumed reaches the limit and the high-strain zone damping gets saturated [19, 20]. However, as the strain continues to increase, the material damping still rises. Mason [21] attributed this damping mechanism to the plastic deformation. At this stage, a larger strain causes significant microplastic deformations the material (generally referred to the plastic deformation before the macroscopic yield point of 0.2% [22,23,24]) and the dislocation line begins to get far from the strong pinners, producing damping associated with the microplastic deformation, which is denoted by $Q_{p}^{ - 1}$.

Penguin [25] proposed a model to explain the damping phenomenon associated with the microplastic deformation and the factors influencing the damping value at this stage, represented as follows:

$$Q_{p}^{ - 1} = \frac{A}{\varepsilon h}\exp (B\varepsilon - B\varepsilon_{p} ),$$

(5)

$$A = \frac{2\rho bv}{\pi f}\exp \left( { - \frac{Q}{KT}} \right),$$

(6)

and

$$B = \alpha \frac{VG}{KT},$$

(7)

where h is a constant, v is the eigenfrequency of the dislocation, ρ is the movable dislocation density, f is the test frequency, Q is the activation energy, α is the orientation factor, G is the shear modulus, ε_p is the strain amplitude at the beginning of the plastic damping, and V is the activation volume.

From Eq. (5), we obtain

$$\ln Q_{p}^{ - 1} \varepsilon = B\varepsilon + C$$

(8)

where ${\text{C}} = \ln A/h - B\varepsilon_{p}$

The microplastic damping curves were fitted according to Eq. (8) and that of the Mg₉₇Zn₁Y₂–xwt%Al alloy (Fig. 5) exhibited a clear turning point (ε_cr3). When the strain amplitude is greater than ε_cr2, the dislocation line begins to deviate from the equilibrium position. At this stage, the material undergoes microplastic deformation, with an associated microplastic damping ($Q_{p}^{ - 1}$).

Fig. 5

The microplastic damping curve of Mg₉₇Zn₁Y₂–Al_x alloy

On the other hand, with the strain amplitude increasingly, more movable dislocation sources inside the material are activated and, hence, $Q_{p}^{ - 1}$ increases. According to Eq. (7), the slope of the curve is proportional to the activation volume of the material dislocation slip. As shown in Fig. 5, when the strain amplitude exceeded ε_cr3, the curve slope dropped significantly; in such cases, the material is usually deformed plastically by slippage, the dislocations on the same basal plant begin to entangle and accumulate, and the back stress generated by this dislocation accumulation makes the activation of dislocation sources more and more difficult [23]. The quenching and entanglement of the dislocations lead to fewer and fewer movable dislocations within the grains; therefore, above ε_cr3, the material hardens and the volume activated by the dislocation slipping decreases.

Table 3 shows the slopes and critical strain amplitudes of the damping curves for the microplastic deformation of the various alloys tested; the one with 0.3 wt% of Al exhibited the maximum V and minimum ε_cr3, corresponding to the damping curve at the high-strain amplitude stage, where its damping value was higher than that of the other alloys. This happened because this alloy had fewer pinners than the others, so that the dislocations were more easily removed, hence, the activation volume of the dislocation slipping was larger. Furthermore, the Al addition refines the grain size, forming the secondary phase and significantly improving the mechanical properties of the alloy, which leads to materials more difficult to plastically deform; therefore, with increasing the Al addition, ε_cr3 increases, which is consistent with the results shown in Table 3. When the alloy is added with 3 wt% of Al at the high-strain amplitude, the damping is greater than that for the one added with 1 wt% of it. This may be due to the effect of the secondary phase (Mg₁₇Al₁₂) on the damping mechanism. Zhang et al. [26] reported that the secondary phase influences the damping behaviors of alloys via some possible mechanisms. One of them is the energy dissipation by the interfaces between primary and secondary phases under cyclic loading, which usually occurs at the weaker interface band. This mechanism of the Mg₉₇Zn₁Y₂–xwt%Al alloy could be considered, if the interface significantly causes energy dissipation, the alloy with more secondary phase will exhibits higher damping, which is consistent with the experimental results. Therefore, the increase in alloy damping with the Al content is mainly due to the increase in interface damping.

Table 3 The slope value and critical strain amplitude of microplastic damping curve of Mg₉₇Zn₁Y₂–Al_x alloy

4 Conclusions

• 1. In the anelastic stage of damping, the classical G–L model can be applied and the mechanism of dislocation damping is dominant. The numbers of weak and strong pinners and the moveable dislocation density in the Mg₉₇Zn₁Y₂–xwt%Al alloy increase with the Al content and the dislocation lines require greater strain to be moved away from the weak pinners.

• 2. In the microplastic stage of damping, the G–L model no longer applies, hence, a new dislocation damping model has been proposed. Based on this new model, the relative activation volume of the dislocation slipping for the microplastic-dependent damping can be obtained from the slope of the straight line.