2.2 Low cycle fatigue

cyclic torsional tests, however, the ASTM standard for strain-controlled axialtorsional fatigue testing with thin-walled tubular specimens [7] can be used. The stress-life (S � N) curve, which is also known as the Wöhler curve, is usually used to represent the relation between applied stress amplitude, σ<sup>a</sup> or τ<sup>a</sup> and fatigue life Nf in a log-log scale. A typical S � N curve shows a linear curve in the

Magnesium - The Wonder Element for Engineering/Biomedical Applications

Understanding that a linear curve in a log-log scale indicates that the relation between the stress amplitude and fatigue life is of a power-type, Basquin [8] was

<sup>f</sup> 2Nf

<sup>f</sup> is the axial fatigue strength coefficient and b is the axial fatigue

Some materials such as steel and magnesium alloys show a distinct plateau commonly called the "fatigue limit". This limit is usually observed between 10<sup>6</sup> and 10<sup>7</sup> cycles. Tests that exceed 107 cycles are usually stopped and are called run-out. This limit used to be called the "endurance limit" that is a stress amplitude below which no failure will occur. However, it has been shown if the testing is continued specimens eventually fail. Therefore, it is recently accepted that endurance limit

<sup>b</sup> (1)

σ<sup>a</sup> ¼ σ<sup>0</sup>

finite life region as shown in Figure 3.

the first to model this curve as

where σ<sup>0</sup>

Figure 3.

Figure 4.

48

Stress life curve for different Mg-alloys [9–13].

A typical stress life curve.

strength exponent.

Similar to the HCF, the low cycle fatigue (LCF) behavior of materials can be characterized for a mode of strain, i.e., normal or shear, by performing straincontrolled experiments. ASTM standard for conducting strain controlled fatigue tests of materials [14] can be followed in tests performed using cyclic axial machine. While controlling the strain, using strain measurement device such as an extensometer, the load signal is also acquired. This allows calculating both the stress and strain that can be represented for a given cycle by a hysteresis loop as shown in Figure 5.

Wrought alloys that are produced by extrusion, rolling or forging processes develop strong texture. This texture is characterized by alignment of the basal plane with the working direction with the c-axis perpendicular to it. The two plastic deformation mechanisms, slipping or twinning, can be activated depending on the loading orientation with respect to the basal plane. An extension along the c-axis activates the tension twins and a subsequent contraction causes detwinning of the lattice. During cyclic loading, twinning occurs in the compression reversal leading to low stress yielding. Detwinning starts as the load is reversed, i.e., during the tension reversal. However, as the tension load increases the hard pyramidal slip is activated in order to accommodate additional strain. This change from detwinning to slip is reflected on the cyclic hysteresis as an inflection point and a concave upward hardening behavior. A representative cyclic hysteresis loop for different magnesium extrusion loaded along the extrusion direction is shown in Figure 6.

The strain-life (ε N) curve is usually used to represent the relation between applied strain amplitude, ε and fatigue life Nf in a log-log scale. Strain life curves for AZ31B, AZ61A and ZK60 magnesium extrusions are presented in Figure 7.

Figure 5. A typical cyclic hysteresis loop [2].

While the positive elastic strain energy density is calculated as

plastic and positive elastic energies

Fatigue of Magnesium-Based Materials DOI: http://dx.doi.org/10.5772/intechopen.85226

Energy components in cyclic hysteresis loop [2].

Figure 8.

Figure 9.

51

Energy life curves of Mg [17–19, 24–27].

sidering the mean stress effect in energy method.

ΔW<sup>þ</sup>

ΔWt ¼ ΔW<sup>þ</sup>

<sup>e</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup> max

The total strain energy density is often calculated as the sum of torsional of the

The inclusion of the positive elastic strain energy density is a method for con-

<sup>2</sup><sup>E</sup> (4)

<sup>e</sup> þ ΔWp (5)

Figure 6. Cyclic hysteresis loops of AZ31B, AZ61A, AM30 and ZK60 magnesium extrusions [15–18].

Figure 7. Strain life curves for AZ31B, AZ61A and ZK60 magnesium extrusions [17–19].

This region is represented by the Coffin-Manson [20, 21] equation as

$$
\varepsilon\_d = \frac{\sigma\_f^{\prime}}{E} \left(2\text{N}\_f\right)^b + \varepsilon\_f^{\prime} \left(2\text{N}\_f\right)^c \tag{2}
$$

where σ<sup>0</sup> <sup>f</sup> and b are defined in Eq. (1) and ε<sup>0</sup> <sup>f</sup> and c are the axial fatigue ductility coefficient and the axial fatigue ductility exponent, respectively.

Ellyin, Golos and Xia [22] energy model is considered as the basis for the application of strain energy in correlating multiaxial fatigue damage. The model considers the plastic and the positive elastic strain energy densities as the damaging variables as shown in Figure 8.

The plastic strain energy density is the area enclosed by the cyclic hysteresis loop which is calculated as [23]:

$$
\Delta \mathcal{W}^p = \int\_{cycle} \sigma d\varepsilon \tag{3}
$$

Fatigue of Magnesium-Based Materials DOI: http://dx.doi.org/10.5772/intechopen.85226

While the positive elastic strain energy density is calculated as

$$
\Delta W\_{\varepsilon}^{+} = \frac{\sigma\_{\text{max}}^{2}}{2E} \tag{4}
$$

The total strain energy density is often calculated as the sum of torsional of the plastic and positive elastic energies

$$
\Delta W\_t = \Delta W\_e^+ + \Delta W\_p \tag{5}
$$

The inclusion of the positive elastic strain energy density is a method for considering the mean stress effect in energy method.

Figure 9. Energy life curves of Mg [17–19, 24–27].

This region is represented by the Coffin-Manson [20, 21] equation as

Cyclic hysteresis loops of AZ31B, AZ61A, AM30 and ZK60 magnesium extrusions [15–18].

Magnesium - The Wonder Element for Engineering/Biomedical Applications

Ellyin, Golos and Xia [22] energy model is considered as the basis for the application of strain energy in correlating multiaxial fatigue damage. The model considers the plastic and the positive elastic strain energy densities as the damaging

<sup>Δ</sup>W<sup>p</sup> <sup>¼</sup>

The plastic strain energy density is the area enclosed by the cyclic hysteresis loop

ð cycle <sup>f</sup> 2Nf

� �<sup>c</sup> (2)

<sup>f</sup> and c are the axial fatigue ductility

σdε (3)

<sup>ε</sup><sup>a</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup> f E 2Nf � �<sup>b</sup> <sup>þ</sup> <sup>ε</sup><sup>0</sup>

Strain life curves for AZ31B, AZ61A and ZK60 magnesium extrusions [17–19].

coefficient and the axial fatigue ductility exponent, respectively.

<sup>f</sup> and b are defined in Eq. (1) and ε<sup>0</sup>

where σ<sup>0</sup>

50

Figure 7.

Figure 6.

variables as shown in Figure 8.

which is calculated as [23]:

#### Magnesium - The Wonder Element for Engineering/Biomedical Applications


Table 1.

Strain and energy fatigue properties of different Mg extruded alloys [16, 18, 28].

The fatigue life is estimated using

$$
\Delta W\_t = a \text{N}\_f^\beta \tag{6}
$$

Another equation suggested by Smith, Watson and Topper [30], commonly

2Nf <sup>2</sup><sup>b</sup> <sup>þ</sup> <sup>σ</sup><sup>0</sup>

Environmental effects such as corrosion or temperature have significant effect on fatigue life. The interactions between these effects and fatigue behavior are complex, therefore, quantitative models for estimating fatigue life may not always be possible. Magnesium is highly reactive and considered as one of the most electrochemically active metals. From structural view point, the reactivity of magnesium, and its alloys, in electrolytic or aqueous environments, is disadvanta-

f ε0 fE 2Nf

<sup>c</sup>þ<sup>b</sup> (11)

known as "SWT parameter", is as follows:

Fatigue of Magnesium-Based Materials DOI: http://dx.doi.org/10.5772/intechopen.85226

2.4 Environmental effects

materials.

Figure 10.

53

fluid "SBF" [33].

σmaxεaE ¼ σ<sup>0</sup>

fluctuating reaching to 1 � <sup>10</sup><sup>6</sup> cycles in a year [31, 32].

by the condition of the corrosive environment.

f <sup>2</sup>

geous because it degrades their mechanical strength. In the contrary, this

high reactivity of magnesium and its alloys make them a potential biodegradable

The pH level of body fluid ranges from 1 to 9 in different tissues. It is also estimated that different parts of the body are subjected to different level of stresses. For example, bones are subjected to a stress of about 4 MPa during regular daily activities. On the other hand, tendons and ligaments experience peak stresses that range from 40 to 80 MPa. Hip joints usually subjected to 3 times the body weight which can increase to 10 times during jumping. These stresses are repetitive and

The poor corrosion resistance of many magnesium alloys is due to internal galvanic corrosion caused by second phases or impurities and the instability of the

Comparison between the stress life curves for biomedical AZ91D and WE43 tested in air and simulated body

hydroxide film [33]. The most common method to assess the influence of corrosion on fatigue behavior is to perform stress-controlled testing at different environmental conditions. Strain-controlled testing is not common because it requires electronic strain measurement device that may not be possible to mount on the sample while being exposed to corrosive substances. Comparisons between stress-life curves for different biomedical magnesium alloys tested at different simulated body fluids "SBF" are shown in Figures 10 and 11. These figures show that the fatigue life of AZ91D at a given stress amplitude is significantly influenced

Figure 9 shows the correlation between the total strain energy density and fatigue life for several magnesium alloys tested at different loading conditions. This figure clearly shows the ability of the energy method to collapse all data in a single and narrow scatter band.

Strain- and energy-based fatigue properties for AZ31B, AZ61A and AM30 magnesium extrusions are listed in Table 1.

#### 2.3 Mean stress effect

Mean stress has a significant effect on fatigue behavior of materials. It is generally observed that tensile mean stresses are detrimental and compressive mean stresses are beneficial. As explained earlier, the positive elastic strain energy density is used to account for the mean stress effect in energy methods. There are different models for stress-based methods that account for mean stress effect [29]:

The Modified Goodman:

$$\frac{\sigma\_a}{\sigma\_f} + \frac{\sigma\_m}{\sigma\_u} = \mathbf{1} \tag{7}$$

Gerber:

$$\frac{\sigma\_a}{\sigma\_f} + \left(\frac{\sigma\_m}{\sigma\_u}\right)^2 = 1 \tag{8}$$

Morrow:

$$\frac{\sigma\_a}{\sigma\_f} + \left(\frac{\sigma\_m}{\sigma\_f'}\right)^2 = 1\tag{9}$$

where σ<sup>f</sup> is the fatigue limit.

Morrow's can be used for strain-based method as:

$$\frac{\Delta \varepsilon}{2} = \varepsilon\_a = \frac{\sigma\_f^{\prime} - \sigma\_m}{E} \left( 2 \text{N}\_f \right)^b + \varepsilon\_f^{\prime} \left( 2 \text{N}\_f \right)^c \tag{10}$$

Another equation suggested by Smith, Watson and Topper [30], commonly known as "SWT parameter", is as follows:

$$
\sigma\_{\max} \varepsilon\_a E = \left(\sigma\_f'\right)^2 \left(2\mathcal{N}\_f\right)^{2b} + \sigma\_f' \varepsilon\_f' E \left(2\mathcal{N}\_f\right)^{c+b} \tag{11}
$$

#### 2.4 Environmental effects

The fatigue life is estimated using

nesium extrusions are listed in Table 1.

and narrow scatter band.

σ0

ε0

E0

E0

Table 1.

2.3 Mean stress effect

Gerber:

Morrow:

52

The Modified Goodman:

where σ<sup>f</sup> is the fatigue limit.

<sup>Δ</sup>Wt <sup>¼</sup> <sup>α</sup>N<sup>β</sup>

Property AZ31B extrusion AZ61A extrusion AM30 extrusion

<sup>f</sup> 723.5 586.1 410.4 b �0.159 �0.153 �0.130

Magnesium - The Wonder Element for Engineering/Biomedical Applications

<sup>f</sup> 0.252 1.823 1.480 c �0.718 �0.832 �0.791

<sup>e</sup> 20.29 7.01 2.995 B �0.440 �0.373 �0.281

<sup>f</sup> 510.74 924.14 1710.69 C �1.052 �1.001 �0.975

Strain and energy fatigue properties of different Mg extruded alloys [16, 18, 28].

Figure 9 shows the correlation between the total strain energy density and fatigue life for several magnesium alloys tested at different loading conditions. This figure clearly shows the ability of the energy method to collapse all data in a single

Strain- and energy-based fatigue properties for AZ31B, AZ61A and AM30 mag-

Mean stress has a significant effect on fatigue behavior of materials. It is generally observed that tensile mean stresses are detrimental and compressive mean stresses are beneficial. As explained earlier, the positive elastic strain energy density is used to account for the mean stress effect in energy methods. There are different

models for stress-based methods that account for mean stress effect [29]:

σa σf

σa σf

σa σf

Morrow's can be used for strain-based method as:

<sup>2</sup> <sup>¼</sup> <sup>ε</sup><sup>a</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup>

Δε

<sup>þ</sup> <sup>σ</sup><sup>m</sup> σu

<sup>þ</sup> <sup>σ</sup><sup>m</sup> σu � �<sup>2</sup>

<sup>þ</sup> <sup>σ</sup><sup>m</sup> σ0 f

<sup>f</sup> � σ<sup>m</sup> <sup>E</sup> <sup>2</sup>Nf

!<sup>2</sup>

� �<sup>b</sup> <sup>þ</sup> <sup>ε</sup><sup>0</sup>

<sup>f</sup> 2Nf

<sup>f</sup> (6)

¼ 1 (7)

¼ 1 (8)

¼ 1 (9)

� �<sup>c</sup> (10)

Environmental effects such as corrosion or temperature have significant effect on fatigue life. The interactions between these effects and fatigue behavior are complex, therefore, quantitative models for estimating fatigue life may not always be possible. Magnesium is highly reactive and considered as one of the most electrochemically active metals. From structural view point, the reactivity of magnesium, and its alloys, in electrolytic or aqueous environments, is disadvantageous because it degrades their mechanical strength. In the contrary, this high reactivity of magnesium and its alloys make them a potential biodegradable materials.

The pH level of body fluid ranges from 1 to 9 in different tissues. It is also estimated that different parts of the body are subjected to different level of stresses. For example, bones are subjected to a stress of about 4 MPa during regular daily activities. On the other hand, tendons and ligaments experience peak stresses that range from 40 to 80 MPa. Hip joints usually subjected to 3 times the body weight which can increase to 10 times during jumping. These stresses are repetitive and fluctuating reaching to 1 � <sup>10</sup><sup>6</sup> cycles in a year [31, 32].

The poor corrosion resistance of many magnesium alloys is due to internal galvanic corrosion caused by second phases or impurities and the instability of the hydroxide film [33]. The most common method to assess the influence of corrosion on fatigue behavior is to perform stress-controlled testing at different environmental conditions. Strain-controlled testing is not common because it requires electronic strain measurement device that may not be possible to mount on the sample while being exposed to corrosive substances. Comparisons between stress-life curves for different biomedical magnesium alloys tested at different simulated body fluids "SBF" are shown in Figures 10 and 11. These figures show that the fatigue life of AZ91D at a given stress amplitude is significantly influenced by the condition of the corrosive environment.

Figure 10.

Comparison between the stress life curves for biomedical AZ91D and WE43 tested in air and simulated body fluid "SBF" [33].

such as AZ31, AZ91, ZK60 and AZ61 to improved their ultimate tensile strength, yield strength and modulus of elasticity composite [36]. Different reinforcements such as aluminum oxide particulate Al2O3 [37], nickel particles [38] or nano-size Y2O3 particles [39] have been tested with magnesium and its alloys and found to

Fatigue behavior of magnesium AZ91 can be improved by adding ceramic rein-

forcements [40]. In addition, the elastic modulus, yield strength and ultimate tensile strength of AZ91 can be increased using SiCp particles as discontinuous reinforcements. Figures 12 and 13 show the effects of two different reinforcements

The effect of different volume fractions of Saffil alumina fibers on the stress-life curve of AZ91D magnesium

Comparison of stress-strain curves of the dual particle reinforced magnesium alloy (AZ31 + Al2O3 + Ni) with the unreinforced matrix alloy (AZ31) cyclically deformed at room temperature (T = 25°C) at R = 0.1

yield different results.

Fatigue of Magnesium-Based Materials DOI: http://dx.doi.org/10.5772/intechopen.85226

Figure 13.

Figure 14.

55

and � 1.0 [38].

metal matrix composite [40].

Figure 11.

Comparison between the stress life curves for biodegradable AZ91D tested in air and different simulated body fluids "SBF" [11].
