**2. Experimental results and analysis**

Jin and Noda [41] verified this situation by determining that the angular distribution function

Also, the studies conducted by Eischen [39] and Jin and Noda [41] have shown that FGMs

In Equation (2), *Kα* (*α* = I, II) represents stress intensity factors, *r* the distance from crack tip, *θ*

modulus at crack tip, and *υ*tip the Poisson ratio at crack tip. These results mean that the stress intensity factor is a determinative fracture parameter for FGMs just like in homogeneous material [67]. Similar results have been found for dynamically propagated cracks by Para‐

( ) ( )

2 2 8 *i i K r u g <sup>E</sup>*

u

tip

a ( ) tip

p

Continuous or stepped grading prevents the abnormal stress behavior of cracks when the interface of two materials is combined [69]. Moreover, Delale and Erdogan [35] found that the effect of spatial variation of the Poisson ratio on the stress singularity can be neglected.

Fracture behavior depending on orientation in the grading region can be defined by consid‐

**1.** Crack propagation is parallel to grading direction: crack does not digress although

**2.** Crack propagation is perpendicular to grading direction: asymmetric crack tip zone

In the cracks that are parallel to the direction, stress at the crack tip region becomes symmetric and it is expected that crack opens toward the grading direction. FGM crack tip stresses in graded regions are significantly lower than in materials with combinations of two material properties [37, 70, 71]. Stress intensity factor of FGMs is found to be greater than that of homogeneous materials. Material grading profile and position of the crack effect stress intensity factor too. When grading step increases, its value increases [35, 37, 42, 72]. Grading format also effects the stress intensity factor. If exponential value *n* is bigger than 1 (*n* > 1), the stress intensity factor is in tendency to decrease compared to *n* < 1 situation [70–72]. Spatial composition changes, which have an important effect on effective fracture toughness of the FGM composites, can effect thermal stress distribution throughout its width. Growing fracture toughness depends on residual stresses partially [53, 64, 73]. As residual stresses change by

a q

<sup>+</sup> » (2)

( ) ( ) 1 1 <sup>2</sup> ( ) <sup>2</sup> *ij ij i jT ij <sup>K</sup> <sup>f</sup> A rw*

(*θ*)) is the same.

a

(*α*)

 p  a

= ++ (1)

 q

) (Eq. (2)) are the same form as homogeneous material.

(*θ*) angular functions, *E*tip the Young's

of the elastic and plastic crack tip area ( *f ij*

78 Advances in Functionally Graded Materials and Structures

s

crack tip stress (*σij*

meswaran and Shukla [68].

ering two limit states [63]:

causes crack to deflect.

effective fracture toughness changes.

( )

q dds

a

*r* a

p

) and displacement (*ui*

the angle with respect to the plane of the crack, *gi*

Two functionally graded aluminum 2014 alloy (1.18% Si, 4.9% Cu, 1.04% Mn, in wt.) matrix materials reinforced with 20% in wt. and 9 μm SiC particles were produced by centrifugal casting. Density of SiC particles is higher than that of aluminum, 3.2 gr/cm3 and 2.8 gr/cm3 , respectively. As a result of the centrifugal force effect, the distribution of SiC particles was varied through the wall thickness of the cylinder. More SiC particles were dispersed to the outer diameter of the FGM cylinder under these circumstances, as expected. By changing casting wall thicknesses, two cylinders having different mechanical properties were produced. The specimens were named FGM1 and FGM2, and SiC-rich and aluminum-rich regions were formed on both of FGM1 and FGM2. FGM1's wall thickness was higher than FGM2's. This result is compatible with the previous studies on the subject [74–83]. Taking into account the wear, fatigue, and fretting behaviors of FGM from previous studies, an aging process was performed (solutionizing at 500 °C, followed by water quenching and reheating for aging at 145 °C for 10 h, followed by water cooling) [75, 82, 83].

Tensile tests were performed using specimens sliced through the wall thickness and were numbered from 1 to 5 for FGM1 and FGM2 according to their position in the cylinder (from the innermost to the outermost layer). Tensile strength was tested on each section to determine the mechanical properties that can be varied such as Young's modulus (*E*), tensile strength (R), and yielding stress (*Re*) (Figures 1–3). Tensile experiments were conducted using video extensometer at 1 mm/min tensile speed. Obtaining tensile specimens from the cylindrical FGM is shown in Figure 4.

**Figure 1.** Young's modulus variations of FGM1 and FGM2 [84].

**Figure 2.** Yielding stress variations of FGM1 and FGM2.

**Figure 3.** Tensile strength variations of FGM1 and FGM2.

An Experimental Crack Propagation Analysis of Aluminum Matrix Functionally Graded Material http://dx.doi.org/10.5772/62428 81

**Figure 4.** Obtaining tensile test samples from cylindrical FGM.

**Figure 1.** Young's modulus variations of FGM1 and FGM2 [84].

80 Advances in Functionally Graded Materials and Structures

**Figure 2.** Yielding stress variations of FGM1 and FGM2.

**Figure 3.** Tensile strength variations of FGM1 and FGM2.

Due to the higher density of SiC particles (relative to aluminum 2014), many more particles were dispersed to the outer diameter of FGM cylinder during centrifugal casting. This produced a gradient in the Young's modulus, from the inside to the outside layer of cylinder (Figure 1) [84]. It was observed that SiC distribution under centrifugal force and wall thickness of cylinders effected Young's modulus variation. As wall thickness of cylinder decreased, it was observed that Young's modulus value (innermost of cylinder) increased from 65 MPa to 84 MPa. Grading functions of FGMs were differed from each other due to the distribution of SiC. Young's modulus values of the outermost region of cylinders were similar (105–106 MPa). It was observed that wall thickness of FGM and manufacturing process had an impact on the composition and mechanical properties of FGM.

As mentioned above, FGMs' properties can vary exponentially, linearly, and according to the rule of force. In this study, Young's modulus variation was calculated according to the rule of force to compare with experimental results using Equations (4) and (5). In these equations, *E*(*x*) represents Young's modulus at *x* point, *E*1 base material Young's modulus, *E*2 reinforcing material Young's modulus, *t* the width/thickness of FGM, *x* the distance from the starting point, *g*(*x*) the distance function, and *p* the gradient exponent. The gradient exponent (*p*) was calculated using equations as 0.1 and 7 for FGM1 and FGM2, respectively.

$$\log\left(x\right) = \left(\frac{x}{t}\right)^p \tag{4}$$

$$E(\mathbf{x}) = E\_1 \left(1 - g\left(\mathbf{x}\right)\right) + E\_2 g\left(\mathbf{x}\right) \tag{5}$$

Tensile and fatigue crack growth tests were performed with a 5-ton capacity Instron fatigue servo-hydraulic test device. Two cylindrical specimens were cut through their wall thickness in four sections via vertical slicing as shown in Figure 4. Tensile tests were carried out to determine the crack opening properties using video extensometer at 1 mm/min tensile speed.

To investigate the effect of variation in mechanical properties and distribution of SiC particles on fatigue crack behavior, fatigue crack growth tests were applied under tensile cyclic load with stress ratio *R* = 0.1. The samples were prepared according to the ASTM E647 (2011) [85] in three separate groups: central notched (middle tension, M(T)), single-edge notched (SE(T)) on SiC-rich side, and single-edge notched (SE(T)) on aluminum-rich side (Figure 5). The test specimens' dimensions are shown in Figure 6. M(T) and (SE(T)) samples are prepared according to the ASTM E647 (2011) [85]. Samples were processed by laser cutting method. According to the ASTM E647 (2011) [85] standard, crack length to be opened must be at least 0.2*W;* sample length *L* > 0.3. *W* represents width belonging to the sample. Notch length opened on M(T) specimens 2*a* = 13 mm, notch length opened on SE(T) *a* = 5.5 mm [86]. Both tensile and fatigue experiments were applied to these samples. Two digital portable microscopes, as shown in Figure 7, were used to determine the crack beginning from notch tip and propagation of that crack. Applied maximum load determined with respect of minimum yielding stress after tensile tests were applied to central and single-edge notched specimens (Eq. (6)).

**Figure 5.** Obtaining fatigue crack growth test samples from cylindrical FGM [84].

An Experimental Crack Propagation Analysis of Aluminum Matrix Functionally Graded Material http://dx.doi.org/10.5772/62428 83

*E x E g x Eg x* ( ) =- + 1 2 (1 ( )) ( ) (5)

Tensile and fatigue crack growth tests were performed with a 5-ton capacity Instron fatigue servo-hydraulic test device. Two cylindrical specimens were cut through their wall thickness in four sections via vertical slicing as shown in Figure 4. Tensile tests were carried out to determine the crack opening properties using video extensometer at 1 mm/min tensile speed.

82 Advances in Functionally Graded Materials and Structures

To investigate the effect of variation in mechanical properties and distribution of SiC particles on fatigue crack behavior, fatigue crack growth tests were applied under tensile cyclic load with stress ratio *R* = 0.1. The samples were prepared according to the ASTM E647 (2011) [85] in three separate groups: central notched (middle tension, M(T)), single-edge notched (SE(T)) on SiC-rich side, and single-edge notched (SE(T)) on aluminum-rich side (Figure 5). The test specimens' dimensions are shown in Figure 6. M(T) and (SE(T)) samples are prepared according to the ASTM E647 (2011) [85]. Samples were processed by laser cutting method. According to the ASTM E647 (2011) [85] standard, crack length to be opened must be at least 0.2*W;* sample length *L* > 0.3. *W* represents width belonging to the sample. Notch length opened on M(T) specimens 2*a* = 13 mm, notch length opened on SE(T) *a* = 5.5 mm [86]. Both tensile and fatigue experiments were applied to these samples. Two digital portable microscopes, as shown in Figure 7, were used to determine the crack beginning from notch tip and propagation of that crack. Applied maximum load determined with respect of minimum yielding stress after tensile tests were applied to central and single-edge notched specimens (Eq. (6)).

**Figure 5.** Obtaining fatigue crack growth test samples from cylindrical FGM [84].

**Figure 6.** Dimension of fatigue crack growth test samples: (a) middle tension M(T)), (b) single-edge notched (SE(T)) [84].

$$
\sigma\_{\text{max}} = 0.3 Re\_{\text{min}} \tag{6}
$$

Stress ratio (*R*) is important for calculations of load increment (∆*P*) and stress concentration factor increment (∆*K*) for homogeneous material according to the ASTM E647 (2011) (Eq. (7– 9)) [85].

$$R > 0 \Longrightarrow \Delta P = P\_{\text{max}} - P\_{\text{min}} \tag{7}$$

$$\begin{aligned} R \le 0 &\Rightarrow \Delta P = P\_{\text{max}}\\ 2a\sqrt{\frac{2a}{W}} < 0.95 &\Rightarrow \Delta K = \frac{\Delta P}{B} \sqrt{\frac{\pi a}{2W} \sec \frac{\pi a}{2}} \end{aligned} \tag{8}$$

$$R = \frac{P\_{\text{min}}}{P\_{\text{max}}} = \frac{\sigma\_{\text{min}}}{\sigma\_{\text{max}}} = \frac{K\_{\text{min}}}{K\_{\text{max}}} \tag{9}$$

For 2*a*/*W* < 0.95 situation, M(T) specimens were accepted homogeneous and Equations (5) and (6) were used in calculations. *a* represents crack half-length. If 2*a*/*W* is greater than or equal to 0.95, it is accepted that crack will be instable and then probably fracture will be occur. The

**Figure 7.** The digital microscopes used in experiments

stress ratio *R* = 0.1 and frequency *f* = 5 Hz were selected as fatigue crack growth experiment parameters. SE(T) specimens' notch was parallel to the grading direction.

Load-crack tip opening diagrams of M(T) specimens obtained from FGM1 after tensile tests are shown in Figure 8. Here, 1–4 refer to specimen numbers from innermost (aluminum-rich side) to outermost (SiC-rich side) regions of cylinder. It can be understood that Al-rich side fractured at a lower load value. Since SiC rate increased, load carrying capacity of specimens increased. SiC-rich side fractured at a higher load value than the others.

**Figure 8.** Load-crack tip opening diagrams of M(T) specimens obtained from FGM1.

Load-crack tip opening diagrams of SE(T) specimens obtained from FGM2 after tensile tests can be seen in Figure 9. The specimen (SE(T)) on SiC-rich side was fractured at a higher load than the (SE(T)) on aluminum-rich side specimen. Curves are almost continued in the same way until opening value is 0.5 mm. However, it is seen that (SE(T)) on aluminum-rich side specimen fractured when load value is up to 12 kN. On the other hand, (SE(T)) on SiC-rich side specimen continues to open till an 1.6 mm opening value at 20 kN load.

**Figure 9.** Load-crack tip opening diagrams of SE(T) specimens obtained from FGM2 [84].

stress ratio *R* = 0.1 and frequency *f* = 5 Hz were selected as fatigue crack growth experiment

Load-crack tip opening diagrams of M(T) specimens obtained from FGM1 after tensile tests are shown in Figure 8. Here, 1–4 refer to specimen numbers from innermost (aluminum-rich side) to outermost (SiC-rich side) regions of cylinder. It can be understood that Al-rich side fractured at a lower load value. Since SiC rate increased, load carrying capacity of specimens

Load-crack tip opening diagrams of SE(T) specimens obtained from FGM2 after tensile tests can be seen in Figure 9. The specimen (SE(T)) on SiC-rich side was fractured at a higher load

parameters. SE(T) specimens' notch was parallel to the grading direction.

**Figure 7.** The digital microscopes used in experiments

84 Advances in Functionally Graded Materials and Structures

increased. SiC-rich side fractured at a higher load value than the others.

**Figure 8.** Load-crack tip opening diagrams of M(T) specimens obtained from FGM1.

Number of cyclic load-crack propagation diagram of M(T) samples obtained from FGM2 with stress ratio *R* = 0.1 condition is shown in Figure 10. Here, 1–4 refer to specimen numbers from innermost (aluminum-rich side) to outermost (SiC-rich side) regions of cylinder, and speci‐ mens fractured at 12,880, 27,000, 34,000, and 48,000 number of cyclic loads, respectively. It can be seen that the fatigue life increases to almost 350% because of increase in SiC rate.

**Figure 10.** Number of cyclic load-crack propagation diagram of M(T) samples obtained from FGM2: 1–4 refers to speci‐ men numbers from innermost (aluminum-rich side) to outermost (SiC-rich side) [84].

Number of cyclic load-crack propagation diagram of SE(T) samples obtained from FGM1 (both of SiC-rich side and aluminum-rich side) with stress ratio *R* = 0.1 is shown in Figure 11. It was determined that the crack growing rate increased after 14,400 cycles for the aluminum-rich side SE(T) sample. However, after 28,800 cycles, the crack growing rate increased dramatically and after 36,850 cycles, sample fractured in a short time. On the other hand, SiC-rich side SE(T) sample's crack growth rate was very slow up to 13,000 cycles under the same cyclic load and after 13,000 cycles, the growth rate increased gradually. Crack growth rate increased dramat‐ ically after 190,000 cycles and sample fractured in a short time after 214,000 cycles. Under the same fatigue load, it was determined that the SiC-rich side SE(T) sample had fatigue life more than 500% compared to the aluminum-rich side SE(T) sample.

**Figure 11.** Number of cyclic load-crack propagation diagram of SE(T) samples obtained from FGM1 [84].

Number of cyclic load-crack propagation diagram of SE(T) samples obtained from FGM1 and FGM2 aluminum-rich side with stress ratio *R* = 0.1 condition is shown in Figure 12. A sample belonging to FGM2 displays a quick fracture behavior compared to FGM1. Whereas FGM2 sample fractured at 36,850cycles, FGM1 sample fractured at 238,000 cycles. FGM1's fatigue life was increased 1.5-fold compared to FGM2's fatigue life, which is explained by Young's modulus variation seen in Figure 1. Graphics show differences in variation of Young's modulus. SiC distribution in FGM2 increases from the innermost region to the outermost region till 0.2 times wall thickness, and then the rise slows down. In FGM1's inner region, distributions of SiC and in parallel with Young's modulus are higher than in the FGM2's. The Young's modulus determined as *E* = 85 GPa innermost of cylinder wall thickness does not visually increase by 0.75 times wall thickness.

After the fatigue crack growth experiments finished, *c* and *m*, which are material-dependent coefficients of the Paris-Erdogan equation (Eq. (3)), were found for each FGM2 M(T) specimens. The crack propagation behavior of the samples, which were from the inner diameter to the outside diameter of the cylinder numbered from 1 to 4, was observed differently in each of the samples. Therefore, the *c* and *m* coefficients were observed different from each other. The FGM's *c* and *m* coefficients are shown in Table 1.

Number of cyclic load-crack propagation diagram of SE(T) samples obtained from FGM1 (both of SiC-rich side and aluminum-rich side) with stress ratio *R* = 0.1 is shown in Figure 11. It was determined that the crack growing rate increased after 14,400 cycles for the aluminum-rich side SE(T) sample. However, after 28,800 cycles, the crack growing rate increased dramatically and after 36,850 cycles, sample fractured in a short time. On the other hand, SiC-rich side SE(T) sample's crack growth rate was very slow up to 13,000 cycles under the same cyclic load and after 13,000 cycles, the growth rate increased gradually. Crack growth rate increased dramat‐ ically after 190,000 cycles and sample fractured in a short time after 214,000 cycles. Under the same fatigue load, it was determined that the SiC-rich side SE(T) sample had fatigue life more

than 500% compared to the aluminum-rich side SE(T) sample.

86 Advances in Functionally Graded Materials and Structures

visually increase by 0.75 times wall thickness.

FGM's *c* and *m* coefficients are shown in Table 1.

**Figure 11.** Number of cyclic load-crack propagation diagram of SE(T) samples obtained from FGM1 [84].

Number of cyclic load-crack propagation diagram of SE(T) samples obtained from FGM1 and FGM2 aluminum-rich side with stress ratio *R* = 0.1 condition is shown in Figure 12. A sample belonging to FGM2 displays a quick fracture behavior compared to FGM1. Whereas FGM2 sample fractured at 36,850cycles, FGM1 sample fractured at 238,000 cycles. FGM1's fatigue life was increased 1.5-fold compared to FGM2's fatigue life, which is explained by Young's modulus variation seen in Figure 1. Graphics show differences in variation of Young's modulus. SiC distribution in FGM2 increases from the innermost region to the outermost region till 0.2 times wall thickness, and then the rise slows down. In FGM1's inner region, distributions of SiC and in parallel with Young's modulus are higher than in the FGM2's. The Young's modulus determined as *E* = 85 GPa innermost of cylinder wall thickness does not

After the fatigue crack growth experiments finished, *c* and *m*, which are material-dependent coefficients of the Paris-Erdogan equation (Eq. (3)), were found for each FGM2 M(T) specimens. The crack propagation behavior of the samples, which were from the inner diameter to the outside diameter of the cylinder numbered from 1 to 4, was observed differently in each of the samples. Therefore, the *c* and *m* coefficients were observed different from each other. The

**Figure 12.** Number of cyclic load-crack propagation diagram of SE(T) samples obtained from FGM1 and FGM2 alumi‐ num-rich side [84].

FRANC2D finite elements program have been developed by the fracture group of Cornell University [87]. The program is usually used for fracture mechanics and fatigue analysis. In this study, FRANC2D analysis was done for FGM2 M(T) specimens. The finite element model can be seen in Figure 13. It was accepted that the M(T) specimen was homogeneous in itself while modeling in FRANC2D. According to the analysis, obtained results from experimental and finite element modeling were similar to each other as seen in Figure 14.


**Table 1.** Calculated *c* and *m* coefficients of FGM2 M(T) specimens: 1–4 refer to specimen numbers from innermost (aluminum-rich side) to outermost (SiC-rich side) regions of cylinder

**Figure 13.** M(T) specimen modeling using FRANC2D program: (a) model, (b) maximum shear stress at crack tip, (c) deformation shape.

**Figure 14.** FRANC2D and experimental analysis results for M(T) specimens obtained from FGM2.
