Effect of cutout on modal properties of edge cracked temperaturedependent functionally graded plates
A. Shahrjerdi^{1} , T. Ezzati^{2}
^{1}Department of Mechanical Engineering, Malayer University, Malayer Branch, Malayer, Iran
^{2}Department of Mechanical Engineering, Payam Noor University, Bushehr Branch, Bushehr, Iran
^{2}Corresponding author
Journal of Vibroengineering, Vol. 17, Issue 4, 2015, p. 20132024.
Received 6 May 2014; received in revised form 4 August 2014; accepted 30 September 2014; published 30 June 2015
JVE Conferences
Modal analysis is employed to analyze the vibration of temperaturedependent of Functionally Graded Plates (FGP) under a thermal environment in order to determine the natural frequencies and mode shapes. Theoretical formulation of various materials’ properties is done using the rule of mixtures. The natural frequencies and mode shapes of simply supported and clamped square plates are investigated as a function of crack, cutout, crack and cutout and temperature dependent properties. The Ansys program is employed for the purpose of analyzing the natural frequency and mode shape of a plate. Nondimensional results are compared for temperaturedependent and temperatureindependent FGP and subsequently validated according to known results obtained from the literature. Numerical results indicate the effect of crack, cutout, gradient index and temperature fields on the vibration characteristics and mode shapes. This study proves that natural frequency decreases with increasing gradient index ($n$) increasing the temperature and simultaneous presence of crackcutout. In addition, clamped plates have a higher frequency than simply supported plates in all cases. Increasing temperatures lead to a maximum decrease in frequency at clamped FGP.
Keywords: FGM, cutout, cracks, temperature dependent properties, natural frequency.
1. Introduction
In recent years, functionally graded materials (FGMs) have become an increasingly important area in applied engineering. FGMs are designed to be thermal barrier materials for aerospace structural applications and fusion reactors (Bayat, Saleem et al. 2009) [1]. One of the most significant aspects of this field is the vibration of FGP. Classical deformation theory was employed by Yang and Shen (2001) [2] to analyze the dynamic response of thin FGP subjected to impulsive loads. Deformation theories with temperature dependent material properties were employed by some researchers.Yang and Shen (2002) [3] studied the vibration of FGP in thermal environments. Shahrjerdi, Mustapha et al. (2011) [4] reported free vibration analysis of solar functionally graded plates (FGPS) with temperature dependent material properties using second order shear deformation theory. Shahrjerdi, Mustapha et al. (2011) [5] investigated the thermal free vibration of FGP. In aerospace structures, cutouts are commonly found as access ports for mechanical and electrical systems, or simplay as weight reduction mechanisms. Cutouts are also made to lighten the loads, provide ventilation and alter the resonant frequency of the structures. In addition, cracks can be present in structures as the result of various causes, such as fatigue and environmental factors like temperature Datta and Biswas (2011) [6]. Cracks present in vibrating components can lead to catastrophic failure as reported by Israr (2011) [7]. For these reasons, there is a need to employ cutout in order to understand the dynamics of cracked structures. The vibration characteristics of structures can be useful for online detection of cracks without the structure actually having to be dismantled. To be specific, the natural frequencies and mode shapes of cracked plates can provide insights into the extent of damage.
Some studies have analyzed the natural frequencies of FGP through the use of cutouts. Most of the investigations on plates with cutouts have been confined to isotropic plates (1973) [8]; and Laminated composites by Ali and Atwal (1980) [3], Huang and Sakiyama (1999) [9], Reddy (1982) [10], Sivakumar, Iyengar et al. (1998) [11]. Recently, Janghorban and Zare (2011) [12] studied the influence of cutouts on the fundamental frequency of FGP in thermal environments using FEM. Saji, Varughese et al. (2008) [13] reported on the analysis of the thermal buckling of FGP using cutouts. Natarajan, Baiz et al. (2011) [14] proposed a method of determining the natural frequencies of cracked FGP.
Several studies have been conducted on the vibration of cracked FGP. The vibration of cracked plates was studied by Lynn and Kumbasar (1967) [15], who used a Green’s function approach. Later, Stahl and Keer (1972) [16] studied the vibration of cracked rectangular plates using elasticity methods. The other numerical methods used to study the dynamic response and instability of plates with cracks or local defects include the following: (1) Finite Fourier series transform (Solecki 1985) [17]; (2) RayleighRitz Method Khadem and (Rezaee 2000) [18]; (3) harmonic balance method (Wu and Shih 2005) [19]; and (4) finite element method (GuanLiang, SongNian et al. 1991) [20]. Lee and Lim (1993) [21]. Recently, Huang, McGee III et al. (2011) [22] proposed solutions for the vibrations of sidecracked FGP based on Reddy thirdorder shear deformation theory using the Ritz technique. Yang and Hao et al. (2010) [23] studied the nonlinear dynamic response of an FGP with a throughwidth crack based on Reddy’s thirdorder shear deformation theory using the Galerkin method.
Most of the abovementioned papers deal with the natural frequencies of temperatureindependent and dependent FGP with cutouts or crack. Temperaturedependent materials in a constant temperature field and temperature variations with surfacetosurface heat flow through the thickness direction were considered in other research without any attention to crack and cutout. Plates with cutouts are extensively used in transport vehicle structures. Cutouts are made to lighten the structure, for ventilation, to provide accessibility to other parts of the structures and to alter the resonant frequency. It is known that cracks or local defects affect the natural frequencies and resonant frequencies of a structural member. Additionally, temperature dependent material properties should be considered in the real conditions when the gradient of temperature cannot be neglected. To the authors’ knowledge, little work has been done in the area of the influence of a centrally located cutout and cracks on the natural frequencies of temperature dependent FGP. In the present work, the FEM analysis is provided to determine the natural frequency of different boundary conditions’ FGP under temperature field. The temperature is assumed to be constant in the plane of the plate. The variation of temperature is assumed to occur only in the thickness direction. The temperature dependent FGP is considered. Material properties follow power law distribution in terms of the volume fractions of the constituents. This research aims to show the effect of material compositions, crack and cutout and temperature fields on the vibration characteristics.
2. Material property of FGP
There are some models in the literature that express the variation of material properties in FGP (Shahrjerdi, Mustapha et al. 2011). FGPs are made of a mixture of two materials, namely a ceramic and a metal generality. It is usually assumed that the top surface of an FGP is ceramic rich while the bottom is metal rich. The region between the two surfaces consists of a blend of the two materials which is assumed in the form of a simple power law distribution.
2.1. Temperature independent material properties
According to a study reported by Shahrjerdi, Mustapha et al. (2011) [5], the material properties of FGPs are assumed to be position dependent and can be expressed as the following:
where ($z$) is coordinate in the thickness direction of an FGP; ${P}_{e}$, ${P}_{c}$, ${P}_{m}$ are effective material properties of the FGP, the properties of the ceramic and the properties of the metal respectively, ${V}_{c}$, ${V}_{m}$ are the ceramic and metal volumes fraction, and ($h$) is the plate thickness. Power n is the volume fraction exponent. Here, a rectangular FGP in Cartesian coordinate system ($x$, $y$, $z$) with constant thickness ($h$), width ($a$), and length ($b$) is considered as shown in Fig. 1.
2.2. Temperature dependent material properties
Material properties of FGPs are assumed to be position and temperature dependent, according to the following relation (Shahrjerdi, Mustapha et al. 2011) [22]:
where $P$ denotes material property and $T$ indicates the environmental temperature; ${P}_{1}$, ${P}_{0}$, ${P}_{1}$, ${P}_{2}$ and ${P}_{3}$ are the coefficients of temperaturedependent material properties unique to the constituent. In this section, mechanical and thermal analysis are carried out for an FGP; this entails a mixture of stainless steel (SUS_{304}) and silicon nitride (Si_{3}N_{4}). The coefficients, ${P}_{0}$,${P}_{1}$, ${P}_{1}$, ${P}_{2}$, ${P}_{3}$ of stainless steel and silicon nitride are indicated by Sharjerdi (Shahrjerdi, Mustapha et al. 2011) [4].
Fig. 1. FGP
Fig. 2. Cracked FGP
3. Crack formulation
Crack may grow in different modes. We begin the crack analysis by considering the idealized case of a straight crack. For an FGP (Fig. 2), each of the displacement and rotation functions are assumed as sets of two functions in Eq. (3) (Huang, McGee III et al. 2011) [22]:
where ${U}_{i}$ are the displacement and rotation components in the $x$, $y$ and $z$ directions, respectively, sets of functions indicated by the subscript “$p$” consist of orthogonal polynomials, which form a mathematically complete set, if an infinite number of terms are used in Eq. (4); (Huang, McGee III et al. 2011) [22]:
where ${P}_{ki}\left(x\right)$ and ${Q}_{kj}\left(y\right)$ are sets of orthogonal polynomials in the ($x$) and ($y$) directions of the rectangular plate reported by Bhat (1985). Sets of functions that describe the crack have been developed by Huang, McGee III et al. (2011). The sets of functions indicated by the subscript “$c$” are to supplement the polynomials by characterizing the important features of the exact solutions along the crack in Eq. (5) (Huang, McGee III et al. 2011) [22]:
$\left.+\sum _{n=1}^{5}\sum _{l=1}^{5}{d}^{\frac{\left(2n1\right)}{2}}\mathrm{s}\mathrm{i}\mathrm{n}\frac{2l+1}{2}\theta \right),\left(k=1,\dots ,5\right),$
where functions ${g}_{k}(x,y)$ are introduced to satisfy the geometric boundary conditions along $x=$ 0, $x=a$, $y=$0, and $y=b$ and the origin of the ($d,\theta $) coordinate system is at the crack tip (Fig. 2).
Substituting Eq. (3), that is depend on the crack function Eq. (5), into Esq. maximum strain energy and maximum kinetic energy that they are reported by (Huang, McGee III et al. 2011) and minimizing the energy functional that is defined in Eq. (6) leads to a standard eigenvalue problem. The Eigen values being the natural frequencies of FGP, which is reported by Huang, McGee III et al. (2011) [22]:
where ${V}_{max}$ is maximum strain energy and the ${T}_{max}$ is maximum kinetic energy and they are a function of ${U}_{i}$ and material properties as reported by (Huang, McGee III et al. 2011) [22]. So FGP’s frequencies depend on the crack’s function, as seen in the Table 1 [22].
Table 1. Natural frequencies for simply supported FGP with side cracks ($h/b=$ 0.1)
$n$

$\frac{d}{a}$

$\theta $

$\frac{{c}_{y}}{b}$

Mode


1

2

3

4


0.2

0

0

0.5

5.363

12.82

12.82

18.56

0.2

45

0.75

5.352

12.79

12.82

18.29

4. Modeling technique
A square plate is modeled by several layers. These layers capture a finite portion of the thickness and are treated like isotropic materials. The “layers” and their associated properties are then layered together to establish the throughthethickness variation of material properties. Although the layered structure does not reflect the gradual change in material properties, a sufficient number of “layers” can reasonably approximate the material gradation. In this research, the modeling and analysis of FGP is carried out using ANSYS software. ANSYS offers a number of elements for the modeling of composite materials. Here eight layers are chosen. Solid186 is employed as type of element and meshing. The study of the behavior of a square FGP is done for a plate whose constituent materials are taken to be (SUS_{304}), and (Si_{3}N_{4}). The top surface of the plate is ceramic (Si_{3}N_{4}) rich while the bottom surface is metal rich (SUS_{304}). The geometry of the plate is chosen to be 0.1×0.1×0.01 m. The radius of circle cutout is 0.005 m and the length of crack is 0.02 m. Table 2 shows different FGP geometries.
Fig. 3. Square plate with 8 layers and meshing model
Table 2. Geometry of FGP
Geometry of FGP


Cracked plate

Plate with central circular cutout

Cracked plate with circular cutout






$a$

$b$

$h$

$\frac{d}{a}$

$\frac{{c}_{y}}{b}$

$\theta $

$a$

$b$

$h$

$\frac{r}{a}$

$a$

$b$

$h$

$\frac{d}{a}$

$\frac{{c}_{y}}{b}$

$\theta $

$\frac{r}{a}$

0.1

0.1

0.01

0.2

0.5

0

0.1

0.1

0.01

0.1

0.1

0.1

0.01

0.2

0.5

0

0.1

Where $a$, $b$, $h$, $d$, $r$ and (${C}_{y},\theta $) are length, width, height and radius of cutout and location of crack respectively.

Fig. 3. shows a Square plate with 8 layers and a meshing model. A recommended geometry of opening crack is shown in Fig. 4.
Fig. 4. Modeling of crack
5. Boundary conditions
5.1. Mechanical boundary conditions
The boundary conditions that can be applied are simply supported Eq. (7), cantilever Eq. (8) and clamped Eq. (9), the displacement and rotation of which are defined in Eq. (7)(9) respectively:
where $u$, $V$, $W$, ${\theta}_{x}$, ${\theta}_{y}$ are displacement in $x$, $y$, $z$ direction and rotation around $x$ and $y$ axes, respectively.
Consider a temperature load at the upper and lower surfaces, such as Eq. (10):
where $T$, ${T}_{0}$, ${T}_{c}$ and ${T}_{m}$ are temperatures of FGP, room, ceramic and metal, respectively. These thermal loads can be applied to solve the steadystate heat transfer Eq. (10) (Wu and Shih 2005):
The temperature distribution along the thickness can be obtained by solving a steady state heat transfer problem (Shahrjerdi, Mustapha et al. 2011):
where ($k$) is profile index and ${K}_{c}$, ${K}_{m}$ are defined as thermal conductively coefficient of ceramic and metal, respectively.
6. Numerical results and validation
6.1. Verification and study
In order to validate the present research, intact FGP is modeled and subsequently compared according to the previous results. The results for temperaturedependent intact FGP, cracked FGP and cracked FGP with cutout obtained by applying FEM are compared by the results (Rahimabadi, Natarajan et al. 2013) [24], (Huang and Shen 2004) [9] and (Natarajan, Baiz et al.) [15]. The dimensionless natural frequency parameter is defined as follows [8]:
where $\omega $, ${\rho}_{m}$, $\upsilon $ and ${E}_{m}$ are natural frequency of FGP, mass density, Young’s modulus and Poisson’s ratio of metal. Table 3 shows the dimensionless frequency of simply supported intact FGP in thermal environment obtained from the present work using Ansys in comparison with Ref. [24]. There is considerable agreement between the presented results and those from the study reported by (Rahimabadi, Natarajan et al. 2013) [24]. The main reasons which can explain the difference depend on the number of layers and the size of meshing. Moreover, the results for cantilevered FGP with a side crack and simply supported FGP with circular cutouts are compared with other results [15, 9]. Table 4 shows the dimensionless frequencies for cantilevered FGP with a side crack ($cy/b=$ 0.5, $d/a=$0.5) as a function of gradient index.
Table 3. Dimensionless frequency of simply supported FGP ($a=b=$ 1; $a/h=$ 8) in thermal environment
Simply supported FGP with central circular cutout $\frac{r}{a}=$ 0.2, $\u2206T=100({T}_{c}=400\text{K},{T}_{m}=300\text{K})$

$\frac{a}{h}$

Gradient index, $k$


0

1


Ref. [24]

Present

Ref. [24]

Present


5

17.4690

18.254

10.5174

11.645


20

17.5380

18.543

10.1776

11.134


25

16.3587

17.006

9.1762

10.321

From Table 5 dimensionless frequency for square FGP with circular cutouts can be seen. The main reasons for the difference between the presented results in Tables 4, 5 and Ref. [9, 15, 24] are the number of layers and meshing size. Table 6 shows the dimensionless frequency of simply supported cracked FGP in a thermal environment obtained from the present work using Ansys in comparison with Ref. [22]. There is considerable agreement between the presented results and those from the study reported by (Huang, McGee III et al. 2011) [22]. The present approach is also compared to a conventional approach obtained by minimizing the energy functional in Eq. (6) in Table 7.
Table 4. Dimensionless frequencies for cantilevered cracked FGP as a function of gradient index
Gradient index, $n$

$a/h$ ($cy/b=$ 0.5, $d/a=$0.5)


10

20


Ref. [13]

Present

Ref. [13]

Present


0

0.9951

1.031

1.0024

1.121

1

0.5973

0.674

0.6016

0.680

2

0.5372

0.601

0.5411

0.6102

Table 5. Dimensionless frequencies for square FGP with circular cutouts
Temperature ${T}_{c}$; ${T}_{m}$

Gradient index

Mode 1

Mode 2


Ref. [24]

Ref. [9]

Present

Ref. [24]

Ref. [9]

Present


${T}_{c}=$400 K, ${T}_{m}=$300 K

0

12.315

12.397

13.362

29.031

29.083

30.10

0.5

8.484

8.615

9.589

19.986

20.215

21.975


1

7.443

7.474

8.980

17.515

17.607

18.050


${T}_{c}=$600 K, ${T}_{m}=$ 300 K

0

11.894

11.984

12.125

28.436

28.504

29.95

0.5

8.269

8.147

9.120

19.535

19.784

21.001


1

7.126

7.171

8.231

17.098

17.213

17.980

Table 6. Dimensionless frequencies for simply supported FGP with side crack ($h/b=$0.1, $\theta =$ 0, ${c}_{y}/b=$0.5, $n=$ 1) as a function of crack function
$\frac{d}{a}$

Mode


1

2

3

4


Ref. [22]

Present

Ref. [22]

Present

Ref. [22]

Present

Ref. [22]

Present


0

3.768

3.702

8.909

8.825

8.909

8.820

12.64

12.11

0.2

3.756

3.698

8.851

8.788

8.867

8.801

12.04

11.53

Table 7. Natural frequencies for simply supported FGP with side cracks are obtained by minimizing the energy functional in Eq. (6) ($h/b=$ 0.1)
$n$

Mode


1

2

3

4


Ref. [22]

Present

Ref. [22]

Present

Ref. [22]

Present

Ref. [22]

Present


0.5

4.4220

4.101

10.620

10.231

10.621

10.233

16.201

15.876

6.2. Numerical results
Tables 810 present the natural frequencies in Si_{3}N_{4}/SUS_{304} for different boundary conditions. In Tables 810, it can be seen that central cutout, side cracked and cracked FGP with cutout have maximum, medium and minimum frequencies at all temperatures, respectively. These results can be justified because the FGP with cutout has less material than do the side cracked ones, which leads to decreased inertia momentums; therefore, the frequency of FGP with cutout is greater than it is for the side crack ones. Furthermore, the natural frequency of cracked FGP with cutout is less than it is in the two other cases. This is due to the decreasing stiffness of FGP. The effect of temperature on the frequencies can be investigated by considering the same value of shape mode and boundary condition. A clamped plate has a higher frequency than does a simply supported plate. The increase in the stiffness is the cause for the increase in frequency when the boundary condition is changed from simply supported to clamped condition, for a fixed aspect ratio and plate thickness. Fig. 5 shows the variation of inplane displacement versus mode number of simply supported FGP for different values of gradient index ($n$). It is obvious that the pattern of in plan displacement in cracked FGP with central cutout is similar to FGP with central cutout. This is due to the small size of crack. As gradient index increases, the inplane displacements decrease slightly because of increasing metal content.
Table 8. Dimensionless natural frequency parameter for temperature dependent simply supported FGP in thermal environments ($n=$ 0.5)
Nondimensional natural frequency ${T}_{m}=$300, ${T}_{o}=$300 K


Mode numbers of FGP (Si_{3}N_{4} and SUS_{304})

${T}_{c}=$300

${T}_{c}=$400

${T}_{c}=$600


Side crack

Central cutout

Crack and cutout

Side crack

Central cutout

Crack and cutout

Side crack

Central cutout

Crack and cutout


1

0.5724

0.6098

0.5600

0.5550

0.6043

0.550

0.5590

0.5957

0.5469

2

1.3792

1.4548

1.3414

1.3277

1.4429

1.3270

1.3496

1.4224

1.3114

3

1.3797

1.4551

1.3429

1.3191

1.4453

1.3190

1.3501

1.4228

1.3128

4

1.4304

1.5827

1.4258

1.3332

1.5811

1.3330

1.4096

1.5619

1.3203

Table 9. Nondimensional natural frequency parameter for temperature dependent cantilever FGP in thermal environments ($n=$0.5)
Nondimensional natural frequency ${T}_{m}=$300, ${T}_{o}=$300 K


Mode numbers of FGP (Si_{3}N_{4} and SUS_{304})

${T}_{c}=$300

${T}_{c}=$400

${T}_{c}=$600


Side crack

Central cutout

Crack and cutout

Side crack

Central cutout

Crack and cutout

Side crack

Central cutout

Crack and cutout


1

0.1102

0.1183

0.1090

0.1100

0.1175

0.1083

0.1077

0.1158

0.1067

2

0.2628

0.2762

0.2548

0.2609

0.2742

0.2529

0.2570

0.2702

0.2492

3

0.6515

0.6898

0.6363

0.6467

0.6847

0.6316

0.6370

0.6746

0.6223

4

0.7188

0.7487

0.6906

0.7125

0.7422

0.6844

0.7001

0.7293

0.6727

Table 10. Dimensionless natural frequency parameter for temperature dependent clamped FGP in thermal environments ($n=$ 0.5)
Nondimensional natural frequency of cantilever FGP, ${T}_{m}=$300, ${T}_{o}=$300 K, ${T}_{c}=$600


Mode numbers of

Side crack

Central cutout

Crack and cutout

1

0.6764

0.8170

0.5054

2

1.3516

1.4623

0.9599

3

1.3727

1.4624

1.2660

4

1.8513

2.3102

1.5929

Fig. 5. Variation of inplane displacement versus mode number of simply supported FGP for different values of gradient index ($n$)
Tables 11, 12 show the contour plots of the transverse and inplane displacement distribution for simply supported and cantilever FGP in different cases. Study of Table 11 indicates that by increasing mode number, variation of counter plot increases. It is obvious that there is no change in the first mode, while a change in the second and third modes is seen. The fourth mode is different for each item. This proves that the simultaneous presence of cracks and cutout make more changes in higher modes. This is due to crack growth. Table 12 identifies transverse displacement distribution for cantilever FGP in different cases (Cracked FGPFGP with cutout and Cracked FGP with cutout). First to third mode shape has no changes, while fourth mode has a variation.
Table 11. Transverse displacement distribution for simply supported FGP in different cases (Cracked FGPFGP with cutoutCracked FGP with cutout)
Mode number

Different cases of FGP


Cracked FGP

FGP with cutout

Cracked FGP with cutout


1




2




3




4




7. Conclusions
Detecting natural frequencies of temperaturedependent cracked FGP with cutout can be an important problem in the mechanical field. Predicting the amount of frequency reduction due to cracks can be important in determining whether and to what extent the resonant frequency should be altered. Natural frequencies of intact FGP, cracked FGP and cracked FGP with cutout subjected linear, temperature fields are investigated using FEM for simply supported, cantilever and clamped boundary conditions. Material properties of FGP are assumed to be temperaturedependent and vary in thickness by a powerlaw distribution in terms of volume fractions of constituents. The results are validated by comparing them with the results of other researchers. There is considerable agreement between the presented results and those from other studies. The main reasons which can explain the difference depend on the number of layer and size of meshing; therefore, using a greater number of layers and finer mesh gives more accurate results in comparing by experimental techniques. Moreover, the results for some important conclusions can be summarized as follows:
Percentage decrease of natural frequency due to crack for clamped boundary conditions has maximum reduction and minimum for simply support at different temperatures, which is due to an increase in the stiffness. The amount of frequencies reduction due to crack is about 30 %, 15 % and 7 % for clamped, cantaliver and simply supported boundary conditions, respectively.
Young’s modulus decreases with the increase in the temperature at different values of temperature of ceramic. A clamped plate has a higher frequency than a simply supported plate in all cases. An increase in stiffness is the cause for the increase in frequency when the boundary condition is changed from simply supported to clamped condition. The frequencies are decreased by increasing the temperature. Decreasing the Young modulus in all conditions leads to such results. The natural frequency of FGP with cutout is greater than it is for side crack ones. A decrease in the moment of inertia can be the main reason for this. Decreasing stiffness causes the natural frequency of cracked FGP with cutout to be less than it is in the two other cases. An increase in temperature leads to a maximum decrease of frequency at clamped FGP. The main reason for such a decrease depends on the decrease in the Young modulus. It can be seen that natural frequency in dependent temperature FGP is smaller than it is in the independent temperature properties. A similar conclusion was reported by (Shahrjerdi, Mustapha et al. (2011)). Stiffness in dependent temperature FGP is smaller than it is in the independent temperature one, which yields the mentioned conclusion. As gradient index $\text{(}n\text{)}$ increases, the inplane displacements reduce slightly. This is due to increasing metal content.
Table 12. Transverse displacement distribution for cantilever FGP in different cases (Cracked FGPFGP with cutoutCracked FGP with cutout)
Mode number

Different cases of FGP


Cracked FGP

FGP with cutout

Cracked FGP with cutout


1




2




3




4




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