Document Type : Research Article
Authors
Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, 3815688349, Iran
Abstract
Keywords
Main Subjects
Static Buckling and Free Vibration Analysis of BiDimensional FG Metal Ceramic Porous Beam
Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, 3815688349, Iran
KEYWORDS 

ABSTRACT 
Porosity; Computational analysis; 2DFGM; Composite materials; Buckling. 
This study presents an analytical solution for static buckling and free vibration analysis of bidimensional functionally graded (2DFG) metalceramic porous beams. To achieve this goal, equations of motion for the beam are derived by using Hamilton's principle and then the derived equations were solved in the framework of Galerkin’s wellknown analytical method for solution of equations. The material properties of the beam are variable along with thickness and length according to the powerlaw function. During the fabrication of functionally graded materials (FGMs), porosities may occur due to technical problems causing microvoids to appear. Detailed mathematical derivations are presented and numerical investigations are performed, while emphasis is placed on investigating the effect of various parameters such as FG power indexes along both directions of thickness and length, porosity, and slenderness ratios (L/h), on the nondimensional frequency and static buckling of the beam based on new higher deformation beam theory. The accuracy of the proposed model is validated based on comparisons of the results with the accepted studies. According to the result in both buckling and vibration analysis, the presented modified transverse shear stress along the thickness has shown closer consequences in comparison with TBT. 
In recent years, a new class of composite materials formed that contained two or more phases which are known as functionally graded materials (FGMs). FGMs are made of two or more substances whose characteristics vary continuously from one direction to another. The performances and advantages of FGMs have led to their increasing use in many sectors including hitech industries, aerospace, naval, automotive, and civil structures[1]. An important application of FGM in structural components is in beams and beam structures. To support the growing use of FGM beams, the study of their vibration behavior becomes necessary. In recent years, the analysis of the free vibration of FGM beams has been performed using different analytical and numerical methods[2]. buckling and free vibration of a curved Timoshenko FG microbeam is studied by Rahmani et al. [3] based on strain gradient theory (SGT) theory. Compared to conventional materials, porous materials possess both remarkable design flexibility and significant applicability across diverse fields. Consequently, numerous researchers have dedicated their efforts to exploring various porous structures. For instance, Ghasemi and Meskini [4] conducted an investigation into the free vibration analysis of porous laminated rotating circular cylindrical shells. Several studies have been carried out on the behaviors of functionally graded material in recent years [510]. Atashipour et al. [11] developed a closedform 2D elasticity solution for stresses and displacements of a curved FG beam subjected to a shear force. Among the earliest investigators in this field, Masjedi et al. [12], conducted a large deflection of functionally graded porous beams based on a geometrically exact theory. The buckling behavior of engineering structures made of advanced materials has been investigated by many researchers[1318]. Buckling of Sandwich Structures with Metamaterials Core Integrated by Graphene Nanoplatelets Reinforced Polymer Composite was analyzed by Shabani and Khorshidi[19]. The analytical solution presented by Dym and Williams [20] can be used to estimate the buckling behavior of curved beams. The nondeterministic vibration frequencies and mode shapes of FG porous beams were investigated by Gao et al. [21] using a hybrid Chebyshev surrogate model with a discrete singular convolution method. Large amplitude nonlinear vibration analysis of functionally graded porous Timoshenko beams is examined by Ebrahimi and Zia [22]. Chen et al. [23] investigated free vibration and transient analyses of functionally graded piezoelectric materials curved Timoshenko beams in their work. Mahmoodabadi et al. [24] used a multiobjective optimization process based on the genetic algorithm to study damping the vibrations of a piezoactuating composite beam. The Peridynamics analytical solutions for the buckling load of beams with various boundary conditions were introduced by Yang et al. [25]. Ly et al. [26] used different hybrid ML models to predict the critical buckling load of Ishaped cellular steel beams with circular openings.
In this study, the effects of variable L/h, porosity coefficient, and gradient indexes in two directions on natural frequency and critical buckling load of 2DFG porous beams have been investigated. Young’s modulus, mass density, and Poisson's ratio of the beam are assumed to vary along the thickness and length of the beam according to powerlaws form and also, they are influenced by the porosity. In this study to achieve better accuracy, modified transverse shear stress has been presented and compared with other theories. In the following, the higherorder governing equations are derived by using Hamilton's principle. In the end, the Galerkin method is employed to solve them, and the nondimensional frequencies and critical buckling loads are obtained.
Consider a beam as shown in Fig. 1. with length L, width b, and thickness h, with Cartesian coordinate system O(x y z) which the origin of coordinate system O is chosen at the left of the beam. The Mechanical properties of the beam, such as Young's modulus E(x, z), shear modulus G(x, z), Poisson's modulus υ(x, z) and mass density ρ(x, z) with the material properties can be varying along length and thickness like Fig. 1.

Fig. 1. Geometry of 2DFG beam. 
The effective material properties(P) can be expressed by using the rule of mixture:

(1) 

(2) 
where P_{c} and P_{m} are epitomes of mechanical properties and R_{c} and R_{m} are volume fractions of ceramic and metal. Substitution of Eq. (2) into Eq. (1), the material properties of the 2DFG beam are obtained as [27]:

(3) 
As shown in Fig. 2, it is assumed that the porosities in the x and z directions are distributed uniformly. The effective material properties of the even porosity are defined as:

(4) 
where is a porous parameter or porosity volume fraction.

Fig. 2. Porosity of beam. 
The assumed beam is graded from metal at the lower left corner edge to ceramic at the top right corner edge (Fig. 1). The volume fraction of ceramic material is given by:

(5) 
The k_{x} and k_{z} are the powerlaw of the beam which ascertain properties in length and thickness direction. The mechanical properties of the 2DFG porous beam can be written as [28]:

(6a) 

(6b) 

(6c) 
The displacement filed the present shear deformation beam theories, are given as [29]:

(7a) 

(7b) 
where u(x) and w(x) respectively represent the axial and transverse displacements for the midaccess, and φ is the rotation of the crosssections. Assuming infinitesimal deformations, straindisplacement relations are[30]:

(8a) 

(8b) 
The stressstrain relations by using Hook's law can be defined as follows:

(9a) 

(9b) 
E and G represent the modulus of elasticity and shear moduli, respectively, while ks denotes the shear correction factor. Where the shear modulus (G) is [31]:

(10) 
Hamilton's principle is employed to extract equations of motion [32, 33].

(11) 
where U and T are the strain and kinetic energy, respectively of the beam. is the variation operator. The strain energy of the beam(U) is calculated as follows [34, 35]:

(12) 
Finally, variation of strain energy with respect to u(x), w(x) and is shown as.

(13) 
where A_{xx}, A_{xz}, B_{xx}, D_{xx}, S_{xz}, and T_{xz} are defined by:

(14a) 

(14b) 

(14c) 
The kinetic energy is obtained as [36]:

(15) 
The inertia coefficients appearing in Eq. 19 can be defined as:

(16) 
Finally, the total variation of kinetic energy associated with the sandwich beam in the integral form as:

(17) 
By substituting strain energy Eq. (13) and kinetic energy Eq. (17) into Hamilton's principal Eq. (11), equations of motion may be expressed as Eqs. (18a) – (18c).

(18a) 

(18b) 

(18c) 
To obtain the theoretical solution, the Galerkin method is considered. According to this method, the displacements functions u(x, t), w(x, t) and are assumed as follows[37]:

(19a) 

(19b) 

(19c) 
where , and are unknown coefficients that will be determined. i = , K denotes the order of series and is the natural frequency. , and are the admissible functions that satisfy the fully clamped boundary conditions.

(20a) 

(20b) 

(20c) 
Free vibration analysis of the bidimensional functionally graded sandwich beam can be computed from Eq. (21a)[38]. Also, the amount of critical buckling loads will be calculated by Eq. (21b):

(21a) 

(21b) 
where M is the global mass matrix, K is the stiffness matrix and unknown coefficients of Eq. (21) is .
In this section, the free vibration of the 2DFG porous beam with respect to porosity coefficients ( ) is studied. Functionally graded material composed of a mixture of alumina and steel as ceramic and metal respectively with the material. Their properties are given in Table 1. The influence of different slenderness ratios, L/h = 5, 10, 15, and 20 on the nondimensional natural frequency are investigated. The shear correction factor for TBT theory is considered as ks = 5/6 [39].
Table 1. Properties of materials[37]
Material 
Elasticity module (E) 
Mass density (ρ) 
Poisson's ratio (υ) 
Alumina 
380 
3800 
0.23 
Steel 
70 
2700 
0.23 
The dimensionless fundamental frequency and critical buckling load are defined as Eq. 22.

(22a) 

(22b) 
where I represent the moment of inertia and A represents the area of the crosssection of the beam.

(23) 
In this research, the solution is calculated for different the powerlaw index between 0 and 10 moreover, porosity coefficients are taken as
0, 0.1, and 0.2. The total thickness of the beam (h) is constant and it is 0.1 m. The function indexes (p0 and q0), are taken as 2 for satisfying the clampedclamped boundary condition. Besides, g(z) and f(z) are related parameters to the beam's theory, as they take as 0 and z for TBT, and z and for NHOBT [40], respectively. Validation of our formulation and the results are obtained and compared with the results of Ref. [41], [42], and [43]. In Table 2, the first frequency of FG porous less beam with
L/h = 5 and k_{x} = 0 are calculated and they will be compared with references. To verify our new theory, the natural frequency of the FG beam with a greater slender ratio in Table 3 is calculated.
In addition, another comparison with ref [41] and [43] is presented to show the accuracy of the new higher shear deformation beam theory to calculate the critical buckling load. The mechanical properties of the pure beam have changed along the z direction and the ratio of beam's length to height is 10. The results are written in Table 4.
The amounts of Ncr of the FG beam with NHOBT and L/h = 5 are reported in Table 5. By checking both tables, it is clear that there is accuracy and agreement between the references and the present study. Thus, the results of natural frequencies and critical buckling loads will be published under effective parameters such as L/h, porosity (η), and functionally graded power indexes (k_{x} and k_{z}).
The results of nondimensional frequencies of a twodimensional functionally graded porous beam with several η, k_{x,} and k_{z} by using two theories are written in Table 6. The amount of ϖ has been reduced by approximately 37% by changing k_{z} from 0 to 8. On the other hand, the variation of k_{x} has affected ϖ by around 34%, indicating that k_{z} has a greater impact on the results. Additionally, by changing η from 0 to 0.2, the amount of ϖ has increased by approximately 3%.
The amount of Ncr under changing k_{x}, k_{z,} and η with TBT and NHOBT is calculated and the consequences are written in Table 7. By changing k_{z} and k_{x} between 0 to 8, the amount of Ncr has reduced by about 76 and 69 % respectively. The factor with the greatest impact on the results is k_{z}, with approximately 7% more effect. Furthermore, the Ncr decreased by about 11% with increasing η from 0 to 0.2.
Table 2. Comparison of the dimensionless natural frequencies (l/h = 5 and η = 0)

kz = 0 
kz = 1 
kz = 2 
kz = 5 
kz = 10 
Present (NHOBT) 
10.2285 
8.0969 
7.3290 
6.6448 
6.3042 
Present (TBT) 
10.2579 
8.1098 
7.3750 
6.7128 
6.4759 
Thuc et al. HOBT [41] 
10.0678 
7.9522 
7.1801 
6.4961 
6.1662 
Thuc et al. FOBT [41] 
9.9984 
7.9015 
7.1901 
6.6447 
6.3161 
Simsek HOBT [42] 
10.0705 
7.9503 
7.1767 
6.4935 
6.1652 
Simsek FOBT [42] 
10.0344 
7.9253 
7.2113 
6.6676 
6.3406 
Table 3. Comparison of the dimensionless natural frequencies (l/h = 20 and η = 0)

kz = 0 
kz = 1 
kz = 2 
kz = 5 
kz = 10 
Present (NHOBT) 
12.2418 
9.4918 
8.6778 
8.2159 
7.9342 
Thuc et al. HOBT [41] 
12.2228 
9.4328 
8.5994 
8.1460 
7.8862 
Thuc et al. FOBT [41] 
12.2202 
9.4311 
8.6047 
8.1698 
7.9115 
Simsek HOBT [42] 
12.2238 
9.4315 
8.5975 
8.1446 
7.8858 
Simsek FOBT [42] 
12.2235 
9.4314 
8.6040 
8.1699 
7.9128 
Table 4. Comparison of the results of critical buckling load (l/h = 10 and η = 0)

kz = 0 
kz = 1 
kz = 2 
kz = 5 
kz = 10 
Present (NHOBT) 
194.4026 
98.8053 
76.8331 
62.9629 
56.4158 
Li and Batra [43] 
195.3400 
98.7490 
76.9800 
64.0960 
57.7080 
Thuc et al. FOBT [41] 
195.3730 
98.7923 
77.0261 
64.1324 
57.7329 
Thuc et al. HOBT [41] 
195.3610 
98.7868 
76.6677 
62.9786 
56.5971 
Table 5. Comparison of the results of critical buckling load (l/h = 5 and η = 0)

kz = 0 
kz = 1 
kz = 2 
kz = 5 
kz = 10 
Present (NHOBT) 
152.1990 
79.8063 
61.2369 
47.116 
41.1128 
Li and Batra [43] 
154.3500 
80.4980 
62.6140 
50.3840 
44.2670 
Thuc et al. FOBT [41] 
154.4150 
80.5480 
62.6616 
50.4207 
44.2946 
Thuc et al. HOBT [41] 
154.5500 
80.6087 
61.7925 
47.7562 
41.8042 
Table 6. First nondimensional frequency (ϖ) of 2DFG porous beam for various kx, kz, and η (L/h = 10).
Theory 
η 
kz 
kx 

0 
1 
2 
4 
6 
8 

TBT 
0 
0 
21.6960 
17.0553 
15.6524 
14.5606 
14.0223 
13.6403 
1 
16.8500 
14.3819 
13.6517 
13.0863 
12.7849 
12.5764 

2 
15.3637 
13.6340 
13.0914 
12.6367 
12.4165 
12.2903 

4 
14.6350 
13.2308 
12.7760 
12.3834 
12.1918 
12.0892 

6 
14.3520 
13.0449 
12.6168 
12.2521 
12.0960 
11.9658 

8 
14.1373 
12.8873 
12.4975 
12.1511 
12.0043 
11.9063 

NHOBT 
0 
0 
21/1618 
16/6429 
15/3731 
14/2928 
13/7558 
13/3948 
1 
16/5051 
14/0453 
13/3743 
12/8050 
12/5325 
12/3574 

2 
15/0571 
13/3139 
12/8158 
12/3854 
12/1817 
12/0524 

4 
14/2740 
12/8894 
12/4747 
12/1139 
11/9551 
11/8567 

6 
13/9665 
12/6941 
12/3101 
11/9907 
11/8451 
11/7547 

8 
13/7419 
12/5464 
12/1924 
11/8892 
11/7570 
11/6799 

TBT 
0.1 
0 
21.9998 
16.7114 
15.0772 
13.6923 
13.0121 
12.5896 
1 
16.5388 
13.6219 
12.7145 
12.0111 
11.6668 
11.4278 

2 
14.6162 
12.6623 
12.0187 
11.4848 
11.2217 
11.0879 

4 
13.6436 
12.1432 
11.6160 
11.1728 
10.9759 
10.8546 

6 
13.3410 
11.9406 
11.4468 
11.0374 
10.8555 
10.7441 

8 
13.1444 
11.7965 
11.3355 
10.9355 
10.7829 
10.6676 

NHOBT 
0.1 
0 
22/3579 
16/9650 
15/3647 
14/0096 
13/3560 
12/9310 
1 
16/7512 
13/8084 
12/9738 
12/2489 
11/9219 
11/7106 

2 
14/8641 
12/8713 
12/2623 
11/7220 
11/4849 
11/3378 

4 
13/9234 
12/3669 
11/8677 
11/4237 
11/2257 
11/1112 

6 
13/6599 
12/1858 
11/7091 
11/2917 
11/1181 
11/0058 

8 
13/4683 
12/0397 
11/5834 
11/1987 
11/0331 
10/9345 

TBT 
0.2 
0 
22.3718 
16.2578 
13.9010 
12.1407 
11.3053 
10.7474 
1 
16.0382 
12.4432 
11.2190 
10.2780 
9.8109 
9.5400 

2 
13.4114 
11.1096 
10.2812 
9.5863 
9.2764 
9.0820 

4 
11.8584 
10.3262 
9.7099 
9.1750 
8.9410 
8.8175 

6 
11.4574 
10.0734 
9.5067 
9.0297 
8.8242 
8.6936 

8 
11.3014 
9.9231 
9.3867 
8.9382 
8.7464 
8.6273 

NHOBT 
0.2 
0 
22/7021 
16/3349 
14/2677 
12/4383 
11/6037 
11/0778 
1 
16/2534 
12/5477 
11/4541 
10/4811 
10/0477 
9/7799 

2 
13/6285 
11/2567 
10/4767 
9/7967 
9/4873 
9/3149 

4 
12/0916 
10/4981 
9/9149 
9/3829 
9/1574 
9/0258 

6 
11/7261 
10/2714 
9/7284 
9/2452 
9/0443 
8/9243 

8 
11/5545 
10/1302 
9/6076 
9/1525 
8/9648 
8/8529 
Table 7. Critical buckling load (Ncr) of 2DFG porous beam for various kx, kz, and η (L/h = 10).
Theory 
η 
kz 
kx 

0 
1 
2 
4 
6 
8 

TBT 
0 
0 
201.3001 
101.2443 
73/3782 
56/8435 
51/2608 
48/3640 
1 
101.8384 
66.6279 
55/4769 
47/7935 
44/9744 
43/4770 

2 
79.3865 
58.3191 
50/8079 
45/2804 
43/1927 
42/0855 

4 
68.2815 
53.5313 
47/9427 
43/6789 
42/0411 
41/1664 

6 
64.3308 
51.4026 
46/5823 
42/8864 
41/4587 
40/6963 

8 
61.6250 
49.8626 
45/5830 
42/2954 
41/0188 
40/3319 

NHOBT 
0 
0 
195/3779 
97/2664 
71/2192 
55/4709 
50/1329 
47/3546 
1 
99/2310 
64/5146 
53/9762 
46/6343 
43/9498 
42/5304 

2 
77/1877 
56/4267 
49/3866 
44/1401 
42/1719 
41/1327 

4 
65/7422 
51/5953 
46/4741 
42/5032 
40/9942 
40/1957 

6 
61/5646 
49/4644 
45/1055 
41/7020 
40/4027 
39/7119 

8 
58/8715 
47/9899 
44/1322 
41/1171 
39/9625 
39/3461 

TBT 
0.1 
0 
189/7017 
86/5973 
58/2271 
42/5622 
37/4739 
34/8816 
1 
88/0537 
52/9688 
41/9996 
34/7625 
32/1893 
30/8440 

2 
64/0969 
44/5446 
37/5045 
32/4321 
30/5622 
29/5826 

4 
52/5693 
39/8829 
34/8210 
30/9839 
29/5379 
28/7694 

6 
49/1145 
38/0454 
33/6697 
30/3341 
29/0618 
28/3879 

8 
46/9595 
36/7694 
32/8503 
29/8570 
28/7108 
28/1008 

NHOBT 
0.1 
0 
184/1132 
82/7353 
56/4400 
41/5326 
36/6694 
34/1715 
1 
85/8718 
51/1773 
40/8558 
33/9401 
31/4780 
30/1892 

2 
62/4375 
43/0432 
36/4529 
31/6364 
29/8622 
28/9356 

4 
50/6270 
38/3560 
33/7335 
30/1575 
28/8153 
28/1133 

6 
46/8667 
36/4948 
32/5662 
29/4938 
28/3337 
27/7230 

8 
44/6357 
35/2597 
31/7613 
29/0213 
27/9825 
27/4327 

TBT 
0.2 
0 
178/0717 
70/4071 
40/9069 
26/8716 
22/6228 
20/5068 
1 
73/6780 
38/2597 
27/4892 
20/9948 
18/8072 
17/6832 

2 
47/4333 
29/7059 
23/3508 
18/9993 
17/4592 
16/6723 

4 
34/6064 
25/0341 
20/8595 
17/7380 
16/5918 
16/0017 

6 
31/5006 
23/5343 
19/9695 
17/2607 
16/2562 
15/7359 

8 
30/0213 
22/6233 
19/3964 
16/9431 
16/0272 
15/5484 

NHOBT 
0.2 
0 
172/8112 
66/3052 
39/5456 
26/1841 
22/1330 
20/0973 
1 
71/9500 
36/7506 
26/6959 
20/4944 
18/4025 
17/3281 

2 
46/3571 
28/6013 
22/6763 
18/5411 
17/0741 
16/3237 

4 
33/4452 
23/9716 
20/1835 
17/2666 
16/1973 
15/6521 

6 
30/0084 
22/4247 
19/2647 
16/7747 
15/8535 
15/3803 

8 
28/3175 
21/5147 
18/6904 
16/4552 
15/6238 
15/1932 
The beam is made of twodimensional functionally graded material; thus, it has been influenced by FG power indexes in x (k_{x}) and z (k_{z}) directions. The critical buckling loads (Ncr) and the natural frequencies (ϖ) of the beam are related to the k_{x} and k_{z}. In Figure 3a, the relation between ϖ and FG power indexes in a porous less beam is shown, the ratio of length to the height of the beam is considered as 10. In Figure 3b, the effect of FG power indexes on Ncr of the porous beam is under new theory (NHOBT) with η = 0.2 and L/h = 10 illustrated. Generally, the amount of Ncr and ϖ have reduced with increasing the FG power indexes. However, Ncr has been affected more than ϖ by changing k_{x} and k_{z}. In the high amount of FG power indexes the variations of Ncr have been moved to near 0. On the other hand, the ϖ has been diminished continuously by reducing the k_{x} and k_{z}. Power indexes affect how material properties vary along the length and thickness directions, which influence their stiffness and mass distribution. A higher FG power index reduces the effective stiffness and increases the effective mass of a beam.

Fig. 3a. Nondimensional frequency of 2DFG porous beam for various k_{z} and k_{x} (TBT, L/h = 10 and η = 0) 

Fig. 3b. Critical buckling load of 2DFG porous beam for various k_{z} and k_{x} (NHOBT, L/h = 10 and η = 0.2) 
The impact of various ratios of the beam's spin to its height on ϖ and Ncr is shown in Figure 4. A beam with n = 0.1 and k_{x} = 1 is used to analyze the three amounts of L/h = 5, 10, and 20 on the ϖ in the first part of the Figure. A direct connection is seen between the amount of L/h and the ϖ. Additionally, the same study on Ncr has been conducted under unique conditions using a new theory. The results of this study are plotted in Figure 4b and the amount of Ncr has been rose by growing the L/h from 5 to 20. Also, the effect of L/h = 5 on the Ncr and ϖ is higher than 10 and 20. Reducing the strain energy will decrease the bending moment of the beam, so when the L/h is going to be diminished, the amount of Ncr and ϖ will be reduced. A higher slenderness ratio means a longer and thinner beam, which is more flexible and less stiff than a shorter and thicker beam.

Fig. 4a. Comparison of nondimensional frequency of 

Fig. 4b. Comparison of Critical buckling load of 
Another important and effective parameter is porosity. Figure 5 shows the effect of η = 0, 0.1, and 0.2 on the Ncr and ϖ. When the amount of η is 0, the beam does not have any porosity. In the constant k_{z} and k_{z}, the ϖ has shown an inverse direction with η so as to by increasing the η, ϖ has fallen. The same relation is seen between η and Ncr, however, the effect on Ncr is bolder than on ϖ. By reducing the stiffness of the beam, the amount of ϖ and Ncr will be influenced and decreased. Porosity affects the material density and strength along the thickness direction of beams, which in turn affects their stiffness and mass. Higher porosity means more voids or holes in the material, which reduces its weight and resistance to deformation. A higher porosity reduces the effective stiffness and increases the effective mass of a beam under both axial and transverse loading, which lowers its critical buckling load and natural frequency.

Fig. 5a. Effect of various porosity(η) on nondimensional frequency of 2DFG porous beam (L/h = 10 and kx = 1) 

Fig. 5b. Effect of various porosity(η) on Critical buckling load of 2DFG porous beam (NHOBT, L/h = 10 and kx = 8) 
In this research paper, we have investigated the impact of various parameters, including porosity (η), slenderness ratios (L/h), and powerlaw indexes (kx and kz) in both axial and thickness directions, on the natural frequencies and critical buckling load of 2DFG porous beams. We have utilized a new higherorder beam theory to analyze these effects.
Our findings demonstrate that the powerlaw indexes (kx and kz) play a significant role in determining the natural frequencies. Increasing the FG power indexes leads to a decrease in the values of the natural frequencies (Ncr) and the critical buckling load (ϖ). This can be explained by the change in material properties along the beam's axial and thickness directions. The powerlaw indexes alter the distribution of the constituent materials, affecting their stiffness and density. Higher powerlaw indexes result in a more significant variation of material properties, leading to decreased natural frequencies and critical buckling loads. In the case of high powerlaw indexes, the variations in Ncr do not exhibit significant changes. Additionally, when considering a constant value for kx and kz, we observe an inverse relationship between the porosity (η) and ϖ. As η increases, ϖ decreases, and the same trend is observed between η and Ncr. This behavior can be attributed to the presence of voids or pores within the material. The presence of voids weakens the overall structural integrity and reduces the effective stiffness of the beam. As a result, the natural frequencies decrease, and the critical buckling load is diminished. Furthermore, frequencies and critical buckling loads are more sensitive to porosity in high powerlaw indexes. A direct connection is seen between the amount of L/h and the ϖ. The amount of Ncr has been rose by growing the L/h from 5 to 20. Also, the effect of L/h = 5 on the Ncr and ϖ is higher than 10 and 20.
These findings provide valuable insights into the behavior of bidimensional FG metal ceramic porous beams and highlight the importance of considering porosity, slenderness ratios, and powerlaw indexes in their design and analysis. Further research in this area can contribute to the development of more efficient and reliable structures with improved performance characteristics.
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript. In addition, the authors have entirely observed the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancy.
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