Impact of Porosity Distribution on Vibration of Porous Functionally Graded Beams

Composite materials are one of the modern materials created to develop the material properties to achieve the requirement of the modern engineering applications. The physical combination of two or more materials is the basic idea of composite material. Several methods can be used to achieve this combination and to get better material properties, and one of these methods is the combination of different materials as layered media, called laminated composite materials and this type of combination leads to the occurrence of stress and temperature discontinuities in the material [1]. In 1980, functionally-graded-material (FGM) is a novel type of materials. FGM was suggested to overcome this problem. The properties of these materials undergo a progressive transformation from the first material to second one based on a specified equation. Nowadays, because of their higher material properties, FG materials are applied in different fields, especially in nuclear, biomedical, aerospace, and optical engineering applications [2-5]. Numerous studies on the mechanical behavior of structures have been done as a result of the widespread uses of FGM [6-13]. However, during the manufacturing process, these materials may display certain flaws like porosity. Therefore, a study on this topic has to be added as soon as possible to have a solid understanding of the porosity influence on the mechanical and structural behaviors of FGM. Because of its various applications, beams—along with plates and shells—have always piqued the interest of researchers among the three types of structures. Simple beam theories such as classical, first, second and higher-order-shear-deformation theory are just a few of the many beam theories that are used to analyze beam structures. On the other hand, researchers can lower the computational cost and keep the resulting error within the permitted range by employing a straightforward model. Furthermore, in order to give the designer an accurate understanding of the mechanical properties, beams composed of FGM with porosity must be thoroughly investigated.

Chen et al. [14] investigated the static bending and elastic buckling of porous-FGB depending on Timoshenko-beam-theory (TBT). The mechanical properties and porosity are gradually distribution along the height of FGB. Two distribution models are used to describe the porosity distribution. They calculated the transverse displacement and critical buckling using Ritz method with various supporting type, i.e., boundary conditions (BC) to investigate the impact of length/height ratio and porosity. Also, Wattanasakulpong and Chaikittiratana [15] analyzed the transverse beam vibration under various boundary conditions considering the impact of inertia and shear effect and using TBT. To determine the material parameters of the FGBs, such as the volume fraction, the revised rule of mixture was utilized of porosity assuming three types of porous distribution models across the beam section. They calculated the natural frequencies of porous-FGBs under various supported ends and they found that "FGM beams with uniform porosity distribution have a stronger influence on natural frequencies compared to uneven porous-FGBs. This method is effective for eigenvalue analysis of structural problems".

In 2016, Al Rjoub and Hamad [16] improved new analytical solution to investigate free vibration of porous-FGB with different boundary conditions using Euler-Bernoulli-theory (EBT) and Timoshenko-beam-theory (TBT). They employed the transfer-matrix method in order to compute non-dimensional frequencies of pure and porous FGBs assuming uniform distribution of porosities along the height of the beam. Their results showed the effects of material index, volume fraction, boundary conditions, length/height ratio, and porosity on dimensionless frequency parameters of pure and porous FGBs. Galeban et al. [17] investigated the free vibration of porous beams by deriving the governing equations using Euler-Bernoulli theory. The mechanical properties are changes in the cross-section porous beam. The results showed that the porosity, mass, fluid compressibility, distribution of pores and supporting types, i.e., boundary conditions (BC) impact the natural frequencies of beams. Hinged-hinged beams exhibit unique natural frequency behavior.

Fouda et al. [18] investigated the porosity influence on the mechanical behavior of FGB assuming Euler-Bernoulli theory and power distribution along the thickness of FG beam to construct the governing equations and the kinematic relations. They concluded that “the static deflection and buckling load affected the porosity and the material distribution parameter, while the frequency is influenced by the material index and has a stronger correlation with porosity in the proposed model”. Akbaş [19] studied the forced vibrations of porous FGBs subjected to dynamic loading. In deep beam, the mechanical properties vary across the height of FGB due to the presence of porosity. The finite element method is utilized to resolve the problems within the "plane solid continua model". Amir et al. [20] analyzed the vibration of a sandwich micro-beam composed of a porous core and face sheets reinforced with FG carbon nanotubes and supported on a Winkler-Pasternak substrate. They considered in their analysis the beam's response to thermal load and employed "sinusoidal-shear-deformation-beam-theory" (SSDBT) to define the displacement components assuming that the properties of the core and facial sheets are dispersed throughout its height. They applied "Hamilton’s principle" and "Navier’s method" to drive the equations of motion to study the influence of porosity value and function, small scale, various kinds of CNTs distribution, and geometrical beam size. The study findings indicate that the frequency decreases when the porosity increases.

In 2020, Zanoosi [21] studied the vibration of a porous-FG micro-beam while considering the influence of thermal effects by utilizing the elasticity of modified strain gradient theory (MSGT). They used MSGT instead of MCST to analyze the size-dependent porous structures and they applied "Hamilton's principle" and the "Navier solution" to determine the natural frequency the micro-beam under simply supported conditions. They investigated the influence of temperature changes, length scale, material index, length/height ratio, and volume fraction of porosity on FG micro-beam natural frequency. They concluded that the temperature increases reduced the micro-beam's frequency. Zghal et al. [22] used a "refined-mixed-finite-element-beam" model for studying static bending analysis of porous-FGBs. They used two distributions of porosity, even and uneven and to ascertain the mechanical properties of porous-FGB, use a modified power-law model. They conducted a parametric study to examine how the boundary conditions, porosity coefficient, types of porosity distributions and power law index affect the displacement and stress of the FGBs. They concluded that “the porosity parameter in modern structural design of cannot be overstated. The percentage of porosity present in the structure can have a profound impact on its overall performance and response”. Rahmani et al. [23] used various theories in order to investigate bending, buckling, and study of vibration of porous-FGBs. They employed the finite element approach to solve problems assuming a power law model to represent the mechanical properties along the height of the FGB, and they used "Hamilton's principle" to drive the equation of motion. They investigated the impact of porosity exponents, power-law index, and supporting type on the bending behavior, buckling characteristics, and natural frequencies of the beam using ANSYS Software. They found that the displacement, critical buckling and frequencies increase when length /height ratio increases. Also, the nondimensional frequencies decrease with decreasing values of the porosity index and power-law index. Ultimately, the FG beam becomes less rigid as porosity and power-law index increase, leading to higher deflection and decreased critical buckling load”. Karamanli and Vo [24], used three alternative models of porosity distribution to examine the size-dependent behavior of porous-FG micro-beams using a quasi-3D theory and a modified-strain-gradient-theory (MSGT). They applied the rule of mixture for determining material properties and material length scale parameters (MLSPs) based on the porosity index, thickness, and material index, thereby enabling a comprehensive understanding of the material characteristics. They checked the accuracy of the suggested model and studied the impacts of varying MLSP, material index, porosity index, and boundary condition on responses of porous-FG micro-beam.

Rahmani [25] modified the "high-order-sandwich-beam-theory" to investigate the frequency responses of clamped-free sandwich beams with homogeneous face sheets and a FG core using a modified power-law model and two porosity distribution models to characterize the material qualities along its height. They used "Hamilton’s principle" and a "Galerkin method" to solve Governing equations of motion. They showed a comparison with specific scenarios outlined in existing literature. They concluded that “as a power-law index, temperature, cross-section, length, and porosity volume fraction increase, the basic frequency parameter falls. Conversely, an increase in the wave number leads to an increase in the frequency”. Anirudh et al. [26] used a "trigonometric-shear-deformation-theory" to examine the transverse vibration and buckling behavior of curved beams composed of porous-FG graphene-reinforced nanocomposites to study the impact of various theories on the static and dynamic performance. They used Lagrangian equations of motion and finite element analysis to derive the equilibrium equations. They conducted a comprehensive study to assess how parameters such as radius of curvature, porosity coefficient, length-to-thickness ratio, distribution pattern of porosity and graphene platelets, platelet geometry, and supporting types affect static bending, elastic stability and free vibration.

Ton [5] studied the impact of porosity on the free vibration behavior of a porous FGB using simple beam theory. He examined the effects of boundary conditions, porosity distribution models, and material distribution on the free vibration problem of porous FGB. He concluded that “the results align well with existing references and confirm the applicability of classical beam theory in analyzing functionally graded porous beams. The mechanical information provided may prove useful to designers for specific purposes”. Adıyaman [1] analyzes the porous-FGB's free vibration behavior utilizing a higher-order-shear-deformation-theory (HOSDT). He used Lagrange’s principle, power law model, and different porosity distribution functions to derive the governing equations and applied FEM to solve these equations. He calculated the dimensionless natural frequencies and studied the influence of material properties, supporting types, and porosity on the nondimensional frequency and mode shape. His results showed that “The mode shapes have comparable traits. despite the influence of porosity on frequencies”. Nguyen et al. [27] investigated three models of porosity distribution using a straightforward two-variable shear-deformation-theory (SDT) to analyze the transverse displacement, bucking, and vibration problems of porous-FGBs. They used Lagrange’s principle, the power law model to derive the governing equations. They applied exponential approximations to determine the buckling. load, displacement., frequency, and. stress of beams under various supporting types using ANSYS model. They studied the influence of boundary conditions, porosity parameters, porous distribution pattern, height-to-span ratio, and shear deformation on the stress., deflection., frequency., and critical buckling load of beams. Turan et al. [28], analyzed the. buckling behavior and porous-FGB's unrestricted vibration under different boundary conditions considering first-order-shear-deformation-theory (FOSDT). They used the power law model, Lagrange's principle, and the Ritz method to derive and solve the equations of motion analytically. While they used Finite-Element-Method (FEM) and artificial-neural-network-method (ANN) to solve the problem numerically. They investigated the critical buckling loads and normalized natural frequencies for different boundary conditions, porosity coefficient, power-law index, length/height ratio, and porosity distribution models. Their results obtained from the analytical, FEM, and ANN methods were found to be in good agreement.

In this paper, the Finite-Element-Model is built using ANSYS software to calculate the free vibration behavior of porous-FGB and study the impacts of porosity and material distribution and supporting types on the natural frequencies and shape mode, in addition to length/height ratio important parameters, which is don’t study previously.

In order to achieve these points, this paper is divided into: Problem Description: in this part, the configuration of the porous-FGB and the variation of mechanical properties due to materials and porosity distributions are described. In the next part, ANSYS or Finite element Model is described in detail, and the third part Validation of the ANSYS Model deals with the comparison between the frequency results of the ANSYS model and that of available literature. Finally, the part Results and Discussion is used to describe the frequency results due to variation of length/height ratio, power-law index K, porosity distribution, and porosity index (α-index).

The ANSYS model used the element “SHELL281” for simulating the variation of the properties across the height of FGB. The characteristics of “SHELL281” are: “Analysis of thin to moderately-thick shell structures are appropriate for SHELL281. The element consists of eight nodes, each of which has six degrees of freedom: x, y, and z axis translations as well as rotations around those axes. The element only has translational degrees of freedom when the membrane option is used. Applications requiring large rotation, large strain, or both are ideally suited for SHELL281. Nonlinear analyses take into consideration changes in shell thickness. The element takes into consideration the impact of scattered stresses on followers, load stiffness. Layered applications such as sandwich building or composite shell modeling can make use of SHELL281. First-order shear-deformation theory (also known as Mindlin-Reissner shell theory controls modeling accuracy for composite shells. True stress measures and logarithmic strain serve as the foundation for the element formulation. Finite membrane strains are possible due to the element's kinematics stretching. On the other hand, it is believed that the curvature variations within a time increment are minimal” [31] (see Figure 4).

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Figure 4. Geometry of SHELL281 [31]

The ANSYS model of porous-FGB was built by applying the following steps [29, 30]:

1. The FGB is drawn as the top area (L*W) as illustrated in Figure 5(a).

2. The required properties of porous-FGB are estimated by the following points:

(a) The height of beam is divided into (N) parts (in this work N=10). Each part is called “layer”.

(b) The height of each layer is calculated as (height of layer= height of beam/N).

(c) From step (a), the required material properties are calculated for N+1 points along the height of porous-FGB using Eq. (2) and assuming [z_i=zo+((h/N) *(i-1))] where zo=-h/2 and (i) is number of point (1≤i≤(N+1)).

(d) The following formula is used to determine the necessary material qualities for each layer:

$\left(E_{layer}\right)_i=\frac{E\left(z_i\right)+E\left(z_{i+1}\right)}{2}$            (4a)

$\left(\mu_{layer}\right)_i=\frac{\mu\left(z_i\right)+\mu\left(z_{i+1}\right)}{2}$            (4b)

$\left(\rho_{layer}\right)_i=\frac{\rho\left(z_i\right)+\rho\left(z_{i+1}\right)}{2}$            (4c)

3. The (N) set of material properties are input into the ANSYS APDL software using the commend “section” assuming the layer is isotropic material as shown in Figure 5(b).

4. The drawing area is meshing using the element “SHELL281” as shown in Figure 5(c) considering the convergence criteria of element size.

5. Three types of supports are considered in this work, clamped-clamped beam (C-C), simply supported beam (S-S) and clamped-free beam (C-F). The boundary conditions of each type of supports are:

(a) Clamped-Clamped beam (C-C): All degree of freedom (UX, UY, UZ, ROTX, ROTY and ROTZ) of the nodes at edges x=0 and x=L are zero.

(b) Simply Supported beam (S-S): The degree of freedom (UX, UY and UZ) of the nodes at edge x=0 is zero. Also, the degree of freedom (UY and UZ) of the nodes at edge x=L are zero.

(c) Clamped-Free beam (C-F): All degree of freedom (UX, UY, UZ, ROTX, ROTY and ROTZ) of the nodes at edge x=0 only is zero.

The model analysis is selected to analyze the free vibration problem of porous-FGB as shown in Figure 5(d).

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Figure 5. General steps of present ANSYS model

In this work, first, second and third natural frequencies of porous-FGB with three supporting types, C-C, S-S and C-F are calculated using ANSYS APDL and the element “SHELL281”. In additional to the supporting types, the impacts of length/height ratio L/h, power-law index K, type of porosity distribution and porosity index α are investigated.

5.1 Even porous-FGB

Figure 9 illustrates the natural frequency parameters results of even porous-FGB when the porosity index α increases from zero to 0.3 for various supporting type and power-law index K. For first mode, the frequency parameter reduces when the power-law index K rises at any porosity index and any supporting types (see Figure 9(a)). In other side, the impact of porosity index is differing and depending on power-law index. For example, the frequency parameter increases when the porosity index increases at K=0, while, for K=10, For any supporting type, the frequency parameter falls as the porosity index rises. In other hand, the frequency parameter of C-C beam is larger than that of S-S and C-F beams (see Figure 9(a)). The same behavior is found in second and third natural frequencies (see Figure 9(b) and (c)). Also, the value of frequency parameter increases with increasing the mode number.

To explain the above results, the frequency of Euler - Bernoulli beam is considered, as an example, and the general equation are written as [35]:

$\omega_i=\left(\beta_i L\right)^2 * \sqrt[2]{\frac{E_{e q} * I}{\rho_{e q^* * * L^4}}}, i= mode \, number $             (6)

The β_iL value depends on the supporting type, A and I are the cross-section area and second moment of area of beam and their values are constant in this work. The value of  $\sqrt[2]{\frac{E_{e q}}{\rho_{e q}}}$  is the effective part in Eq. (6). This part $\sqrt[2]{\frac{E_{e q}}{\rho_{e q}}}$ depends on modulus ratio of FG beam $\frac{E_{\text {bottom }}}{E_{\text {top }}}$, density ratio of FG beam $\frac{\rho_{\text {bottom }}}{\rho_{\text {top }}}$, power-law index and porosity index. In this work, the modulus and density ratio are constant and smaller than 1 as illustrated in Table 1: modulus ratio $=\frac{70}{380}$ and density ratio $=\frac{2707}{3960}$. According Eq. (1), when $\mathrm{K}=0$ and infinity the equivalent material properties are the material properties of metal and ceramic respectively. When the power-law increases (larger than zero) the equivalent material will be larger than the metal material properties and smaller than the ceramic material properties. According to modulus and density ratio: modulus ratio $=\frac{70}{380}$ and density ratio $=\frac{2707}{3960}$, the equivalent modulus increases with rate larger than that of equivalent density for the same power-law index, therefor, the part ${ }^2 \sqrt{\frac{E_{e q}}{\rho_{e q}}}$ increases with increasing power-law index and this leads to increases frequency.

In order to investigate the impact of porosity index, the porosity in porous-FGB is assumed to effect averagely on the material properties [1]. This means the porosity is found in metal and ceramic materials at the same time and the impact of porosity depends on porosity index and function of porosity distribution in additional to average of the modulus and density of metal and ceramic (see Eq. (2)). In even porous-FGB, the function of porosity distribution is constant and don’t depend on the position in height of beam, i.e., z direction. Generally, the porosity existence causes reduction in material properties elastic modulus and density. For pure metal beam, the reducing rate of modulus due to porosity is larger than the reducing rate of density, therefore, the part $\sqrt[2]{\frac{E}{\rho}}$ reduces and this leads to increase the frequency of pure beam. But for pure metal beam, the reducing rate of modulus due to porosity is smaller than the reducing rate of density, therefore, the part $\sqrt[2]{\frac{E}{\rho}}$ reduces and this leads to decrease the frequency of pure beam. For porous-FGB, i.e., $0 \leq \mathrm{K} \leq \infty$, the increase of porosity index affects reversely on the frequency of beam while increase of the power-law index affects proportionally on the frequency of beam. In other words, the impact of power-law index is opposite to the impact of porosity index and the part $\sqrt[2]{\frac{E_{e q}}{\rho_{e q}}}$ depends on the combination effect of these two indices. Therefore, the part $\sqrt[2]{\frac{E_{e q}}{\rho_{e q}}}$ decreases at high porosity index and at any power-law index. It can see that the frequency decreases when the porosity index increases at high power-law index.

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(a) 1st MODE

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(b) 2nd MODE

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(c) 3rd MODE

Figure 9. The variation of the first, second and third frequency parameters of even porous-FGB due to variation of porosity index (α-index) for different supporting types when L/h=40

Figure 10 shows the natural frequency parameters results of even porous-FGB when the power-law index K increases from zero to 10 for different porosity index and supporting type. At every power-law index K, the frequency parameter falls as the porosity index rises and any supporting types. In other hand, the frequency parameter of C-C beam is larger than that of S-S and C-F beams (see Figure 10(a)). The same behavior is found in second and third natural frequencies (see Figures 10(b) and (c)). Also, the value of frequency parameter increases with increasing the mode number.

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(a) 1st MODE

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(b) 2nd MODE

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(c) 3rd MODE

Figure 10. The variation of the first, second and third parameters of even porous-FGB due to variation of power-law index K for different supporting types when L/h=40

5.2 Uneven-I porous-FGB

Figures 11 and 12 show the impacts of the porosity index and power-law index on the natural frequency parameters of porous-FGB for Uneven-I porosity distribution function and different supporting type. From Figure 11, the first frequency parameter reduces slightly when the porosity index rises at any power-law index and any supporting types (see Figure 11(a)). In other side, the influence of power-law index on the natural frequency parameters appears sharply and take the same profile in first, second and third natural frequency (see Figures 11(b) and (c)). Also, the value of frequency parameter increases with increasing the mode number. For Uneven-I porosity distribution function, the influence of porosity depends on z coordinate and concentrates at the center of FG beam. The influence of Uneven-I porosity distribution function is less than that of even one, therefore, the slight variation in natural frequency parameter is appear as shown in Figure 12.

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(a) 1st MODE

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(b) 2nd MODE

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(c) 3rd MODE

Figure 11. The variation of the first, second and third frequency parameters of uneven-I porous-FGB due to variation of porosity index (α-index) for different supporting types when L/h=40

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(a) 1st MODE

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(b) 2nd MODE

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(c) 3rd MODE

Figure 12. The variation of the first, second and third frequency parameters of uneven-I porous-FGB due to variation of power-law index K for different supporting types when L/h=40

5.3 Uneven-II porous-FGB

Figures 13 and 14 show the impacts of the porosity index and power-law index on the natural frequency parameters of porous-FGB for uneven-II porosity distribution function and different supporting type. From Figure 13, the first frequency parameter decreases when the porosity index increases at any power-law index and any supporting types (see Figure 13(a)). In other side, the effect of power-law index on the natural frequency parameters appears sharply and take the same profile in first, second and third natural frequency (see Figures 13(b) and (c)). Also, the value of frequency parameter increases with increasing the mode number. For uneven-II porosity distribution function, the influence of porosity depends on z coordinate and concentrates at the edges of FG beam. The influence of uneven-II porosity distribution function is larger than that of Uneven-I porosity distribution function and smaller than that of even one. Therefore, Figure 14 shows sharply the variation in natural frequency parameter due to porosity distribution function, supporting type and mode number.

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(a) 1st MODE

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(b) 2nd MODE

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(c) 3rd MODE

Figure 13. The variation of first, second and third frequency parameters of uneven-II porous-FGB due to variation of porosity Index (α-index) for different supporting types when L/h=40

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(a) 1st MODE

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(b) 2nd MODE

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(c) 3rd MODE

Figure 14. The variation of the first, second and third frequency parameters of uneven-II porous-FGB due to variation of power-law index K for different supporting types when L/h=40

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(a) α-index=0

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(b) α-index=0.1

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(c) α-index=0.2

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(d) α-index=0.3

Figure 15. The impact of length-to height ratio on the first natural frequency parameter of the even porous-FGB under different supporting types for different power-law Index and different porosity index

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(a) α-index=0

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(b) α-index=0.1

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(c) α-index=0.2

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(d) α-index=0.3

Figure 16. The impact of length-to-height ratio on the first natural frequency parameter of the uneven-I porous-FGB under different supporting types for different power-law index and different porosity index

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(a) α-index=0

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 (b) α-index=0.1

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(c) α-index=0.2

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(d) α-index=0.3

Figure 17. The impact of length-to-height ratio on the first natural frequency parameter of the uneven-II porous-FGB under different supporting types for different power-law index and different porosity index

5.4 Effect of length/height ratio

The impact of length/height ratio on the first natural frequency parameter of porous, Even, Uneven-I and Uneven-II. FG beam under different supporting types for different power-law and porosity index are shown in Figures 15-17 respectively. Generally, the effect of length-to-height ratio on the first frequency parameter vanishes when the length-to-height ratio is greater than 20 for any supporting type, power-law index and porosity index. Also, the effect of length- to-height ratio on the first frequency parameter of C-C porous-FGB is greater than that of S-S and C-F porous-FGB respectively for any power-law index and porosity index. The reason of this behavior is similar to that described previously. The shear effect due to variation in material properties is small when the length-to-height ratio of porous-FGB increases, therefore the variation in the first frequency parameter decreases.