Introduction

Functionally graded materials (FGM) are considered as special race of composites but different from normal ones, where layers with different properties are combined to form a single structure. The main difference is that due to the techniques used in developing FGMs, the material properties vary following a particular function. The layers are combined on a microscopic level, making it seem like new material. For a long time, it was believed that functionally graded materials are prone to cracks and delamination, unlike normal composite materials, due to their microscopically distributed properties. Still, in recent years, it has been observed that these materials can also develop cracks type defects in severe environments. Therefore, a deep study of cracked FGM structures under different loading conditions becomes essential. In this study, the authors are going to find the variation in natural frequency of a simply supported cracked bidirectional FGM plate with a variation in different parameters such as plate geometry, crack length, porous nature of the material, as well as the variation profile of the material using different functional laws. Also, the buckling behavior of a simply supported cracked bidirectional FGM plate with different parameters such as crack length, plate geometry, material variation profile, and porosity value is studied in the present article since buckling can occur quite a lot in thin plates, such a mode of crisis can often causes the failure of strength. Both studies are performed using ABAQUS software for finite element analysis, in which the plate with multiple layers having different properties is constructed using a python script. As the plate is bidirectionally graded, the material properties vary along the x and z-directions with grading index nx and nz.

FGM has been a hot topic of study in the past few years due to their wide variety of use in places where normal materials cannot hold their own. A study in [1] presented the solution for vibration along with buckling problems for a simple plate cracked from one edge and a plate which would have an crack located internally in the center of the plate. These problems were formulated as dual series equations and diminished to second-type homogenous Fredholm integral equations. Buckling loads for cracked plates at the edge or the center are estimated [2] using hierarchical trigonometric functions to outline the displacement function of the damaged plate. They used a Discrete Kirchoff-Mindlin Triangular element in the analysis of the plate with cracks of different sizes. Buckling response of various cracked elastic plates exposed to tension or compressive load while taking care of the effects of mechanical, geometrical, as well as different boundary conditions, and varying crack lengths and orientations are analyzed [3]. In this case, the nonlinear analysis was performed with crack faces interaction taken into account. Buckling analysis of curvilinearly laminated composite plates for arbitrarily shaped cutouts is examined utilizing isogeometric analysis [4, 5], and the same analysis is performed to periodically supported beams and verified using ABAQUS simulation software [6]. Thermal post-buckling properties of porous composite nanoplates made of an FGM with a central cutout of various shapes are investigated [7] utilizing a surface elastic-based 3D nonlinear formulation. The buckling analysis of homogeneous, isotropic, regular hexagonal thin sheets under in-plane hydrostatic and uniaxial compression is investigated in [8].

Some researchers [9, 10] used the Bernoulli–Euler beam theory, as well as the rotational spring model, to investigate frequency responses as well as elastic buckling of functionally graded materials for different types of boundary conditions, number, and locations of cracks, etc. Similarly, a few studies [11, 12] investigated the buckling behvaiour of the unidirectional FGM plate exposed to thermo-mechanical load using an improved higher-order shear deformation plate theory to compensate for the transverse shear strains by keeping top and bottom faces stress-free. Reddy’s third-order plate theory was used in [13] along with a novel Ritz procedure including additional functions on a unidirectional FGM plate according to P-FGM to find the effect of parameters such as slenderness ratio, power-law coefficient, etc. on the natural frequency of side crack containing FGM plate. Using the extended Q4 formulation [14] showed the FEA of functionally graded crack containing plate to find the natural frequencies. Classic plate theory was used by [15] through the FE approach to find the critical buckling load of cracked FGM plate under uniaxial and biaxial tension and varying parameters such as grading index. Combining the extended isogeometric method, as well as HSDT [16], natural frequencies at different modes of cracked FGM plates of different geometries and the properties varying in the thickness direction according to E-FGM type were found out. The functionally graded structure’s vibration and thermal buckling responses were reported in [17] using FEA and first-order theory. The developments of vibrational studies of small structures utilizing nonlocal/nonclassical continuum theories are reviewed, mainly emphasizing small beam-like structures [18]. The nonlinear large-amplitude natural frequency responses of nanoshells made of porous FGM by taking vibrational mode interactions and surface stress size effects into account are analyzed [19]. A light sandwich plate’s damped vibrational behavior under periodic load for a finite amount of of time is investigated [20]. To estimate the minor effects on the nonlinear instability characteristics of hydrostatic pressurized FGM microsized shells, a nonlocal strain gradient shell model is applied [21]. A detailed review is carried on natural frequency analysis of small-scaled plate-based structures [22]. Lazreg Hadji et al. focused on the eigenvalue responses of an imperfect FG sandwich plate [23] and FG Porous Sandwich Plates [24], which consists of an isotropic homogeneous core that rests on an elastic foundation and porous face sheets made of FGMs. Natural frequencies of imperfect S-FGM and P-FGM beams [25] and porous FG nanobeams using hyperbolic shear deformation beam theory [26] are investigated.

An 8-noded degenerated shell element was used for the extended FEA [27] for buckling analysis of crack containing composite plates, also involving transverse shear deformation effects. Rectangular and circular cracked composite plates were analyzed with varying crack sizes and locations, as well as a number of cracks. Using the extended FE method, [28] incorporated the discrete shear gap method to remove the shear locking on an accurately extended 3-node triangular plate element to study the buckling type failure analysis of crack containing FGM plates exposed to either uniaxial or biaxial compression loads. Singular convolution approach was used [29] in to discretize the differential equation for buckling of the laminated panel. Another study [30] presented the XFEM formulation of vibrational behavior of five different FGM shell models of different geometry with cracks present in them and the effects of different parameters on the natural frequencies of these models. Using Berger’s formulation to include nonlinearity, as well as Galerkin’s method to solve governing differential equation, [31] presented a nonlinear analytical model to show the effect of crack orientation, size, as well as aspect ratio in the vibrational analysis of FGM plates containing a crack. Based on FSDT and Von-Karman’s theory, [32, 33] used an extended Ritz formulation on multilayered composite cracked plates for buckling and post-buckling analysis.

Extended isogeometric analysis (XIGA) [34] was used through the cracked plate, with the help of Reddy’s HSDT using a Ritz approach, [35] presented the buckle and post-buckling analysis of simple cracked plates. In a study, [36] the transient response of a unidirectional porous FGM plate with patterns was presented and the different types of functions generally used in FGM technology were explained. Bifurcation buckling of the FGM plate has been presented [37, 38], and critical buckling loads have been found for all the different boundary conditions for which bifurcation buckling is possible. In another study [39], the vibration and cracked identification of P-FGM plates with through width edge cracks were presented. The crack was modeled as a massless rotational spring, and the SIF was calculated using the Mindlin plate theory and differential quadrature method to determine the spring stiffness. Artificial neural network method was used [40] to predict natural frequencies of simple supported P-FGM plates of different materials, crack lengths, porosity values, crack orientation and power-law index using Navier’s solution after using Hamilton's principle to derive equations of motion. Another study [41]- presented the vibrational response of multidirectional FGM plate using different grading functions, as well as even or uneven porosity values. The material variation in multidirectional grading is explained with the equations of all three laws, also, in [42], the transient response of multidirectional FGM plate using ANSYS, MATLAB and Python code in ABAQUS are computed.

Using the asymptotic numerical method and thin plate kinematics of von Karman, [43] proposed buckling and post-buckling analysis of an FGM plate. In one study [44], a side cracked thin rectangular plate was separated into sub-plates, which were then put into the Hamiltonian system and analytically solved using the symplectic superposition approach. Another study [45] proposed quantification and localization approaches to detect the damage in FGM plates employing Frequency Response Function and ANN, as well as Arithmetic Optimization Algorithm for damage quantification. Some researchers [46, 47] investigated the natural frequency and critical buckling load of multidirectional porous FGM sandwich plates where material properties varied based on Voigt’s micro-mechanical model according to P-FGM distribution. One study [48] reviewed the effect of the formation of a crack on the eigen frequency responses of the FGM component.

It is observed from the review of the articles that a lot of research in the field of vibration and buckling is available on FG structure. The majority of the research focused on the structural analysis of unidirectional FGM with a P-FGM type of grading pattern without damage. Moreover, none of the analyses performed to study the frequency and buckling characteristics of bidirectional FG panels associated with the crack and porosity type of defects. Hence, to address the same, the goal of the present research is to develop a generalized computational facility for the eigenvalue (vibration and buckling) of the graded structure with and without crack and porosity type of defects. Also, the present model utilizes all three kinds of grading patterns (P-FGM, S-FGM and E-FGM). The accuracy of the developed model is verified using a validation test. Using a validation test, the created model’s accuracy is checked. The next step is to conduct the parametric analysis to delibirate the influence of various design factors on the eigenvalue charactaristics of the bidirectional FG panel.

Methodology

The geometry of the bidirectional FG plate is shown in Fig. 1. The grading along thickness and length direction is shown in Fig. 2. Figure 3 displays the crack geometry, and Fig. 4 shows the porosity distribution inside the FG plate. Here a and b denote the length and width of the FG plate, respectively. h denotes the thickness of FG plate. Crack length is denoted by d. The material variation in the FG plate follows various grading patterns as discussed in the below subsection.

Fig. 1
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The geometry of bidirectional FG plate

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Fig. 2
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a The grading along the thickness direction, b The grading along the length direction

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Fig. 3
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Crack geometry

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Fig. 4
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Porosity distribution inside the FG plate

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Material Properties of the Graded Panel

Variation of material properties in the FG structure follows a particular function in a specific direction. Three different forms of material grading patterns are used in the current work considering uni/bidirectional material property variation.

E-FGM

The properties change according to the mathematical exponential function in this material variation profile. The mathematical formulation for the same can be given in terms of material one and material two with varying volume fractions. The mathematical formula for any property, G(z, x), considering bidirectional grading [49], is as follows:

$$G\left( {z,x} \right) = G_{1} e^{{\left( {\frac{ - 1}{2}\ln \left( {\frac{{G_{1} }}{{G_{2} }}} \right)\left( {1 - \frac{2z}{h}} \right)\left( \frac{x}{a} \right)^{{n_{x} }} } \right)}}$$
(1)

Similarly, for bidirectional grading with even porosity (having porosity index λ):

$$G\left( {z,x} \right) = G_{1} e^{{\left( {\frac{ - 1}{2}\ln \left( {\frac{{G_{1} }}{{G_{2} }}} \right)\left( {1 - \frac{2z}{h}} \right)\left( \frac{x}{a} \right)^{{n_{x} }} - 0.5\lambda \ln \left( {\frac{{G_{1} }}{{G_{2} }}} \right)} \right)}}$$
(2)

where, G1 and G2 are the properties of metal and ceramic, respectively. nx is the material variation profiles in the x-direction.

P-FGM

The exponential function provides only one pattern for specific materials combination, but P-FGM provides a number of grading patterns. This grading method uses the volume fraction index to define the property of a layer. The volume fraction has a material variation profile factor. This factor is also known as the volume fraction exponent, or simply, the power-law coefficients. The mathematical formulation [50] of any property, G(z, x), at any location is given by:

$$G(z,x) = (G_{1} - G_{2} )V_{f} + G_{2}$$
(3)

Here the volume fraction (Vf) is defined for P-FGM as follows:

$$V_{f} = \left( {\frac{1}{2} + \frac{z}{h}} \right)^{{n_{z} }} \left( \frac{x}{a} \right)^{{n_{x} }}$$
(4)

where, nz is the volume fraction exponents in the thickness direction.

The expression of G(z, x) for bidirectional grading with even porosity [51] is given by:

$$G(z,x) = (G_{1} - G_{2} )V_{f} + G_{2} - 0.5\lambda (G_{1} + G_{2} )$$
(5)

S-FGM

When a single power-law function FGM is added to multilayered composite structures, the stress concentrations happen at a particular interface where the structure’s material variation is smooth yet changes quickly.

Due to this problem, Chung and Chi in 2006 [50] formulated a new volume fraction using two power-law functions, as expressed in Eq. (6). It is done to make the stress distribution smooth all over the structure.

$$V_{f} = \left\{ \begin{gathered} \begin{array}{*{20}c} {\left[ {1 - 0.5\left( {1 - \frac{2z}{h}} \right)^{{n_{z} }} \left( \frac{x}{a} \right)^{{n_{x} }} } \right]} & {for} & {0 \le z \le \frac{h}{2}} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\left[ {0.5\left( {1 + \frac{2z}{h}} \right)^{{n_{z} }} \left( \frac{x}{a} \right)^{{n_{x} }} } \right]} & {for} & { - \frac{h}{2} \le z \le 0} \\ \end{array} \hfill \\ \end{gathered} \right.$$
(6)

For unidirectional grading, the value of nx will become zero, i.e. the power-law exponent in the length direction will become null.

Methodology

A simulation tool, ABAQUS, provided by Dassault System, is used to perform the FEA of the FGM plate with or without crack and porosity. In ABAQUS, specific steps are required to follow to complete the analysis, which are described below:

  • A 3D planar shell-type element was created in the ‘create part’ section, forming a square plate.

  • Selecting the ‘create partition’ option, partitions along the x-axis are created equal to the number of layers.

  • Created the required number of materials with calculated properties using the materials section.

  • Selecting each partition created, composite layups equal to the number of layers along the x-axis were created with number of plys equal to the number of layers along the z-axis. In this way, a number of layers (side layers × down layers) are created in total, each having its properties according to the function.

  • In the Interaction section, a particular option was selected from the menu bar. Then, the crack \(\to\) assign seam option was selected, and the crack region was chosen for assigning the seam.

  • In the assembly section, create new instance option was selected, and assembly was created.

  • Based on the type of analysis, steps were created in the create step section.

  • SSSS boundary conditions were given to the plate in the load section, and the mechanical load was applied.

  • The mesh section defined mesh geometry and setting, and meshing was created according to the required mesh size.

  • In the job section, a new job was created and then submitted.

  • In the visualization section, the deformed plate was visible. By selecting the results option in the menu bar, eigenvalues for different modes can be found, and the plate orientation on each mode can be seen.

A flow chart procedure for the static and transient deflection analysis of the FG plate is presented in Fig. 5. The layers of the bidirectional FG plate along the length and transverse direction can be seen in Fig. 6 and Fig. 7, respectively.

Fig. 5
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Simulation procedure for eigenvalue analysis of FG plate

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Fig. 6
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Bidirectional graded FGM layers along the x-axis in ABAQUS

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Fig. 7
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Layers along the z-axis in Bidirectional graded plates in ABAQUS

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Automation

The following steps are followed to make the script for the analysis of the FG plate:

  • Using the macro manager option available in ABAQUS, a work macro created and started making the model. Macro for a simple plate, having a small number of layers and SSSS boundary conditions was developed.

  • The composite layup feature was used to provide different properties to each layer.

  • The functions are utilized (Eqs. 1–6) to create materials according to the required grading pattern. The materials are assigned to each layer with their location only.

  • Different scripts were created for different analysis types, such as vibration and buckling analysis, with and without crack. Also, the unidirectional and multidirectional grading formulas were incorporated into other scripts.

  • After the FGM plate was created and analyzed, a simple plate was made, and then a crack was generated in the plate. The coding part for crack generation was recorded using the macro manager.

  • The program for generating cracks in a simple plate was incorporated into different analysis scripts of simple FGM plates.

  • Using the Python scripts, results were computed and validated for the natural frequency and buckling analysis of the cracked multidirectional FGM plate. The new results are then generated for the different parameters.

Results and Discussion

After successfully making the FEA model of cracked bidirectional FGM plate in ABAQUS using python code, the structural responses are studied. To check the correctness of our present model, a verification study is carried out for both the vibration and buckling behavior of the plate. After validation, the natural frequency and buckling characteristics of the composite plate are studied for unidirectionally and bidirectionally graded FGM plates with different grading patterns, and the effect of various parameters such as porosity and power-law indices, aspect ratio, thickness ratio etc. is studied.

Validation

The present method has been validated by relating the current results and available articles. Validation has been done for vibration and buckling conditions separately. For validating the vibrational and buckling responses, the current results are compared with the published values in [14] and [52], respectively.

Validation for Vibration Analysis

The validation of the present model for vibration study has been performed here by comparing the obtained results with the available data in [14] and presented in Table 1. The table shows that the obtained results matches the results in the published data is followed. The first three modes of vibration in ABAQUS are shown in Figs. 8, 9, 10. The natural frequency responses are presented in non-dimensional form for the validation purpose.

$$\Omega = \frac{{\omega a_{{}}^{2} }}{{h\sqrt {\frac{{\rho_{c}^{{}} }}{{E_{c}^{{}} }}} }}$$
(7)

where, ω is the natural frequency of the plate, ρc and Ec, respectively, are the mass density and Young’s modulus of ceramic material. The materials used for validation are Aluminum (Al)/Alumina (Al2O3), having a square shape and thickness ratio of 10. The plate is SSSS at all the edges. The material properties for Al are νm = 0.3, ρm = 2702 kg/m3 and Em = 70 GPa. Similarly, for Al2O3 νc = 0.3, ρc = 3800 kg/m3 and Ec = 380 GPa.

Table 1 The first three non-dimensional natural frequencies of a SSSS square FG plate with a horizontal side crack (a/b = 1, h/a = 0.1, d = 0.5a)
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Fig. 8
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First Mode of vibration in cracked FGM plate

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Fig. 9
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Second mode of vibration in cracked FGM plate

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Fig. 10
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Third mode of vibration in cracked FGM plate

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Validation for Buckling Analysis

The validation of the existing model for buckling analysis has been done by comparing the current results with the published responses [52] and presented in Table 2. The charted numbers indicated the accuracy of the present model as the error among the current and reference values lies within the acceptable range. The materials used are Al/Al2O3, having a square shape and thickness ratio of 10. The plate is SSSS at all the edges. The material properties for Al and Al2O3 are specified in the above subsection. The critical buckling load values are presented in non-dimensional form for the validation study.

$$\lambda_{{{\text{cru}}}} = \lambda_{{{\text{crb}}}} = \frac{{Nb^{2} }}{{\pi^{2} D}}$$
(8)

where, \(\lambda_{{{\text{cru}}}} \,{\text{and}}\,\lambda_{{{\text{crb}}}}\) denote unidirectional and bidirectional critical buckling load, N denote externally applied in-plane mechanical load and D is expressed as:

$$D = \frac{{E_{c} h^{3} }}{{12\left( {1 - \nu^{2} } \right)}}$$
(9)
Table 2 Validation for buckling analysis of FGM plate
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Vibration Analysis

After the verification study, the parametric study for vibration and buckling analysis of FG plate structures is carried out. In the vibration analysis, Aluminum and Alumina are used as the FG materials. SSSS boundary condition is used for all the results, and the natural frequency is converted to non-dimensional form using the formula given in Eq. (7).

Effect of Crack Length and Porosity Index

The frequency responses of bidirectional FG plate are computed for a variety of crack lengths and porosity index and presented in Table 3. Tabulated data show that the values of the frequency declines as the rise in crack length. Also, as the porosity index upsurges, the natural frequency value declined for all the FG plates. With an introduction of crack or porosity, the stiffness of the plate declines, which leads to a decrease of frequency data.

Table 3 Non-dimensional natural frequencies of cracked SSSS bidirectional FG plate (a = b = 1, a/h = 10, nx = 1, nz = 5)
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Effect of Grading Indices (Unidirectional and Bi-Directional)

In Table 4, the impact of power exponents on the non-dimensional frequency of the cracked FG plate is shown. From table, it can be noted that as the grading index increases, the value of frequency decreases. This is due to the fact that as the grading index rises, the volume of metal in the FGM also rises, lowering the stiffness of the structure. Also, the comparison of unidirectional and bidirectional grading is shown here, which indicates that the frequency value decreases with the bidirectional grading as it impacts the stiffness more.

Table 4 Non-dimensional natural frequency of cracked SSSS uni/bidirectional FG plate (a = b = 1, a/h = 10, d = 0.5a)
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Effect of Aspect Ratio

In Table 5, non-dimensional frequency of the bidirectional FG plate are provided for different aspect ratios. The relationship between the structure’s length and stiffness is inverse. The structural stiffness reduces as length increases. Decreased stiffness leads to a fall in the frequency responses. The tabular data follow the same trend, i.e., frequency declines with an rise in aspect ratio (reduction in the width of the plate). Figures 11, 12 show the vibration in the FGM plate with an aspect ratio of 0.5 and 2, respectively.

Table 5 Non-dimensional natural frequency of cracked SSSS bidirectional FG plate (a = 1, a/h = 10, d = 0.5a, nx = 1, nz = 5)
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Fig. 11
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Vibration in FGM plate with aspect ratio 0.5

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Fig. 12
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Vibration in FGM plate with aspect ratio 2

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Effect of Thickness Ratio

In Table 6, the variation of non-dimensional frequency responses for different values of thickness ratio is shown. The stiffness of the FG plate lowered down with an increase in thickness ratio (a/h) or decrease in thickness. Further, the frequency responses of the bidirectional FG plate must decrease with an increase in thickness ratio (reduction of thickness). But tabulated data follows the reverse trend due to the non-dimensional form Eq. (7).

Table 6 Non-dimensional critical buckling load of cracked SSSS bidirectional FG plate (a = b = 1, d = 0.5a, nx = 1, nz = 5)
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Buckling Analysis

After performing the frequency analysis, the model has been expanded to calculate the impact of various geometrical and material characteristics on the mechanical buckling strength,.To do so, a few numerical experimentations are carried out and discussed in the following lines. Aluminum and Alumina are used in the buckling analysis of FG materials. A UDL of magnitude 1 N/m2 is applied, and the SSSS boundary condition is used to compute all the critical buckling load of the plate, which was then converted to non-dimensional form using Eq. (8).

Effect of Crack Length

The effect of crack length on the critical buckling load of the FG plate is presented in Table 7. With an upsurge in crack length, the stiffness of the FG plate decreases, which leads to a decrease in critical buckling load. But tabular data follow the reverse trend due to the non-dimensional form of the critical buckling load. Also, as the porosity index rises, the stiffness is decreased, and the critical buckling load also reduced for all the FG plates.

Table 7 Non-dimensional critical buckling load of cracked SSSS bidirectional FG plate (a = b = 1, a/h = 10, nx = 1, nz = 5)
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Effect of Grading Indices

The critical buckling load for the FG plate with respect to change in power exponents is computed and presented in Table 8. Due to a rise in the power-law exponent, the structure’s stiffness decreases. Decreased stiffness leads to the decrease in the critical buckling load values of the FG plate, and tabulated data also show the same trend. The comparison of unidirectional and bidirectional grading is also shown in Table 8. Critical buckling load is less with bidirectional grading when compared to the unidirectional one.

Table 8 Non-dimensional critical buckling load of cracked SSSS uni/bidirectional FG plate (a = b = 1, a/h = 10, d = 0.5a)
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Effect of Aspect Ratio

The critical buckling loads evaluated by changing the aspect ratio, grading type and porosity index values presented in Table 9. With an increase in length, the stiffness of any structure decreases. The stiffness and buckling load are directly proportional to each other. Therefore, the buckling load should follow decreasing trend with an increase in aspect ratio, but tabular data depicts the reverse trend due to the non-dimensional form.

Table 9 Non-dimensional critical buckling load of cracked SSSS bidirectional FG plate (a = 1, a/h = 10, d = 0.5a, nx = 1, nz = 5)
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Effect of Thickness Ratio

In Table 10, the variation of critical buckling loads for various values of thickness ratio and porosity index is shown. With a rise in thickness ratio (a/h) or reduction in total thickness, the structural stiffness falls down, which leads to the reduction in critical buckling load. But critical buckling load in tabular data increases with an increase in thickness ratio due to the non-dimensional form expressed in Eq. (8). It can also observed from the tabular data that the increase in critical buckling load due to an increase in thickness ratio is far less than that due to increase in porosity index.

Table 10 Non-dimensional critical buckling load of cracked SSSS bidirectional FG plate (a = b = 1, d = 0.5a, nx = 1, nz = 5)
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Conclusion

The python code has been utilized to develop a batch module to compute the eigenvalues (natural frequency responses and critical buckling load) of the bidirectional graded cracked porous structure. The model’s efficacy has been proved by comparing the results with the data available from different domains. A few generalized understandings are obtained, i.e., the cracks cause a substantial decrement in total structural stiffness, which, in turn, reduces the final results adequately. Thin structural components are affected due to the presence of a crack in specific orientations, like in the lengthier edge of the rectangular shape. Also, it is possible to cause the propagation of the damage, which may further reduce the structural integrity and strength enormously.

Similarly, the porosity decreases the total stiffness and associated responses like the frequency and the critical buckling load parameters. The study indicates a minor influence of porosity for the exponential type of graded structure compared to all other grading patterns. Additionally, the final failures may affect factors like grading index and type of grading, as both cause an extreme change in the overall effective material properties. Finally, the study helps the designer choose the type of grading, porosity and other relevant parameters with damage for a suitable grading structure. The results show that the P-FGM will be the most flexible structural configuration compared to different types, including the damage and/or porosity.