Sigmoid Law Research Articles

Printable functionally graded tibial implant for TAR: FE study comparing implant materials, FGM properties, and implant designs

Tibial implants with functionally graded material (FGM) for total ankle replacement (TAR) can provide stiffness similar to the host tibia bone. The FGM implants with low stiffness reduce stress shielding but may increase implant-bone micromotion. A trade-off between stress shielding and implant-bone micromotion is required if FGMs are to substitute traditionally used Ti and CoCr metal implants. The FGM properties such as material gradation law and volume fraction index may influence the performance of FGM implants. Along with the FGM properties, the design of FGM implants may also have a role to play. The objective of this study was to examine FGM tibial implants for TAR, by comparing implant materials, FGM properties, and implant designs. For this purpose, finite element analysis (FEA) was conducted on 3D FE models of the intact and the implanted tibia bone. The tibial implants were composed of CoCr and Ti, besides them, the FGM of Ti and HA was developed. The FGM implants were modelled using exponential, power, and sigmoid laws. Additionally, for power and sigmoid laws, different volume fraction indices were taken. The effect of implant design was observed by using keel type and stem type TAR fixation designs. The results indicated that FGM implants are better than traditional metal implants. The power law is most suitable for developing FGM implants because it reduces stress shielding. For both power law and sigmoid law, low values of the volume fraction index are preferrable. Therefore, FGM implant developed using power law with 0.1 vol fraction index is ideal with the lowest stress shielding and marginally increased implant-bone micromotion. FGM implants are more useful for keel type fixation design than stem type design. To conclude, with FGMs the major complication of stress shielding can be solved and the longevity and durability of TAR implants can be enhanced.

A numerical investigation for the development of functionally graded Ti/HA tibial implant for total ankle replacement: Influence of material gradation law and volume fraction index.

Stress shielding is one of the major concerns for total ankle replacement implants nowadays, because it is responsible for implant-induced bone resorption. The bone resorption contributes to the aseptic loosening and failure of ankle implants in later stages. To reduce the stress shielding, improvements can be made in the implant material by decreasing the elastic mismatch between the implant and the tibia bone. This study proposes a new functionally graded material (FGM) based tibial implant for minimizing the problem of stress shielding. Three-dimensional finite element (FE) models of the intact tibia and the implanted tibiae were created to study the influence of material gradation law and volume fraction index on stress shielding and implant-bone micromotion. Different implant materials were considered that is, cobalt-chromium, titanium (Ti), and FGM with Ti at the bottom and hydroxyapatite (HA) at the top. The FE models of FGM implants were generated by using different volume fractions and the rule of mixtures. The rule of mixtures was used to calculate the FGM properties based on the local volume fraction. The volume fraction was defined by using exponential, power, and sigmoid laws. For the power and sigmoid law varying volume fraction indices (0.1, 0.2, 0.5, 1, 2, and 5) were considered. The geometry resembling STAR® ankle system tibial implant was considered for the present study. The results indicate that FGMs lower stress shielding but also marginally increase implant-bone micromotion; however, the values were within the acceptable limit for bone ingrowth. It is observed that the material gradation law and volume fraction index influence the performance of FGM tibial implants. The tibial implant composed of FGM using power law with a volume fraction index of 0.1 was the preferred option because it showed the least stress shielding.

Winkler‐Pasternak foundation effect on the buckling loads of arbitrarily rigid or restrained supported nonlocal beams made of different FGM and porosity distributions

AbstractThe present research investigates lateral stability of a functionally graded nanobeam using Eringen's differential nonlocal elasticity model under rigid (clamped, pinned, free) and deformable (lateral, rotational restraints) boundary conditions. Sigmoid and power law have been employed as grading laws to study the influence of the material distribution on the snap‐buckling analysis of a nanobeam with arbitrary boundary conditions. Moreover, Fourier sine series with Stokes’ transformation are employed to investigate the effects of boundary conditions on the stability response of nanobeams embedded in a Pasternak foundation. A parametric study has been performed to investigate the effect of deformable boundaries, Pasternak foundation and small‐scale parameters on the stability response of the nanobeam and the results have been presented in a series of tables and figures. It has been observed that consideration of the small‐scale parameter, Pasternak foundation, deformable boundaries and functionally grading index (of sigmoid and power‐law) are essential while analyzing the static stability response. The obtained analytical results may be used as benchmarks in future researches of functionally graded nanobeams embedded in an elastic medium.

Computational modeling of sigmoid functionally graded material (SFGM) plate

The first-order shear deformation theory (FSDT) was used to explore the natural frequency response of functionally graded piezoelectric plates subjected to static electrical and mechanical strain in this present study. A monomorph model for a functionally graded piezoelectric plate with material properties that change according to sigmoid law with respect to plate thickness has been considered. A three-dimensional finite element model with a free tetrahedral element mesh was created using COMSOL 4.2 Multiphysics® software, with each node having three degrees of freedom. Variations in the FGPM plate’s free vibration boundary conditions, composition, and geometry have all been investigated. In free vibration analysis, non-dimensional natural frequency of FGPM plate initially decreasing considerably and then remaining almost constant with the increase in volume fraction index when material property graded by power law. When material properties are varied by sigmoid law, with an increase in volume fraction index, the non-dimensional natural frequency of FGPM plates remains virtually constant. FGPM plates have a lower non-dimensional natural frequency if the thickness to width ratio is greater. Non-dimensional natural frequencies of Clamped-Clamped FGPM plates (C–C–C–C) are greater compared to Clamped-Free FGPM (C–F–C–F) and Simply Supported Free FGPM (S–F–S–F).

Traveling Wave Characteristics of a Rotating Functionally Graded Laminated Cylindrical Shell with General Boundary Conditions

The traveling wave modes of rotating functionally graded laminated cylindrical shells with general boundary conditions are studied by using different admissible displacement functions. Firstly, the effective temperature-dependent material properties of the functionally graded materials (FGMs) are described by the Voigt model and the Sigmoid-type volume fraction. Subsequently, considering Love’s thin shell theory, the motion equations of the FGMs shell subjected to the Coriolis force, the centrifugal inertial force, the hoop initial tension and the internal thermal force are derived. Thereafter, the displacement functions are expanded by six different sets of admissible displacement functions. The Rayleigh–Ritz method is utilized to derive the eigenvalue equations of the rotating FGMs laminated shells. In the end, the convergence rate and computational efficiency of the six polynomial functions are compared and parameter study is carried out. Results indicate that the Legendre orthogonal polynomials (LOP) are the best ones in terms of convergence rate and computational efficiency for predicting the traveling wave modes. The patterns of [Formula: see text] expressed by Sigmoid law are more sensitive to the modal frequency of the shell than the ceramic volume fraction exponents. The stiffness of the axial springs plays a more significant role in the traveling wave modes than the stiffness of other springs.

Nonlinear static and dynamic (deflection/stress) responses of porous functionally graded shell panel and experimental validation

The static and dynamic characteristics of functionally graded (FG) structures have been investigated in this article, considering full geometrical nonlinearity. The finite element (FE) solutions are obtained for the graded panel by modelling through higher-order kinematics (HSDT) and Green-Lagrange strain-displacement relations (GLNST). The desired graded panel properties are obtained through Voigt’s micromechanical approach, considering different grading patterns (exponential, EPL; sigmoid, SGM and power law, POL). Additionally, to maintain the generality of the graded structure, different porosity distribution types (even and uneven, EVP and UEP) are incorporated into the proposed mathematical model. Further, the numerical solutions are obtained using Newmark’s method’s direct iterative technique and constant integration steps. The solution sensitivities are verified for the developed algorithm via adequate convergence and comparison tests. In addition, the experimental validation is performed using the layerwise fabricated luffa-fibre reinforced FG plates to validate the proposed theoretical model’s accuracy. Lastly, the responses are computed using the newly derived model for the variable design-dependent parameters associated with the FG structural geometry and properties. The final deliverables, that is the nonlinear deflection responses (NDFR)/stress values, are discussed under mechanical loadings (static and dynamic).

Dynamic analysis of porosity dependent functionally graded sigmoid piezoelectric (FGSP) plate

This paper analyzes the vibration behaviour of the functionally graded sigmoid piezoelectric (FGSP) plate subjected to electrical and mechanical loading of a porous and non-porous plate. The sigmoid law is used for the variations of material properties along with the direction of thickness with different porosity types, i.e. even and uneven types. The plate incorporates the geometrical nonlinearity with von Kármán strains and First-order shear deformation theory (FSDT) based displacement field. For getting the nonlinear governing equation from Hamilton's principle or energy principle and higher-order numerical formulation with nine nodded isoperimetric elements and the six degrees of freedom per node. The perfect FGSP plate shows multi-periodic excitation with two strong attractors for SSSS boundary conditions, while quasi-periodic responses and 3-T periodic responses are observed for CCCC and CFFF boundary conditions. For evenly distributed porous SSSS FGSP square plate, the center displacement increases with an increase of the porosity from 0.2 to 0.4, while for un-evenly distributed porous center displacement reduces with a rise of the porous exponent from 0.2 to 0.4. The current analysis outcomes can be used for perfect and porous functionally graded piezoelectric smart structure applications under different electromechanical loading environments.

Vibration Response Analysis of Tapered Porous FGM Plate Resting on Elastic Foundation

This paper attempts to demonstrate the vibration response of porous Functionally Graded Material (FGM) plate with variable thickness. For the first time, the porous tapered FGM plate being considered is mathematically modeled and assumed to be resting on a linear, parabolic, sinusoidal, and exponential varying Winkler’s elastic foundation. The even and uneven porosity distribution as a micro-defect is assumed in a tapered FGM plate that varies according to well-defined mathematical rules. The tapered FGM plate, across the edges, is supported with various boundary conditions. Simple Power (P-) and Sigmoid (S-) Law have been chosen for the homogenization of material properties that are tailored in the thickness direction. First-order shear deformation theory (FSDT) is applied to describe the displacement function for computing the strain field. The variational approach has been used to establish the formulation for free vibration response. The equation of motion has been derived using Hamilton’s principle and solved by implementing Galerkin Vlasov’s method. Parametric studies on elastic foundations have been done to explore the results and relevance to the real problems. It was observed that variable foundation has a significant effect on the tapered FGM plate rather than the tapered homogenous plate. In addition, it was witnessed that the effect of variable foundation effect diminishes as a constraint on the edges of the tapered FGM plate increases. Also, some benchmark results based on the porosity effect and the influence of variable elastic foundations are exhibited in this study for future reference.

Electroelastic Static and Vibration Response Analysis of Sigmoid PZT-5A/ Pt-Based Smart Functionally Graded Plate

The current study is focused on the static and vibration analysis of a sigmoid smart functionally graded (SFG) plate resting on the two parameters elastic foundation exposed to electromechanical loading at different boundary conditions. The sigmoid law (S-Law) is used to evaluate the distribution of SFG properties along the thickness direction. The governing equation of the motion of the SFG plate is obtained using first-order shear deformation theory (FSDT) and Hamilton’s principle. The derived governing equation is then solved employing the higher-order finite element method (HFEM) using 9-noded Lagrange quadrilateral elements with 54 degrees of freedom (DOFs) per element. Convergence and comparative tests are carried out for the effectiveness and accuracy of the present approach. Non-dimensional frequency increases negative electric loading [Formula: see text] and reduces positive electric loading at a constant side-to-thickness ratio. The effect of the shear layer parameter on the stress and deflection of the SFG plate is more compared to the Winkler foundation. The obtained results are helpful for the accurate design of the SFG-based structures under electromechanical loading.

Static and vibration response analysis of sigmoid function-based functionally graded piezoelectric non-uniform porous plate

This paper has done static and vibration response analysis of the sigmoid functionally graded piezoelectric (S-FGP) tapered plate under thermo-electric load with different even/uneven type porosity. The governing equations of plate motion are derived using first-order shear deformation theory (FSDT) based displacement fields with Hamilton’s principle, and material property distribution is considered as per sigmoid law in the thickness direction of the FGP plate. The obtained governing equations are solved using the higher-order finite element method (FEM) with nine noded interpolation functions with 63 degrees of freedom (DOFs) per element. Convergence and the comparison study have been performed to exhibit the efficacy of the present study. It is observed that parameters like the non-dimensional center deflection, non-dimensional stress, and non-dimensional frequency are decreased as the material gradient index increases. The positive and negative types of electric loading have considerable effects on the pre/post type buckling configuration of the porous S-FGP tapered plate under thermal effect. The present analysis results can be used in the various smart structure application made of porous FGP materials.

Effect of Porosity on the Thermal Buckling Analysis of Power and Sigmoid Law Functionally Graded Material Sandwich Plates Based on Sinusoidal Shear Deformation Theory

This work examines the effect of porosity distributions on thermal buckling analysis of functionally graded material (FGM) sandwich plates. To consider the porosity effect, five different types of distribution models, even, uneven, logarithmic uneven, linear uneven, and sinusoidal uneven are considered. It is assumed that the FGM faces of the sandwich plate are porous while the ceramic core is nonporous. To investigate the thermal buckling behavior of porous FGM sandwich plates, four different types of thermal loads, such as uniform, linear, nonlinear, and sinusoidal temperature rise along the thickness direction are considered. Effective material properties and thermal expansion coefficients of FGM sandwich plates are evaluated based on Voigt’s micromechanical model considering power law FGM (P-FGM) and sigmoid function FGM (S-FGM). The analytical solution is carried out using Hamilton’s variational principle considering the von Karman nonlinearity. The equilibrium and stability equations are derived based on sinusoidal shear deformation theory (SSDT). Numerical results are obtained to observe the influence of different porosity distributions, porosity coefficients, thermal loadings, and geometrical parameters over critical thermal buckling temperature.

Thermal buckling behavior of power and sigmoid functionally graded material sandwich plates using nonpolynomial shear deformation theories

PurposeThe purpose of this article is to carry out the thermal buckling analysis of power and sigmoid functionally graded material Sandwich plate (P-FGM and S-FGM) under uniform, linear, nonlinear and sinusoidal temperature rise.Design/methodology/approachThermal buckling of FGM Sandwich plates namely, FGM face with ceramic core (Type-A) and homogeneous face layers with FGM core (Type-B), incorporated with nonpolynomial shear deformation theories are considered for an analytical solution in this investigation. Effective material properties and thermal expansion coefficients of FGM Sandwich plates are evaluated based on Voigt's micromechanical model considering power and sigmoid law. The governing equilibrium and stability equations for the thermal buckling analysis are derived based on sinusoidal shear deformation theory (SSDT) and inverse trigonometric shear deformation theory (ITSDT) along with Von Karman nonlinearity. Analytical solutions for thermal buckling are carried out using the principle of minimum potential energy and Navier's solution technique.FindingsCritical buckling temperature of P-FGM and S-FGM Sandwich plates Type-A and B under uniform, linear, non-linear, and sinusoidal temperature rise are obtained and analyzed based on SSDT and ITSDT. Influence of power law, sigmoid law, span to thickness ratio, aspect ratio, volume fraction index, different types of thermal loadings and Sandwich plate types over critical buckling temperature are investigated. An analytical method of solution for thermal buckling of power and sigmoid FGM Sandwich plates with efficient shear deformation theories has been successfully analyzed and validated.Originality/valueThe temperature distribution across FGM plate under a high thermal environment may be uniform, linear, nonlinear, etc. In practice, temperature variation is an unpredictable phenomenon; therefore, it is essential to have a temperature distribution model which can address a sinusoidal temperature variation too. In the present work, a new sinusoidal temperature rise is proposed to describe the effect of sinusoidal temperature variation over critical buckling temperature for P-FGM and S-FGM Sandwich plates. For the first time, the FGM Sandwich plate is modeled using the sigmoid function to investigate the thermal buckling behavior under the uniform, linear, nonlinear and sinusoidal temperature rise. Nonpolynomial shear deformation theories are utilized to obtain the equilibrium and stability equations for thermal buckling analysis of P-FGM and S-FGM Sandwich plates.

Vibration response analysis of sigmoidal functionally graded piezoelectric (FGP) porous plate under thermo-electric environment

In the present work, the vibration response of the porous functionally graded piezoelectric plates with electro-thermal loading using finite element formulations has been studied. A sigmoid law is used for the material property deviation along the thickness direction. The first-order shear deformation theory (FSDT) with von Karman nonlinear strains and Hamilton's principle is used to obtain the governing equations. The governing equations are solved by finite element methods with 9-noded iso-parametric rectangular elements with 7 degrees of freedom (DOFs) per node. The accuracy of the results is evaluated by comparing them with the results available in the literature. The analysis shows that the uneven type porosity distribution gives a larger nondimensional frequency than the even type porosity distribution FGP plate. The non-dimensional frequency decreases with increases in the ratio. Among all the conditions, CCCC boundary conditions have a larger frequency than SSSS, SFSF, SFCF, and CFFF. The obtained frequency under applied electric loading depends on the magnitude and the applied voltage sign. As the temperature change increases with the porosity exponent of the FGP plate, the nondimensional frequency is also increasing. The obtained results can design a porous FGP material-based smart composite structure under a thermo-electric environment.

Natural frequency analysis of sigmoid functionally graded sandwich beams in the framework of high order shear deformation theory

The current study aims to analyze the natural frequencies of sigmoid functionally graded (FG) sandwich beams having various configurations employing the high-order shear deformation theory. To do this, three diverse types of sandwich layup schemes have been constructed. The material characteristics of FG components of sandwich beams are presumed to vary continuously along the thickness direction, as indicated by the volume fraction of constituents specified with sigmoid law distribution, whilst the other components are assumed to be isotropic ceramic/metal. The governing equations are derived for higher-order shear deformation theory, i.e., hyperbolic shear deformation theory (HSDT). Navier's method is used to solve the governing differential equations analytically for simply supported boundary conditions at both ends. To show the efficiency of the current theory, the numerical results are obtained not only for HSDT but also for first-order shear deformation theory (FSDT), in the special case. Similarly, power law distributions of FG sandwich beams are also presented for emphasizing the effectiveness of sigmoid law distribution. Detailed comparisons are performed with available results of open literature to confirm the current formulation. Finally, the effects of the HSDT, variations of material properties, layup schemes, and geometrical characteristics of sandwich beams on the non-dimensional natural frequencies are discussed.

A numerical solution to thermo‐mechanical behavior of temperature dependent rotating functionally graded annulus disks

PurposeThe purpose of this study is to investigate Thermo-mechanical limit elastic speed analysis of functionally graded (FG) rotating disks with the temperature-dependent material properties. Three different material models i.e. power law, sigmoid law and exponential law, along with varying disk profiles, namely, uniform thickness, tapered and exponential disk was considered.Design/methodology/approachThe methodology adopted was variational principle wherein the solution was obtained by Galerkin’s error minimization principle. The Young’s modulus, coefficient of thermal expansion and yield stress variation were considered temperature-dependent.FindingsThe study shows a substantial increase in limit speed as disk profiles change from uniform thickness to exponentially varying thickness. At any radius in a disk, the difference in von Mises stress and yield strength shows the remaining stress-bearing capacity of material at that location.Practical implicationsRotating disks are irreplaceable components in machinery and are used widely from power transmission assemblies (for example, gas turbine disks in an aircraft) to energy storage devices. During operations, these structures are mainly subjected to a combination of mechanical and thermal loadings.Originality/valueThe findings of the present study illustrate the best material models and their grading index, desired for the fabrication of uniform, as well as varying FG disks. Finite element analysis has been performed to validate the present study and good agreement between both the methods is seen.

Limit Elastic Analysis of Functionally Graded Rotating Disks Under Thermo-Mechanical Loading

In this work, the thermo-mechanical limit elastic speed analysis of functionally graded rotating disk has been reported. Three different material models, i.e., power law (P-FGM), sigmoid law (S-FGM), and exponential law (E-FGM), along with varying disk profiles, namely uniform, tapered, and exponential disk profiles, are considered. The methodology adopted is variational principle wherein the solution has been obtained by Galerkin’s error minimization principle. Halpin–Tsai method was used to estimate the modulus, modified rule of mixture for yield strength, and the rule of mixture for density and coefficient of thermal expansion. This study aims to analyze the effects of material models, grading indices, aspect ratio, and disk geometry on disk performance when subjected to combined thermal and mechanical loadings. Finite element analysis has been performed to validate this study and good agreement between both the methods is seen. The study shows a substantial difference in the limit speed for different disk profiles changing from uniform thickness to exponentially varying thickness. The von Mises stress distribution and location of yielding at limit speed are shown for different indices, material models, and disk profiles. In P-FGM, limit speed decreases with the increase in grading indices whereas in E-FGM, limit speed decreases with the decrease in grading indices. For increase in aspect ratio, limit elastic speed decreases in all the cases.

Exact solution for free vibration analysis of linearly varying thickness FGM plate using Galerkin-Vlasov’s method

The present paper investigates the free vibration analysis for functionally graded material plates of linearly varying thickness. A non-polynomial higher order shear deformation theory is used, which is based on inverse hyperbolic shape function for the tapered FGM plate. Three different types of material gradation laws, specifically: a power (P-FGM), exponential (E-FGM), and sigmoid law (S-FGM) are used to calculate the property variation in the thickness direction of FGM plate. The variational principle has been applied to derive the governing differential equation for the plates. Non-dimensional frequencies have been evaluated by considering the semi-analytical approach viz. Galerkin-Vlasov’s method. The accuracy of the preceding formulation has been validated through numerical examples consisting of constant thickness and tapered (variable thickness) plates. The findings obtained by this method are found to be in close agreement with the published results. Parametric studies are then explored for different geometric parameters like taper ratio and boundary conditions. It is deduced that the frequency parameter is maximum for S-FGM tapered plate as compared to E- and P-FGM tapered plate. Consequently, it is concluded that the S-FGM tapered plate is suitable for those engineering structures that are subjected to huge excitations to avoid resonance conditions. In addition, it is found that the taper ratio is significantly affected by the type of constraints on the edges of the tapered FGM plate. Some novel results for FGM plate with variable thickness are also computed that can be used as benchmark results for future reference.

Vibration characteristics of porous FGM plate with variable thickness resting on Pasternak's foundation

In the paper, free vibration analysis of tapered Functionally Graded Material (FGM) plate with the inclusion of porosity has been performed. The tapered porous FGM plate is considered resting on a two-parameter (Winkler and Pasternak) elastic foundation. The displacement model of the kinematic equation for the plates in the present formulation is based on the First-order shear deformation theory (FSDT). The governing equation for free vibration analysis of FGM plates is obtained using Hamilton's principle. Simple power-law, Exponential Law, and Sigmoid law are used for tailored the material properties in the thickness direction of FGM plates. The solution of the resulting partial differential equation is obtained by using Galerkin-Vlasov's method with different boundary conditions. The solutions for uniform and uniform varying thick plates are investigated, and a comparative study is examined by comparing the results obtained with FSDT and Higher-order shear deformation theory (HSDT). The findings of the comparative study with the present approach provide pertinent outcomes for the vibration analysis of tapered FGM plates. The analytical solution for vibration analysis is presented to reveal the effects of porosity parameter, volume exponent, span ratio, aspect ratio, porosity distribution, and boundary conditions. Also, the elastic foundation parameter on tapered FGM plate increases the non-dimensional frequency, and the Pasternak foundation effect always dominates over the Winkler foundation.

Thermal buckling of porous symmetric and non-symmetric sandwich plate with homogenous core and S-FGM face sheets resting on Pasternak foundation

The effect of porosity and thermal environment on buckling responses for the sandwich plates with different boundary conditions is analyzed and discussed in the present study. The author’s recently developed sandwich plate using a new modified sigmoid function based functionally graded material (S-FGM) plate of different symmetric and asymmetric configurations is considered. A new temperature distribution through the thickness based on modified sigmoid law is proposed in which temperature is assumed to be conducted in the thickness direction. Three different types of porosity are considered viz. even, uneven but symmetric and uneven but non-symmetric. A new uneven non-symmetric porosity model is used in which micro-voids are varied in accordance with material property variation in the thickness direction to capture the accurate distribution of voids on the plate. The principle of virtual work is employed to derive equilibrium equations. An exact solution is obtained using the assumed solution with shape functions satisfying the edge boundary conditions. This is implemented using Galerkin Vlasov’s method. The analysis has been performed to study the effect of porosity, geometric configurations, inhomogeneity parameter, and foundation parameter. It is observed that the elastic foundation parameters have the least effect on non-symmetric plate configuration subjected to nonlinear temperature distribution in comparison to other symmetric and non-symmetric plate configurations and temperature distribution. Besides, the non-symmetric plate configuration possesses maximum thermal buckling temperature in comparison to a symmetric configuration. The volume fraction exponent with value 2 is observed as a changeover point where the effect of porosity distribution is null and void for 2-1-1 plate configuration irrespective of boundary conditions. The calculated outcomes and interpretations can be useful as a validation study for the imminent investigation of sandwich S-FGM plates having porosities in the thermal environment.

Vibration of porous functionally graded plates with geometric discontinuities and partial supports

Vibrational study of the porous functionally graded plate with geometric discontinuities and partial supports has been presented in the present paper. The kinematics of functionally graded plate is based on the refined exponential shear deformation theory. The displacement field has been refined by dividing the in-plane and out of the plane displacements into bending and shear components. The theory accounts for the nonlinear transverse shear stress variation along with the thickness with only four unknowns. The closed-form solution (Navier’s solution), as well as FEM-based solution, have been used for the vibration analysis of functionally graded plate. The geometric discontinuities have been incorporated in terms of a circular cut-out of different sizes at the center of the plate. Modified rule of mixtures, modified sigmoid law, and trigonometric law have been used to compute the effective material properties of the functionally graded plate. A C0 continuous iso-parametric FEM formulation has been used to attain the results in the case of FEM solution, and the efficacy of the present solution is demonstrated by comparing the results with the available literature. The results reflect that the porosity inclusion, circular cut-out, and position of the boundary constraints have a notable influence on the fundamental frequency of the functionally graded plate. It is also concluded that after a specific radius of circular cut-out, the vibration response of functionally graded plate exhibits nonlinearity in nature.