Bi-directional Functionally Graded Research Articles

Electroelastic wave dispersion in the rotary piezoelectric NEMS sensors/actuators via nonlocal strain gradient theory

This article introduces a computational means for investigating the electroelastic nonlinear wave dispersion traits of the nano-dimension sandwich pipe, which is composed of a core formed of a bi-directional functionally graded (Bi-FG) material, together with a piezoelectric sensor/actuator. A combination of Hamilton’s principle, first-order shear deformation, along with Von-Karman nonlinearity, is used for modeling and obtaining the nonlinear governing equations of the nano-sized sandwich pipe connected to a piezoelectric part. The nonlinear governing equations to obtain the nonlinear phase velocity of the current system are determined using a combination of analytical and multiple scales approaches. Due to some computational cost for choosing the precise values of both the nonlocality factor and length scale of the nanopipe in the laboratory, for the first time in this research, with the aid of COMSOL multi-physics finite element simulation, the results are verified, and new outcomes for obtaining the exact functions for nonlocal and length scale factors is presented. In addition, an artificial neural network (ANN) is utilized in this study for the prediction of the results. The mathematical and finite element results were used to train the ANN. A newly presented optimization algorithm is exploited for the first time for optimizing the ANN parameters concluding higher accuracy of the ANN predictions. Consequently, to explore the influences of the location of the piezoelectric patch, nonlocality and length scale factors, and applied voltage parameter on the phase velocity characteristics of the nano-dimension sandwich pipe made of a Bi-FG core and electrically patch, an effort is performed. As an applicable result that can be useful in the related nano-industries, the current work presents exact nonlocal and length scale functions for different conditions.

Symmetric/Asymmetric Vibrational Characteristics of Bi-Directional Functionally Graded Annular Microplates

In this study, a free vibration model of bi-directional functionally graded (BD-FG) annular microplates is established based on the modified couple stress theory (MCST) and the first-order shear deformation theory (FSDT), and the corresponding governing equations are derived from Hamilton’s principle. The annular microplates are assumed to be composed of metal and ceramic materials, and the equivalent material properties are a combination of power-law function and exponential ones. The generalized differential quadrature method (GDQM) is used to obtain the vibration frequencies and mode displacements for the C–C, S–C, C–S, and S–S (S: simply supported; C: clamped) annular microplates. The convergence and validity of the proposed model are verified through selective numerical examples. Further, the effects of gradient index, material length scale parameter, boundary conditions, and geometrical dimensions on the symmetric/asymmetric vibration frequencies of the annular microplate are explored. The modal assurance criterion is applied to estimate the influence of the radial gradient index and material length scale parameter on different-order mode displacements of the annular microplate. The results show that: (1) the vibration frequencies of the annular microplate are sensitive to changes in the geometry and boundary conditions; (2) compared with the axial gradient index, the radial gradient index not only affects the vibration frequencies of the annular microplate, but also impacts the distribution of the highest and lowest points of the vibration mode displacements as well as the vibration mode nodal lines; (3) as the material length scale parameter increases, the vibration frequencies are significantly increased, and the higher-order vibration modes of the annular microplate also change.

Nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator: Introducing machine learning approach to measure nonlinear phase velocities

In today's rapidly changing corporate environment, business executives are turning to cutting-edge technology to solve complex problems efficiently. A machine learning approach allows for a more thorough understanding and quantification of decision-making outcomes, leading to increased confidence in business decisions. In this work for the first time, machine learning approach and mathematical simulation are used to model nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator. Using mathematical modeling, the current work studies the nonlinear phase velocity in a sandwich nanoshell composed of a bi-directional functionally graded (BD-FG) core and piezoelectric patch. The modeling uses first-order shear deformation theory with the fairly thick hypothesis to describe the displacement fields of piezoelectric layers. For core layer, a higher order shear deformation theory with terms of higher order in the Taylor series expansion is presented. The nanosystem is accurately modeled using nonlocal strain gradient theory, incorporating nonlocal and length scale characteristics. The solution approach section outlines the multiple scale method for the time domain and the harmonic solution methodology for the displacement domain to address nonlinear equations. To reduce the computational expenses, a machine-learning approach is introduced to address the nonlinear vibration issue in the current study. Validating the conclusions of the current study by comparing them with machine-learning solutions and another published paper can confirm their accuracy. The results section presents the impacts of factors like area of piezoelectric patch, length scale, applied voltage, nonlocality, geometry condition and FG power index on the nonlinear phase velocity of the BD-FG nanoshell coupled with piezoelectric patch.

THERMAL ANALYSIS OF BI-DIRECTIONAL FUNCTIONALLY GRADED PLATES SUBJECTED TO HEAT GENERATION

The two-dimensional steady-state heat transfer problem is analyzed under the most general boundary conditions in a heterogeneous environment. It is assumed that the internal heat generation varies arbitrarily in two directions, while the thermal conductivity coefficient of the material is bi-directly (thickness-length) graded with the Mori-Tanaka model. The linear inhomogeneous partial differential equations produced under these conditions may not be solved by conventional methods. The governing equation of the system is numerically solved with high accuracy using the 2D pseudospectral Chebyshev method. Benchmark problems are used to check the accuracy of the method. It has been observed that the effect of volumetric heat generation on the temperature distribution is more than the heat transmission coefficient. Besides, this method is simple and effective and can be easily applied to such problems.

Dynamic stability and frequency responses of the tilted curved nanopipes in a supersonic airflow via 2D hybrid nonlocal strain gradient theory

In the presented work, vibrational behavior related to a curved nanopipe which is under supersonic airflow and conveying fluid flow, is examined. To model the size-dependent nanopipe, the Quasi-2D hybrid type of nonlocal strain gradient theory (QHNSGT) is presented. Formulations are obtained by means of Hamilton’s principle for bi-directional functionally graded (Bi-FG) nanopipe. Also, formulations are solved by means of the generalized differential quadrature method (GDQM). By taking into account that the fluid flow is infinite, incompressible, uniform flow, Newtonian, laminar, as well, as viscous, and with the help of the Navier-Stokes equation, the fluid-structure interaction is obtained. A quasi-2D hybrid type of higher-order shear deformation theory is employed to introduce the displacement fields. The verification section shows that the results of this paper are very near to the results of the published article in the literature. One of the important findings of the current research is critical values of the Mach number could have increased with the aid of increasing the rigidity of edges and decreasing fluid flow velocity. Another marvelous output is that opening angle and airflow stability have an indirect relation, and increasing the opening angle provides an expansion in the unstable area.

HSDT mesh-free method for free vibration analysis of sandwich structures

The current study presents a meshless collocation method based on radial basis functions (MC-RBF) to analyze the vibrational response associated with a sandwich circular cylindrical nanopipe in a supersonic airflow and conveying fluid flow. The quasi-2D hybrid type of nonlocal strain gradient theory (QHNSGT) as a higher order shear deformation theory (HSDT) has been presented to model the size-dependent nanopipe equations extracted using the energy method for a sandwich nanopipe. The sandwich nanostructure is made of metal, ceramic, and bi-directional functionally graded (Bi-FG) material, respectively, lower, topper, and middle surfaces. By taking into account that the fluid flow is infinite, incompressible, uniform flow, Newtonian, laminar, as well as viscous, and with the help of the Navier-Stokes equation, the fluid-structure interaction is obtained. Also, the displacement fields are defined via the Quasi-2D hybrid type of high-order shear deformation theory. The governing equations are discretized via the MC-RBF method to obtain the natural frequencies. The verification section shows that the results of the current research are very near to the results of the published article in the literature. By using COMSOL multi-physics software, the results are verified, and the outputs show that the velocity of fluid flow has a significant role in the vibrational responses of the current composite structure.

Data-driven solution for nonlinear wave dispersion of sandwich nanoshell with piezoelectric patch layers

It’s all about speed and scale in the quickly evolving corporate world of today, and business executives are looking to cutting-edge technology to help them find answers to their most difficult problems. A data-driven strategy provides the means to more thoroughly comprehend, confirm, and quantify decision-making outcomes. It is possible to feel greater confidence in the decisions made by the designers in a data-driven business. So, using this essential technique, and via mathematical modeling, the nonlinear wave dispersion of a sandwich nanoshell made of a Bi-directional functionally graded (BD-FG) core, and piezoelectric patch is investigated. In the mathematical modeling, first-order shear deformation theory with moderately thick hypothesis is used to model the displacement fields of the current work for both layers. After that, compatibility conditions are presented to model interface equations between the layers. For correct and exact modeling of the current nanosystem, nonlocal strain gradient theory with length scale and nonlocal parameters is presented. For solving the nonlinear equations, the harmonic solution technic for the displacement domain, and the multiple scale method for the time domain are presented in the solution procedure section. After that, to decrease the computational cost of the current work, the data-driven solution is presented for solving the nonlinear vibration problem. Comparing the results of current work with the outcomes of data-driven solutions and another published article can verify the results. Finally, in the results section, the influences of various factors such as length scale, nonlocal, applied voltage, area of piezoelectric patch, FG power index, and geometry condition of the structure on the nonlinear phase velocity of the BD-FG nanoshell coupled with piezoelectric patch is presented.

Nonlinear free vibration of bi-directional functionally graded porous plates

Understanding the nonlinear behavior of advanced engineering structures is of great importance in supporting the analysis, design, and manufacturing processes. The primary objective of this paper, therefore, is to focus on exploring the nonlinear free vibrational characteristics of bi-directional functionally graded (FG) plates considering internal pores under various conditions. To accomplish this, we present an approximate numerical model derived based on the framework of NURBS-based isogeometric analysis (IGA), in conjunction with four-variable refined plate theory (RPT) and the von Kármán assumption, to reckon the displacement field. The mechanical properties of bi-directional FG plate models can smoothly and continuously vary along two in-plane directions according to a power law. Additionally, it is assumed that internal porosities in the matrix materials can be dispersed into two independent patterns, either the even or uneven porosity distributions. The nonlinearity in free vibration, assessed by means of the nonlinear-to-linear frequency ratio related to the central deflection amplitude, can be obtained using an iterative scheme with a displacement control strategy. The accuracy and effectiveness of the current numerical model are demonstrated through a comparison with existing solutions. Comprehensive parametric investigations are subsequently carried out to provide insight into the impact of various factors on the nonlinear free vibration characteristics of plate structures under different conditions. The new insights obtained from this research could serve as valuable benchmark outcomes as well as contribute to a more comprehensive understanding of the nonlinear responses in future analysis and design processes.

On the Free Vibration Behavior of Temperature-Dependent Bidirectional Functionally Graded Curved Porous Beams

Based on a quasi-3D beam formulation, study on frequencies of bidirectional functionally graded (BDFG) curved porous beam in thermal environment is carried out. In severe thermal conditions, where BDFGs are considered to be highly efficient, temperature affects the properties of the BDFG beams. Consequently, this study focuses on the free vibration of porous BDFG curved beams by considering the effective temperature-dependent properties as a function of position across the thickness and thermal rise. The displacement field used contains indeterminate terms and involves a few variables to define. The mechanical characteristics of the curved beam are supposed to be temperature-dependent and graded in both axial and transverse direction depending on various porosity patterns. The governing equations of the simply supported curved porous beam are derived using the principal of virtual works and are then solved utilizing the Navier solution. The accuracy of the current formulation is tested by checking its results with other relevant publications found in the literature. Through a parametric study, we examine the impact of materials properties temperature-dependence, grading indexes, porosity distribution, radius of curvature and other parameters on the frequencies of curved bidirectional functionally graded porous beams. The results reveal that these parameters have a great influence on the free vibration response of porous BDFG curved beams. These results can serve as reference solutions for future investigations.

Dynamic characteristics of cracked simply supported bidirectional functionally graded Rayleigh beam

This paper investigates the dynamic behavior of cracked Rayleigh beams constructed from bidirectional functionally graded (BDFG) materials under simple boundary conditions. A torsional massless spring is employed to model the beam's open crack type. The vibration equations are obtained using Hamilton's principle. The graded beam material properties are varied throughout the thickness based on the power-law distribution and in the longitudinal direction using the exponential material distribution. To solve the dynamic equations, Galerkin's approach is employed. The paper evaluates the impacts of the axial index, gradient property index, beam modulus ratio, and crack parameters on the natural frequencies of the FG beam. The results indicate that the dimensionless natural frequencies of intact graded beams decrease with increased gradient index k. In contrast, they increase with a rise in the modulus ratio. Additionally, the results demonstrate that an increase in the crack depth ratio decreases dimensionless natural frequencies.

Vibration Analysis of Bidirectional Functionally Graded Porous Plates Using MITC3+

The article presents the vibration behavior of bidirectional functionally graded (BDFG) porous plates using the new triangular 3-node element (a cubic bubble function for the interpolation of the rotations to enrich the bending displacements) which is called the MITC3+ element. The BDFG porous plates have material properties varying in two directions with even and uneven porosity distribution. Based on Hamilton’s principle, the motion equations for BDFG plates are derived. Then, some examples are performed to verify the accuracy and reliability of the proposed method. Finally, the new numerical results on the free vibration response of BDFG porous plates are provided.

Nonlinear free vibrations analysis of truncated conical shells made of bidirectional functionally graded materials

For the first time, present paper addresses the nonlinear vibrations analysis of a truncated conical shell made of bidirectional functionally graded (BDFG) materials whose mechanical properties varied continuously through the thickness and the length directions of the conical shell based on the power law distribution. Accordingly, the set of nonlinear equation of motions of the conical shell are derived based on first-order shear deformation theory (FSDT) and von Karman strain-displacement relations using Hamilton’s principle. System of partial differential equations of the BDFG conical shell are transformed into time-dependent ordinary differential equations using Galerkin approach. Afterward, the nonlinear equations are analytically solved by means of modified Poincaré–Lindstedt method, and the nonlinear frequency of the BDFG conical shell is obtained. For verification purpose, the present outcomes are compared with those available in previous researches. Eventually, the effect of longitudinal and transverse gradient indexes, vibration amplitude, and geometrical parameters on the nonlinear frequency of the BDFG conical shell is investigated. The present research may give benchmark solutions and some guidelines for designing BDFG truncated conical shells.

Novel incremental procedure in solving nonlinear static response of 2D-FG porous plates

The manuscript presents a nonlinear mathematical model capable to investigate the nonlinear bending response of Bi-directional functionally graded (BDFG) plate resting on elastic foundations including large deformations in the sense of von Kármán. A unified higher order shear plate theory is used to implement the shear influence with parabolic distribution through thickness direction. The gradation of materials is portrayed by power law function through axial (x-direction) and thickness (z-direction). The governing equations consist of a set of four nonlinear coupled partial differential equations. The assumption that the material properties change in the x- direction complicates the governing equations since they have variable-coefficients. Winkler–Pasternak model is exploited to present the elastic foundations with normal and shear deformations. In this work, a novel incremental–iterative method is proposed, and some tricky computational issues are performed to ensure stable and fast convergence rates even with large incremental load steps. The differential/integral quadrature method (DIQM) is developed to numerically discretize the variable-coefficient governing equations on a rectangular plate with different boundary conditions. Parametric studies are presented to illustrate the impact of nonlinear strains, gradation indices, geometrical properties, and boundary conditions on the transverse displacement, normal and shear stresses.

Thermo-mechanical vibration and stability behaviors of bi-directional FG nano-pipe conveying fluid

This work focuses on the thermo-mechanical modeling of a size-dependent pipe made of bi-directional functionally graded (FG) materials and conveying fluid through its inner wall. The material properties of this FG nano-scale pipe along radial and axial directions are assumed to be separately varying according to the power-law and exponential distributions. The governing equations of the FG nano-scale pipe are derived according to nonlocal strain gradient theory and Hamilton’s principle, and interactions between the size-dependent pipe and its inner fluid are revealed via size-dependent constitutive relation. The frequency response of the pipe and its critical flow velocity are calculated via generalized differential quadrature method. After validation of the proposed model by comparing with previous works, a detailed parametric study is conducted to fully examine the coupling effect of material distribution and various parameters including small size, aspect ratio, boundary constraint, temperature change and flow velocity on the frequency response of the FG nano-pipe. Simultaneously, the critical flow velocity of inner fluid under which the divergence phenomenon appears is also predicted and physically explained in detail. Numerical results show that bi-directional FG design is especially beneficial to improve the stability of FG nano-pipe, which can provide a guide to design advanced micro/nanofluidic devices for bio-engineering applications.

Analytical solutions for vibration of Bi-directional functionally graded nonlocal nanobeams

This article provides analytical solutions for the vibration of bidirectional functionally graded (BDFG) nanobeams. The material characteristics of BDFG nanobeams vary along the axial and the thickness directions. The nonlocal elasticity theory of Eringen is followed to model the small-scale effects. The nanobeam's vibrational behavior is formulated by Euler-Bernoulli and Timoshenko beam theories. Hamilton's principle is used to derive the governing equations of motion. Using the Laplace transform, analytical solutions to the governing equations are established, and sets of closed-form frequency equations for four boundary conditions of nonlocal nanobeams are presented. The effects of material constants in both the axial and the thickness directions are investigated. The accuracy of the presented frequency equations is demonstrated by comparisons with a few numerical results available in the literature. It is discovered that when the material parameter along the axis increases, clamped nanobeams have a higher natural frequency, but simply supported, cantilevered, and propped cantilevered nanobeams have a lower. Nonetheless, a rising material parameter along the thickness direction exhibited identical decreasing patterns in natural frequency for all boundary conditions.

Free vibration analysis of bi-directional functionally graded cylindrical shells with varying thickness

The article presents a semi-analytical solution for the free vibration problem in bi-directional functionally graded (FG) cylindrical shells with varying thickness, utilizing a modified shear deformation shell theory (MSDT). In this model, the mechanical properties of the cylindrical shells change along both the axial and thickness directions. Classical analytical methods for shell structures often face limitations in terms of boundary conditions, transverse shear deformation, and rotary inertia, making the use of MSDT and Galerkin procedure necessary for solving the problem of free vibration analysis in this study. The boundary conditions of the bi-directional FG cylindrical shells are chosen to be simply supported and clamped, and the presented method is compared to relevant results in the literature to validate its effectiveness. Moreover, a comprehensive parametric investigation is conducted to evaluate the influence of material and geometrical parameters on the natural frequencies. The results show that the natural frequency of the FG cylindrical shells is significantly affected by the material and geometrical parameters, including the power-law index, the thickness variation, the aspect ratio, and the boundary conditions.

Nonlocal strain gradient finite element procedure for hygro-thermal vibration analysis of bidirectional functionally graded porous nanobeams

ABSTRACT This paper presents the hygro-thermal vibration analysis of bidirectional functionally graded (BDFG) porous nanobeams resting on elastic foundation based on a nonlocal strain gradient theory and the finite element method. The effects of temperature and moisture on structures are assumed to cause tension load in the plane and do not change the material's mechanical properties. Nanobeams are made from functionally graded (FG) materials which the mechanical properties change along with the thickness (z-axis) and length (x-axis) of nanobeams according to the power-law model. The porosity distribution is considered to be even and uneven. For the first time, a two-node beam element with nine degrees of freedom at each node is developed to analyze the hygro-thermal vibration behavior in the nanoscale. The elastic foundation is used in this work to be the Winkler-Pasternak type. Applying Hamilton's principle based on the refined higher-order shear deformation beam theory (RBT) and the~nonlocal strain gradient model, governing equations of nanobeams are derived. The accuracy of the proposed method is verified by comparing the obtained numerical results with those of the published works in the literature. The nonlocal coefficient makes the nanobeam softer, while the strain gradient coefficient makes it stiffer. In addition, the effects of the nonlocal coefficient, porosity coefficient, foundation stiffness, boundary conditions on the natural frequency of nanobeams are investigated in detail.

Effect of material distribution on bending and buckling response of a bidirectional FG beam exposed to a combined transverses and variable axially loads

This article considers a bidirectional functionally graded (BDFG) beam with various distributions of volume fraction. The impact of these distributions on the buckling and bending response of BDFG beam with general boundary conditions is investigated via a quasi-3D solution. It is assumed that the beam is exposed to a set of in-plane varying compressive loads and there mechanical characteristics change in both directions. Also, the BDFG beam is considered to be exposed different external mechanical loads. Hamilton’s principle is employed to derive the governing equations, which are then solved using analytical solution to obtain buckling and bending characteristics. The precision of the current formulation is investigated via a comparison with available data in literature. Some numerical results are presented and discussed to assess the effect of different parameters such type of volume fraction, boundary condition, varying load, and beam geometry on the buckling and bending of BDFG beam.

Size-dependent vibrations of bi-directional functionally graded porous beams under moving loads incorporating thickness effect

This article analyzes the forced and free vibrations of bi-directional functionally graded (BDFG) porous nanobeams under moving loads by considering the thickness effect based on the nonlocal strain gradient theory (NSGT). It is assumed that the material properties of the system are graded across the transverse and axial directions according to power and exponential laws, respectively. Various porosity models, including even, uneven, and asymmetric porosity distribution patterns, are considered to model the porosity effects. The dynamic equation of the system is derived by implementing the physical neutral surface concept. Analytical and numerical approaches are adopted to acquire the vibration response of the system. Mathematical closed-form expressions are provided for various dynamic phenomena, such as two types of cancelation, two types of resonance, maximum resonance, cancelation disappearance, and resonance disappearance. In addition, for the first time in this article, it is demonstrated that BDFG porous nanobeams under moving loads are prone to experience the double cancelation phenomenon. Several comparison studies with published data are conducted to validate the results. Also, comprehensive parametric studies are carried out to elucidate the impacts of system parameters such as thickness power index, axial gradient parameter, scale parameter ratio, axial force, porosity coefficient, and porosity distribution pattern on the dynamic magnification factor, critical load speed, and maximum free-response. The outcomes established that by considering the thickness effect, the stiffening behavior of the system is amplified depending on the slenderness ratio. Also, it is concluded that, among the considered porous models, the critical load speed for the system with an even porosity distribution is more dependent on the thickness power index.

Three-dimensional natural frequency investigation of bidirectional FG truncated thick hollow cone

PurposeThe authors examined the numerical natural frequency analysis of a 2D functionally graded (FG) truncated thick hollow cone using 3D elasticity theory.Design/methodology/approachThe material properties of the 2D-FGM (two dimensional-functionally graded materials) cone are graded along the radial and axial axes of the cone using a power–law distribution. The eigenvalue problem was solved using finite element analysis (FEA) employing graded hexahedral elements, and the verification of the finite element approach was assessed by comparing the current solution to earlier experimental studies.FindingsThe effects of semivertex angle, material distribution and the cone configuration on the natural frequencies have been analyzed. For various semivertex angles, thickness, length and power law exponents, many results in the form of natural frequencies and mode shapes are presented for the 2D-FGM cone. As a result, the effects of the given parameters were addressed, and the results were compared, demonstrating the direct efficiency of raising the power–law exponents and cone thickness on the rise of natural frequencies.Originality/valueFor the first time, the numerical natural frequency analysis of a 2D-FG truncated thick hollow truncated cone based on 3D equations of elasticity has been investigated. The material properties of the truncated cone have been distributed along two directions, which has not been considered before in any research for the truncated thick cone. The reason for using these innovative volume fraction functions is the lack of accurate coverage by functions that are available in the literature (Asemi et al., 2011; Babaei et al. 2021).