Generalized Shear Deformation Theory Research Articles

Thermal postbuckling and thermally induced postbuckled flutter of tri-directional functionally graded plates in yawed supersonic flow

The current work examines the thermal postbuckling, aero-elastic flutter and thermally induced postbuckled flutter about a static equilibrium state of tri-directional functionally graded (TFGM) rectangular and tapered plates in yawed supersonic flow. Based on the von Kármán nonlinear strains, the general higher-order shear deformation theory (GHSDT) and the first-order piston theory, the governing equations of motion for TFGM plates in yawed supersonic flow are established via the Hamilton's principle. The isogemetric analysis (IGA), the load continue strategy associated with the Newton Raphson iterative technique are exploited synthetically to capture the postbuckling paths, and then the postbuckled flutter behaviors are acquired with the static equilibrium state being obtained. Comparative and convergence studies are provided to improve reliabilities of the present material model, formulae and code implementations. Subsequently, detailed parametric investigations are carried out to evaluate the influences of material volume fractions, boundary conditions and airflow angles on the thermal postbuckling, aero-elastic flutter and postbuckled flutter behaviors of TFGM plates. The results show that it is the in-plane volume fractions, rather than the thickness volume fraction, that exert a greater influence on the thermal postbuckling and flutter behaviors of TFGM plates. Furthermore, TFGM plates exhibit more diverse postbuckling paths and more complex deformations due to the inhomogeneity of the in-plane material properties compared with z-directional FGM plates.

Isogeometric 3D optimal designs of functionally graded triply periodic minimal surface plates

Recent research attention has been drawn to the interdisciplinary engineering applications of functionally graded triply periodic minimal surface (FG-TPMS) lattice structures due to their lightweight yet high mechanical properties. These features are especially important in the dynamic structural behaviors yet the TPMS gradation has not been studied in multi-dimensions. This paper presents the establishment of the optimization problem finding the 3D optimal designs of FG-TPMS plates under free vibration by assigning the relative density to each control point and applying the concept of NURBS-based interpolation for the continuous effective mechanical properties. The general 3D case of the constraint on total material volume is derived from the existing 1D case, which is required for the 3D optimization problem. Isogeometric analysis and generalized shear deformation theory are utilized to analyze the free vibration behavior of the FG-TPMS plates. Verification with the 1D case and a parametric study of the 3D FG-TPMS plate are conducted. Optimization examples include both basic and complex geometries. Unseen-before optimal 3D FG-TPMS designs with significantly elevated natural frequencies are obtained. The physical meaning of the optimal solutions can be explained by the similarity to porous structures in nature: less dense in the center or at free end, and more dense at the constrained boundaries and surfaces.

Simultaneous optimal tri-directional distribution of material and porosity in functionally graded plates under free vibration

This paper for the first time attempts to find the simultaneously optimal three-dimensional distribution of porosity and material phases in functionally graded materials (FGMs) model that altogether maximize the natural frequency of porous FG plates. By using the same concept of non uniform rational B-splines (NURBS) based interpolation for the material distribution, the problem of finding the optimal porosity distribution is enabled. An upper bound of porosity is pointed out, which is necessary for considering the porosity design variables. Generalized shear deformation theory (GSDT) in the framework of isogeometric analysis (IGA) for free vibration analysis of the porous FG plates is verified with previous works. The findings are: in the optimal design of material only, the uniformly distributed pores have a neutral or detrimental effect on 4-side constrained plates but slightly increase the frequency for the cantilever plate; and with that same amount of porosity, the simultaneously optimal distributions of pores, similar to those in bio-materials or natural bones in the way that are most foam-like in the center or at the furthest from the boundaries, result in lighter yet of significantly elevated vibration frequency structures.

Accelerating tri-directional material distribution optimization in functionally graded plates with an adaptive design control point variable selection

In this paper, the problem of optimizing tri-directional material distribution in functionally graded (FG) plates under static loads to minimize the compliance is studied. Generalized shear deformation theory (GSDT) in the framework of isogeometric analysis is employed as the analyzer for efficiency and accuracy, and a non-uniform rational B-spline function is used for depicting the material distribution. As the number of control points in fine 3D design meshes, i.e., the number of optimization variables, can be too large for an algorithm to find a practically feasible design, we propose an adaptive variable selection mechanism. It gradually selects control points at important regions as variables by considering neighbor material variance, hence accelerates the optimization process. Various plate geometries are considered to validate the present approach. The findings confirm that the design mesh should be fine, and that the proposed variable selection strategy enables the optimization algorithm to find much more refined designs even in cases of large dimensions while also reducing computation.

A general higher-order shear deformation theory for buckling and free vibration analysis of laminated thin-walled composite I-beams

A general higher-order shear deformation theory for buckling and free vibration analysis of laminated thin-walled composite I-beams is proposed in this paper. It is based on a unified nonlinear variation of shear strains in the wall thickness and then theoretical formulation is derived in the general form which can recover the previous conventional theories such as classical and first-order thin-walled beam theory. Series-type solutions with hybrid shape functions are developed for different boundary conditions. Numerical examples are performed on laminated thin-walled composite I-beams with arbitrary lay-ups. Effects of transverse shear strains, fiber angles and slenderness ratio on the critical buckling loads and natural frequencies of the beams are reported.

Dynamic response of porous E-FGM thick microplate resting on elastic foundation subjected to moving load with acceleration

For the first time, dynamic analysis of exponentially functionally graded material (E-FGM) including porosities is investigated. As a first endeavor, forced vibration response of thick microplates made of porous E-FGM when subjected to moving loads with an acceleration in speed is studied in this research study. The role of a two-parameter elastic foundation is also included. A powerful mathematical formulation supported by the assumptions of the general third-order shear deformation theory (GTSDT) as well as the modified couple stress theory (MCST) is developed to find an accurate model for thick size-dependent microplates. The effective material properties of imperfect FGM, which change exponentially along the z-direction, are established via developing a modified rule of mixture which includes the porosity imperfection. The governing equations for simply-supported microplate is found using a virtual work of Hamilton principle, and afterwards they are solved by developing a state space method (SSM) in conjunction with a set of mathematical series. The numerical results attempt to indicate the influence of variation in volume fraction index, porosity index, elastic foundations, and small-scale parameter, highlighted by different load speeds, acceleration ratios, and time histories of moving load under upper surface of E-FGM thick microplate. The parametrical studies can be used in better designing of micro/nanostructures made of porous FGMs under accelerated moving loads.

Small size-effect isogeometric analysis for linear and nonlinear responses of porous metal foam microplate

The main objective of this paper is to explore the linear and nonlinear responses of porous metal foam micro-plate using a modified couple stress theory coupled with isogeometric analysis. The material properties of metal foam micro-plate are derived from three porosity variation models, namely uniform, symmetric and asymmetric variation across the thickness of the microplate. The stress-strain and the modified couple stress constitutive relations are obtained using a seventh order distributed function based on the general shear deformation theory. Furthermore, the Hamilton’s principle is utilized to establish the size-effect equilibrium equations for metal foam micro-plate. The presented numerical solution is able to capture the small size effect of microplate by considering the material length scale parameter in the classical continuum theory. The reported results demonstrate the effect of small size-effect, porosity variation, porosity coefficient, slenderness ratio on free vibration, linear and nonlinear static bending of porous metal foam micro-plate. The role of pattern porosity variation in the nonlinear responses becomes more important by increasing the porosity coefficient. The pattern of symmetric porosity presents the best performance of linear and nonlinear responses.

Thermomechanical analysis of snap-buckling phenomenon in long FG-CNTRC cylindrical panels resting on nonlinear elastic foundation

Snap-buckling phenomenon in an infinitely long nanocomposite cylindrical panel subjected to uniformly distributed transverse pressure loading is analyzed in this research. It is assumed that the functionally graded (FG) carbon nanotube reinforced composite (CNTRC) shell is resting on nonlinear elastic foundation. The uniform thermal field is also included into the formulation. Thermomechanical properties of the shell are assumed to be temperature dependent. The CNTs distribution pattern can be uniform or functionally graded through the shell thickness. The shell is formulated using a general high-order shear deformation theory. Governing equilibrium equations of the shell which undergoes thermal–mechanical coupling loading are established via the virtual displacement principle. To capture the large deflections, the von Kármán type of kinematic assumptions is utilized. Two types of edge conditions are considered for the shell with infinite length which are immovable simply supported and clamped–clamped. To obtain the limit buckling load of the shell, the two-step perturbation technique and Galerkin procedure are used. Results of this study reveal that the snap-buckling phenomenon can take place in a shallow cylindrical panel subjected to lateral pressure loading. This attractive behavior is observed due to the immovability of the flat edges of the shell. It is shown that the snap-buckling phenomenon in a long cylindrical panel is highly dependent upon the elastic foundations, thermal environments, geometrical parameters and material properties.

Free vibration and snap-through instability of FG-CNTRC shallow arches supported on nonlinear elastic foundation

This paper analytically investigates the nonlinear free vibration and snap-through instability of nanocomposite shallow arches reinforced with carbon nanotubes (CNTs). The functionally graded (FG) carbon nanotube reinforced composite (CNTRC) arch with shallow curvature is analysed under uniformly distributed transverse loading. The arch is assumed to rest on a three-parameter nonlinear elastic foundation in a uniform temperature field. Thermo-mechanical properties of the arch are graded through the thickness and are considered to be temperature-dependent. The equations of motion are established based on a general high-order shear deformation theory. These nonlinear equations are derived by Hamilton’s principle within the framework of the von Kármán assumption. The static equilibrium equations and dynamic equations of motion are solved analytically for the arch with immovable simply supported edges. The two-step perturbation technique and Galerkin method are implemented to obtain the closed-form solutions. The novel results illustrate the influences of CNT distribution pattern, foundation stiffness and geometrical parameters on the linear/nonlinear frequency and snap-through instability of the arch. It is shown that the minimum value of limit loads/frequencies belongs to the FG-O type of nanocomposite arches and the maximum ones are obtained in FG-X pattern.

Free Vibration of Functionally Graded Carbon Nanotube-reinforced Doubly-curved Shells

Background:: Carbon nanotubes (CNTs) reinforced structures are the main elements of structural equipment. Hence a wide range of investigations has been performed on the response of these structures. A lot of studies covered the static and dynamic phenomenon of CNTs reinforced beams, plates and shells. However, there is no study on the free vibration analysis of a doubly-curved nano-size shell made of CNTs reinforced composite materials. Methods:: This work utilized a general third-order shear deformation theory to model the nanoshell where the general strain gradient theory is used in order to capture both nonlocality and strain gradient size-dependency. The Navier solution solving procedure is adopted to solve the partial differential equations (PDEs) and get the natural frequency of the system which is obtained through the Hamilton principle. Results:: The current study shows the importance of small-scale coefficients. The natural frequency increases with rising the strain gradient-size dependency which is because of stiffness enhancement, while the natural frequency decreases by increasing the nonlocality. In addition, the numerical examples covered the CNTs distribution patterns. Conclusion:: This work also studied the importance of shell panel’s shape. It has been observed that spherical shell panel has a higher frequency compared to the hyperbolic one. Furthermore, the frequency of the system increases with growing length-to-thickness ration.

Study on nonlinear vibrations of temperature- and size-dependent FG porous arches on elastic foundation using nonlocal strain gradient theory

A nonlocal strain gradient theory is developed in this paper to study the large amplitude vibrations of arches made of functionally graded (FG) porous material. The case of shallow arches resting on nonlinear elastic foundation is modeled via a general higher-order shear deformation theory. The third-order model of Reddy, the first-order model of Timoshenko, and the classical model of Euler–Bernoulli are analyzed. Thermomechanical properties of the arch exposed to the uniform thermal field are assumed to be temperature dependent. The nonlinear motion equations of the arch are established by employing Hamilton’s principle and the von Karman type of geometric nonlinearity. The two-step perturbation technique and the Galerkin method are utilized to solve the nonlinear governing equations. The size-dependent linear and nonlinear frequencies of the arch are obtained for the immovable pinned-pinned boundary conditions. The comparison studies are performed to verify the present solution method with the provided data in the literature, and a good agreement is observed. The novel parametric studies covered in this research include the effects of several parameters such as elastic foundation, nonlocal and length scale parameters, porosity, temperature field, power law index, and geometrical parameters on the frequencies of FG porous arches in detail.

A novel general higher-order shear deformation theory for static, vibration and thermal buckling analysis of the functionally graded plates

This paper proposes a new general framework of higher-order shear deformation theory (HSDT) and solves the structural responses of the functionally graded (FG) plates using novel exponential shape functions for the Ritz method. Based on the fundamental equations of the elasticity theory, the displacement field is expanded in a unified form which can recover to many different shear deformation plate theories such as zeroth-order shear deformation plate theory, third-order shear deformation plate theory, various HSDTs and refined four-unknown HSDTs. The characteristic equations of motion are derived from Lagrange’s equations. Ritz-type solutions are developed for bending, free vibration and thermal buckling analysis of the FG plates with various boundary conditions. Three types of temperature variation through the thickness are considered. Numerical results are compared with those from previous studies to verify the accuracy and validity of the present theory. In addition, a parametric study is also performed to investigate the effects of the material parameters, side-to-thickness ratio, temperature rise and boundary conditions on the structural responses of the FG plates.

Analysis of functionally graded thick-walled cylinders with high order shear deformation theories under non-uniform pressure

This paper intends to investigate the static behavior and stress analysis of functionally graded (FG) thick cylinders under constant and non-uniform distributed internal pressures based on a general shear deformation theory. Third-order and sinusoidal shear deformation theories in particular are employed to indicate this achievement that these two theories can capture the behavior of shear deformable thick-walled cylinders. The nonlinear governing differential equations are discretized by generalized differential quadrature method, and then the discretized equations are solved by the Newton–Raphson iterative method. The results obtained in the present analysis are compared to those acquired by the first-order shear deformation theory. The results indicate that considering higher-order shear deformation effects leads to significant changes compared with the first order shear deformation theory. In addition, this study shows that the results obtained by Reddy and Touratier shear deformation theories are the same. This reveals that both of them are convenient to use for FG thick cylinders.

Material optimization of tri-directional functionally graded plates by using deep neural network and isogeometric multimesh design approach

The paper is aimed at enhancing computational performance for optimizing the material distribution of tri-directional functionally graded (FG) plates. We exploit advantages of using a non-uniform rational B-spline (NURBS) basis function for describing material distribution varying through all three directions of functionally graded (FG) plates. Two-dimensional free vibration and buckling behaviors of multi-directional (1D, 2D and 3D) FG plates analyzed by using a combination of generalized shear deformation theory (GSDT) and isogeometric analysis (IGA) is first proposed. This approach can help to save a significant amount of computational cost while still ensure the accuracy of the solutions. The effectiveness and reliability of the present method are demonstrated by comparing it to other methods in the literature. The obtained results are in excellent agreement with the reference ones. More importantly, data sets consisting of input-output pairs are randomly generated from the analysis process through iterations for the training process in deep neural networks (DNN). DNN is utilized as an analysis tool to supplant finite element analysis to reduce computational cost. By using DNN, behaviors of the multi-directional FG plates are directly predicted from those material distributions. Optimal material distributions of tri-directional FG plates under free vibration or compression in various volume fraction constraints are found by using modified symbiotic organisms search (mSOS) algorithm for the first time. Moreover, an isogeometric multimesh design technique is also used to diminish a large number of design variables in optimization. Optimal results obtained by DNN are compared with those of IGA to verify the effectiveness of the proposed method.

A reliability-based optimization approach for material and thickness composition of multidirectional functionally graded plates

A reliability-based design optimization (RBDO) methodology is first proposed to simultaneously optimize material and thickness distribution of multidirectional functionally graded (MFG) plates for compliance minimization under uncertainties of design variables and system parameters. The modified sequential optimization and reliability assessment (MSORA) method is integrated into the recently suggested adaptive hybrid evolutionary firefly algorithm (AHEFA) to release a novel RBDO approach which is the so-called MSORA-AHEFA. The MSORA is refined from its original for computational cost savings by eliminating the iterative process of finding most probable points (MPPs). An isogeometric multimesh design (IMD) approach is presented to form two separate non-uniform rational B-spline (NURBS) surfaces via the k−refinement strategy. In which, a coarser design NURBS one is employed to define design variables of the ceramic volume fraction and z−axis coordinate of the top side coincidentally assigned at each of control points. A finer analysis NURBS one constructed by the isogeometric analysis (IGA) and a generalized shear deformation theory (GSDT) is used for bending analyses of MFG plates with arbitrary thickness. Applying such two separately defined surfaces dramatically lessens the number of design variables and the computational cost in optimization problems, yet still manifesting optimal profiles properly. Meanwhile, mechanical behavior of the plate in analysis ones is precisely simulated as well. For the validation of the suggested MSORA-AHEFA, its results for the benchmark welded beam are compared with those of formerly published work. Several numerical examples regarding RBDO of MFG plates are then executed to attest the effectiveness of the current paradigm.

Vibrational behavior of doubly curved smart sandwich shells with FG-CNTRC face sheets and FG porous core

As a first endeavor, the free flexural vibration behavior of doubly curved complete and incomplete sandwich shells with functionally graded (FG) porous core, FG carbon nanotube reinforced composite (FG-CNTRC) face sheets and integrated piezoelectric layers is investigated. The variable radii shells with the three most common types of geometries, i.e., elliptical, cycloid and parabolic, are considered. The system equations are derived based on the general higher-order shear deformation theory and Maxwell's equation. The generalized differential quadrature (GDQ) method is employed to discretize the governing partial differential equations subjected to different boundary conditions. The accuracy and reliability of the approach are verified by comparing the results with the existing solutions in open literature. The effects of porosity parameter and porosity distribution through the thickness direction, carbon nanotube (CNT) volume fraction, different boundary conditions and various shell geometrical parameters on the flexural vibrational behavior of the smart sandwich shell structures are investigated and useful results are presented.

An isogeometric multimesh design approach for size and shape optimization of multidirectional functionally graded plates

This article firstly presents a novel numerical methodology to concurrently optimize material distribution (size) and thickness variation (shape) of multidirectional functionally graded (MFG) plates under free vibration within the isogeometric analysis (IGA) framework. An isogeometric multimesh design (IMD) approach is proposed to generate two distinct non-uniform rational B-spline (NURBS) surfaces via the k-refinement strategy. A finer analysis one relied upon a combination of the IGA and a generalized shear deformation theory (GSDT) is utilized for the unknown solution approximation in finite element analyses (FEAs). Whilst the other coarser design one is employed for the exact geometry representation as well as the optimal material and thickness depiction. Size and shape design variables are in turn the ceramic volume fraction and z-axis coordinate of the top side of the MFG plate coincidentally assigned to each of control points on this surface. Flexibly utilizing such two surfaces helps diminish a large number of design variables and considerably save the computational cost in optimization problems, yet still appropriately manifesting optimal material and thickness profiles. Additionally, this definition accurately simulates mechanical behavior of MFG plates in analysis ones as well. A recently developed derivative-free adaptive hybrid evolutionary firefly algorithm (AHEFA) is used to solve constrained frequency maximization problems. Several numerical examples are executed to verify the effectiveness and robustness of the present paradigm.

Vibration of Triangular Functionally Graded Carbon Nanotubes Reinforced Composite Plates with Elastically Restrained Edges in Thermal Environment

As a first attempt, the vibrational behavior of triangular functionally graded carbon nanotubes reinforced composite (FG-CNTRC) isosceles plates with elastically restrained edges under thermal environment is studied. The governing equations are derived based on a unified generalized shear deformation theory, which can easily be degenerated to the other two-dimensional theories. The Chebyshev–Ritz method together with a mapping technique is employed to derive the eigenfrequency equations. The fast rate of convergence of the method is demonstrated numerically, and its accuracy is verified by comparing the results in the limit cases with some existing solutions in the literature. Then, the effects of the carbon nanotubes (CNTs) distribution in thickness direction together with the geometrical parameters, temperature rise and the elastic coefficients of the edge restraints on the frequency parameters are investigated. It is shown that the induced thermal stresses increase the impact of the spring elastic coefficients on the frequency parameters.

Bending and free vibration analyses of in-plane bi-directional functionally graded plates with variable thickness using isogeometric analysis

This article is firstly concerned with bending and free vibration analyses of in-plane bi-directional functionally graded (IBFG) plates with variable thickness in the framework of isogeometric analysis (IGA). The plate thickness is smoothly altered in both x- and y-axes by a predetermined power law. Two types of power-law material models with the symmetrical and asymmetrical volume fraction distribution are suggested to characterize the in-plane material inhomogeneity. A non-uniform rational B-spline (NURBS) surface for simultaneously representing both variable thickness and volume fraction distribution of each constituent is employed. By using the k-refinement strategy, the C0-continuous requirement at symmetrical material interfaces can be achieved, yet still ensuring material gradations elsewhere owing to the prominent advantage of NURBS basis functions in easily controlling continuity. Effective material properties are then evaluated by either the rule of mixture or the Mori-Tanaka scheme. An analysis NURBS surface separately created with the foregoing NURBS surface is utilized to exactly describe geometry and approximately solve unknown solutions in finite element analysis (FEA) based on the IGA associated with a generalized shear deformation theory (GSDT). The Galerkin C1-continuous isogeometric finite element model is therefore simply achieved due to the possibility of flexibly meeting high-order derivatives and continuity of analysis NURBS functions. In addition, no shear correction factors exist in the present formulation, although shear deformation effects are still considered. The influences of variable thickness, material property, length-to-thickness ratio, boundary condition on bending and free vibration responses are investigated and discussed in detail through several numerical examples.

NURBS-based modeling and analysis for free vibration and buckling problems of in-plane bi-directional functionally graded plates

ABSTRACTIn this article, a novel idea of utilizing a non-uniform rational B-spline (NURBS)-based material mesh independently defined with the analysis mesh to represent the material distribution is introduced to free vibration and buckling problems of in-plane bi-directional functionally graded (IBFG) plates. Two power-law material models with the symmetrical and asymmetrical volume fraction distribution are proposed as the first experiment. By applying the refinement scheme, the continuous condition at symmetrical interfaces of material profiles can be easily achieved, while material gradations are still guaranteed elsewhere due to the outstanding advantage of material NURBS basis functions in controlling continuity. Either the rule of mixture or the Mori–Tanaka scheme is then used to estimate effective material properties. The analysis mesh is constructed by generalized shear deformation theory (GSDT)-based isogeometric analysis (IGA) for exactly modeling geometrical domains and approximately solving unknown solutions in finite element analysis (FEA). Accordingly, the continuous requirement of the Galerkin isogeometric finite element model is simply met owing to the possibility of flexibly fulfilling high-order derivatives and continuity of analysis NUBRS functions. Additionally, the present formulation is also completely free from shear correction factors, yet still considering shear deformation influences. Several numerical examples are presented to demonstrate the performance and effectiveness of the proposed method. The effects of material gradations, aspect ratios, and different boundary conditions on IBFG plate responses are examined in detail as well.