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The authors propose an intelligent computational method using the deep feedforward neural network integrated with an improved balance composite motion optimization algorithm to formulate a so-called iBCMO-DNN for solving the stochastic thermal buckling problems of functionally graded porous microplates. In the present approach, the deterministic behaviors of the functionally graded porous microplates were firstly analyzed by the combination of a unified higher-order shear deformation theory, modified strain gradient theory, and Ritz-type series solutions, and then stochastic responses under uncertainty of material properties are obtained by the iBCMO-DNN algorithm. The deep neural network with the long short-term memory model is used as a surrogate method to replace the time-consuming computational model, while the improved balance composite motion optimization is used to search for the optimal solutions. The obtained numerical results for various boundary conditions, uncertainty parameters, and three types of temperature distribution indicated that the proposed method achieves its accuracy and effectiveness in predicting stochastic thermal buckling of the functionally graded porous microplates. The improved balance composite motion optimization algorithm demonstrates a computational time ∼1.7 times faster than its predecessor. Furthermore, integrating a deep neural network into the improved algorithm reduces computational time to about 2/5 compared to the method without it. [ABSTRACT FROM AUTHOR]

• An efficient numerical framework for magneto-electro-elastic functionally graded porous nanoplate. • The generalized weak form formulation for nonlinear bending of magneto-electro-elastic functionally graded porous nanoplate. • The magneto-electro-elastic model based on Maxwell's equations is established. • The proposed model can high-efficiently predict mechanical, magnetic and electrical coupling. • Some novel benchmark numerical results are illustrated and introduced. We propose an effective approach for geometrically nonlinear analysis of magneto-electro-elastic functionally graded (MEE-FG) porous nanoplate. The key idea relies on a size-dependent isogeometric approach, which is considered as a reliable and interesting technique in this area. A generalized model for MEE-FG nanoplates with porosities satisfies assumptions of the nonlocal Eringen's theory based on von Kármán strains, isogeometric approach and the higher-order shear deformation theory, inherently addressing third derivatives in the approximations. By employing the nonlocal theory, the proposed model effectively illustrates the reduction in nanoplate stiffness under the influence of the nonlocal parameter. We also explore porous distributions across the plate thickness, employing both even and uneven functions, and note that an increase in the porous parameter leads to a more pronounced nonlinear deflection. The electric and magnetic potentials in accordance with Maxwell's equation are denoted. The generalized weak formulation of MEE-FG porous nanoplates is derived by the principle of extended virtual displacement. The influence of small-scale parameter, power law exponent, porosity coefficient and porous distributions on the nonlinear deflection of MEE-FG porous nanoplates is investigated. These findings have significant implications for future research in this field. [ABSTRACT FROM AUTHOR]

The purpose of this study is to present a quasi-three-dimensional (quasi-3D) shear deformation theory for static bending and free vibration analyses of porous sandwich functionally graded (FG) plates with graphene nanoplatelets (GPLs) reinforcement. In addition, we propose a novel sandwich plate model with various outstanding features in terms of structural performance. The quasi-3D theory-based isogeometric analysis (IGA) in conjunction with refined plate theory (RPT) is first exploited to capture adequately the thickness stretching effect for porous sandwich FG plate structures reinforced with GPLs. The Non-Uniform Rational B-Splines (NURBS)-based IGA is employed in order to describe exactly the geometry models as well as approximate the unknown field with higher-order derivatives and continuity requirements while the RPT model includes only four essential variables. The sandwich FG plates consist of a core layer containing internal pores reinforced by GPLs and two functionally graded materials (FGMs) skin layers. Effective mechanical properties can be evaluated by employing the Halpin-Tsai model along with the rule of mixture. Various combinations of two porosity distributions and three GPL dispersions in the core layer are thoroughly investigated. Several numerical investigations are conducted to examine the effects of several key parameters on the static bending and free vibration behaviors of sandwich FG plate structures. [ABSTRACT FROM AUTHOR]

Abstract We in this study present a mixed solid-shell finite element formulation for laminated shell analysis. With the commonly adopted truncations applied to the strain components and the Jacobian determinant, a generalized stiffness matrix is established. Discrete shear gap (DSG) technique together with assumed natural strain (ANS) method is employed in order to alleviate shear locking and trapezoidal locking. Stiffness matrix formulation is analytically calculated based on the edge-based cell backgrounds, which result in simple and efficient computation. The proposed formulation does not require using cross-diagonal meshes, which can be an obstacle for several existing three-node degenerated shell elements. This aspect makes the present method much more appealing than others through examples tested. Numerical validations show that the present method performs well for both structured and unstructured triangular meshes. [ABSTRACT FROM AUTHOR]

Functionally graded porous plates have been validated as remarkable lightweight structures with excellent mechanical characteristics and numerous applications. With inspiration from the high strength-to-volume ratio of triply periodic minimal surface (TPMS) structures, a new model of porous plates, which is called a functionally graded TPMS (FG-TPMS) plate, is investigated in this paper. Three TPMS architectures including Primitive (P), Gyroid (G), and wrapped package-graph (IWP) with different graded functions are presented. To predict the mechanical responses, a new fitting technique based on a two-phase piece-wise function is employed to evaluate the effective moduli of TPMS structures, including elastic modulus, shear modulus, and bulk modulus. In addition, this function corresponds to the cellular structure formulation in the context of relative density. The separated phases of the function are divided by the different deformation behaviours. Furthermore, another crucial mechanical property of porous structure, i.e, Poisson's ratio, is also achieved by a similar fitting technique. To verify the mechanical characteristics of the FG-TPMS plate, the generalized displacement field is modelled by a seventh-order shear deformation theory (SeSDT). A NURBS-based isogeometric analysis (IGA) is then employed to capture the C 1 continuity in approximations. Numerical examples regarding static, buckling, and free vibration analyses of FG-TPMS plates are illustrated to confirm the reliability and accuracy of the proposed approach. The influences of various porosity distributions, average relative densities, thickness-to-length ratios, aspect ratios, and boundary conditions on the plate responses are achieved. Consequently, these FG-TPMS structures can provide much higher stiffness than the same-weight isotropic plate. The greater stiffness-to-weight ratio of these porous plates compared to the full-weight isotropic ones should be considered the most remarkable feature. Thus, these complex porous structures have numerous practical applications because of these high ratios and their fabrication ability through additive manufacturing (AM) technology. [ABSTRACT FROM AUTHOR]

A Reissner-Mindlin plate formulation on arbitrary polygonal meshes is proposed for plate analysis. We consider four barycentric shape function types named Wachspress, mean-value, Laplace and piecewise-linear and show its properties in numerical computation for Reissner-Mindlin plate problems. We then generalize an assumed strain field along sides of polygons under the enforcement of the Timoshenko’s beam assumption. The present approach is numerically verified by the bending patch test. It offers a general yet simple form for Reissner-Mindlin plate elements that is not only implementable for arbitrary polygonal meshes but also avoids transverse shear locking phenomenon at thin plate limit. The performance of the proposed elements is found through numerical examples. Our key contribution to this work is that the present formulation is established in a compact (or unified) form which is valid for triangular, quadrilateral and arbitrary polygonal meshes. [ABSTRACT FROM AUTHOR]

This paper presents a generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates. We exploit a higher-order shear deformation theory in each layer such that the continuity of the displacement and transverse shear stresses at the layer interfaces is ensured. Thanks for enforcing the continuity of the displacement and transverse shear stresses at an inner-laminar layer, the minimum number of variables is retained from the present theory in comparison with other layerwise theories. The method requires only five variables, the same as what obtained from the first- and higher-order shear deformation theories. In comparison with the shear deformation theories based on the equivalent single layer, the present theory is capable of producing a higher accuracy for inner-laminar layer shear stresses. The free boundary conditions of transverse shear stresses at the top and bottom surfaces of the plate are fulfilled without any shear correction factors. The discrete system equations are derived from the Galerkin weak form, and the solution is obtained by isogeometric analysis (IGA). The discrete form requires the C continuity of the transverse displacement, and hence NURBS basis functions in IGA naturally ensure this condition. The laminated composite and sandwich plates with various geometries, aspect ratios, stiffness ratios and boundary conditions are studied. The obtained results are compared with the 3D elasticity solution, the analytical as well as numerical solutions based on various plate theories. [ABSTRACT FROM AUTHOR]

Abstract: We present in this paper a simple and effective approach that incorporates isogeometric finite element analysis (IGA) with a refined plate theory (RPT) for static, free vibration and buckling analysis of functionally graded material (FGM) plates. A new inverse tangent distributed function through the plate thickness is proposed. The RPT enables us to describe the non-linear distribution of shear stresses through the plate thickness without any requirement of shear correction factors (SCF). IGA utilizes basis functions namely B-splines or non-uniform rational B-splines (NURBS) which reach easily the smoothness of any arbitrary order. It hence satisfies the C 1 requirement of the RPT model. The present method approximates the displacement field with four degrees of freedom per each control point allowing an efficient solution process. [Copyright &y& Elsevier]

Abstract: We present in this paper a simple and effective formulation based on a fifth-order shear deformation theory (FiSDT) in combination with isogeometric finite element analysis (IGA) for composite sandwich plates. The FiSDT yields non-linear distribution of the transverse shear stresses through the plate thickness and ensures a prior tangential stress-free boundary condition. The IGA uses same basis functions, namely B-splines or non-uniform rational B-splines (NURBS), for preserving the precisely geometric representation and providing the numerical solution. It enables to achieve easily the smoothness with arbitrary continuity order and in the present method the C 1-continuity requirement of higher order shear deformation models is fulfilled. The static, dynamic and buckling analysis of rectangular and circular plates is investigated for different boundary conditions. Numerical examples are given to show high accuracy of the proposed method. [Copyright &y& Elsevier]

The cell-based strain smoothing technique is combined with the well-known three-node Mindlin plate element (MIN3) to give a so-called the cell-based smoothed MIN3 (CS-MIN3) for static and free vibration analyses of plates. In the process of formulating the system stiffness matrix of the CS-MIN3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the stabilized MIN3 is used to compute the strains and to avoid the transverse shear locking. Then the strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles. The numerical examples demonstrated that the CS-MIN3 is free of shear locking, passes the patch test and shows four superior properties such as: (1) be a strong competitor to many existing three-node triangular plate elements in the static analysis, (2) can give high accurate solutions for problems with skew geometries in the static analysis, (3) can give high accurate solutions in free vibration analysis, (4) can provide accurately the values of high frequencies of plates by using only coarse meshes. [ABSTRACT FROM AUTHOR]

Abstract: An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap (DSG) technique using triangular meshes (ES-DSG) was recently proposed to enhance the accuracy of the existing FEM with the DSG for analysis of isotropic Reissner/Mindlin plates. In this paper, the ES-DSG is further formulated for static, free vibration and buckling analyses of functionally graded material (FGM) plates. The thermal and mechanical properties of FGM plates are assumed to vary across the thickness of the plate by a simple power rule of the volume fractions of the constituents. In the ES-DSG, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains associated with the edges of the elements. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to demonstrate the performance of the present formulation for FGM plates. [Copyright &y& Elsevier]

Abstract: An edge-based smoothed finite element method (ES-FEM) for static, free vibration and buckling analyses of Reissner–Mindlin plates using 3-node triangular elements is studied in this paper. The calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements. In order to avoid the transverse shear locking and to improve the accuracy of the present formulation, the ES-FEM is incorporated with the discrete shear gap (DSG) method together with a stabilization technique to give a so-called edge-based smoothed stabilized discrete shear gap method (ES-DSG). The numerical examples demonstrated that the present ES-DSG method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature. [Copyright &y& Elsevier]

This study presents a combination approach of the higher-order shear deformation theory, nonlocal strain gradient theory (NSGT) and isogeometric analysis (IGA) for the free vibration of carbon nanotube-reinforced (CNT) magneto-electro-elastic (MEE) nanoplates. To account size-dependent effects at the nanoscale, the classical theory model is extended with two additional scale parameters. However, this extended model necessitates at least the third derivative of the approximation function, which is incompatible with the standard finite element method. So, IGA with NURBS offers higher-order continuity through its basis functions, making it well-suited for this size-dependent model. To simplify computations, a power-law scheme is employed to represent the material properties. Various distribution types of carbon nanotubes (CNTs) including UD, FG-X, FG-O and FG-V are incorporated to investigate their effects on mechanical behaviors of CNT-MEE nanoplates. The governing equations of motion are derived in their weak form using the principle of extended virtual displacement and then solved by isogeometric analysis (IGA). The impact of the magnetic, electric and elastic fields on the coupling behaviors of CNT-MEE nanoplates are studied. Specially, parametric studies are conducted to analyze the influence of geometrical parameters, CNT distributions, CNT volume fraction, matrix volume fraction, electric voltage, magnetic potential, nonlocal and strain gradient parameters on the natural frequencies of the CNT-MEE nanoplates. Comparisons between the results obtained using NSGT and the classical theory reveal significant findings. The natural frequencies calculated by NSGT exhibit dependence on the relative values of the nonlocal and strain gradient parameters. • Investigation of carbon nanotube reinforced magneto-electro-elastic nanoplates. • The exploration of phenomena related to both stiffness reduction and enhancement. • Accurate prediction of the magnetic and electrical coupling and size effects. • Novel benchmark numerical results are introduced. [ABSTRACT FROM AUTHOR]

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. [Display omitted] • Establishment of 3D FG-TPMS optimization problem by NURBS interpolation. • Generalized shear deformation theory with IGA is employed and verified. • Various geometries including curved plates are investigated. • Unseen before optimal designs with significantly elevated natural frequencies. [ABSTRACT FROM AUTHOR]

Nowadays, sandwich plates with cellular core structures are gaining the attention of researchers owing to their outstanding features. For a better insight into auxetic structures, in this study, we propose an excellent computational approach based on polygonal meshes to comprehensively examine the free vibration, buckling and dynamic instability behaviors of the auxetic honeycomb sandwich plate structures. A generalized C 0 -type higher-order shear deformation theory ( C 0 -HSDT) in conjunction with both Laplace and quadratic serendipity shape functions is employed to approximate the strain fields for polygonal plate elements. The sandwich plate structures are constituted by an auxetic honeycomb core layer with negative Poisson's ratio and two skin layers reinforced by graphene nanoplatelets (GNPs). Ultra-light features of the plate structures can be obtained by using the auxetic honeycomb cells with negative Poisson's ratio while the GNPs are embedded into skin layers to enhance the structural stiffness. In order to determine the dynamic instability region of the sandwich plate, Bolotin's approach is utilized in the current research. Several numerical examples are carried out to investigate the influences of geometrical parameters of auxetic cell and GNPs reinforcement on the structural behaviors. The results obtained in the current research can be considered as benchmark ones to investigate auxetic sandwich plate structures. [ABSTRACT FROM AUTHOR]

The goal of this study is to utilize a higher order shear deformation theory (HSDT) and the moving Kriging meshfree method for analyzing bending, free vibration, and buckling behaviors of functionally graded (FG) triply periodic minimal surface (TPMS) plates. The FG-TPMS plates are modeled using porous structures of Primitive (P), Gyroid (G) and wrapped package-graph (IWP) patterns with six different volume distribution cases for each pattern. The mechanical properties such as elastic modulus, shear modulus, and Poisson's ratio, are estimated using a fitting technique based on a two-phase piece-wise function. The governing equations of the FG-TPMS plates are established using the virtual work principle and then solved using the moving Kriging meshfree method. The study examines various geometries including square, circular, annular, and square with a cutout heart, to investigate the displacement, natural frequency, and critical buckling load parameters of the FG-TPMS plates. Additionally, those parameters are also analyzed with respect to different length-to-thickness ratios, TPMS types, volume distribution cases, and boundary conditions. The numerical results are compared to the original reference ones obtained by isogeometric analysis in the literature. [ABSTRACT FROM AUTHOR]

• A novel size-dependent meshfree approach for magneto-electro-elastic functionally graded nanoplates is developed. • The generalized weak form formulation for magneto-electro-elastic functionally graded nanoplates is proposed. • The magneto-electro-elastic model based on Maxwell's equations is established. • The proposed model can high-efficiently predict magnetical and electrical coupling characteristics and size effect. • Some novel benchmark numerical results are illustrated and introduced. In this paper, we introduce a novel size-dependent meshfree approach for analyzing free vibrations of magneto-electro-elastic (MEE) functionally graded (FG) nanoplates. This approach combines the nonlocal strain gradient theory (NSGT), the higher-order shear deformation theory (HSDT), and meshfree method for the first time. MEE materials are essentially mixed by the barium titanate and cobalt ferrite which are definitely coupled by both piezoelectric and piezomagnetic effects. Our approach captures both the nonlocal and strain gradient effects in nanoplates, which result in the reduced stiffness and increased stiffness, respectively, by adjusting two parameters. The kind of effective material properties of MEE-FG nanoplates are expressed using a power-law scheme. The coupled governing equations, based on the principle of extended virtual displacement, are solved using the moving Kriging (MK) meshfree method. Numerical examples are given to investigate the effect of geometrical parameters, initial electric voltage, power index, initial magnetic potential, strain gradient parameter, and nonlocal parameter on the natural frequency of MEE-FG nanoplates. [ABSTRACT FROM AUTHOR]

In this study, a nonlocal strain gradient isogeometric model, which includes the higher-order shear deformation theory, nonlocal strain gradient theory and isogeometric analysis method, for free vibration analysis of functionally graded (FG) nanoplates made of magneto-electro-elastic (MEE) materials is the first time presented. The mechanism of reduction and increase in the stiffness of MEE-FG nanoplates are shown in the proposed model under the influence of two scale parameters. For simplicity of calculation, the material properties of the MEE-FG nanoplates are described by the power-law scheme. A weak form of governing equations of motion of the MEE-FG nanoplates is obtained by applying the principle of extended virtual displacement. The natural frequency of the MEE-FG nanoplates is evaluated through the change of power index, geometrical parameter, nonlocal parameter, strain gradient parameter, electric voltage and magnetic potential. The results obtained through nonlocal strain gradient theory (NSGT) are lower than those predicted by classical theory when the nonlocal parameter is larger than or equal to the strain gradient parameter. On the contrary, natural frequencies obtained by NSGT are higher than those estimated by classical theory as the nonlocal parameter is smaller than the strain gradient parameter. [ABSTRACT FROM AUTHOR]

This study brings to readers the generalized formulation of three-variable plate theory and an efficient computational approach for analyzing plates. The theory not only has three degree of freedoms (DOFs) per node, which complies with three dimensional space of full plate model as classical plate theory (CPT) but also accounts for the effect of shear deformation without any requirement of shear correction factors (SCF). A complete set of strong forms, weak form as well as classical and non-classical boundary conditions (BCs) for linear and geometrically nonlinear analysis are consistently derived in this paper through a variational approach. The strong forms are sixth order differential equations, resulting in the symmetrical fourth order differential weak form. It is known that Isogeometric Analysis (IGA) arguably outweighs classical finite element method in terms of high continuity and high order differentiability. Thanks to its advantage, an IGA framework for the generalized three-variable plate theory is formulated with completely locking-free and only three DOFs per node. The classical BCs are strongly enforced to system equations as usual whilst the non-classical BCs are weakly imposed by a penalty approach. The new plate theory with only three-variable is thereafter used for static linear and nonlinear analysis of isotropic and functionally graded material (FGM) plates to demonstrate its ability. The reliability and accuracy of the present approach are ascertained by comparing the obtained results with other existing ones. Based on a robust formulation devoted in the paper, the proposed approach can be further extended for numerous problems related to the shear deformable effect in the literature. [ABSTRACT FROM AUTHOR]

This paper presents an effective and simple computational formulation based on isogeometric analysis (IGA) and generalized higher-order shear deformation theory (GHSDT) to study size-dependent analysis of functionally graded carbon nano-reinforced composite (FG-CNTRC) nanoplates. The material properties of FG-CNTRC are assumed to be graded through the thickness direction according to four special distributions of carbon nanotubes (CNTs). The differential nonlocal equations are utilized to take into account size effects. The nonlocal governing equations are approximated according to IGA based on GHSDT, which satisfies naturally the higher-order derivatives continuity requirement in weak form of FG-CNTRC nanoplates. Carbon nanotube volume fraction and nonlocal effects on responses of FG-CNTRC nanoplates with different volume fractions are studied. Numerical results prove high accuracy and reliability of the present method in comparison with other available numerical approaches. [ABSTRACT FROM AUTHOR]

A BCMO-ANN algorithm for vibration and buckling optimization of functionally graded porous (FGP) microplates is proposed in this paper. The theory is based on a unified framework of higher-order shear deformation theory and modified couple stress theory. A combination of artificial neural network (ANN) and balancing composite motion optimization (BCMO) is developed to solve the optimization problems and predict stochastic vibration and buckling behaviors of functionally graded porous microplates with uncertainties of material properties. The characteristic equations are derived from Hamilton's principle and approximation of field variables under Ritz-type exponential series. Numerical results are obtained to investigate the effects of the material distribution, material length scale, porosity density and boundary conditions on natural frequencies and critical buckling loads of functionally graded porous microplates. The novel results derived from this paper can be used as future references. • A BCMO-ANN algorithm for stochastic vibration and buckling analysis of functionally graded porous microplates is presented. • A unified framework of higher-order shear deformation theory and modified couple stress theory is presented. • Ritz method with hybrid shape functions yields fast convergence and accurate results. • Effects of materials and geometry on stochastic behaviors of functionally graded porous microplates are investigated. [ABSTRACT FROM AUTHOR]

This paper brings to the readers a unified framework on higher order shear deformation theories (HSDTs), modelling and analysis of laminated composite plates. The major objective of this work is to (1) unify all higher order shear deformation theories in a unique formulation by a polynomial form; (2) propose the new higher shear deformation models systematically based on a unified formulation. In addition, the effect of thickness stretching is taken into account by considering a quasi-3D theory. The principle of virtual displacements is exploited to derive a weak form based on the generalized displacement fields of the higher order shear deformation theories. Numerical results are computed by using isogeometric analysis and verified to show the accuracy and reliability of the present approach. It is found that the unique formulation of a polynomial form can theoretically cover all existing HSDTs models and is thus sufficient to describe the nonlinear and parabolic variation of transverse shear stress. Moreover, the proposed higher order shear deformation theories predict the proper responses for laminated composite plates in comparison with the available ones in the literature. [ABSTRACT FROM AUTHOR]

• Nonlinear static stability problem of FG-TPMS shallow spherical shells is established. • Nine FG-TPMS designs by combining three TPMS structures and three porosity distributions are obtained. • Governing equations are derived using HSDT incorporating von Kármán's geometrical nonlinearities. • The Ritz energy minimization method is employed to achieve an efficient solution for the governing equations. • Buckling and postbuckling behaviors of FG-TPMS shallow spherical shells are analyzed numerically. An analytical solution for the nonlinear static stability problem of functionally graded triply periodic minimal surface (FG-TPMS) shallow spherical shells is studied in the current research for the first time. Three common TPMS structures including Primitive (P), Gyroid (G), and I-graph and Wrapped Package-graph (IWP) with three models of functionally graded porosity distribution along the thickness are considered. The shallow spherical shells (shallow SSs) are subjected to combined thermo-mechanical loadings and rested on a nonlinear elastic foundation. The fundamental formulas are expressed based on the higher-order shear deformation theory (HSDT) and von Kármán's geometrical nonlinearities. Employing the Ritz energy minimization method, the explicit relationship between load and deflection is derived. Subsequently, the static stability behavior of FG-TPMS shallow SSs is investigated. Numerical illustrations are investigated to show the superior thermo-mechanical load-carrying performance of the FG-TPMS SSs compared to corresponding isotropic structures of the same weight. The significant effects of geometric parameters, nonlinear elastic foundation parameters, and the type of FG-TPMS structures on the nonlinear static stability behavior of shallow SSs are further considered. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a novel and effective formulation based on isogeometric approach (IGA) and higher-order deformation plate theory (HSDT) to study the behavior of functionally graded material (FGM) plates. HSDT model using C 1 continuous element is able to improve the accuracy of solution and describe exactly the shear stress distribution without shear correction factors. IGA utilizes the non-uniform rational B-spline (NURBS) functions which allow us to achieve easily the smoothness with arbitrary continuity order. The present method hence fulfills the C 1 – requirement of HSDT model. The effective material properties of the FGM plates, which property varies only through the thickness of plate, are calculated using the rule of mixture and the Mori–Tanaka homogenization technique. The static, dynamic and buckling analysis of rectangular and circular plates is investigated for different boundary conditions. Numerical results show high effectiveness of the present formulation. [Copyright &y& Elsevier]

Abstract: An alternative alpha finite element method coupled with a discrete shear gap technique for triangular elements is presented to significantly improve the accuracy of the standard triangular finite elements for static, free vibration and buckling analyses of Mindlin–Reissner plates. In the , the piecewise constant strain field of linear triangular elements is enhanced by additional strain terms with an adjustable parameter which results in an effectively softer stiffness formulation compared to the linear triangular element. To avoid the transverse shear locking, the discrete shear gap technique (DSG) is utilized and a novel triangular element, the is obtained. Several numerical examples show that the achieves high reliability compared to other existing elements in the literature. Through selection of , under or over estimation of the strain energy can be achieved. [Copyright &y& Elsevier]

A nonlocal strain gradient meshfree model based on the nonlocal strain gradient theory (NSGT) and higher-order shear deformation theory (HSDT) is presented and developed to examine the bending and free vibration behaviors of functionally graded (FG) nanoplates. Regarding two scale parameters, nonlocal and strain gradient effects in nanostructures are appeared in the current model. The power index and Mori-Tanaka schemes are utilized to compute material properties. A weak form of governing equations of motion for NSGT can be obtained by using the principle of virtual displacements. Later, the deflections and natural frequencies of the FG nanoplates are resolved by using the moving Kriging (MK) meshfree method. A detailed study to assess effects of exponential factor, length-to-thickness ratio, geometry, boundary condition, strain gradient parameter and nonlocal parameter on the deflections and natural frequencies of the FG nanoplates is revealed by numerical results. It is shown that large different results of deflections and natural frequencies between NSGT and classical theory are indicated through numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we present for the first time a size-dependent model based on the modified couple stress theory (MCST) and isogeometric analysis (IGA) for the static and free vibration behaviors of functionally graded carbon nanotube reinforced composite (FG-CNTRC) nanoplates. By using higher order shear deformation theory for displacement fields, the shear correction factor is omitted when determining the stiffness matrix. Based on the rule of mixture, the effective Young’s and shear moduli of carbon nanotube (CNT) are established. For verifying the accuracy and trustworthiness of the proposed method, the present results are compared with those of analytical solutions, and excellent agreement is obtained. The proposed model can capture the small scale effect for FG-CNTRC nanoplates. The effect of length scale on stresses and natural frequencies of FG-CNTRC nanoplates is discussed in details. [ABSTRACT FROM AUTHOR]

This paper proposes a simple and efficient approach based on a moving Kriging interpolation-based (MKI) meshfree method and a two-variable refined plate theory for static, free vibration and buckling analyses of isotropic plates. A generalized formulation through various higher-order distributed functions is presented. Shear correction factors are not required due to zero-shear stresses satisfied at the top and bottom surfaces of plates. The governing partial differential equations are discretized by a weak Galerkin form and numerically solved by using MKI basis functions. The present theory considers the transverse, shear deflections and their derivations while only the deflections are included in approximate solutions. A new correlation function is proposed to construct MKI shape functions so that the underlying solution becomes stable. In addition, a rotation-free technique based on isogeometric analysis is presented to enforce boundary conditions of normal slopes for clamped plate cases, which is simpler and more efficient than several existing approaches. Numerical results show excellent performance of the present method. [ABSTRACT FROM AUTHOR]

This paper presents a new simple four-unknown shear and normal deformations theory (sSNDT) for static, dynamic and buckling analyses of functionally graded material (FGM) isotropic and sandwich plates. The fully three-dimensional material matrix is used in the relation between stress and strain. The present theory uses only four independent unknowns although it is additionally accounted for a deformation in the axial direction. In comparison with the first and higher order shear deformation theories, the number of independent unknowns of the present theory retains four degrees of freedom per node. The shear stress free surface conditions are naturally satisfied so that the shear correction factors are no longer required. The discrete system of equations is derived from the Galerkin weak form and numerically solved by isogeometric analysis (IGA). This discrete form requires the C 1 -continuity of the displacement field. Therefore, NURBS basis functions in IGA can easily satisfy this condition. Several examples are given to demonstrate the efficiency of the present method. [ABSTRACT FROM AUTHOR]

A novel and effective formulation that combines the eXtended IsoGeometric Approach (XIGA) and higher-order shear deformation theory (HSDT) is proposed to study the free vibration of cracked functionally graded material (FGM) plates. Herein, the general HSDT model with five unknown variables per node is applied for calculating the stiffness matrix without needing shear correction factor (SCF). In order to model the discontinuous and singular phenomena in the cracked plates, IsoGeometric Analysis (IGA) utilizing the Non-Uniform Rational B-Spline (NURBS) functions is incorporated with enrichment functions through the partition of unity method. NURBS basis functions with their inherent arbitrary high order smoothness permit the C 1 requirement of the HSDT model. The material properties of the FGM plates vary continuously through the plate thickness according to an exponent function. The effects of gradient index, crack length, crack location, length to thickness on the natural frequencies and mode shapes of simply supported and clamped FGM plate are studied. Numerical examples are provided to demonstrate the performance of the proposed method. The obtained results are in close comparison with other published solutions in the literature. [ABSTRACT FROM AUTHOR]

This paper presents a simple and effective formulation based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) to investigate the static and dynamic behavior of functionally graded carbon nano-reinforced composite plates. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded through the thickness direction according to several linear distributions of the volume fraction of carbon nanotubes. The governing equation is approximated according to the HSDT model using isogeometric elements based on Non-Uniform Rational B-Spline (NURBS) basis functions. This achieves naturally any desired degree of continuity through the choice of the interpolation order, so that the method easily fulfils the C 1 -continuity requirement of the HSDT model. The accuracy and reliability of the proposed method is verified by comparing its numerical predictions with those of other available numerical approaches. [ABSTRACT FROM AUTHOR]

Highlights: [•] CS-DSG3 is extended to the layerwise deformation theory. [•] Dynamic response of laminated plates on viscoelastic foundation is considered. [•] A moving mass with specified velocity on triangular elements at any time is defined. [Copyright &y& Elsevier]

Abstract: An edge-based smoothed stabilized discrete shear gap method (ES-DSG3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. In this paper, the ES-DSG3 is extended and incorporated with a layerwise theory for static and free vibration analyses of composite and sandwich plates. In the layerwise theory, the behavior of each layer follows the first-order shear deformation theory and the condition of displacement continuity is imposed at the interfaces of layers. This hence does not require shear correction factors and improves significantly the accuracy of transverse shear stresses. The stiffness formulation of the ES-DSG3 is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements for each layer. The accuracy and reliability of the proposed method are confirmed in several numerical examples. [Copyright &y& Elsevier]

Abstract: This paper presents a new inverse tangent shear deformation theory (ITSDT) for the static, free vibration and buckling analysis of laminated composite and sandwich plates. In the present theory, shear stresses are vanished at the top and bottom surfaces of the plates and shear correction factors are no longer required. A weak form of the static, free vibration and buckling models for laminated composite and sandwich plates based on ITSDT is then derived and is numerically solved using an isogeometric analysis (IGA). The proposed formulation requires C 1-continuity generalized displacements and hence basis functions used in IGA fulfill this requirement. Numerical examples are provided to show high efficiency of the present method compared with other published solutions. [Copyright &y& Elsevier]

Highlights: [•] CS-DSG3 is extended to static and free vibration analyses of FGPs. [•] Formulation is based on the C0-type HSDT. [•] Thermal and mechanical loads are considered and a two-step procedure is performed. [ABSTRACT FROM AUTHOR]

Abstract: A cell-based smoothed discrete shear gap method (CS-DSG3) for static and free vibration analyses of Reissner–Mindlin shells is formulated by combining the cell-based strain smoothing technique with the discrete shear gap method (DSG3) using three-node triangular elements. In the CS-DSG3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the stabilized DSG3 is used to compute the strains and to avoid the transverse shear locking. Then the strain smoothing technique on whole of the triangular element is used to smooth the strains on these three sub-triangles. The CS-DSG3 hence not only overcomes the drawback of the DSG3 but also improves the accuracy as well as the stability of the DSG3. The numerical examples demonstrated that the CS-DSG3 is free of shear locking and achieves the high accuracy compared to other existing shell elements. [Copyright &y& Elsevier]