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The development of Micro-Electro-Mechanical Systems (MEMS) and portable electronic devices have facilitated the application of energy harvesters in self-powered microelectromechanical devices. This work presents an explicit topology optimization framework for the design of multi-material flexoelectric composite structures. It aims to achieve flexoelectric energy harvesting structures with enhanced energy conversion efficiency by optimizing the distribution of elastic and flexoelectric materials concurrently. The proposed method utilizes a set of groups of moving morphable components (MMC) to characterize the distribution of flexoelectric and elastic materials. The influence of different material overlapping schemes is also investigated in this work. The combination of isogeometric analysis (IGA) and MMC enables an efficient solution of flexoelectric high-order partial differential equations (PDEs). Numerical examples and experiments verify the effectiveness of the proposed method. Compared to other methods, the component-based MMC method not only facilitates the formation of efficient structures, but also directly produces the geometric model required for manufacturing. [ABSTRACT FROM AUTHOR]

In the present work, the fracture analysis of functionally graded (FG) porous structure containing an edge cracked is carried out in the presence of different types of porosity distributions using extended isogeometric analysis (XIGA). Firstly, the different types of porosity distribution functions are mathematically modeled in the porous FG structure across the length of the domain. The effective properties of the porous FG structure are computed using power law. Also, an additional term of porosity is incorporated in the power law to include the effect of porosity in the FG structure. The effective properties are computed across the length of the structure in the presence of different types of porosity distributions. Further, a pre-existing crack is modeled in the domain to study its influence on the fracture behaviour of porous FG structure using XIGA. To validate the accuracy, the results for the non-porous FG structure are compared with the available results in the literature (with the analytical and numerical solution), and they are found in good agreement (percentage error in the range of 0.04–––1.78%). Moreover, the comparative study is performed to investigate the influence of different types of porosity distributions on the fracture behaviour of FG structure. [ABSTRACT FROM AUTHOR]

Laminated curved composite parts, used, e.g., in the spar and ribs in aircraft and wind turbine blades, are typically subjected to high interlaminar stresses. This work focuses on a two-step procedure to study laminated Euler–Bernoulli curved beams discretized via Isogeometric Analysis (IGA). First, we solve a (planar) Euler–Bernoulli curved beam formulation in primal form to obtain the tangential and transverse displacements. This formulation features high-order PDEs, which we can straightforwardly approximate using either an IGA-Galerkin or an IGA-collocation approach. Starting from the obtained displacement solution, which accounts for bending-stretching coupling, we can directly compute the normal stress only, while we do not have information concerning the transverse shear stress state, typically responsible for delamination. However, by imposing equilibrium in strong form in a curvilinear framework which eases the post-processing, eliminating the need for coordinate changes, we can easily recover interlaminar transverse shear stresses at locations of interest. Such a posteriori step requires calculating the high-order displacement derivatives in the equilibrium equations and, therefore, demands once again higher-order regularity that can be easily fulfilled by exploiting the high-continuity properties of IGA. Extensive numerical tests prove the effectiveness of the proposed approach, which is also aided by the IGA's superior geometric approximation. • Two-step local equilibrium-based stress recovery for laminated Euler-Bernoulli beams. • 3D constitutive model reduction with exact integration through the beam thickness. • High-order PDEs in primal form and recovery requirements easily handled via IGA. • Coupled membrane-bending problem solved via IGA Galerkin and Collocation approaches. [ABSTRACT FROM AUTHOR]

We present an adaptive multi-patch isogeometric analysis on the basis of truncated hierarchical B-splines (THB-splines) for solving two-dimensional complex isotropic/orthotropic elasticity. The THB-splines have local refinement property and their basis functions exhibit linear independence, so these features are highly applicable for adaptive isogeometric analysis. Guided by a posterior error estimator based on stress recovery, the adaptive algorithm is utilized in isogeometric analysis. In order to further extend the proposed method to solve complex geometry problems, the multi-patch technique is adopted to achieve exact modeling with Nitsche's method as a multi-patch coupling approach. An isotropic numerical example with exact analytical solutions and three orthotropic numerical examples are presented to verify the effectiveness and accuracy of the developed method. Numerical solutions show that the developed adaptive isogeometric analysis method has high computational efficiency. • An adaptive multi-patch isogeometric analysis method on the basis of THB-splines is presented. • Multi-patch technique is used to accurately describe complex geometric shapes. • The Nitsche's method is employed as a multi-patch coupling approach. • An adaptive local refinement method based on posterior error estimation is proposed. • The proposed method is suitable for solving complex isotropic/orthotropic elasticity. [ABSTRACT FROM AUTHOR]

Isogeometric Analysis is a numerical method that integrates the concepts of geometric modeling and structural analysis. It approximates the displacement field using the same basis functions employed by CAD systems to describe the structure's geometry. This work proposes an isogeometric formulation for analysis of functionally graded panels based on rational Bézier triangles, allowing the exact geometry representation and automatic discretization of topologically complex models. The formulation is applied to the free vibration and stability analysis of functionally graded plates and curved panels. Monotonic convergence under mesh refinement was observed in all examples. Furthermore, results show that curved functionally graded panels display a complex nonlinear behavior and can present bifurcation buckling before reaching the limit load. [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]

An isogeometric formulation is presented for fiber path optimization of tow-steered composite laminates (TSCL) with minimal parametric instabilities. Here, the fiber path functions of a TSCL is expressed smoothly by C p − 1 continuous non-uniform rational B-splines (NURBS) for engineering shapes, providing accurate field solutions. In this way, the motion equations of a TSCL plate is developed based on the third-order shear deformation theory (TSDT) and the Bolotin's approximation is used to determine the parametric instability region. Primarily, the accuracy and efficiency of presented framework is measured for a TSCL example with respect to the existing solutions. Then, the optimal fiber paths of TSCL plates are searched using the genetic algorithm (GA) for various geometries, force component combinations, and dynamic-to-static force ratios. Results reveal that the optimal fiber paths are not necessarily symmetric even for regular domains, demonstrating the non-uniform coupling of bending and twisting stiffness in TSCL plates. The optimal design of TSCL fiber paths necessitates the consistency of local stiffness distribution and the resultant bending–twisting mode shape, specially when the dynamic force is high with respect to the static one. Geometry, force component combination, and dynamic-to-static force ratio impose case-specific fiber paths for optimal TSCL plates, requiring both minimal deformation and instability opening. • NURBS express smooth fiber paths and are used for genetic-based design optimization. • Practical optimal fiber paths are found for force cases and dynamic-to-static ratios. • Significant drops in instability openings of optimal TSCL plates are achieved. • Optimal paths have mode-consistent stiffness form, expressly for high dynamic forces. [ABSTRACT FROM AUTHOR]

• A new computational approach for bending and vibration analysis of axially functionally graded graphene platelet-reinforced composite (AFG-GPLRC) curved beams is proposed. • The key idea is to exploit advantages of isogeometric finite element formulation with third-order shear deformation beam theory. • Excellent performance of the present method is confirmed by numerical validations. • Brought out the effect of the physical and geometric properties of GPLs on the nonlinear bending and vibration responses of the AFG-GPLRC curved beams. The linear and nonlinear isogeometric finite element models of an axially functionally graded graphene platelet-reinforced composite (AFG-GPLRC) curved beam are established within the framework of the third-order shear deformation beam theory (TSDT) and von-Kármán's nonlinear geometric relation. The AFG-GPLRC curved beams can be seen as composite structures in which the graphene platelets (GPLs) are continuously distributed in the matrix along the length direction of the curved beam according to different patterns. The modified Halpin-Tsai parallel model and the rule of mixture are implemented to predict the effective Young's modulus and mass density as well as Poisson's ratio, respectively. Hamilton's principle, TSDT, and von-Kármán's strain-displacement relation are combined to derive the governing partial differential equation of motion and corresponding boundary conditions. Furthermore, the Non-Uniform Rational B-splines (NURBS)-based isogeometric analysis (IGA) approach together with a direct iterative technique are utilized to solve the nonlinear governing equation. The accuracy and efficiency of the proposed IGA framework are confirmed by comparing corresponding numerical solutions with other available results. The parametric investigations, such as the curved beam's geometric parameters, boundary conditions, and GPL's distribution patterns, on the nonlinear bending and vibration responses of the AFG-GPLRC curved beams are carried out by some illustrative examples. [ABSTRACT FROM AUTHOR]

In this research, a comprehensive Nitsche-based isogeometric analysis framework is presented for studying the free vibration and buckling characteristics of laminated plates with cutouts under uniform and non-uniform thermal loads. The approach utilizes non-overlapping non-uniform rational B-Splines patches to discretize the perforated plate and seamlessly connects them using the Nitsche method. The technique is then applied to investigate the free vibration and buckling behaviors of laminated plates with cutouts under varying thermal conditions. Three tests with different element sizes are used to analyze composite plate under uniform thermal load, demonstrating good convergence and accuracy of the present method. Rectangular laminated plates without cutouts, considering various boundary conditions, fiber orientations, and temperature variations under uniform or non-uniform thermal loads are analyzed. The natural frequencies and critical buckling temperatures are found to be in excellent agreement with existing literature. Additionally, the analysis is expanded for laminated plates with cutouts under diverse boundary conditions, considering variations in fiber orientations, cutout shapes and numbers, and types of thermal loads. Thorough investigations are carried out to understand the influence of these factors on the vibrational and buckling behaviors of the plates. Moreover, numerical study with non-conforming mesh demonstrates strong capability of the proposed method. [Display omitted] • A Nitsche-based IGA for studying vibration and thermal buckling of laminated plates with cutouts under thermal loads. • Both uniform and nonuniform temperature distribution are considered. • The novel method produces good convergence and accuracy. • The proposed method is applicable for studying vibration of heated perforated laminates meshed with non-conforming elements. • It can be easily extended for perforated laminated plate with arbitrary complicated geometries subjected to thermal loads. [ABSTRACT FROM AUTHOR]

The free vibration of different kinds of stepped coupled doubly-curved shell structures with elastically constrained edges are investigated by adopting the Jacobi-Ritz method for the first time. Coupled structures are comprised of substructures, where paraboloidal, hyperbolical, elliptical, and cylindrical stepped shells are typical ones. The Flügge's thin shell theory is utilized to construct the analytical model, together with the multi-segment partitioning strategy. For each shell segment, despite of various boundary conditions, the displacement components along the meridional directions are expressed by Jacobi polynomials and those along the circumferential directions are represented by Fourier series. Then the unknown coefficients of the displacements are obtained by introducing the Rayleigh-Ritz method. The solutions proposed here for coupled structures have two main advantages: first, there is no need to vary the displacement or the motion equations; and secondly, the efficiency of modeling can be notably enhanced. By comparison with (Finite Element method) FEM and others' results, the reliability of current method can be validated. At last, the free vibrations of different kinds of coupled structures containing stepped shell are analyzed by presenting several numerical examples, the results of which may be served as reference data. [ABSTRACT FROM AUTHOR]

Due to wide application of anisotropic composite plates in modern engineering structures and they were studied rare in literature, the main goal of this work is to study dynamic stability analysis of general anisotropic composite plates. To this end, here, we use the advantages of isogeometric analysis (IGA) to develop a higher-order shear deformation theory (HSDT) framework. In this work, force-frequency curves are obtained for general anisotropic composite plates using novel IGA-HSDT approach which have been previously presented using conventional finite element for specially orthotropic plates. Based on observation, the developed method is higher-order accurate, stable for wide spectral frequency range of anisotropic plates, and efficient in capturing the mode-converging phenomenon. IGA-HSDT model affirmed that the thick plates are more sensitive to frequency convergence prior to divergence with respect to thin ones. Furthermore, C 3 NURBS capture the discrete spectrum accurately which is important for explicit dynamic applications of anisotropic plates. Specifically, anisotropic plates with clamped boundaries and low slenderness ratios have mode-converging phenomenon in dynamic stability curves prior to fundamental mode divergence which is not detected in previous works. [ABSTRACT FROM AUTHOR]

In this study, a NURBS formulation based on the four-variable refined plate theory (RPT) for free vibration, buckling and static bending analyses of multilayer functionally graded graphene platelets reinforced composite (FG GPLRC) plates, for the first time, is proposed. The distributions of graphene platelets (GPLs) in the polymer matrix either uniformly or non-uniformly including different patterns are considered. The Young's modulus of the nanocomposites is predicted by the modified Halpin–Tsai model, while the Poisson's ratio and density mass are implemented by the rule of mixtures. Governing equations are derived and the NURBS formulation is employed to obtain natural frequencies, critical buckling loads and deflections of multilayer FG GPLRC plates. Thanks to continuous higher-order derivatives of NURBS basis functions in isogeometric analysis (IGA), the present approximation is easy to satisfy the C 1-continuty requirement of the RPT model. In addition, a rotation-free technique is applied to eliminate the bending and shear slopes in the case of clamped boundaries. Effects played by GPLs weight fraction, GPLs distribution patterns, number of layers, thickness-to-length ratio are investigated. Numerical results indicate that the inclusion of GPLs can significantly improve the stiffness of plates. [ABSTRACT FROM AUTHOR]

Abstract Compared with traditional composite panels, the modeling, analysis and design of variable-stiffness panels with curvilinear fibers are much more complicated. Although the design flexibility is greatly enhanced, the design of variable-stiffness structures must meet manufacturing constraints to ensure that the designed structures can be fabricated finally. In order to design variable-stiffness composite panels that is very challenging due to its nonlinearity and non-convexity, a novel multi-stage design method for variable-stiffness panels is developed based on lamination parameters. First, lamination parameters are taken as design variables, and the stiffness distribution is obtained efficiently by few iterations. Next, the lamination parameters are transformed into actual layups. Finally, the realistic fiber path is regenerated by considering manufacturing constraints. In addition, the isogeometric analysis, which is more suitable for variable-stiffness structure, is used instead of the traditional finite element analysis to improve the analysis accuracy. Illustrative example demonstrates the high computational efficiency and optimization capacity of proposed method, compared to gradient-based algorithm and evolutionary algorithm. [ABSTRACT FROM AUTHOR]

Abstract In this study, we for the first time present an isogeometric Bézier finite element formulation for bending and transient analysis of functionally graded porous (FGP) plates reinforced by graphene platelets (GPLs) embedded in piezoelectric layers. We name it as PFGP-GPLs for short. The plates are constituted by a core layer, which contains the internal pores and GPLs dispersed in the metal matrix either uniformly or non-uniformly according to three different patterns, and two piezoelectric layers perfectly bonded on the top and bottom surfaces of host plate. The modified Halpin–Tsai micromechanical model is used to estimate the effective mechanical properties which vary continuously along thickness direction of the core layer. In addition, the electric potential is assumed to vary linearly through the thickness for each piezoelectric sublayer. A generalized C 0 -type higher-order shear deformation theory ( C 0 -HSDT) in association with isogeometric analysis (IGA) based on Bézier extraction is investigated. Our approach allows performing all computations the same as in the conventional finite element method (FEM) yet the present formulation shows more advantages. The system of time-dependent equations is solved by the Newmark time integration scheme. The effects of weight fractions and dispersion patterns of GPLs, the coefficient and distribution types of porosity as well as external electrical voltages on structure's behaviors are investigated through several numerical examples. These results, which have not been published before, can be considered as reference solutions for future works. [ABSTRACT FROM AUTHOR]

Abstract A two-directional functionally graded microbeam model is developed for natural frequency and mechanical bending analysis by using NURBS–based isogeometric analysis combined with a non-classical quasi-3D beam theory (NCQ3BT). Material parameters of microbeam continuously and smoothly change along thickness and axial directions. The small size effects are seized with the modified couple stress theory. In the NCQ3BT, both normal and shear deformations are considered without the need for the shear correction factor. The high-order continuity of NURBS directly meets the demand of C1-continuity in the NCQ3BT. Numerical results prove the superior performance and accuracy of the developed approach. The influences of material gradient factors along the axial and thickness directions, material length scale factor, boundary condition, and other aspect ratios of two-directional FG microbeams on mechanical behavior are investigated. [ABSTRACT FROM AUTHOR]

Abstract In this paper the isogeometric method is used to present a shear refined composite beam model through the concept of dimensional reduction method. A one-dimensional (1D) beam model is extracted from three-dimensional (3D) elasticity problem. The 1D beam model is developed using cross-sectional properties from two-dimensional (2D) beam sectional analysis. The 2D analysis of the cross-section is presented by implementing the transverse shear effects in isogeometric analysis. In beam cross-sectional problem, the influence of linear parameterization is investigated for isogeometric modeling of beam cross-section. Moreover, the effect of delamination on cross-sectional stiffness constants is discussed. Using isogeometric analysis (IGA), less degrees of freedom is needed in contrast to classical finite element method and automatic mesh refinement capability is attained. The present composite beam model eliminates the expensive use of 3D finite element analysis with its high precision and fidelity to 3D problem. [ABSTRACT FROM AUTHOR]

In this study, a volume parametric modeling method of lattice structure is proposed, and an efficient multiscale topology optimization framework is realized based on isogeometric analysis (IGA) to construct the gradient lattice structure. The skeleton model is constructed, which can accurately describe the topology structure and improve data utilization and computational efficiency. Based on the skeleton model, a uniform volume parameter lattice structure with an arbitrary topology of multiple types of unit cells is constructed, which is suitable for IGA. Moreover, multiscale topology optimization based on IGA is realized to construct the gradient lattice structure. The same data model is used in modeling, analysis, and optimization, which can accurately represent the geometric shape without discretization errors. At the same time, the multiscale topology optimization iteration is realized by adjusting the density of control points. The optimized model can be directly analyzed and re-optimized, thus realizing the integrated design of lattice structure modeling, simulation, and optimization. The effectiveness and robustness of the algorithm are verified by several mechanical parts and freeform models. These examples show that the gradient lattice structure has higher strength and better stress distribution than the uniform lattice structure under the same boundary conditions. • The gradient lattice structure is designed based on multiscale topology optimization. • Various types of volume parametric lattice structures are constructed. • The model can be analyzed directly by using isogeometric analysis. [ABSTRACT FROM AUTHOR]

This study investigates an effective design methodology for the optimal material distribution of bi-directional functionally graded plates (2D-FGPs) with complex shape. In mechanical analysis, a multi-patch isogeometric method is used to analyze the statics of 2D-FGP, which is based on the third-order shear deformation theory and Nitsche's technology. This method can effectively avoid the use of duplicate nodes in the parameterization of IGA for complex shapes to obtain C 1 -continuity of 2D-FGPs. In the optimal design problem, we have constructed a rectangular material design mesh (RMDM) based on the shape of 2D-FGPs, which can map the material distribution on the surface of plate to achieve the implementation of the optimization process. This two-dimensional B-spline control points of RMDM are set as the design variable with mass reduction and the first-order natural frequency maximization as optimization objectives, and limited arbitrary deflection as constraint conditions. In addition, an improved multi-objective particle swarm optimization algorithm (IMOPSO) is used to obtain a series of Pareto optimization solutions that meet the needs of the designer. The validity and applicability of this innovative combination of multi-patch isogeometric analysis and IMOPSO are demonstrated through several numerical examples in integrated design. This approach further accomplishes the numerical unified CAD/CAE optimization design of 2D-FGPs across multiple non-smooth boundaries. [ABSTRACT FROM AUTHOR]

We present an isogeometric homogenization theory (IGH) for efficiently identifying homogenized and local creep and relaxation response of linearly viscoelastic polymer composites with different microstructural parameters. The principal idea is to construct exact geometric representations of both two- and three-dimensional unit cell microstructures for periodic materials by utilizing multiple conforming NURBS patches that are also employed for the displacement field interpolation function at the local scale. The IGH-based unit cell formulation is then converted to the viscoelastic solution with the Laplace-Carson space parameters via the correspondence principle. Subsequently, we leverage the Zakian formula to reverse the transformed IGH solution and obtain the homogenized creep and relaxation response of the composite in the original time space. The modelling and predictive capabilities of the IGH theory have been extensively validated vis-à-vis the elasticity-based and conventional finite-element homogenization techniques, and the advantages of the proposed technique over the reference techniques were demonstrated. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the vibration problem of beams with axial functionally graded materials (FGMs) and variable thickness is firstly investigated by isogeometric analysis (IGA) in conjunction with three-dimensional (3D) theory. Based on guaranteeing geometry exactness of the strength of non-uniform rational B-splines (NURBS), the curves of non-uniform thicknesses of beams are exactly described. Two beams models (slender model and plump model) are taken into account. Then the material properties smoothly varying in axial direction are calculated by two types of material distributions, power-law and exponential law. The requirement for the weak form of FGMs beams is easily satisfied as NURBS can provide higher order derivative. In numerical results, the convergence is demonstrated, then the accuracy of the current work is validated through comparing solutions with those from the commercial package ANSYS. Moreover, the effects of geometrical proprieties, material parameters as well as boundary conditions on the frequency are also examined. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates the thermal and mechanical buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates by using isogeometric analysis (IGA) based on modified couple stress theory (MCST). A refined hyperbolic shear deformation theory is used for buckling analysis, which satisfies free transverse shear stress conditions on the top and bottom surfaces of plate without a need for shear correction factor. The material properties of carbon nanotube reinforced composite plates are assumed to be temperature-dependent. For numerical analysis the IGA method using B-Spline or Non-Uniform Rational B-Spline (NURBS) functions is employed. The obtained results are compared with those available in the literature. Also, the influence of different parameters on mechanical and thermal buckling analysis is investigated. These parameters include material length scale parameter, boundary conditions, aspect and length-to-thickness ratios of plate, different types of FG-CNTRC distribution, volume fraction of CNTs and temperature dependency. [ABSTRACT FROM AUTHOR]

Abstract For composite panels with cutouts, curvilinear fiber path can adjust the in-plane stiffness distribution to increase the buckling resistance, but it results in huge computational cost for the buckling analysis when FEA is employed. In this study, variable-stiffness panels with cutouts are analyzed via isogeometric method, where cutouts are represented by the level set method. The method for suppressing artificial buckling modes is proposed to improve the prediction accuracy. Moreover, the analytical sensitivity is derived to facilitate fiber path optimization, and a new bi-level optimization framework considering manufacturing constraints is established. Finally, the proposed method is verified by variable-stiffness aircraft panel with multiple cutouts, which can not only provide an accurate prediction of buckling load, but also exhibit high convergence rate and low computational cost for fiber path optimization. [ABSTRACT FROM AUTHOR]

Abstract This paper studies buckling and post-buckling behaviours of multi-layered plates under in-plane compression based on Reissner-Mindlin plate theory. The governing equations are derived from a kinematic nonlinearity based on the von-Kármán assumptions and are thereafter discretized by isogeometric analysis (IGA), which utilizes the NURBS basis functions. For the symmetrically laminated plates, stability analysis consists of three steps: pre-buckling, buckling and post-buckling analyses. In fact, the pre-buckling stresses must be first determined in the pre-buckling analysis and become an important factor in accurate estimation of the critical buckling and post-buckling loads. Otherwise, in the imperfect or unsymmetrically stacked plates, there is no buckling bifurcation phenomenon. The Newton-Rapshon method is hence adopted to solve the geometrically nonlinear problem. Numerical examples are supplied to investigate the effect of an initially geometrical imperfection, which is possible imperfection type such as sine-type, global-type or local-type imperfection, on the post-buckling response of the plates. [ABSTRACT FROM AUTHOR]

Abstract A new hyperbolic tangent shear deformation theory (HTSDT) for the static, free vibration and buckling analysis of laminated composite plates is presented. In the present theory, shear stresses disappear at the top and bottom surfaces of the plates and shear correction factors are no longer required. Weak forms of the static, free vibration and buckling analysis for laminated composite plates based on the HTSDT are then derived and are numerically solved using the isogeometric analysis (IGA). The proposed formulation requires C1 continuity generalized displacements, whereas the basis functions used in IGA can perfectly fulfill this requirement. Based on the available solutions in the literature, the present method shows high accuracy and efficiency when numerical examples are solved. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates the static linear elasticity, natural frequency, and buckling behaviour of functionally graded porous plates reinforced by graphene platelets (GPLs). Both first- and third-order shear deformation plate theories are incorporated within the Isogeometric Analysis (IGA) framework. The pores and the GPLs within the plates are distributed in the metal matrix either uniformly or non-uniformly according to different patterns. The graded distributions of porosity and nanocomposite are achieved by material parameters varying across the thickness direction of plate. The Halpin-Tsai micromechanics model is implemented to establish the relationship between porosity coefficient and Young's modulus, as well as to obtain the mass density of the nanocomposite. The variation of Poisson's ratio is determined by the mechanical properties of closed-cell cellular solids under Gaussian Random Field scheme. A comprehensive parametric study is accomplished to investigate the effects of weight fraction, distribution pattern, geometry, and size of the GPLs reinforcement on the static linear elasticity, natural frequency, and buckling behaviour of the nanocomposite plates with diverse metal matrices and porosity coefficients. The outcome of numerical investigation shows that the inclusion of the GPLs can effectively improve the stiffness of functionally graded porous plate. [ABSTRACT FROM AUTHOR]

This paper presents size-dependency effects on nonlinear transient dynamic response of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates under a transverse uniform load in thermal environments. To consider the length scale and size-dependency effect of nanostructures, a nonlocal continuum theory of Eringen is adopted. The nonlocal governing equations for nanoplate theory are derived from the Hamilton’s principle and approximated by using isogeometric analysis associated with the higher-order shear deformation theory. A numerical model based on the von Kármán strains and Newmark time integration scheme is employed to solve geometrically nonlinear transient problems. The material properties of the FG-CNTRC nanoplate are assumed to be graded and temperature-dependent in the thickness direction, which are expressed through a micromechanical model. Effects of nonlocal parameter, carbon nanotube volume fraction, length-to-thickness ratio, distributions of carbon nanotubes and temperatures through thickness are investigated in detail. Several numerical results show the reliability of the present method. [ABSTRACT FROM AUTHOR]

Focusing on planar isotropic petal-shaped auxetics, an isogeometric design framework is presented to achieve tunable effective properties. Specifically, the design framework includes (i) a NURBS-based parametric modelling scheme that characterizes petal-shaped auxetics with a small number of design variables; (ii) a systematic consideration of petal form, component widths and base material properties; (iii) a semi-analytical sensitivity analysis method based on material derivatives; and (iv) constraints for effective stiffness and target Poisson ratio. Three cases are considered: Case A with the same component width, Case B with different component widths, and Case C for composite designs with multiple base materials. For each case, a design limit curve is obtained for the effective Poisson ratio over a range of effective stiffness constraints, to give a quick overview on the properties attainable for each design setting. The optimization framework is next demonstrated for designing composite petal-shaped auxetics with target effective properties. [ABSTRACT FROM AUTHOR]

This study aims to investigate the postbuckling response of functionally graded (FG) nanoplates by using the nonlocal elasticity theory of Eringen to capture the size effect. In addition, Reddy’s third-order shear deformation theory is adopted to describe the kinematic relations, while von Kámán’s assumptions are used to account for the geometrical nonlinearity. In order to calculate the effective material properties, the Mori-Tanaka scheme is adopted. Governing equations are derived based on the principle of virtual work. Isogeometric analysis (IGA) is employed as a discretization tool, which is able to satisfy the C 1 -continuity demand efficiently. The Newton-Raphson iterative technique with imperfections is employed to trace the postbuckling paths. Various numerical studies are carried out to examine the influences of gradient index, nonlocal effect, ratio of compressive loads, boundary condition, thickness ratio and aspect ratio on the postbuckling behaviour of FG nanoplates. [ABSTRACT FROM AUTHOR]

This paper develops a three-dimensional (3D) isogeometric analysis (IGA) and meshfree coupling approach to investigate the static, dynamic and buckling behaviors for plates and shells of functionally graded material (FGM). The meshfree method and IGA are coupled using the higher-order consistency condition in the physical domain so that the higher-order continuity of basis functions is guaranteed, and the topological complexity of the global volumetric parameterization for IGA to build the 3D geometry can be overcome. By employing IGA elements on the domain boundary and meshfree nodes in the interior domain, the approach preserves the advantages of the exact geometry and flexible discretization in the problem domain. Based on the coupling approach, the analyses for FGM plates and shells are carried out, and the effects of the material volume fraction, the side-to-thickness ratio and the curvature of the cylindrical shell on the deflection, natural frequency, and buckling load are investigated. The coupling approach is verified by comparing with the solutions obtained from other existing theories. [ABSTRACT FROM AUTHOR]

This paper presents a novel formulation based on an improved moving Kriging (iMK) meshfree method and a refined plate theory (RPT) for geometrically nonlinear static and dynamic analyses of functionally graded material (FGM) plates. The nonlinearity of plates is formed in the total Lagrangian approach based on the von Karman strain assumptions. The present plate formulation has the following advantages: only four variables for each node, the tangential stress-zero boundary condition at the top and bottom surfaces of plates, and the stringent continuity requirement of the generalized displacements being simply treated without any additional variables. The novelty of the iMK meshfree method is to propose a quartic spline correlation function for establishment of basic shape functions. Thanks to this improvement, the solution of the iMK meshfree method is stable and no longer depends on an uncertain adjustment of correlation parameter θ . In addition, a simple rotation-free technique originated from isogeometric analysis (IGA) is successfully applied to the iMK meshfree method for enforcing the slopes of clamped boundary condition. Several numerical examples are provided to demonstrate high performance of the present method in comparison with other numerical methods. [ABSTRACT FROM AUTHOR]

This paper investigates the free vibration behaviors of functionally graded (FG) plates considering in-plane material inhomogeneity. Isogeometric analysis (IGA) in conjunction with a refined plate theory (RPT) is employed for fulfilling the investigation. In RPT, the displacement field is described only with four unknown variables and shear correction factor is not necessary any more. A new transverse shear stress function is proposed by combining linear function with cosine function and the shear stress has parabolical distribution through the plate thickness. The governing and discretized equations are solved using IGA rooted in non-uniform rational B-splines (NURBS) basis functions, which have high-order continuous derivatives and facilely satisfy the C 1 - continuity condition of the RPT. The excellent efficiency and accuracy of the current method are testified in the analysis of skew and elliptical plates by comparison with other researchers’ solutions. Furthermore, the influence of the geometric parameter, boundary condition and material inhomogeneity on dynamic characteristics is discussed in detail. [ABSTRACT FROM AUTHOR]

This paper presents a size-dependent four-unknown shear deformable model for static bending, free vibration and buckling analyses of isotropic and sandwich functionally graded (FG) microplates based on the modified strain gradient theory (MSGT). The MSGT requires three material length scale parameters instead of five parameters as well known in the original strain gradient theory. The material parameters of isotropic and sandwich FG microplates are directly derived from a rule of mixture. The governing equations are derived from the principle of virtual work. Since the present method contains the higher-order gradients in the weak form, NURBS-based isogeometric analysis is suitable for the solution procedure. The effects of geometrical parameters, boundary conditions, volume fraction and material length scale parameters are investigated through isotropic and sandwich FG microplate examples. Obtained results indicate that the consideration of strain gradients leads to a rise of the plate stiffness, and a reduction of displacement and an increase in natural frequency as well as critical buckling load of FG microplates are therefore remarked. Moreover, the present model can be degenerated into the modified couple stress model or classical model when a few material length-sale parameters are neglected. [ABSTRACT FROM AUTHOR]

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 C 0 -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 C 1 -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. [ABSTRACT FROM AUTHOR]

An attempt has been made here for the isogeometric static analysis of the laminated composite plane beams by using refined zigzag theory. In this study; instead of the standart finite elements, which use polynomial shape functions, an isogeometric refined zigzag finite element (IGRZF) has been developed, that gives the opportunity to get the exact beam geometries directly from a computer aided design (CAD) software, Rhinoceros. To provide less computational effort, the refined zigzag theory has been introduced, that make IGRZF independent from the number of layers considered. The aforementioned finite element has been implemented in an in-house Mathematica code, which can handle both thin and thick beams without the problem of shear locking and does not require shear correction factors. Using this approach, various sandwich beams have been analyzed and the obtained results are compared with other reliable published results for various aspect ratios and support types. [ABSTRACT FROM AUTHOR]

The inverse finite element method (iFEM) has been used to achieve the shape sensing of small displacements based on linear elastic theory. However, with the development of smart structure technology, the formulation is not suitable for the anisotropic composite structures with large deformation in practical engineering. Therefore, a nonlinear iFEM algorithm is proposed to monitor the linear and nonlinear deformation of the anisotropic composite beams. The formulation not only involves the effect of orientation of composite fibers on strain distribution into the shape sensing model, but also accounts for the effect of shear deformation without any requirement of shear correction factor. Initially, the third-order shear deformation theory (TSDT) is reviewed along with deriving the nonlinear strain field based on von-Karman strain theory. Considering the problem that the couple term of the shear strain and bending strain is unmeasurable, the relationship between shear and bending displacements is established according to the derived constitutive equations. Then, the proposed nonlinear iFEM method reconstructs the deformed structural shape, where isogeometric analysis (IGA) approach is used to construct the displacement functions and the experimental section strains are calculated from the discretized surface strains. Finally, several examples are solved to verify the proposed methodology. Numerical results demonstrate that the nonlinear iFEM algorithm can improve the reconstruction accuracy by 4% with respect to the linear iFEM method for beam structures. Hence, the proposed approach can be used as a viable tool to predict nonlinear deformation of composite structure. [ABSTRACT FROM AUTHOR]

In the present work an Equivalent Single Layer (ESL) formulation is proposed for the static analysis of doubly-curved anisotropic structures of arbitrary geometry and variable stiffness resting on a Winkler elastic foundation. In-plane and out-of-plane general distributions of linear elastic springs are provided for the implementation of general external constraints along the edges of the structure. The structure is geometrically described accounting for the principal curvatures of the shell object of analysis. A generalized set of blending functions based on Non-Uniform Rational Basis Spline (NURBS) curves is adopted so that arbitrary shaped structures can be modelled with the same approach. The fundamental governing equations are obtained in terms of displacement field unknowns, which has been effectively described accounting for a unified formulation based on the minimum potential energy principle. General anisotropic lamination schemes are considered, setting a general orientation of each lamina, as well as all possible material symmetries. The numerical implementation is performed by means of the Generalized Differential Quadrature (GDQ) method, thus allowing a strong formulation of the structural problem. A series of validation examples is performed on shells with zero, single and double curvatures in which the static structural response provided with the proposed formulation has been compared to that obtained from a refined three-dimensional finite element model, showing a great accordance between these different approaches. The research shows that the employment of higher order theories, together with the GDQ method, allows to obtain very accurate results with a reduced computational cost, compared to finite element simulations. [ABSTRACT FROM AUTHOR]

This paper presents a new, simplified approach to model the delamination phenomenon in laminated composites, where the drawbacks of the existing theories might be avoided. To this end, a phase-field model (PFM) is implemented to capture the irregularities in the crack fronts of the bond interface and an isogeometric analysis (IGA) is used to discretize the interface domain. The IGA methodology provides a higher-order continuity in the solution domain, leading to enhanced accuracy and significantly reduced computational cost. Several verifications and engineering examples are presented to validate and investigate the efficiency of the developed formulation in predicting the delamination behavior of laminated composites. Results reveal that the present approach can provide a smoothly-varying damaged interface with less computational cost than other classical methods. Moreover, the developed phase-field method reveals higher sensitivity of the interface fracture patterns to the stacking configurations than the cohesive finite element one. [ABSTRACT FROM AUTHOR]

Variable-stiffness (VS) laminated shells with curved fibers have gradually become a promising lightweight structural concept in the aerospace field. However, as the complexity of the shell geometry and the fiber angle increases, the traditional finite element method (FEM) requires mesh refinement to guarantee accurate discretization, which significantly increases the computational cost, especially in the post-buckling analysis with multiple solving requirements. In this study, a non-uniform rational B-Splines (NURBS)-based degenerated solid shell formulation in the total Lagrangian framework is proposed for the post-buckling analysis of VS laminated shells. Since the nonlinear function related to the rotational degrees of freedom (DOFs) is additionally introduced into the displacement field expression, the restriction on the magnitude of the nodal rotations is effectively eliminated. In addition, the paper adopts the NURBS surface reconstruction technology based on the point-line-surface features to build B-spline CAD models from the point cloud with geometric imperfect features. Then, an integrated modeling-analysis framework for predicting the post-buckling behavior of VS structures with geometric imperfections is proposed. A series of numerical examples systematically demonstrate that the framework can effectively conduct the post-buckling analysis of shell structures with geometric imperfections. Meanwhile, the accuracy and robustness of the solutions by IGA are proved to be better than FEM. [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]

A multi-layered shell formulation is developed based on a layerwise deformation theory (Reddy, 2004) within the framework of isogeometric analysis (IGA). IGA utilizes Non-Uniform Rational B-splines (NURBS) to represent the geometry as well as to describe the field variables (Hughes et al., 2005). The high-order smoothness of NURBS offered the opportunity of capturing the structural deformation efficiently in a rotation-free manner. The derivation also follows a layerwise theory, which assumes a separate displacement field expansion within each layer, and considers transverse displacement component as C 0 -continuous at layer interfaces, thus resulting in a layerwise continuous transverse strain states. Since the in-plane and through-thickness integrations are carried out individually, this approach is capable of capturing the complete three-dimensional stress states in a two-dimensional setting, which improves the computational efficiency. A knot insertion technique is utilized for the discretization in the through-thickness direction, and C 0 -continuity is enforced by means of knot repetition at dissimilar material interfaces. The performance of the proposed model is demonstrated using multiple laminated composites and sandwich plates (including functionally graded material core) as examples. Numerical results prove the accuracy of the proposed formulation and show that the isogeometric layerwise shell is superior to its finite element counterpart. [ABSTRACT FROM AUTHOR]

Nitsche’s method had recently demonstrated its potential for a seamless and continuity-preserving coupling of NURBS patches in isogeometric shell analysis. Trimmed and non-conforming patches and even blended models including classical thin-shell and solid-shell formulations can be coupled successfully while preserving a continuous flux along the coupling interfaces. In this paper, we take advantage of the method’s modeling flexibility for an extension to NURBS-based laminate composite shell structures. We introduce a global-local model coupling approach to enrich laminate composite thin-shell Kirchhoff-Love models locally with a full three-dimensional continuum solution. The approach provides a reliable coupling of the classical lamina theory with a layerwise formulation revealing the full interlaminar stress state at reduced computational cost. The layerwise shell model is based purely on a mid-surface NURBS description of the shell body thus supporting the direct application of a surface-defined CAD data representation of shells. We apply a refined finite cell approach to handle efficiently trimmed geometries, as common in CAD models. We demonstrate reliability, accuracy of the coupling approach and efficiency in terms of numerical costs and modeling effort of our approach with several examples. [ABSTRACT FROM AUTHOR]

Isogeometric collocation has emerged as an efficient numerical technique for solving boundary value problems and is a potential alternative for Galerkin based Isogeometric methods. In this investigation, Isogeometric collocation has been proposed for the linear static bending analysis of laminated composite plates governed by Reissner–Mindlin theory. Three formulations are presented in this paper namely, standard primal formulation, mixed formulation and a locking-free primal formulation. The standard primal formulation adopts displacements and rotations as unknown field variables. Mixed formulation considers displacements, rotations and transverse shear forces as the unknown field variables. Locking-free primal formulation is a rotation-free formulation with displacements and transverse shear strains as the unknown field variables. Results for benchmark problems on bending of rectangular laminated composite plates are obtained and compared with the ones existing in the literature. The three formulations of Isogeometric collocation presented in this paper are assessed in terms of accuracy and the computational time required to assemble and solve the problem. [ABSTRACT FROM AUTHOR]

The paper presents an effective thin plate formulation based on isogeometric analysis (IGA) and a non-classical Kirchhoff plate theory to study static bending, free vibration, and buckling behaviors of homogeneous and functionally graded microplates. The small scale effects are captured using a non-classical Kirchhoff plate theory which is developed based on a modified couple stress theory. The requirement for C 1 -continuity in terms of the non-classical Kirchhoff plate theory is straightforwardly possessed with the aid of inherent high-order continuity of non-uniform rational B-spline (NURBS). Studies on convergence and comparison with reference solutions are demonstrated in order to show the effectiveness and accuracy of the proposed method. Numerical examples are presented to illustrate the effects of small scale on the mechanical response of homogeneous and functionally graded microplates. The results reveal that the small scale effects lead to a reduction of deflection and an increase in frequency and buckling loads because of an increase in plate stiffness, and more importantly the small scale effects are significant for the thin plates. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to present a new numerical method to solve the weak formulation of the governing equations for the free vibrations of laminated composite shell structures with variable radii of curvature. For this purpose, the integral form of the stiffness matrix is computed numerically by means of the Generalized Integral Quadrature (GIQ) method. A two-dimensional structural model is introduced to analyze the mechanical behavior of doubly-curved shells. The displacement field is described according to the basic aspects of the general Higher-order Shear Deformation Theories (HSDTs), which allow to define several kinematic models as a function of the free parameter that stands for the order of expansion. Since an Equivalent Single Layer (ESL) approach is considered, the generalized displacements evaluated on the shell middle surface represent the unknown variables of the problem, which are approximated by using the Lagrange interpolating polynomials. The mechanical behavior of the structures is modeled through only one element that includes the double curvature in its formulation, which is transformed into a distorted domain by means of a mapping procedure based on the use of NURBS (Non-Uniform Rational B-Splines) curves, following the fundamentals of the well-known Isogeometric Analysis (IGA). For these reasons, the presented methodology is named Weak Formulation Isogeometric Analysis (WFIGA) in order to distinguish it from the corresponding approach based on the strong form of the governing equations (Strong Formulation Isogeometric Analysis or SFIGA), previously introduced by the authors. Several numerical applications are performed to test the current method. The results are validated for different boundary conditions and various lamination schemes through the comparison with the solutions available in the literature or obtained by a finite element commercial software. [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]

Variable-stiffness panel with curvilinear fibers is a promising structural concept compared to constant-stiffness designs. However, for the traditional finite element analysis (FEA), there is no guarantee that the fiber angle is continuous and smooth due to element discretization. In this study, on the basis of Mindlin plate theory, the buckling behavior of composite variable-stiffness panels is investigated based on isogeometric analysis (IGA), whose main feature is that the continuity of fiber angle on the whole panel is guaranteed. In particular, since geometric stiffness matrix has a significant influence on the buckling behavior, it is obtained by performing a static analysis prior to the buckling analysis herein, which can further improve the prediction accuracy of current methods. Different fiber path functions, ply number, geometric parameter, as well as various boundary and loading conditions are adopted to verify the proposed buckling analysis method. Finally, the prediction accuracy, total degree-of-freedom and CPU time are compared with the traditional FEA, which indicates that the isogeometric buckling analysis method can provide an adequate accuracy in a more efficient manner. [ABSTRACT FROM AUTHOR]

This paper presents a first know study of the effect of orientation angle of carbon nanotubes (CNTs), embedded in a single-ply polymer-based CNT-reinforced composite, on buckling behavior of skew plates using an isogeometric analysis. The plate model is based on Reddy’s higher-order shear deformation theory (HSDT). Four in-plane loading conditions together with simply supported and cantilevered boundary conditions have been considered. The optimum CNT fiber orientations for CNT-reinforced composite rhombic plates with varying skew angles and boundary conditions are presented. Moreover, the effect of width-to-thickness ratio on optimum CNT orientation is explored. The results show that the efficiency of the skew plate structures can be significantly improved by simply placing the CNTs in the correct orientation. [ABSTRACT FROM AUTHOR]

Practical applications such as airplane wings are usually subjected to combined thermal and mechanical loads, and they hence are prone to buckling failure. Preceding works on the buckling of advanced materials, e.g., functionally graded materials, under combined thermal and mechanical loads are rather rare in literature. In this paper, we report new numerical results of thermal-mechanical buckling of functionally graded rectangular and skew plates (FGPs) under combined thermal and mechanical loads. The numerical responses of buckling are computed using isogeometric analysis (IGA) based on the first-order shear deformation plate theory (FSDT) without shear-locking effect. We present formulations and then provide validation of numerical results computed by the proposed formulation against reference existing solutions. Parametric study is also performed to explore insight into the effects of various numerical aspect ratios such as gradient index, plate aspect ratio, loading type, skew angle, and boundary condition, etc. on mechanical response of FGPs. The stability diagrams are also presented. [ABSTRACT FROM AUTHOR]

The paper presents a new numerical approach for buckling analysis of non-uniform thickness nanoplates in an elastic medium using the isogeometric analysis (IGA). By ignoring the van der Waals interaction between two adjacent plates, non-uniform thickness nanoplates are described as a single-layered graphene sheet. The governing differential equation of the nanoplates is derived by the nonlocal theory in which the nonlocal stress-strain relation is used to capture the nonlocal mechanics caused by small size effects. The governing equation is then discretized into algebraic equations and solved by using IGA procedure to determine the critical buckling load. By using the non-uniform rational B-splines, IGA easily satisfies the required continuity of the partial differential equations in buckling analysis. Several numerical examples are solved and compared with those of previous publications to illustrate the performance of IGA for buckling analysis of nanoplates. [ABSTRACT FROM AUTHOR]