Multi-scale modelling of sandwich structures using hierarchical kinematics

The aim of this paper is to present a multi-scale method for the mechanical modelling of sandwich structures. Low- and high-order sandwich elements are formulated on the basis of Carrera’s Unified Formulation (CUF) and bridged within the Arlequin framework. According to CUF, an N-order polynomials approximation is assumed on the beam cross-section for the unknown displacements, being N a free parameter of the formulation. Low-order, computationally cheap elements are used to describe the global mechanical response. High-order, computationally demanding elements are used to capture the local effects in the boundary layers. CUF framework is here enhanced by the assumption of the Constrained Variational Principle (CVP) in order to derive a new class of layered beam finite elements with an independent kinematic field for each lamina. Results are assessed towards two- and three-dimensional finite element solutions. The numerical results show that, in the context of CUF, the Arlequin method effectively couples sub-domains modelled via variable order finite elements. The proposed coupled models yield accurate results, being able to predict both the global solution and the local effects, with a reduced computational cost.     

Unified formulation of geometrically nonlinear refined beam theories

By using the Carrera Unified Formulation (CUF) and a total Lagrangian approach, the unified theory of beams including geometrical nonlinearities is introduced in this article. According to CUF, kinematics of one-dimensional structures are formulated by employing an index notation and a generalized expansion of the primary variables by arbitrary cross-section functions. Namely, in this work, low- to higher-order beam models with only pure displacement variables are implemented by utilizing Lagrange polynomial expansions of the unknowns on the cross section. The principle of virtual work and a finite element approximation are used to formulate the governing equations, whereas a Newton-Raphson linearization scheme along with a path-following method based on the arc-length constraint is employed to solve the geometrically nonlinear problem. By using CUF and three-dimensional Green-Lagrange strain components, the explicit forms of the secant and tangent stiffness matrices of the unified beam element are provided in terms of fundamental nuclei, which are invariants of the theory approximation order. A symmetric form of the secant matrix is provided as well by exploiting the linearization of the geometric stiffness terms. Various numerical assessments are proposed, including large deflection analysis, buckling, and postbuckling of slender solid cross-section beams. Thin-walled structures are also analyzed in order to show the enhanced capabilities of the present formulation. Whenever possible, the results are compared to those from the literature and finite element commercial software tools.     

Buckling of composite thin walled beams by refined theory

This work presents the buckling analysis of laminated composite thin walled structures by the 1D finite element based unified higher-order models obtained within the framework of the Carrera Unified Formulation (CUF). In the present study, the refined beam theories are obtained on the basis of Taylor-type expansions. The finite element analysis has been chosen to easily handle arbitrary geometries as well as boundary conditions. Buckling behavior of laminated composite beam and flat panels are analyzed to illustrate the efficacy of the present formulation and various types of buckling modes are observed depending on the geometrical and material parameters. It is observed that the lower order models are unable to deal with torsion.     

Elastoplastic analysis of compact and thin-walled structures using classical and refined beam finite element models

The paper presents results on the elastoplastic analysis of compact and thin-walled structures via refined beam models. The application of Carrera Unified Formulation (CUF) to perform elastoplastic analysis of isotropic beam structures is discussed. Particular attention is paid to the evaluation of local effects and cross-sectional distortions. CUF allows formulation of the kinematics of a one-dimensional (1D) structure by employing a generalized expansion of primary variables by arbitrary cross-section functions. Two types of cross-section expansion functions, TE (Taylor expansion) and LE (Lagrange expansion), are used to model the structure. The isotropically work-hardening von Mises constitutive model is incorporated to account for material nonlinearity. A Newton–Raphson iteration scheme is used to solve the system of nonlinear algebraic equations. Numerical results for compact and thin-walled beam members in plastic regime are presented with displacement profiles and beam deformed configurations along with stress contour plots. The results are compared against classical beam models such as Euler–Bernoulli beam theory and Timoshenko beam theory, reference solutions from literature, and three-dimensional (3D) solid finite element models. The results highlight: (1) the capability of the present refined beam models to describe the elastoplastic behavior of compact and thin-walled structures with 3D-like accuracy; (2) that local effects and severe cross-sectional distortions can be detected; (3) the computational cost of the present modeling approach is significantly lower than shell and solid model ones.     

Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates

Accurate free-vibrations and linearized buckling analysis of anisotropic laminated plates with different lamination schemes and simply supported boundary condition are addressed in this paper. Approximation methods, such as Rayleigh-Ritz, Galerkin and Generalized Galerkin, based on Principle of Virtual Displacement are derived in the framework of Carrera’s Unified Formulation (CUF). CUF widely used in the analysis of composite laminate beams, plates and shells, have been here formulated both for the same and different expansion orders, for the displacement components, in the thickness layer-plate direction. An extensive assessment of advanced and refined plate theories, which include Equivalent single Layer (ESL), Zig-Zag (ZZ) and Layer-wise (LW) models, with increasing number of displacement variables is provided. Accuracy of the results is shown to increase by refining the theories. Convergence studies are made in order to demonstrate that accurate results are obtained examining thin and thick plates using trigonometric approximation functions. The effects of boundary terms, upon frequency parameters and critical loads are evaluated. The effects of the various parameters (material, number of layers, fiber orientation, thickness ratio, orthotropic ratio) upon the frequencies and critical loads are discussed as well. Numerical results are compared with 3D exact solution when available from the open literature.     

基于行波方法的智能悬臂梁振动控制

从振动波传播的观点，基于行波理论研究了一维悬臂梁的振动控制方法。以Bernoulli-Euler梁模型为研究对象，研究了行波理论建模和分析方法。在此基础上，基于横向位移、横向转角、应变这三种不同测量信号，应用波传递和反射的关系式，分别设计了行波控制器，并推导了时域和频域的控制力。最后对一悬臂梁进行了振动控制，数值算例表明了行波控制是有效的。     

Static analysis of reinforced thin-walled plates and shells by means of finite element models

In this paper, variable kinematic one-dimensional (1D) structural models have been used to analyze thin-walled structures with longitudinal stiffeners and static loads. These theories have hierarchical features and are based on the Carrera Unified Formulation (CUF). CUF describes the displacement field of a slender structure as the product of two function expansions, one over the cross-sectional coordinates, Taylor (TE) or Lagrange (LE) expansions were used here, and one along the beam axis. The results obtained using the refined 1D models have been compared with those from classical finite element analyses that make use of plates/shells and solids elements. The performances of classical and refined structural models have been compared in terms of accuracy and computational costs. The results show that the use of the LE over the cross-section allows the strain/stress fields to be evaluated accurately for all the structural components. The comparisons with the results obtained using the classical models highlight how, the use of 1D refined models, allows the number of degrees of freedom (DOF) to be reduced, meanwhile, the accuracy of the results can be preserved.     

Free vibration analysis of reinforced thin-walled plates and shells through various finite element models

Abstract

This article deals with free vibration analysis of thin-walled structures reinforced by longitudinal stiffeners using refined one-dimensional (1D) models.The 1D theory, which is used in the present article, has hierarchical features and it is based on the Carrera Unified Formulation (CUF). The displacement field over the cross section is obtained by means of Taylor (TE) or Lagrange (LE) expansions. Finite element (FE) method is applied along the beam axis to obtain weak form solutions of the related governing equations. The obtained results are compared with those from classical finite element formulations based on plate and shell (2D), beam (1D), and solid (3D) elements that are available in commercial software. When solid formulation is used to build the FE solutions, stringers and skin are modeled with only 3D elements while, in the 2D-1D FE models, shell and beam elements are used for skin and stringers, respectively. Three benchmark problems are analyzed: a flat plate, a curved panel, and a thin-walled cylinder. When TE models are used, different orders of expansion, N, are considered, where N is a free parameter of the formulation. As far as Lagrange expansions are concerned, four-node (LE 4) and nine-node (LE 9) elements are used to build different meshes on the cross section. The results show that the present 1D models are able to analyze the dynamic behavior of complex structures and can detect 3D effects as well as very complex shell-like modes typical of thin-walled structures. Moreover, the 1D-CUF elements yield accurate results with a low number of degrees of freedom.     

Nonlocal piezoelasticity based wave propagation of bonded double-piezoelectric nanobeam-systems

Piezoelectric nanobeam (PNB) offer the possibility of being used in micro-electromechanical systems and nano-electromechanical systems and the dynamic testing of such structures often produces stress wave propagation in them. This work concerns with the size-dependent wave propagation of double-piezoelectric nanobeam-systems (DPNBSs) based on Euler–Bernoulli beam model. The two piezoelectric nanobeams are coupled by an enclosing elastic medium which is simulated by Pasternak foundation. Nonlocal piezoelasticity theory is used to derive the general differential equation based on Hamilton’s principal to include those scale effects. Particular attention is paid to the wave propagation piezoelectric control of the coupled system in three cases namely in-phase wave propagation, out-of-phase wave propagation and wave propagation when one PNB is stationary. In three mentioned cases, an analytical method is proposed to obtain phase velocity; cut-off and escape frequencies of the DPNBSs. Results indicate that the imposed external voltage is an effective controlling parameter for wave propagation of the coupled system. Furthermore, the phase velocity of in-phase wave propagation is independent of elastic medium stiffness.     

Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates

Accurate free-vibrations and linearized buckling analysis of anisotropic laminated plates with different lamination schemes and simply supported boundary condition are addressed in this paper. Approximation methods, such as Rayleigh-Ritz, Galerkin and Generalized Galerkin, based on Principle of Virtual Displacement are derived in the framework of Carrera’s Unified Formulation (CUF). CUF widely used in the analysis of composite laminate beams, plates and shells, have been here formulated both for the same and different expansion orders, for the displacement components, in the thickness layer-plate direction. An extensive assessment of advanced and refined plate theories, which include Equivalent single Layer (ESL), Zig-Zag (ZZ) and Layer-wise (LW) models, with increasing number of displacement variables is provided. Accuracy of the results is shown to increase by refining the theories. Convergence studies are made in order to demonstrate that accurate results are obtained examining thin and thick plates using trigonometric approximation functions. The effects of boundary terms, upon frequency parameters and critical loads are evaluated. The effects of the various parameters (material, number of layers, fiber orientation, thickness ratio, orthotropic ratio) upon the frequencies and critical loads are discussed as well. Numerical results are compared with 3D exact solution when available from the open literature.     

Wave propagation of coupled double-DWBNNTs conveying fluid-systems using different nonlocal surface piezoelasticity theories

This work is concerned with the size-dependent wave propagation of coupled double-walled boron nitride nanotubes (DWBNNTs) conveying nanoflow-systems based on Timoshenko beam theory. The two DWBNNTs are coupled by an enclosing visco-Pasternak medium. The small-scale effects are captured applying different surface piezoelasticity theories, including stress gradient, strain gradient, and strain inertia gradient. An analytical method is proposed to obtain phase velocity, cut-off, and escape frequencies of the system. Three cases of in-phase wave propagation, out-of-phase wave propagation, and wave propagation with one DWBNNT fixed are considered. Results indicate that ignoring surface and small-scale effects lead to inaccurate results.     

Axiomatic/asymptotic analysis of refined layer-wise theories for composite and sandwich plates

This article deals with layer-wise (LW) models for composite and sandwich plates. Refined layer-wise models are built according to the Carrera Unified Formulation (CUF), which has been developed over the last decade for beams, plate, and shell theories. CUF allows the hierarchical implementation of refined models based on any-order expressions of the unknown variables. In this article, displacement variables are expanded along the layer thickness through Legendre polynomials. Comparisons with previous analysis based on equivalent single layer (ESL) approaches are given. The effect of each term of the expansion on the accuracy of stress/displacement components for the static response of composite and sandwich plates is analyzed. Ineffective terms are discarded from the expansion in order to save computational cost. The reduced models obtained, which are denoted as mixed axiomatic/asymptotic models, are as accurate as full expansion models. Numerical analysis is restricted to closed-form solutions via Navier-type solutions. A number of problems related to laminated and sandwich structures are solved and related reduced models are built by varying geometrical, lay-up, and mechanical parameters. Results show that in some cases (in particular those related to sandwich plates) reduced layer-wise models can save up to 50% of the degrees-of-freedom of the full models without significant accuracy losses. It is found that the significant terms related to reduced models are very much subordinated to the problems considered and that from that point of view the use of a framework that can generate any theory, such as CUF, appears very suitable to build reduced models for plates.     

Extension of MITC to higher‐order beam models and shear locking analysis for compact,thin‐walled,and composite structures

A class of mixed interpolated beam elements is introduced in this paper under the framework of the Carrera Unified Formulation to eliminate the detrimental effects due to shear locking. The Mixed Interpolation of Tensorial Components (MITC) method is adopted to generate locking‐free displacement‐based beam models using general 1D finite elements. An assumed distribution of the transverse shear strains is used for the derivation of the virtual work, and the full Gauss‐Legendre quadrature is used for the numerical computation of all the components of the stiffness matrix. Linear, quadratic, and cubic beam elements are developed using the unified formulation and applied to linear static problems including compact, laminated, and thin‐walled structures. A comprehensive study of how shear locking affects general beam elements when different classical integration schemes are used is presented, evidencing the outstanding capabilities of the MITC method to overcome this numerical issue. Refined beam theories based on the expansion of pure and generalized displacement variables are implemented making use of Lagrange and Legendre polynomials over the cross‐sectional domain, allowing one to capture complex states of stress with a 3D‐like accuracy. The numerical examples are compared to analytic, numerical solutions from the literature, and commercial software solutions, whenever it is possible. The efficiency and robustness of the proposed method demonstrated throughout all the assessments, illustrating that MITC elements are the natural choice to avoid shear locking and showing an unprecedent accuracy in the computation of transverse shear stresses for beam formulations.     

Comparative study of different nonconserving time integrators for wave propagation in hyperelastic waveguides

This paper addresses the formulation and numerical efficiency of various numerical models of different nonconserving time integrators for studying wave propagation in nonlinear hyperelastic waveguides. The study includes different nonlinear finite element formulations based on standard Galerkin finite element model, time domain spectral finite element model, Taylor–Galerkin finite element model, generalized Galerkin finite element model and frequency domain spectral finite element model. A comparative study on the computational efficiency of these different models is made using a hyperelastic rod model, and the optimal computational scheme is identified. The identified scheme is then used to study the propagation of transverse and longitudinal waves in a Timoshenko beam with Murnaghan material nonlinearity.     

Elastic wave propagation in a microstructured acoustic fiber

Elastic wave propagation along the structure of hollow cylinders in a linear isotropic medium is investigated. The multipole method for modeling elastic waves propagation in such structures is formulated and implemented. Using the multipole method, dispersion dependencies of the structures (microstructured fibers) containing 3, 6, and 7 hollow cylinders are calculated. Comparison with wave dispersion properties along one cylinder is made. Also, an approximate physical model based on an equivalent coaxial waveguide and multipole method is proposed. Exploiting this model, wave dispersion of the wave propagating along a structure with 18 hollow cylinders is calculated. Validation of the model is also proposed.     

Aerodynamic and mechanical hierarchical aeroelastic analysis of composite wings

Abstract

The coupled bending-torsion flutter is here investigated through Carrera Unified Formulation (CUF). The hierarchical capabilities of CUF offer a procedure to obtain refined one-dimensional models that, by going beyond the assumptions of classical theories, accurately describe the kinematics of structures. Aerodynamic loadings have been determined according to Theodorsen theory, from which the steady formulation can be easily obtained. The displacement variables over the cross section (x-z plane) are approximated by x,z polynomials of any order, N. The finite element method is used to solve the governing equations, which are derived in a weak form through the principle of virtual displacements. The equations are written in terms of “fundamental nuclei,” which do not vary with the theory order, N. Several wing configurations have been studied, giving great attention to thin-walled box beams made of orthotropic material. The effects of sweep angle and lamination scheme on flutter conditions have been investigated, and the results have been compared with solutions obtained from two-dimensional theories, experimental tests, and aeroelastic analyses carried out with the doublet lattice method (DLM). The unsteady theory, combined with advanced beam theories, represents a computationally cheap tool for preliminary aeroelastic studies of complex wing structures.     

Study of radial basis collocation method for wave propagation

A numerical method based on radial basis functions and collocation method is proposed for wave propagation. Standard collocation and weighted boundary collocation approaches yield significant errors in wave problems. Therefore, a new method based on explicit time integration scheme that can correct the inaccuracy in the solutions and the errors accumulated in time integration is developed. This method can be easily applied for low and high dimensional wave problems. The stability conditions are obtained and the relationships between control parameters and stability are evaluated. Requirement of collocation points in numerical dispersion is studied and nondispersion condition is formulated. Eigenvalue analysis is investigated to evaluate the effectiveness of radial basis collocation method for solving wave problems. Eigenvalue study with and without imposing the boundary conditions are compared. The influences of shape parameters and distribution of collocation points and source points are presented. Numerical examples are simulated to examine and validate the proposed method.     

Refined shell elements for the analysis of functionally graded structures

The present paper considers the static analysis of plates and shells made of Functionally Graded Material (FGM), subjected to mechanical loads. Refined models based on the Carrera’s Unified Formulation (CUF) are employed to account for grading material variation in the thickness direction. The governing equations are derived from the Principle of Virtual Displacement (PVD) in order to apply the Finite Element Method (FEM). A nine-nodes shell element with exact cylindrical geometry is considered. The shell can degenerate in the plate element by imposing an infinite radius of curvature. The Mixed Interpolation of Tensorial Components (MITC) technique is extended to the CUF in order to contrast the membrane and shear locking phenomenon. Different thickness ratios and orders of expansion for the displacement field are analyzed. The FEM results are compared with both benchmark solutions from literature and the results obtained using the Navier method that provides the analytical solution for simply-supported structures subjected to sinusoidal pressure loads. The shell element based on refined theories of the CUF turns out to be very efficient and its use is mandatory with respect to the classical models in the study of FGM structures.     

Stability analysis of FGM sandwich plates by using variable-kinematics Ritz models

Abstract

A linearized buckling analysis of functionally graded material (FGM) isotropic and sandwich plates is carried out by virtue of the Hierarchical Trigonometric Ritz Formulation (HTRF). Quasi-3D Ritz models based on equivalent single layer (ESL) and zig zag (ZZ) plate theories are developed within the framework of the Carrera Unified Formulation (CUF). Several in-plane loading conditions accounting for axial, biaxial, and shear loadings are taken into account. Parametric studies are carried out in order to evaluate the effects of significant parameters, such as volume fraction index, length-to-thickness ratio, sandwich plate type, and loading type, on the critical buckling loads.     

Variable Kinematic Shell Elements for the Analysis of Electro-Mechanical Problems

The present article considers the linear static analysis of both composite plate and shell structures embedding piezoelectric layers by means of a shell finite element with variable through-the-thickness kinematic. The refined models used are grouped in the Unified Formulation by Carrera (CUF) and they permit to accurately describe the distribution of displacements and stresses along the thickness of the multilayered shell. The shell element has nine nodes and the mixed interpolation of tensorial components (MITC) method is employed to contrast the membrane and shear locking phenomenon. The governing equations are derived from the principle of virtual displacement (PVD) and the finite element method (FEM) is employed to solve them. Cross-ply multilayered plates and cylindrical shells embedding piezoelectric layers are analyzed with simply-supported boundary conditions and subjected to sensor and actuator configurations. Various thickness ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in literature and the analytical solutions obtained using the CUF and the Navier’s method. From the analysis, one can conclude that the shell element based on the CUF is very efficient and its use is mandatory with respect to the classical models in the study of multilayered structures embedding piezo-layers.