Carrera Unified Research Articles

Hierarchical beam finite elements for geometrically nonlinear analysis coupled with Asymptotic Numerical Method

This paper couples the Carrera's Unified Formulation (CUF) with the Asymptotic Numerical Method (ANM) for investigating geometrically nonlinear behaviors of beam structures. On the basis of CUF, a family of advanced one-dimensional beam models is firstly established by deriving a fundamental nucleus. The ANM that is an efficient and robust nonlinear solver is then used to solve the resulted nonlinear systems. Several typical nonlinear problems in thin/thick beam structures are carried out, and numerical results demonstrate the validity and efficiency of the proposed framework. Besides, the importance of using high-order kinematics for beam structures undergoing large deformations is emphasized.

Modeling of simultaneous shape memory and pseudoelastic effects of shape memory alloys on nonlinear dynamic response of multilayer composite plate embedded with pre-strained SMA wires under thermal condition

ABSTRACTShape memory alloys as a kind of smart materials, have two unique characteristics entitled shape memory and pseudoelastic effects. In this paper, a nonlinear dynamic analysis of multilayered composite plate embedded with pre-strained SMA wires under thermal condition is implemented considering both effects of SMAs simultaneously, for the first time. The constitutive equation proposed by Brinson is used for modelling the behaviors of SMA wires. In this work, equations of motion are derived in the framework of Carrera's Unified Formulation, for robust analysis of the problem. In this regard, the nonlinear strains are employed for modelling the thermal effects and also the effect of recovery stresses in the SMA wires. In this study not only the material properties are instantaneous variable with respect to the time and location, but they are also unknown, which make the problem more complicated. For this aim, a transient nonlinear finite element in conjunction with incremental iterative algorithm proposed by the author is employed, in order to solve the equations. Results show that upon the thermal condition, the recovery stress is generated in the SMA wires and therefore reduce the amplitude of a damped response of the plate. One conclusion from this study is that, in contrast with many studies reported before, it cannot be always say that the stress recovery in SMA wires is increased with increasing the pre-strain of SMA wires. In other words, for a specified increasing in temperature, only a limit amount of pre-strain comes back to its original shape and not more. Also, several numerical examples upon effect of different thermal condition, different volume fraction of SMA wires and different boundary conditions have been analyzed.

Analysis of composite layered beams using Carrera unified formulation with Legendre approximation

Abstract This paper presents some numerical investigations related to the use of a new approximation function to be applied in Carrera's Unified Formulation (CUF). The main objective is to study and assess the efficiency of the CUF, when Legendre approximation functions are used at the section level, on 1D element, using an equivalent single layer (ESL) formulation. Previous experimental and analytical results, obtained in the case of GFRP composite bridge decks and sandwich panels, are used to further validate the numerical results being reported. 3D finite elements are also used, in order to compare and validate the numerical structural outputs. The main conclusion is that the classical Legendre polynomial functions are suitable and provide accurate results. The use of this type of functions does not require any stabilization procedure of the resulting governing system for the case of 1D elements, in clear contrast to what typically happens when other approximation functions are used with an equivalent single layer formulation.

A new robust quadrilateral four-node variable kinematics plate element for composite structures

This paper presents a new four-node quadrilateral finite element for composite plates. A large number of displacement-based, variable kinematics plate models are formulated in the framework of Carrera's Unified Formulation, which encompass Equivalent Single Layer as well as Layer-Wise models with a displacement field that is defined by polynomials up to 4th order along the thickness direction z. The main novelty consists in the formulation of a field compatible approximation for the transverse shear strain field, referred to as QC4 interpolation, which eliminates the shear locking pathology by constraining only the z− constant transverse shear strain terms. Extensive numerical studies are proposed that demonstrate the absence of spurious modes and of locking problems as well as the enhanced robustness with respect to distorted element shapes in comparison to classical isoparametric approaches. The new QC4 variable kinematics plate element displays excellent convergence rates under different boundary and loading conditions, and it yields accurate displacement and stresses for both, thick and thin composite plates.

A boundary-discontinuous-displacement based Fourier analysis of thick laminated beams via a robust 1D-CUF model

This paper presents an analytical solution for the static analysis of thick laminated rectangular beams with clamped boundary conditions at either or both of the beam's edges. A unified formulation known as Carrera's Unified Formulation (CUF) is used in order to consider shear deformation theories of arbitrary order. The governing equations are obtained by using the principle of virtual work. The main novelty is the use of the boundary-discontinuous Fourier approach for laminated beams in the framework of a unified formulation. Unlike Navier-type solutions, the present development can obtain analytical solutions for beams with clamped boundary conditions. A 3D finite element solution is used to validate the obtained results. The present theory can analyze clamped beams accurately so benchmark results are provided.

Free vibration and buckling analysis of cross-ply laminated composite plates using Carrera's unified formulation based on Isogeometric approach

In this paper, The Isogeometric approach (IGA) and Carrera's Unified Formulation (CUF) are employed for free vibration and linearized buckling analysis of laminated composite plates. The non-uniform rational B-spline (NURBS) basis functions utilized in IGA, are employed as higher order smooth functions to approximate field solution leading to enhance precision of analysis. CUF presents an effective formulation to employ any order of Taylor expansion to analyze two-dimensional plate models. Higher order NURBS basis functions attenuate the shear locking properly and higher order theories supposed by CUF are free from Poisson locking phenomenon and they do not need the use of any shear correction factor. Therefore, combining IGA and CUF ends in a suitable methodology to analyze laminated plates.

Analysis of viscoelastic sandwich laminates using a unified formulation and a differential quadrature hierarchical finite element method

In this paper we present a layerwise differential quadrature hierarchical finite element (DQHFE) model for the analysis of sandwich laminated plates with a viscoelastic core and laminated anisotropic face layers. The stiffness and mass matrices of elements are obtained by Carrera's Unified Formulation (CUF). The dynamic problem is solved in the frequency domain with viscoelastic frequency-dependent material properties for the core. The dynamic behavior of the model is compared with solutions found in literature. The DQHFE model was shown to be able to present highly accurate results using much less degrees of freedom than low order schemes like the finite element method.

Shell Finite Elements for the Analysis of Multifield Problems in Multilayered Composite Structures

This paper deals with the analysis of layered structures under thermal and electro-mechanical loads. Constitutive equations for multifield are considered and the Principle of Virtual Displacements (PVD) is employed to derive the governing equations. The MITC9 shell finite element based on the Carrera's Unified Formulation (CUF) has been applied for the analysis. The models grouped in the CUF have variable through-the-thickness kinematic and they provide an accurate distribution of displacements and stresses along the thickness of the laminate. The shell element has nine nodes and the Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomenon. The finite element analysis of multilayered plates and shells has been addressed. Variable kinematics, as well as layer-wise and equivalent single layer descriptions, have been considered for the presented FEs, according to CUF. A few problems are analyzed to show the effectiveness of the proposed approach. Various laminations, thickness ratios and curvature 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.

Reissner's Mixed Variational Theorem and variable kinematics in the modelling of laminated composite and FGM doubly-curved shells

The present article proposes a mixed displacements/transverse stresses approach for the free vibration analysis of laminated composite and FGM doubly-curved shells. The theoretical formulation is derived by combining Reissner's Mixed Variational Theorem (RMVT), Carrera's Unified Formulation (CUF) and the Ritz method. With the application of the RMVT the interlaminar equilibrium of the transverse normal and shear stresses is fulfilled a priori by exploiting the use of Lagrange multipliers. The transverse normal and shear stresses become primary variables within the formulation and are modelled with a Layer-Wise (LW) kinematics description. However, on the other hand, displacement variables, which describe the kinematics of the shell structures, are defined using Equivalent Single Layer (ESL), Zig-Zag (ZZ) and LW shell theories. The Mixed Hierarchical Trigonometric Ritz Formulation (MHTRF) is then used as solution technique to compute the natural frequencies of laminated composite and FGM doubly-curved shells. Several study-cases are addressed and the proposed RMVT-based shell models are assessed by comparison with both 3D elasticity and 3D Ritz solutions. The effect of significant parameters such as orthotropic ratio, stacking sequence, aspect ratio, lamination angle, length-to-thickness and radius-to-length ratios as well as volume fraction index on the dimensionless frequency parameters is discussed.

Non-linear transient dynamic analysis of sandwich plate with composite face-sheets embedded with shape memory alloy wires and flexible core- based on the mixed LW (layer-wise)/ESL (equivalent single layer) models

In this study, a nonlinear dynamic analysis of sandwich plate with flexible core and laminated composite face sheets embedded with shape memory alloy (SMA) wires is investigated using the mixed LW/ESL models in the framework of Carrera's Unified Formulation. The instantaneous phase transformation effects are considered for every point on the face sheets. Since, this research deals with the transient nonlinear problem and the structure is thick plate with flexible core, employing a model with high accuracy and low computational cost is vital in this work. For this aim, the new mixed LW/ESL models proposed are employed in this study. The constitutive equation proposed by Brinson is used for modeling the nonlinear behavior of SMA wires. The governing equations are derived employing the Reissner Mixed Variational Theorem (RMVT) in order to satisfy the interlaminar continuity of transverse stresses between the layers. The nonlinear governing equations are solved based on the transient finite element along with the iterative incremental method. Some parametric studies such as intensity of impulsive pressure, location of SMA wires, plate aspect ratio, ratio of face sheet's thickness to the total thickness, volume fraction of the SMA wires and also boundary conditions upon the loss factor are investigated.

Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core

In this paper, the Ritz minimum energy method, based on the use of the Principle of Virtual Displacements (PVD), is combined with refined Equivalent Single Layer (ESL) and Zig Zag (ZZ) shell models hierarchically generated by exploiting the use of Carrera's Unified Formulation (CUF), in order to engender the Hierarchical Trigonometric Ritz Formulation (HTRF). The HTRF is then employed to carry out the free vibration analysis of doubly curved shallow and deep functionally graded material (FGM) shells. The PVD is further used in conjunction with the Gauss theorem to derive the governing differential equations and related natural boundary conditions. Donnell–Mushtari's shallow shell-type equations are given as a particular case. Doubly curved FGM shells and doubly curved sandwich shells made up of isotropic face sheets and FGM core are investigated. The proposed shell models are widely assessed by comparison with the literature results. Two benchmarks are provided and the effects of significant parameters such as stacking sequence, boundary conditions, length-to-thickness ratio, radius-to-length ratio and volume fraction index on the circular frequency parameters and modal displacements are discussed.

Accurate Buckling Analysis of Composite Layered Plates with Combined Thermal and Mechanical Loadings

This article is devoted to the linearized buckling analysis of laminated composite plates subjected simultaneously to mechanical and thermal loading. Orthotropic and anisotropic plates are considered. The Principle of Virtual Displacement (PVD) is applied and the Finite Element Method (FEM) is adopted in order to approximate the solution. The same case-studies are handled using different kinds of plate Finite Elements (FEs) and the accuracy of the analyses is discussed. The considered set of FEs is hierarchical in the sense that a variable kinematical approach is employed: both Equivalent Single Layer (ESL) and Layer-Wise (LW) variable descriptions are implemented, keeping the expansion order generic for the thickness variables. FE matrices are obtained in compact form by referring to Carrera's Unified Formulation (CUF). When available, exact solutions are referred to in order to assess the FEM results, which are all obtained with the academic code MUL2.

Radial basis functions collocation for the bending and free vibration analysis of laminated plates using the Reissner-Mixed Variational Theorem

Abstract In this paper, we combine the Carrera's Unified Formulation CUF ( E. Carrera. Theories and Finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarking. Arch. Comput. Meth. Eng., 10:215–297, 2003 ) and a radial basis function (RBF) collocation technique for predicting the static deformations and free vibrations behavior of thin and thick cross-ply laminated plates. For the first time, the Reissner-Mixed Variational Theorem is used together with the RBF collocation to achieve a highly accurate technique. The accuracy and efficiency of this collocation technique for static and vibration problems are demonstrated through numerical examples.

Buckling analysis of sandwich plates with functionally graded skins using a new quasi‐3D hyperbolic sine shear deformation theory and collocation with radial basis functions

AbstractA hyperbolic sine shear deformation theory is used for the linear buckling analysis of functionally graded plates. The theory accounts for through‐the‐thickness deformations. The buckling governing equations and boundary conditions are derived using Carrera's Unified Formulation and further interpolated by collocation with radial basis functions. The collocation method is truly meshless, allowing a fast and simple discretization of equations in the domain and on the boundary. A numerical investigation has been conducted considering and neglecting the thickness stretching effects on the buckling of sandwich plates with functionally graded skins. Numerical results demonstrate the high accuracy of the present approach.

Free vibration analysis for layered shells accounting of variable kinematic and thermo-mechanical coupling

The free vibration analysis of one-layered and two-layered metallic cylindrical shell panels is evaluated in this work. The free frequency values are investigated for both thermo-mechanical and pure mechanical problems. Thermo-mechanical frequencies are calculated by means of a fully coupled thermo-mechanical model where both the displacement and temperature are primary variables in the considered governing equations. Pure mechanical frequencies are obtained from a mechanical model where the effect of the temperature field is not included in the stiffness matrix and the displacement is the only primary variable of the problem. The inclusion of the thermal part in the stiffness matrix gives larger frequencies. Both thermo-mechanical and pure mechanical models are developed in the framework of Carrera's Unified Formulation (CUF) in order to obtain several variable kinematic models. Both equivalent single layer and layer wise approaches are considered for multilayered shells. The use of refined two-dimensional theories for shells permits the evaluation of the effects of the thermo-mechanical coupling for lower and higher order modes, higher frequency values, multilayered configurations, thick and thin shells and several values of the radius of curvature of the shell geometry. It has mainly been concluded that the thermo-mechanical coupling is not influenced by the curvature of the shells, therefore, the main conclusions already given for the plate geometry are here confirmed: - the thermo-mechanical coupling is correctly determined if both the thermal and mechanical parts are correctly approximated; - it is small for each investigated case; - it influences the various vibration modes in different ways; - it has a limited dependence on the considered case, but this dependence vanishes if a global coupling is considered.

Free Vibration Analysis for Layered Shells Accounting of Variable Kinematic and Thermo-Mechanical Coupling

The free vibration analysis of one-layered and two-layered metallic cylindrical shell panels is evaluated in this work. The free frequency values are investigated for both thermo-mechanical and pure mechanical problems. Thermo-mechanical frequencies are calculated by means of a fully coupled thermo-mechanical model where both the displacement and temperature are primary variables in the considered governing equations. Pure mechanical frequencies are obtained from a mechanical model where the effect of the temperature field is not included in the stiffness matrix and the displacement is the only primary variable of the problem. The inclusion of the thermal part in the stiffness matrix gives larger frequencies. Both thermo-mechanical and pure mechanical models are developed in the framework of Carrera's Unified Formulation (CUF) in order to obtain several variable kinematic models. Both equivalent single layer and layer wise approaches are considered for multilayered shells. The use of refined two-dimensional theories for shells permits the evaluation of the effects of the thermo-mechanical coupling for lower and higher order modes, higher frequency values, multilayered configurations, thick and thin shells and several values of the radius of curvature of the shell geometry. It has mainly been concluded that the thermo-mechanical coupling is not influenced by the curvature of the shells, therefore, the main conclusions already given for the plate geometry are here confirmed: – the thermo-mechanical coupling is correctly determined if both the thermal and mechanical parts are correctly approximated; – it is small for each investigated case; – it influences the various vibration modes in different ways; – it has a limited dependence on the considered case, but this dependence vanishes if a global coupling is considered.

Refined Shell Models for the Vibration Analysis of Multiwalled Carbon Nanotubes

The free vibration response of double-walled carbon nanotubes is studied in this article. A continuum approach is used and refined shell models are obtained through the application of Carrera's Unified Formulation (CUF). In order to apply the continuum models, the multi-walled carbon nanotube is assumed to be a multilayer cylindrical shell with an equivalent thickness and Young's modulus. Equivalent single-layer and layer-wise approaches in CUF are modified to consider the nested tubes of a multi-walled carbon nanotube as independent and separately vibrating. The van der Waals interaction between two adjacent tubes is estimated by using the Lennard-Jones model. The proposed results emphasize the differences that exist among available continuum models and present shell approaches. In particular, the attention is devoted to van der Waals interaction and a comparison has been made between models that take into account the van der Waals forces and models that neglect them.

Two higher order Zig-Zag theories for the accurate analysis of bending, vibration and buckling response of laminated plates by radial basis functions collocation and a unified formulation

In this article, we combine the Carrera's Unified Formulation, CUF (Carrera E. Theories and Finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarking. Arch. Comput. Methods Eng., 2003; 10: 215–297.) and a radial basis function collocation technique for predicting the static deformations, free vibrations and buckling behavior of thin and thick cross-ply laminated plates. We develop by the CUF two Zig-Zag theories according to Murakami's Zig-Zag function. Both theories account for through-the-thickness deformations, allowing the analysis of thick plates. The accuracy and efficiency of this collocation technique for static, vibration, and buckling problems are demonstrated through numerical examples.

THERMOMECHANICAL EFFECT IN VIBRATION ANALYSIS OF ONE-LAYERED AND TWO-LAYERED PLATES

The free vibration problem of one-layered and two-layered metallic plates is investigated in this work. The thermomechanical effect is evaluated using a fully coupled thermomechanical model. The free frequency values of fully coupled problems are compared to the values of the pure mechanical problems. In pure mechanical models, the displacement is the only primary variable of the problem, while in fully coupled thermomechanical models, the temperature is also considered as a primary variable and the effect of the thermomechanical stiffness is evaluated. The thermoelastic coupling usually provides higher frequencies with respect to the pure mechanical case because it acts like a thermal source, which is proportional to the strain rate, which leads to a bigger global stiffness of the structure. Both thermomechanical and mechanical models are developed in the framework of Carrera's Unified Formulation (CUF). CUF permits several refined two-dimensional theories to be obtained with orders of expansion in the thickness direction, from linear to fourth-order, for both displacements and temperature. Both equivalent single layer and layer-wise approaches are considered for the multilayered plates. The thermomechanical effect is investigated, in terms of frequencies, for thick and thin one-layered and two-layered plates, and for lower and higher modes. It has mainly been concluded that the thermomechanical coupling: (a) Is correctly determined if both the thermal and mechanical parts are correctly approximated; (b) Is small for each investigated case; (c) Influences the various vibration modes in different ways; and (d) Has a limited dependence on the considered case, but this dependence vanishes if a global coupling is considered. Only fully coupled thermomechanical models allow to analyze this type of problem. The effect of the thermomechanical coupling on higher-order modes can only be investigated using refined two-dimensional theories.

Accurate Vibration Analysis of Multilayered Plates Made of Functionally Graded Materials

Closed form solutions of free vibration problems in simply supported multilayered plates made of Functionally Graded Material (FGM) are dealt with in the present paper. Various refined plate theories are compared in the cases of both equivalent single-layer and layer-wise variable descriptions. Classical theories are compared to advanced ones, in which higher orders of expansion in the thickness direction are used for the transverse displacement. The plate governing equations are provided by applying Carrera's Unified Formulation (CUF) which, in this paper, is extended to the dynamics and free vibration response of plates embedding FGM layers. CUF permits one to handle the governing equations of the whole theory in a unified manner and a hierarchical evaluation of the natural frequencies of plates is therefore obtained. The results are given for one-layer and multi-layer plates embedding FGM layers. Conclusions are drawn regarding the accuracy of classical and advanced plate modelling for various vibration modes.