Variable Kinematic Models Research Articles

Multi-patch NURBS formulation for anisotropic variable angle tow composite plates

Conventional methods result in sharp variation of stiffness in analysis of variable angle tow (VAT) composite plates, which can be enhanced using high-order continuity of non-uniform rational B-spline (NURBS). To this end, a Bézier extraction-based isogeometric (IGA) model based on the higher-order shear deformation theory (HSDT) is developed for the parametric instability study of anisotropic VAT composite plates. Bézier extraction-based IGA approach expresses exactly the orientation function of VAT composites for multi-patch geometries, which provides accurate solution for general-shape plates. The accuracy and convergence of the developed framework is compared with recently formulated methods. Primary results shows that the developed concept solves VAT problems with much lower number of equations than variable kinematic models. Further, response characteristics of analyzed anisotropic VAT plates are presented and discussed considering various fiber orientation functions, boundary conditions and multi-patch geometry. Based on results, Bézier control mesh is more accurate than the NURBS-based IGA solution due to considering the continuity of strain field along coupled boundaries. Anisotropic VAT plates have severe deformation couplings and the current concept accurately captures this feature even with courser meshes. The interaction between boundary conditions and fiber orientations along edges affects mainly the dynamic instability regions (DIR) of modeled VAT plates, dislike to straight fiber ones. The dynamic instability opening (DIO) of VAT plates relies on the fiber orientation functions and boundaries. In anisotropic VAT plates, maximum frequency case and maximum buckling force case would not necessarily be the same. Modes shapes are variable of external force intensity and therefore the Rayleigh-Ritz based methods are not appropriate for dynamic stability analysis of VAT plates. It is essential to change the NURBS control mesh to the Bézier type to offer the required accuracy for the multi-patch domains.

Analysis of variable angle tow composites structures using variable kinematic models

This work presents a refined beam model based on the Carrera Unified Formulation for the free-vibration analyses of variable stiffness composite laminate characterized by layers with curvilinear fibers. These models introduce a refined kinematic description over the cross-section to obtain a 3D displacement field. Taylor and Lagrange polynomials have been used to describe the cross-sectional variables that is, equivalent single layer and layer-wise approaches have been considered. Variable stiffness composite materials have been introduced considering a continuous variation of the lamination angle thanks to an ad hoc integration scheme. Extensive validation of the models has been performed including convergence analyses and comparisons with commercial codes. The results obtained using the present models have been compared with those from open literature considering composites with different values of thickness, lamination, and boundary conditions. The results show that the use of refined models is mandatory for the analysis of such structures where complex laminations may create strong mechanical couplings.

A variable ESL/LW kinematic plate formulation for free-vibration thermoelastic analysis of laminated structures

In this article, a fully-coupled thermoelastic formulation is developed to deal with the free-vibration analysis of multilayered plate composite and sandwich structure. Some advanced theories are obtained by expanding the unknown displacement variables along the thickness direction and using equivalent-single-layer (ESL) models, layer-wise (LW) models, and variable-kinematic models. The variable-kinematic models permit to reduce the computational cost of the analyses grouping some layers of the multilayered structure with ESL models and keeping the LW models in other zones of the multilayer. This model is here extended for the free-vibration analysis of fully-coupled thermomechanical problems. The used refined models are grouped in the Carrera unified formulation (CUF), and they accurately describe the displacement field and the temperature distributions along the thickness of the multilayered plate. The governing equations are derived from the principle of virtual displacement, and the finite element method (FEM) is employed to solve them. Isotropic plates, cross-ply composite plates, and sandwich structures with composite skins and simply-supported edges are analyzed. Various aspect ratios are considered. The results, obtained with different theories within CUF context, are compared with the elasticity solutions, and other higher-order analytical and FEM solutions given in the literature. From the results, it is possible to conclude that the plate element based on the CUF is very efficient in the study of thermomechanical problems of composite structures. The variable-kinematic models combining the ESL with the LW models, permit to have a reduction of the computational costs, respect with the full LW models, preserving the accuracy of the results.

Electro-mechanical analysis of composite and sandwich multilayered structures by shell elements with node-dependent kinematics

ABSTRACTIn this work, a new class of finite elements for the analysis of composite and sandwich shells embedding piezoelectric skins and patches is proposed. The main idea of models coupling is developed by presenting the concept of nodal dependent kinematics where the same finite element can present at each node a different approximation of the main unknowns by setting a node-wise through-the-thickness approximation base. In a global/local approach scenario, the computational costs can be reduced drastically by assuming refined theories only in those zones/nodes of the structural domain where the resulting strain and stress states, and their electro-mechanical coupling present a complex distribution. Several numerical investigations are carried out to validate the accuracy and efficiency of the present shell element. An accurate representation of mechanical stresses and electric displacements in localized zones is possible with reduction of the computational costs if an accurate distribution of the higher-order kinematic capabilities is performed. On the contrary, the accuracy of the solution in terms of mechanical displacements and electric potential values depends on the global approximation over the whole structure. The efficacy of the present node-dependent variable kinematic models, thus, depends on the characteristics of the problem under consideration as well as on the required analysis type.

Analysis of laminated composite structures with embedded piezoelectric sheets by variable kinematic shell elements

In this article, the static analysis of multilayered shell structure embedding piezoelectric layers is performed using some advanced theories, obtained by expanding the unknown variables along the thickness direction using equivalent single-layer models, layer-wise models, and variable kinematic models. The variable kinematic models permit to reduce the computational cost of the analyses by grouping some layers of the multilayered structure with equivalent single-layer models and keeping the layer-wise models in other zones of the multilayer. This model is here extended to the static analysis of electro-mechanical problems. The used refined models are grouped in the Carrera Unified Formulation, and they accurately describe the displacement field, the stress distributions, and the electric potential along the thickness of the multilayered shell. The shell element has nine nodes, and the mixed interpolation of tensorial components method is used to contrast the membrane and shear locking phenomenon. The governing equations are derived from the principle of virtual displacement, and the finite element method is employed to solve them. Cross-ply plates and shells, with piezoelectric skins and simply supported edges, subjected to bi-sinusoidal mechanical or electrical load are analyzed. Various aspect ratios and radius-to-thickness ratios are considered. The results, obtained with different theories within Carrera Unified Formulation context, are compared with the elasticity solutions given in the literature. From the results, it is possible to conclude that the shell element based on Carrera Unified Formulation is very efficient in the study of electro-mechanical problems of composite structures. The variable kinematic models combining the equivalent single-layer with the layer-wise models permit to have a reduction of the computational costs, with respect to the full layer-wise theories, preserving the accuracy of the results in localized layers.

Refined finite element solutions for anisotropic laminated plates

This paper presents some solutions for mechanical responses of angle-ply laminated plates under transverse distributed loads, which are obtained by using refined finite element models adopting variable kinematics based on Carrera’s Unified Formulation (CUF). Plates with several types of stacking sequence under different boundary conditions are considered. Layer-wise (LW) models based on Chebyshev polynomials (first kind) and Equivalent Single Layer (ESL) models based on Trigonometric series are used in the analysis. To compare the performances of different displacement-based kinematic models, a set of simply supported boundary conditions and mixed clamped-free boundaries are adopted in the numerical study. A nine-node MITC (Mixed Interpolated of Tensorial Components) plate element is employed to contrast the shear locking phenomenon of thin plates. CUF-based variable kinematic models are used in the numerical study and the number of expansion terms in the thickness direction is increased until the requisite numerical accuracy is achieved. By comparing the numerical results obtained with CUF-based refined models and ABAQUS 3D models as well as reference solutions from literature, the effectiveness of the adopted models is verified. The newly studied numerical cases can be taken as benchmarks for future research.

Shell elements with through-the-thickness variable kinematics for the analysis of laminated composite and sandwich structures

Abstract Several efforts have been made in the last years to improve the efficiency and the effectiveness of structural models for the analysis of laminated shell structures. Among the others, many recent and past works in the literature have been aimed at formulating theories of structures that maximize the accuracy of analysis meanwhile reducing the computational costs. In this paper, this objective is pursued by implementing advanced shell theories with through-the-thickness variable kinematic capabilities. By employing the Carrera Unified Formulation (CUF), the proposed shell model is obtained by expressing the displacement field as an arbitrary and, eventually, hierarchical expansion of the primary unknowns along the thickness. Thus, Equivalent-Single-Layer (ESL), Layer-Wise (LW) models as well as variable kinematic models which combine ESL and LW approaches within the shell thickness can be obtained in a straightforward and unified manner. After the unified shell model is formulated, the governing equations and the related finite element arrays are obtained by employing the principle of virtual work. A nine node finite element is implemented to approximate the solution field, and the Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomena. Some numerical examples are discussed, including three- and ten-layered cross-ply shells under bi-sinusoidal load and simply-supported boundary conditions, a multilayered spherical panel subjected to bi-sinusoidal load and a sandwich cylinder undergoing bi-sinusoidal pressure. Moreover, various thickness and radius-to-thickness ratios are considered. Whenever possible, the results are compared with those from the literature and from exact elasticity solutions. The analysis of the results clearly shows the enhanced capabilities of the present variable-kinematic shell element, which allows the analyst to opportunely reduce the computational costs and enhance the accuracy of the model only in those regions of the thickness domain where an accurate evaluation of the stress/strain field is needed.

A variable kinematic shell formulation applied to thermal stress of laminated structures

ABSTRACTIn this article, the thermoelastic static analysis of multilayered shell structure is performed using some advanced theories, obtained by expanding the unknown displacement variables along the thickness direction using equivalent-single-layer (ESL) models, layer-wise (LW) models, and variable-kinematic models. The variable-kinematic models permit to reduce the computational cost of the analyses grouping some layers of the multilayered structure with ESL models and keeping the LW models in other zones of the multilayer. This model is here extended for the static analysis of uncoupled thermomechanical problems. The results obtained with the classical assumed linear temperature profile along the thickness of the shell are compared with those achieved with the calculated temperature profile solving the Fourier heat conduction equation. The used refined models are grouped in the Carrera’s unified formulation (CUF), and they accurately describe the displacement field and the stress distributions along the thickness of the multilayered shell. The shell element has nine nodes, and the mixed interpolation of tensorial components method is used to contrast the membrane and shear locking phenomenon. The governing equations are derived from the principle of virtual displacement, and the finite element method is used to solve them. Cross-ply plates and shells with simply supported edges, subjected to bisinusoidal thermal load are analyzed. Various aspect ratios and radius to thickness ratios are considered. The results, obtained with different theories within CUF context, are compared with the elasticity solutions given in the literature. From the results, it is possible to conclude that the shell element based on the CUF is very efficient in the study of thermomechanical problems of composite structures. The variable-kinematic models combining the ESL with the LW models permit to have a reduction of the computational costs, with respect with the full LW models, preserving the accuracy of the results in localized layers.

Nonhomogeneous atherosclerotic plaque analysis via enhanced 1D structural models

The static analysis of structures with arbitrary cross-section geometry and material lamination via a refined one-dimensional (1D) approach is presented in this paper. Higher-order 1D models with a variable order of expansion for the displacement field are developed on the basis of Carrera Unified Formulation (CUF). Classical Euler-Bernoulli and Timoshenko beam theories are obtained as particular cases of the first-order model. Numerical results of displacement, strain and stress are provided by using the finite element method (FEM) along the longitudinal direction for different configurations in excellent agreement with three-dimensional (3D) finite element solutions. In particular, a layered thin-walled cylinder is considered as first assessment with a laminated conventional cross-section. An atherosclerotic plaque is introduced as a typical structure with arbitrary cross-section geometry and studied for both the homogeneous and nonhomogeneous material cases through the 1D variable kinematic models. The analyses highlight limitations of classical beam theories and the importance of higher-order terms in accurately detecting in-plane cross-section deformation without introducing additional numerical problems. Comparisons with 3D finite element solutions prove that 1D CUF provides remarkable three-dimensional accuracy in the analysis of even short and nonhomogeneous structures with arbitrary geometry through a significant reduction in computational cost.

Variable Kinematic One-Dimensional Finite Elements for the Analysis of Rotors Made of Composite Materials

Abstract This paper deals with the dynamic response of rotors made of anisotropic, laminated composite materials. It is a sequel to the authors’ previous work, which was devoted to the rotordynamics of metallic structures. The used variable kinematic one-dimensional models describe any cross-sectional deformation of the rotor and go beyond the plane strain assumptions of classical Euler–Bernoulli and Timoskenko beam theories. Refined theories are obtained by applying the Carrera unified formulation, which is extended here to the rotordynamics of multilayered composites. The displacement variables over the rotor cross section x-z plane are approximated by x,z polynomials of any order N. Thin-walled cylindrical shafts and boxes are analyzed. These structures are made of unidirectional layers, whose fiber orientation can vary with respect to the rotor–axis as well as in the x-z plane. Several analyses have been carried out to determine the vibrational response as a function of the rotating speed. Classical beam theories are obtained as particular cases and results available in the literature, including shell results, are used to assess the presented theory. The proposed refined models are very effective in investigating the dynamic behavior of laminated composite rotors.

Dynamic response of thin-walled structures by variable kinematic one-dimensional models

This paper investigates the accuracy capabilities of using variable kinematic modeling in compact and thin-walled beam-like structures with dynamic loadings. Carrera Unified Formulation (CUF) is employed to introduce refined one-dimensional (1D) models with a variable order of expansion for the displacement unknowns over the beam cross-section. Classical Euler–Bernoulli and Timoshenko beam theories are obtained as particular cases of these variable kinematic models while a higher order expansion permits the detection of in-plane cross-section deformation, since it leads to shell-like solutions. Finite element (FE) method is used to provide numerical results and the Newmark method is implemented as a time integration scheme. Some assessments with closed form solutions are discussed and comparisons with shell-type results obtained with commercial FE software are made. Further analyses address both compact and thin-walled cross-sections. In particular, the case of a deformable thin-walled cylinder loaded by time-dependent internal forces is discussed. The results clearly show that finite elements which are formulated in the CUF framework do not introduce additional numerical problems with respect to classical beam theories. Comparisons with elasticity solutions prove that the present 1D CUF model offers an accuracy in analyzing thin-walled structures which is typical of shell or three-dimensional models with a remarkable reduction in the computational cost required.

Variable kinematic plate elements coupled via Arlequin method

SUMMARYIn this work, plate elements based on different kinematic assumptions and variational principles are combined through the Arlequin method. Computational costs are reduced assuming refined models only in those zones with a quasi‐three‐dimensional stress field, whereas computationally cheap, low‐order elements are used in the remaining parts of the plate. Plate elements are formulated on the basis of a unified formulation (UF). Via UF, higher‐order, layer‐wise and mixed theories can be easily formulated. Classical theories, such as Kirchhoff's and Reissner's models, can be obtained as particular cases. UF is extended to the Arlequin method to derive the matrices that account for the coupling between different theories. Multi‐layered composite plates are investigated. Variable kinematic multiple models solutions are assessed towards mono‐model results and three‐dimensional exact results. Numerical investigation has shown that Arlequin method in the context of UF effectively couples sub‐domains having finite elements based upon different theories, reducing the computational costs without loss of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.

Ritz analysis of vibrating rectangular and skew multilayered plates based on advanced variable-kinematic models

A variable-kinematic Ritz formulation based on two-dimensional higher-order layerwise and equivalent single-layer theories is described in this paper to accurately predict free vibration of thick and thin, rectangular and skew multilayered plates with clamped, free and simply-supported boundary conditions. The main result is the derivation at a layer level of so-called Ritz fundamental nuclei for the stiffness and mass matrices which are invariant with respect to both the assumed kinematic model and the type of Ritz functions. In this work, products of Chebyshev polynomials and boundary-compliant functions are chosen as admissible trial set. After studying the convergence of the method, its accuracy is evaluated, in terms of frequency parameters and through-the-thickness distribution of modal displacements, by comparison with some reference results available in the literature. Results for sandwich plates with soft core are given for the first time, which may serve as benchmark values for future research.

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.

Variable kinematic models applied to free-vibration analysis of functionally graded material shells

Closed-form solutions of free-vibration problems of simply supported multilayered shells made of Functionally Graded Material have been examined in the present paper. A variable kinematic shell model, which is based on Carrera’ sU ni fied Formulation is extended, in this work, to dynamic shell cases. Classical shell theories are compared to refined ones as well as to layer-wise kinematics and mixed assumptions based on the Reissner mixed variational theorem. A comparison with the few results available in the open literature is presented and conclusions are drawn regarding the accuracy of classical and advanced shell modeling to evaluate lower and higher vibration modes as well as the behavior of these modes in the shell thickness direction.

Variable Kinematic Model for the Analysis of Functionally Graded Material plates

This work addresses the static analysis of functionally graded material plates subjected to transverse mechanical loadings. The unified formulation and principle of virtual displacements are employed to obtain both closed-form and finite element solutions. The use of the unified formulation permits a large variety of plate models with variable kinematic assumptions to be compared in the same framework. These differ according to the order of the expansion in the thickness direction and the variable description: the order of the expansion ranges from 1 to 4, thus covering first-order as well as higher-order theories; the description of the unknowns can be equivalent single layer or layerwise. The dependence of the material data on the thickness direction was introduced by employing thickness functions that are derived from the Legendre polynomials that are used in the layerwise case. The proposed approach is independent of the used transition function, and as a result, any continuous variations of the material properties in the thickness direction may be easily implemented. The obtained solutions (closed form and finite element) are validated through comparison with three-dimensional exact solutions and other available solutions. These show the limitations of classical plate theories as well as the convenience of the use of the proposed variable kinematic models for the analysis of functionally graded material plates.