Variable Kinematics Research Articles

Assessment of MITC plate elements based on CUF with respect to distorted meshes

This paper discusses the robustness of plate elements based on Mixed Interpolation of Tensorial Components (MITC) technique and the variable kinematics approach of Carrera Unified Formulation (CUF) with respect to the problem of distorted meshes. MITC was originally proposed for Reissner-Mindlin type plates to develop shear locking free plate elements. In the present framework, refined plate elements are obtained by referring to high-order Equivalent Single Layer as well as Layer-Wise models expressed in CUF for the analysis of multilayered anisotropic structures. Four-node and nine-node elements are considered and some applications are developed for both isotropic and multilayered composite plates. Results related to the MITC approach are compared to the isoparametric elements, including selectively reduced quadrature schemes, for both, the static and free-vibration analysis. They show that CUF-MITC elements maintain their effectiveness also in the case of distorted meshes, for all the materials studied and kinematic models considered: the obtained elements are robust as free from shear locking and spurious zero-energy modes.

Progressive delamination of laminated composites via 1D models

This paper presents a novel numerical framework to simulate the progressive delamination in laminated structures based on 1D component-wise models. The proposed numerical tool is a part of the virtual testing platform built within the Carrera Unified Formulation, a hierarchical, higher-order structural framework to generate theories of structures via a variable kinematic approach. Formulated within the Lagrange polynomial CUF models, the component-wise approach models the components of a complex structure through 1D CUF models at reduced computational costs and 3D-like accuracies. The effectiveness of CUF-CW models to capture accurate 3D transverse fields are of interest to solve delamination problems by integrating a class of higher-order cohesive elements to simulate the cohesive mechanics among the various components of the structure. The present framework adopts a bilinear constitutive law based on the mixed-mode delamination propagation and an efficient arc-length solver based on an energy-dissipation constraint. The numerical results aim to verify the accuracy and computational efficiency of CUF-CW models through benchmark composite delamination problems including multiple delamination fronts and comparisons with reference literature solutions and standard 3D FEM models. The outcomes show multi-fold improvements in the analysis times, good matches with experimental results, and promising enhancements of the meshing process due to the absence of aspect ratio constraints.

Computationally Efficient Concurrent Multiscale Framework for the Linear Analysis of Composite Structures

This paper presents a novel multiscale framework based on higher-order one-dimensional finite element models. The refined finite element models originate from the Carrera unified formulation (CUF): a novel and efficient methodology to develop higher-order structural theories hierarchically via a variable kinematic approach. The concurrent multiscale framework consists of a macroscale model to describe the structural level components interfaced with efficient CUF micromechanical models. Such micromechanical models can take into account the detailed architecture of the microstructure with high fidelity. The framework derives its efficiency from the capability of CUF models to detect accurate three-dimensional-like stress fields at reduced computational costs. This paper also shows the ability of the framework to interface with different classes of representative volume elements and the benefits of parallel implementations. The numerical cases focus on composite and sandwich structures and demonstrate the high fidelity and feasibility of the proposed framework. The efficiency of the framework stems from comparisons with the analysis time and memory requirement against traditional multiscale implementations.

Elastic buckling and vibration analysis of FG polymer composite plates embedding graphene nanoplatelet reinforcements in thermal environment

The present article proposes a comprehensive buckling and vibration analysis of functionally graded (FG) polymer composite plates reinforced with graphene platelets (GPLs). The effective material properties of the FG-GPLRC plates are accurately computed by using various ad-hoc micromechanical models. Several graphene nanoplatelets distribution patterns, within the polymer matrix, are investigated. The governing differential equations (GDEs) are derived by using Hamilton’s principle combined with the method of the power series expansion of the displacement components and the Gauss theorem. They are, then, solved in an exact sense by using the Navier-type closed-form solution. Various case-studies are addressed to show the high level of accuracy of the proposed formulation. Moreover, the effect of length-to-thickness ratio, volume fraction, aspect ratio, variable kinematics as well as GPLs distribution pattern is discussed.

Influence of the abrasive grain trajectory on the machining of thin-walled cylinder liners of internal combustion engines with variable kinematics of honing

Honing with variable kinematics has a number of advantages in comparison to traditional honing [1-19], such as less tool wear [12, 13], lower processing temperature [13], better surface quality [12, 13] and less deviation of cylindricity [13]. The article discusses the influence of the variable kinematics on the machining process.

Classical, higher-order, zig-zag and variable kinematic shell elements for the analysis of composite multilayered structures

Abstract In the present work, a shell finite element with a variable kinematic field based on a new zig-zag power function is proposed for the analysis of laminated shell structures . The kinematic field is written by using an arbitrary number of continuous piecewise polynomial functions. The polynomial expansion order of a generic subdomain is a combination of zig-zag power functions depending on the shell thickness coordinate. As in the classical layer-wise approach, the shell thickness can be divided into a variable number of mathematical subdomains. The expansion order of each subdomain is an input parameter of the analysis. This feature enables the solution to be locally refined over generic regions of the shell thickness by enriching the kinematic field. The advanced finite shell elements with variable kinematics are formulated in the framework of the Carrera Unified Formulation. The finite element arrays are formulated in terms of fundamental nuclei, which are invariants of the theory approximation order and the modelling technique (Equivalent-Single-Layer, Layer-Wise). In this work, the attention is focused on linear static stress analyses of composite laminated shell structures. The governing equations are obtained by applying the Principle of Virtual Displacements, and they are solved using the Finite Element method . Furthermore, the Mixed Interpolated Tensorial Components (MITC) method is employed to contrast the shear locking phenomenon . Several numerical investigations are carried out to validate and demonstrate the accuracy and efficiency of the present shell element.

Wave propagation in compact, thin-walled, layered, and heterogeneous structures using variable kinematics finite elements

ABSTRACTThe article investigates wave propagation characteristics for a class of structures using higher-order one-dimensional (1D) models. 1D models are based on the Carrera Unified Formulation (CUF), a hierarchical formulation which provides a framework to obtain refined structural theories via a variable kinematics description. Theories are formulated by employing arbitrary expansions of the primary unknowns over the beam cross-section. Two classes of beam models are employed in the current work, namely Taylor Expansion (TE) and Lagrange Expansion (LE) models. Using the principle of virtual work and finite element method, the governing equations are formulated. The direct time integration of equation of motion is carried through an implicit scheme based on the Newmark method and a dissipative explicit method based on the Tchamwa–Wielgosz scheme. The framework is validated by comparing the response for the stress wave propagation in an isotropic beam to an analytical solution available in the literature. The capabilities of the proposed model are demonstrated by presenting results for wave propagation analysis of a sandwich beam and a layered annular cylinder structure. The ability of CUF models to detect 3D-like behavior with a reduced computational overhead is highlighted.

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.

The Ritz – Sublaminate Generalized Unified Formulation approach for piezoelectric composite plates

ABSTRACTThis paper extends to composite plates including piezoelectric plies the variable kinematics plate modeling approach called Sublaminate Generalized Unified Formulation (SGUF). Two-dimensional plate equations are obtained upon defining a priori the through-thickness distribution of the displacement field and electric potential. According to SGUF, independent approximations can be adopted for the four components of these generalized displacements: an Equivalent Single Layer (ESL) or Layer-Wise (LW) description over an arbitrary group of plies constituting the composite plate (the sublaminate) and the polynomial order employed in each sublaminate. The solution of the two-dimensional equations is sought in weak form by means of a Ritz method. In this work, boundary functions are used in conjunction with the domain approximation expressed by an orthogonal basis spanned by Legendre polynomials. The proposed computational tool is capable to represent electroded surfaces with equipotentiality conditions. Free-vibration problems as well as static problems involving actuator and sensor configurations are addressed. Two case studies are presented, which demonstrate the high accuracy of the proposed Ritz-SGUF approach. A model assessment is proposed for showcasing to which extent the SGUF approach allows a reduction of the number of unknowns with a controlled impact on the accuracy of the result.

Multidimensional Model for the Stress Analysis of Reinforced Shell Structures

The present paper proposes an approach that can be used to mix one-, two-, and three-dimensional refined models, derived using the Carrera Unified Formulation, to build a variable kinematic model that is able to deal with the static analysis of complex thin-walled structures. The adopted formulation, which only has displacements as degrees of freedom, allows these models to easily be connected to each other; that is, a variable kinematics model can be derived without ad hoc techniques. The refined models used in the present paper ensure high accuracy and low computational costs. The displacement continuity at the interface is guaranteed by the formulation, and no stress singularities appear in the kinematic model transition. The mixed interpolation tensorial component approach has been used, in a unified sense, for one-, two-, and three-dimensional models to avoid the shear locking effect. The accuracy of the present approach has been confirmed by comparing the results with those from the literature and with those obtained using commercial finite element codes. The static response of a reinforced panel and a section of an aircraft fuselage have been investigated to show the capabilities of the present approach. The use of refined structural models makes it possible to overcome the limits of classical structural models and, at the same time, to reduce the computational costs.

Static and free-vibration analyses of dental prosthesis and atherosclerotic human artery by refined finite element models.

Static and modal responses of representative biomechanical structures are investigated in this paper by employing higher-order theories of structures and finite element approximations. Refined models are implemented in the domain of the Carrera unified formulation (CUF), according to which low- to high-order kinematics can be postulated as arbitrary and, eventually, hierarchical expansions of the generalized displacement unknowns. By using CUF along with the principle of virtual work, the governing equations are expressed in terms of fundamental nuclei of finite element arrays. The fundamental nuclei are invariant of the theory approximation order and can be opportunely employed to implement variable kinematics theories of bio-structures. In this work, static and free-vibration analyses of an atherosclerotic plaque of a human artery and a dental prosthesis are discussed. The results from the proposed methodologies highlight a number of advantages of CUF models with respect to already established theories and commercial software tools. Namely, (i) CUF models can represent correctly the higher-order phenomena related to complex stress/strain field distributions and coupled mode shapes; (ii) bio-structures can be modeled in a component-wise sense by only employing the physical boundaries of the problem domain and without making any geometrical simplification. This latter aspect, in particular, can be currently accomplished only by using three-dimensional analysis, which may be computationally unbearable as complex bio-systems are considered.

Hygrothermal analysis of multilayered composite plates by variable kinematic finite elements

ABSTRACTAdvanced plate models with variable kinematics for steady state hygrothermal analysis of composite laminates are proposed. The refined models discussed include both layer-wise (LW) and equivalent single layer (ESL) models, and the Carrera Unified Formulation (CUF) is used. The mixed interpolation of tensorial component (MITC) method is applied to a nine-node element to contrast the shear locking phenomena. The governing equations are derived from the principle of virtual displacement (PVD) taking into account elastic mechanical, thermal and hygroscopic effects. Through-the-thickness variations of temperature and moisture concentration are calculated by solving the Fourier equation and the Fick law, respectively. Cross-ply plates with symmetrical lamination and simply supported edges subjected to bisinusoidal thermal/hygroscopic loads are analyzed considering various thickness ratios. Results obtained with assumed linear and calculated temperature/hygroscopic profiles are compared. Variable kinematics with a variety of thickness functions are compared regarding both accuracy and computational costs. The results show that all the kinematics proposed can approximate the transverse shear stress distribution through the thickness with satisfactory accuracy when sufficient expansion terms are adopted. In some cases, miscellaneous expansions can lead to significant reductions in computational costs. The results presented here can be used as benchmark solutions for future works.

Variable kinematic shell elements for composite laminates accounting for hygrothermal effects

ABSTRACTThis article presents advanced shell models for the steady-state hygrothermal analysis of composite laminates. The Carrera Unified Formulation is used to derive refined models that include both layer-wise and equivalent single layer models. The governing equations are derived from the principle of virtual displacement taking into account thermal and hygroscopic effects. The geometrical relations for the exact cylindrical geometry are here considered. Through-the-thickness variations of temperature and moisture concentration are calculated by solving the Fourier equation and the Fick law, respectively. The mixed interpolation of tensorial component method is applied to a nine-node shell element to contrast the membrane and shear locking phenomena. Simply supported cross-ply cylindrical shells with antisymmetrical lamination subjected to bisinusoidal thermal/hygroscopic loads are analyzed considering various thickness/curvature ratios. Results obtained with assumed linear and calculated temperature/hygroscopic profiles are presented. Variable kinematics are compared regarding both accuracy and computational costs. The results show that all the kinematics can approximate the transverse shear stress distribution through the thickness with satisfactory accuracy when sufficient expansion terms are adopted. In some cases, miscellaneous expansions can lead to significant reductions in computational costs. The results presented here can be used as benchmark solutions for future works.

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.

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.

Analysis of selected dynamic features of a two-wheeled transmission system

The paper presents results of analysis of selected dynamic features of an uneven-running two- -wheeled strand transmission. The characteristic features of such type of transmissions is the ability of obtaining variable kinematic and dynamic features by the application of wheels with a noncircular envelope. Particularly, high interest of application of uneven-running transmissions was done in the previous century. At that time, a concept of application of such transmissions for process and machine control was invented. The applications in textile and engineering industry are widely known. A mathematical model of the multiplying transmission is presented. Few versions of dynamic features of the transmission for different loads and solid moments of inertia are examined. The presented results can be very helpful for designers of classical and uneven-running strand transmissions.

Component-wise vibration analysis of stiffened plates accounting for stiffener modes

This work has the aims to assess and compare the behavior of different finite element models in a free-vibration analysis of reinforced structures. The effects of the aspect ratio and cross-sectional shape of the stringers have been analyzed using both classical and refined FE. One- and two-dimensional classical finite element models, as well as refined one-dimensional elements, derived using the Carrera Unified Formulation, have been considered in an analysis of reinforced structures. The accuracy and efficiency of these models have been investigated. A three-dimensional model has been used as a reference solution. Three different modeling approaches have been considered in the present work. Two approaches are based on classical models provided by commercial tools: the first uses two-dimensional elements for both a plate and stringers, while the second uses two-dimensional elements to simulate the plate and the beam elements for the stringers. The third approach uses a refined one-dimensional model, based on the Carrera Unified Formulation. This refined one-dimensional model considers a variable kinematic displacement field over the beam cross section. In the present work, Lagrange polynomials are used to describe the cross-sectional displacement field. The use of a component-wise modeling approach allows the stiffeners to be modeled as independent entities. The component-wise approach was first assessed and convergence was then evaluated. The performances of these models in the analysis of reinforced structure were then compared with those from classical models. The use of refined one-dimensional models allows stiffener modes to be investigated. The results show that this approach is comparable to a full three-dimensional solution. The use of classical one- and two-dimensional models does not allow the local deformation of reinforcements to be taken into account. Therefore, these models cannot be considered accurate when the aim of the analysis is to investigate the local mode.

Improving Prediction of Flapping-Wing Motion By Incorporating Actuator Constraints With Models of Aerodynamic Loads Using In-Flight Data

Flapping-wing flight is a challenging system integration problem for designers due to tight coupling between propulsion and flexible wing subsystems with variable kinematics. High fidelity models that capture all the subsystem interactions are computationally expensive and too complex for design space exploration and optimization studies. A combination of simplified modeling and validation with experimental data offers a more tractable approach to system design and integration, which maintains acceptable accuracy. However, experimental data on flapping-wing aerial vehicles which are collected in a static laboratory test or a wind tunnel test are limited because of the rigid mounting of the vehicle, which alters the natural body response to flapping forces generated. In this study, a flapping-wing aerial vehicle is instrumented to provide in-flight data collection that is unhindered by rigid mounting strategies. The sensor suite includes measurements of attitude, heading, altitude, airspeed, position, wing angle, and voltage and current supplied to the drive motors. This in-flight data are used to setup a modified strip theory aerodynamic model with physically realistic flight conditions. A coupled model that predicts wing motions is then constructed by combining the aerodynamic model with a model of flexible wing twist dynamics and enforcing motor torque and speed bandwidth constraints. Finally, the results of experimental testing are compared to the coupled modeling framework to establish the effectiveness of the proposed approach for improving predictive accuracy by reducing errors in wing motion specification.

Design and analysis of a metamorphic mechanism cell for multistage orderly deployable/retractable mechanism

Normal kinematic joint can be transformed to variable kinematic joint(VKJ), which refers to a kinematic joint that is capable of topological variation in a mechanism, by utilizing effect of friction and self-locking. In this paper a metamorphic mechanism cell which can realize deploying, self-locking, unlocking, retracting and interlocking with other cells, is designed by incorporating VKJs. Self-locking margin is proposed to estimate the self-locking capability. The variable topology configurations of this cell are presented and the mobility of this cell is analyzed. With the new metamorphic cell, a cable-driven telescopic model with 3 tubes is built, and its motion simulations are conducted to verify the design method. The results demonstrate that the impact in transformation is mitigated and multistage orderly deployable/retractable mechanisms can be built by this method.

Free-vibration analysis of space vehicle structures made by composite materials

This work investigates the effects of composite materials and non-structural masses on the dynamic behavior of space structure components and whole space vehicle. A refined one-dimensional model has been used in the analyses, and the effects of composite materials and of the fuel mass introduced as non-structural masses have been considered. The adopted refined one-dimensional Finite Element Model has been developed using the Carrera Unified Formulation. This numerical tool allows to develop a variable kinematic displacement field over the beam cross-section, that is, a set of Lagrange (LE) expansions polynomials was adopted for the cross-sectional displacement field approximation. The use of such one-dimensional models leads to the so-called component-wise (CW) approach in which stiffeners and plate are modeled using the same one-dimensional kinematic. Static and free vibration analysis of space structural components and complete space structures have been performed. Both compact and thin-walled structural configurations have been considered. The results have been assessed using analytical solutions or refined three-dimensional Finite Element Models. Composite materials and non-structural masses, e.g. the fuel mass or payload, have been included in the analysis. The results show the capability of the present model to provide a quasi three-dimensional solution with a low computational cost. The refined kinematic allows composite materials to be investigated accurately.