Variable Kinematics Research Articles

C0 Fixed Degrees of Freedom Zigzag Model with Variable In-Plane and Out-of-Plane Kinematics and Quadrilateral Plate Element

Physically based, multilayered zigzag models have the merit of providing accurate stress predictions with few fixed unknowns, but unfortunately they involve derivatives of functional degrees of freedom (e.g., u,x, w,x, w,xxx, and so on). Here, a variable kinematics zigzag model with up to third-order derivatives always giving accurate predictions is considered in order to apply a technique that enables the conversion of derivatives of the unknowns. Such technique relies on energy updating concepts for finding closed form, modified expressions of displacements constituting the equivalent C0 model. From this model, a simple quadrilateral plate element with standard C0 interpolation functions is developed and tested. The first objective is finding a priori modified displacement fields that make the C0 zigzag model equivalent from the energy standpoint to its initial counterpart containing derivatives. Then a finite element is developed, whose structural model is equivalent from the energy standpoint to its counterpart with derivatives as variables. Comparisons with three-dimensional (3D) elasticity solutions of benchmark test cases and to an experiment show that the present element obtains accurate results.

Indentation of sandwiches using a plate model with variable kinematics and fixed degrees of freedom

A recent 3-D, 5 degrees of freedom, variable kinematics zig-zag homogenised model is used for studying sandwich indentation. This model cannot describe in a point-wise sense cells collapse, buckling, debonding failure, plastic yielding, or brittle fracture. Therefore, the progressive folding followed by densification of the core is described through mechanical properties variable with loading, which are computed apart once at a time through a 3D FEA discretisation of the core structure. Onset of damage is determined using stress-based criteria; degraded properties of failed regions are obtained from a mesoscale damage model. The results show efficiency and accuracy of this approach.

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.

Layer-wise and equivalent single layer models for smart multilayered plates

Layer-wise and equivalent single layer plate models for magneto-electro-elastic multiphysics laminates are presented in a unified framework. They are based on variable kinematics and quasi-static behavior of the electromagnetic fields. The electromagnetic state of each single layer is preliminary determined by solving the corresponding governing equations coupled with the proper interface continuity and external boundary conditions. By so doing, the electromagnetic state is condensed into the plate kinematics and the layer governing equations are inferred by the principle of virtual displacements. This approach identifies effective mechanical layers, which are kinematically equivalent to the original smart layers. These effective layers are characterized by stiffness, inertia and load properties which take the multifield coupling effects into account as their definitions involve the electromagnetic coupling material properties. The layers governing equations are finally assembled enforcing the mechanical interface conditions. This allows to obtain the smart plate resolving system, which involves primary mechanical variables only. Results for thick simply-supported multilayered plates are obtained by an exact closed-form Navier-type solution and compared with benchmark 3D solutions to investigate the features and accuracy of the proposed modeling approach.

Variable kinematic beam elements for electro-mechanical analysis

This paper proposes a refined electro-mechanical beam formulation. Lagrange-type polynomials are used to interpolate the unknowns over the beam cross section. Three- (L3), four- (L4), and nine-point(L9) polynomials are considered which lead to linear, bi-linear, and quadratic displacement field approximations over the beam cross-section. Finite elements are obtained by employing the principle of virtual displacements in conjunction with the Carrera Unified Formulation (CUF). The finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not depend on the assumptions made. Additional refined beam models are implemented by introducing further discretizations, over the beam cross-section. Some assessments from bibliography have been solved in order to validate the electro-mechanical formulation. The investigations conducted show that the present formulation is able to detect the electro-mechanical interaction.

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 Kinematics Layerwise Model for Analysis of Bonded Joints

Variable Kinematics Layerwise Model for Analysis of Bonded Joints

Analysis of bonded joints with laminated adherends by a variable kinematics layerwise model

A displacement-based zig-zag plate model with variable in-plane and through-the-thickness representation and fixed degrees of freedom is developed and applied to the analysis of bonded joints. Just the in-plane displacements and the shear rotations of the middle plane are used as primary variables. The continuity functions enable an a priori fulfilment of the out-of-plane stress contact conditions at the interfaces between adjacent layers. High-order contributions to displacements are included to meet the stress boundary conditions at the upper and lower faces and to allow the in-plane representation to be varied, e.g. from the adherends to the overlap. In this way, it is possible to better simulate the variation of solutions and the stress boundary conditions at the ends of the overlap. Closed form expressions of these quantities are obtained using symbolic calculus, which, once incorporated in the model, simplify the obtainment of governing equations and speed-up computations. As the representation can vary from point to point, the present model permits an accurate analysis of laminates with general boundary conditions and of bonded joints under a unified approach. Applications are presented to sample cases of bonded joints with laminated adherends using appropriate series expansions of displacements. Linear and nonlinear benchmark test cases for single- and double-lap joints taken from the literature are considered. The results show that accurate stress predictions are computed with a low computational effort in all the cases considered. A good accuracy is achieved just using one component in the series expansion, which implies solving a 3×3 system.

Refined equivalent single layer formulations and finite elements for smart laminates free vibrations

A family of 2D refined equivalent single layer models for multilayered and functionally graded smart magneto–electro-elastic plates is presented. They are based on variable kinematics and quasi-static behavior for the electromagnetic fields. First, the electromagnetic state of the plate is determined by solving the strong form of the electromagnetic governing equations coupled with the corresponding interface continuity conditions and external boundary conditions. The electromagnetic state is then condensed into the plate kinematics, whose governing equations can be written using the generalized principle of virtual displacements. The procedure identifies an effective elastic plate kinematically equivalent to the original smart plate. The effective plate is characterized by inertia, stiffness and loading properties which take the multifield coupling effects into account through their definitions, which involve the electromagnetic coefficients appearing in the smart materials constitutive law. The proposed model extends the techniques and tools available for the assessment of the mechanical behavior of multilayered composite plates to smart laminates. Additionally, finite elements for the proposed single layer models are formulated and validated against available benchmark 3D solutions.

Indentation of Sandwiches Using a Refined Zig-Zag Model and a Mesoscale Damage Model

An accurate and cost effective zig-zag plate model with variable kinematics and fixed degrees of freedom recently developed by the authors, which a priori fulfills the stress and displacement continuity requirements, is applied to study indentation of sandwiches with honeycomb/foam core. The variable elastic properties of the core during crushing are evaluated apart and once at a time through a 3D finite element analysis. Shell elements with elastic-plastic isotropic properties are adopted to discretize honeycomb cores and faces, instead solid elements with nonlinear material behavior are used for foam cores. To keep low the computational burden, the structural response is computed in closed form by the zig-zag model using these elastic properties. The damage analysis is carried out through a mesoscale model that determines the degraded properties of failed regions in a physically consistent way. A good agreement with experiments taken from the literature being shown, the present simulation procedure with a low computational effort is proven to be an efficient alternative to the ones currently employed.

Influence of Non-Structural Localized Inertia on Free Vibration Response of Thin-Walled Structures by Variable Kinematic Beam Formulations

Variable kinematic beam theories are used in this paper to carry out vibration analysis of isotropic thin-walled structures subjected to non-structural localized inertia. Arbitrarily enriched displacement fields for beams are hierarchically obtained by using the Carrera Unified Formulation (CUF). According to CUF, kinematic fields can be formulated either as truncated Taylor-like expansion series of the generalized unknowns or by using only pure translational variables by locally discretizing the beam cross-section through Lagrange polynomials. The resulting theories were, respectively, referred to as TE (Taylor Expansion) and LE (Lagrange Expansion) in recent works. If the finite element method is used, as in the case of the present work, stiffness and mass elemental matrices for both TE and LE beam models can be written in terms of the same fundamental nuclei. The fundamental nucleus of the mass matrix is opportunely modified in this paper in order to account for non-structural localized masses. Several beams are analysed and the results are compared to those from classical beam theories, 2D plate/shell, and 3D solid models from a commercial FEM code. The analyses demonstrate the ineffectiveness of classical theories in dealing with torsional, coupling, and local effects that may occur when localized inertia is considered. Thus the adoption of higher-order beam models is mandatory. The results highlight the efficiency of the proposed models and, in particular, the enhanced capabilities of LE modelling approach, which is able to reproduce solid-like analysis with very low computational costs.

Comparisons between 1D (Beam) and 2D (Plate/Shell) Finite Elements to Analyze Thin Walled Structures

This paper compares 1D and 2D assumptions for the analysis of isotropic thin-walled structures. Refined 1D (beam) and 2D (shell) models were exploited. These models are based on a variable kinematic approach which means that the displacement field assumed can be refined to any extent since the orders of the polynomial expansions are taken as free parameters. The Finite Element (FE) approach was used to provide numerical results. FEM is, in fact, the most popular technique used in the analysis of structures in engineering. The Carrera Unified Formulation (CUF) was adopted in order to write finite element matrices of 1D and 2D models in a concise form based on the so-called fundamental nuclei. In this work, 1D models are used to assessing well-established benchmarks which are often used to solve shell problems, such as the pinched shell problem. 1D and 2D models were compared in terms of accuracy and computational costs. Furthermore, locking phenomena are discussed. It is concluded that some of the well-known shell problems, including those that are usually used as 'benchmarks' in FE shell developments, can be dealt with 1D refined models. The use of 1D refined models results in minor numerical locking, much lower computational efforts and comparable accuracy with shell models

Evidence for an east–west regional gravity trend in northern Tunisia: Insight into the structural evolution of northern Tunisian Atlas

The Atlas orogeny in northern Algeria and Tunisia led to the destruction of Tethys oceanic lithosphere and cumulated in a collision of microplates rifted off the European margin with the North African continental margin. The location of the boundary between African plate and Kabylian microplate is expressed in northern Algeria by a crustal wedge with double vergence of thrust sheets, whereas in northern Tunisia the geologic environment is more complex and the location of the plate boundary is ambiguous. In this study, we analyzed gravity data to constrain the crustal structure along the northern margin of Tunisia. The analysis includes a separation of regional and residual gravity anomalies and the application of gradient operators to locate density contrast boundaries. The horizontal gradient magnitude and directional gradient highlight a prominent regional E–W gravity gradient in the northern Tunisian Atlas interpreted as a deep fault (active since at least the Early Mesozoic) having a variable kinematic activity depending on the tectonic regime in the region. The main E–W gravity gradient separates two blocks having different gravitational and seismic responses. The southern block has numerous gravity lineaments trending in different directions implying several density variations within the crust, whereas the northern block shows a long-wavelength negative gravity anomaly with a few lineaments. Taking into account the geologic context of the Western Mediterranean region, we consider the E–W prominent feature as the boundary between African plate and Kabylian microplate in northern Tunisia that rifted off Europe. This hypothesis fits most previous geological and geophysical studies and has an important impact on the petroleum and mineral resource prospection as these two blocks were separated by an ocean and they did not belong to the same margin.

Use of Lagrange multipliers to combine 1D variable kinematic finite elements

This paper deals with finite element problems that require different formulations in different subregions of the problem domain. Attention is focused on a variable kinematic, one-dimensional, finite element formulation which was recently introduced by the first author. Finite elements with different order of expansion over the cross-section plane are employed in different regions of the 1D domain. Lagrange multipliers are used to “mix” different elements. Constraints are imposed on displacement variables at a number of points whose location over the cross-section is a parameter of the method. The number and the location of the connection points can be modified until convergence is reached. The method is first assessed by encompassing sample problems and then it is applied to analyze a number of structures which requires different formulations in different regions. Compact, thin-walled and bridge-like sections are considered to show the effectiveness of the methodology proposed as well as its advantages to solve practical problems.

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.

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.

Vibration Analysis of Anisotropic Simply Supported Plates by Using Variable Kinematic and Rayleigh-Ritz Method

This work deals with accurate free-vibration analysis of anisotropic, simply supported plates of square planform. Refined plate theories, which include layer-wise, equivalent single layer and zig-zag models, with increasing number of displacement variables are take into account. Linear up to fourth N-order expansion, in the thickness layer-plate direction have been implemented for the introduced displacement field. Rayleigh-Ritz method based on principle of virtual displacement is derived in the framework of Carrera’s unified formulation. Regular symmetric angle-ply and cross-ply laminates are addressed. Convergence studies are made in order to demonstrate that accurate results are obtained by using a set of trigonometric functions. The effects of the various parameters (material, number of layers, and fiber orientation) upon the frequencies and mode shapes are discussed. Numerical results are compared with available results in literature.

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.