Reissner Mixed Variational Theorem Research Articles

Analysis of laminated shells using pseudospectrals and the Reissner-Mixed Variational Theorem

In this paper, we combine the Carrera’s Unified Formulation CUF and a pseudospectral technique for predicting the static deformations and free vibrations behaviour of thin and thick cross-ply laminated shells. For the first time, the Reissner-Mixed Variational Theorem is used together with pseudospectrals to achieve a highly accurate technique. The accuracy and efficiency of this numerical technique for static and vibration problems are demonstrated through numerical examples.

A mechanical analysis of variable angle-tow composite plates through variable kinematics models based on Carrera’s unified formulation

Variable Angle-Tow (VAT) laminates offer a promising alternative to straight fiber composites. By varying fibers orientation within the structure plane, ambitious design and performance goals can be achieved. However, the wider design space results in a more complex problem with more parameters to consider. Carrera’s Unified Formulation (CUF) has been used in previous works performing buckling, vibrational and stress analyses of VAT plates. Usually, one-dimensional (1D) CUF beam models are used, while two-dimensional (2D) plate models are obtained as a particular case of shells by considering a null curvature. In most cases, a linear law is considered to describe the variation of fibers orientation in the main plane of the structure. The purpose of this article is to extend the CUF 2D plate finite elements family to the mechanical analysis of composite laminated plate structures with curvilinear fibers. The main contribute consists in the development of a CUF FE model within the Reissner’s Mixed Variational Theorem (RMVT) context for an improved calculation of the out-of-plane stress components. Results show that RMVT can predict in-plane stresses and satisfy the though-the-thickness transverse stresses continuity due to inter-layer equilibrium. The accuracy of RMVT-based models is also investigated using two different approximation points distributions along the plates thickness.

Properties and layerwise modeling of the Harlequin variational theorem for composite structures

The analysis of composite structures requires an accurate prediction of the displacements, strain, and (especially) stress fields. To meet this requirement, the Harlequin Variational Theorem of the First Family (HVTFF) was recently formulated by adopting the Partitioned Parametrized Variational Procedure (PPVP). This work introduces for the first time layerwise (LW) axiomatic models that simultaneously represent displacements, stresses, and strains within HVTFF framework. In addition, the physical meaning of the user-selected variational parameter appearing in the functional is assessed and its effects on the accuracy of the fields’ representations is determined. It is also shown that HVTFF can lead to pure displacement formulations or displacement-stress approaches typical of the Hellinger-Reissner Principle (HRP) and Reissner’s Mixed Variational Theorem (RMVT).

A FEM Free Vibration Analysis of Variable Stiffness Composite Plates through Hierarchical Modeling

Variable Angle Tow (VAT) laminates offer a promising alternative to classical straight-fiber composites in terms of design and performance. However, analyzing these structures can be more complex due to the introduction of new design variables. Carrera’s unified formulation (CUF) has been successful in previous works for buckling, vibrational, and stress analysis of VAT plates. Typically, one-dimensional (1D) and two-dimensional (2D) CUF models are used, with a linear law describing the fiber orientation variation in the main plane of the structure. The objective of this article is to expand the CUF 2D plate finite elements family to perform free vibration analysis of composite laminated plate structures with curvilinear fibers. The primary contribution is the application of Reissner’s mixed variational theorem (RMVT) to a CUF finite element model. The principle of virtual displacements (PVD) and RMVT are both used as variational statements for the study of monolayer and multilayer VAT plate dynamic behavior. The proposed approach is compared to Abaqus three-dimensional (3D) reference solutions, classical theories and literature results to investigate the effectiveness of the developed models. The results demonstrate that mixed theories provide the best approximation of the reference solution in all cases.

Electro-mechanical analysis of functionally graded graphene reinforced composite laminated plate with macro fiber composite actuator

Graphene is considered to be a desirable reinforcement material for composite materials due to its exceptional mechanical properties. However, there is a lack of research on the interlaminar stress analysis of graphene reinforced composite plates bonded with a macro fiber composite (MFC) actuator subjected to electro- mechanical loads. Inaccurate descriptions of interlaminar shear deformations can significantly affect the mechanical performance of the MFC piezoelectric graphene reinforced composite plate due to the electro-mechanical coupling feature and abrupt change in material properties at the interfaces of neighboring layers. To address this issue, this study proposes an attractive plate model for the analysis of MFC piezoelectric laminated plate. The three-dimensional (3D) elasticity equations and the Reissner mixed variational theorem (RMVT) are employed to improve the precision of interlaminar shear stresses. The 3D elasticity solutions and results obtained from other models are used to evaluate the proposed model. Furthermore, this study thoroughly investigates the effect of several significant parameters on the deformations and stresses of the MFC piezoelectric plates with reinforcements.

Free Vibration Analysis of Functionally Graded Graphene-Reinforced Composite-Laminated Plates

Graphene is recognized as one of the most alluring reinforcements for composite materials due to its exceptional mechanical capabilities. However, the existing models disregarding the compatible requirements of interlaminar stresses in the literature may meet severe challenges for the dynamic analysis of functionally graded graphene-reinforced composite (FG-GRC) laminated thick plates. If transverse shear deformations cannot be described accurately, the prediction of natural frequencies for FG-GRC laminated plates will be seriously and directly affected. Accordingly, an attractive plate theory in conjunction with the analytical approach and finite-element formulation is proposed for free vibration analysis of FG-GRC plates. An improved transverse shear stress field is acquired in terms of the three-dimensional (3D) elasticity equations and the Reissner mixed variational theorem (RMVT). The performance of the suggested model was evaluated using 3D elasticity solutions and the results obtained from other models. Numerical results showed that the suggested model can produce promising results for the thick laminated and sandwich plates. Additionally, a thorough investigation was conducted to study the effects of several graphene parameters, including volume fractions, distribution patterns, span-to-thickness ratios, aspect ratios, boundary conditions, and stacking sequences, on the dynamic responses of FG-GRC laminated plates.

Dynamic analysis of graphene-reinforced sandwich plates with piezoelectric macro fiber composite face sheets

Graphene is considered as an ideal candidate for composite materials owing to the outstanding physical characteristics. However, existing higher-order shear deformation theories in the literature may meet severe challenges for the accurate prediction of natural frequencies of graphene-reinforced sandwich plates with piezoelectric macro fiber composite (MFC) face sheets. If transverse shear deformations cannot be described accurately, the dynamic behaviors of piezoelectric sandwich plates with graphene reinforcements will be significantly impacted by the large differences of material properties at interfaces of adjacent layers and electro-mechanical coupling characteristic. Thereby, a novel electro-mechanical coupling theory is to be proposed for the free vibration of graphene-reinforced sandwich plates with MFC face sheets. Based on the Reissner mixed variational theorem (RMVT), the precision of the interlaminar shear stresses including the electro-mechanical coupling effect can be improved. The modified interlaminar shear stress field is absorbed in the strain energy, which can enhance the accuracy of the free vibration response of piezoelectric sandwich plates. Additionally, the second-order derivatives of in-plane displacements have been eliminated from the interlaminar shear stress components, which can substantially simplify the finite element implementation. Thus, a simple C0-type finite element formulation is developed for the free vibration of piezoelectric sandwich plates with graphene reinforcements. The 3D elasticity solutions and the results obtained from other theories are used to evaluate the performance of the proposed theory. In comparison with the existing higher-order theories, the proposed theory can produce more accurate results. In addition, a comprehensive parametric study is conducted to explore the impacts of significant parameters on the free vibration behaviors of piezoelectric graphene-reinforced sandwich plates.

Layerwise theories for composite beams with continuous and discontinuous stresses

Due to the unique characteristics of composite materials, the study of composite beams is far more complex than the study of homogeneous beams. The finite element method has proven to be a powerful approach to analyze composites subjected to the most distinctive situations. In the present work, two element solutions using cubic polynomials are considered: with continuous stresses and with discontinuous stresses along the transverse direction. Both converge to the analytical solution as the number of elements increase, i.e. with a finer mesh. Besides satisfying the boundary conditions at the surfaces and interfaces, the first solution gives better outcomes close to the center of the beam. On the other hand, the second solution gives better outcomes close to the borders of the beam, but it has a larger number of nodal parameters. The results are compared to a zig-zag element solution which has a number of nodal parameters independent of the number of layers. An element based on the Reissner mixed variational theorem is also included for additional comparisons. It is concluded that the cubic polynomials used to expand the cross section functions, must be different in each layer in order to achieve a reasonable agreement between the analytical and the calculated transverse normal stress.

A new zigzag shell model in terms of Reissner’s mixed variational theorem for composite laminates with curvature

In this study, a new mixed-field zigzag shell model is proposed in terms of Reissner’s Mixed Variational Theorem (RMVT) for the composite laminates with curvature. In this model, refined zigzag theory (RZT) is employed to improve the accuracy and efficiency by introducing through-the-thickness piecewise linear warping functions. Moreover, the shear correction factors are eliminated. It is also different from the published models for composite laminated shells, the continuity conditions of interlaminar transverse shear stresses (TSSs) are satisfied a priori based on RMVT. The equilibrium equations and related boundary conditions are also derived from RMVT. Subsequently, the static performance, natural frequencies, and critical loads of composite laminates with curvature are verified by comparing them with the 3D elastic solutions and other existing ones. It is observed that the present model accounts for not only the efficiency of the low-order models but also the accuracy of the high-order models. Finally, it is shown that the ply stacking, side-to-thickness ratios, and side-to-radius ratios play an important role in percentage errors between the present model and the previous ones.

A new electro-mechanical finite formulation for functionally graded graphene reinforced composite laminated thick plates with piezoelectric actuator

Graphene is regarded as one of the most attractive reinforcements for composite materials attributing to the extraordinary physical characteristics. However, existing models in the literatures might meet severe challenges for the interlaminar stress prediction of functionally graded graphene reinforced composite (FG-GRC) laminated thick plate integrated with piezoelectric fiber reinforced composite (PFRC) actuator under electro-mechanical loadings. If transverse shear deformations cannot be described accurately, the mechanical performance of FG-GRC laminated plate with PFRC actuator will be significantly impacted by the electro-mechanical coupling effect and the sudden change of material characteristics at the interfaces. Thereby, a new electro-mechanical coupled plate model with only seven independent displacement variables is to be proposed in this paper. Employing the Reissner mixed variational theorem (RMVT), the precision of the transverse shear stresses considering the electro-mechanical coupling effect can be improved. Moreover, the second-order derivatives of in-plane displacement parameters have been removed from the transverse shear stress components, which can greatly simplify the finite element formulation. Thus, based on the proposed electro-mechanical coupled plate model, a simple C0-type finite element formulation is developed for interlaminar shear stress analysis of FG-GRC laminated thick plate with a PFRC actuator. The 3D elasticity solutions and the results obtained from other models are used to assess the performance of the proposed finite element formulation. Additionally, comprehensive parametric studies are achieved for the influences of graphene volume fraction, distribution pattern, electro-mechanical loading, boundary conditions, lamination scheme and geometrical parameters of the plate on the deformations and stresses of the FG-GRC laminated plate with a PFRC actuator.

Reissner’s Mixed Variational Theorem and triangular finite element discretizations: an energetic interpretation

Reissner’s Mixed Variational Theorem (RMVT) presents both displacements and transverse stresses as primary variables. This property allows the a-priori fulfillment of both interlaminar compatibility and equilibrium with potentially excellent numerical performance. However, the triangular finite element based on RMVT has never been assessed from an energetic perspective. This aspect is investigated in the present contribution for the first time: the functional reconstitution technique is extended to RMVT-based triangular elements (retaining transverse normal stress) and it is demonstrated that the elements exhibit significantly lower errors than the corresponding displacement-based formulation. Moreover, the percentage errors on the approximation of the semi-complementary energy is shown to be invariant with the plate thickness-to-width ratio.

A semi-analytical FE method for the 3D bending analysis of nonhomogeneous orthotropic toroidal shells

Based on Reissner's mixed variational theorem (RMVT), the authors develop a semi-analytical finite element (FE) method for a three-dimensional (3D) bending analysis of nonhomogeneous orthotropic, complete and incomplete toroidal shells subjected to uniformly-distributed loads. In this formulation, the toroidal shell is divided into several finite annular prisms (FAPs) with quadrilateral cross-sections, where trigonometric functions and serendipity polynomials are used to interpolate the circumferential direction and meridian-radial surface variations in the primary field variables of each individual prism, respectively. The material properties of the toroidal shell are considered to be nonhomogeneous orthotropic over the meridian-radial surface, such that homogeneous isotropic toroidal shells, laminated cross-ply toroidal shells, and single- and bi-directional functionally graded toroidal shells can be included as special cases in this work. Implementation of the current FAP methods shows that their solutions converge rapidly, and the convergent FAP solutions closely agree with the 3D elasticity solutions available in the literature.

Interlaminar stress analysis of functionally graded graphene reinforced composite laminated plates based on a refined plate theory

Nowadays, graphene is considered as one of the most ideal reinforcements for composite materials due to the outstanding mechanical properties. From the literature survey, it is found that the researches on interlaminar stress analysis of functionally graded graphene reinforced composite (FG-GRC) laminated structures are scarce in literature. If transverse shear deformations are unable to be described accurately, the reasonable design of FG-GRC laminated structures will meet severe challenges due to the differentiation of material properties at the adjacent layers. Thereby, such issue is less studied by using the efficient models, so an advanced mixed-form plate theory is to be proposed for transverse shear stress analysis of FG-GRC laminated plates. The proposed theory can meet beforehand compatible conditions of transverse shear stresses at the interfaces of adjacent laminates and only contains seven independent displacement variables. By applying the three- dimensional (3D) equilibrium equations and the Reissner mixed variational theorem (RMVT), the accurate transverse shear stresses are obtained. The 3D elasticity solutions and the results computed by using the chosen models are selected to appraise the capability of the proposed model. Numerical results show that the proposed plate model can yield accurately transverse shear stresses without any post-processing procedure. In addition, the effects of graphene volume fraction, graphene distribution pattern, lamination sequence, span-to-thickness ratio and aspect ratio on the displacements and stresses of the FG-GRC laminated plates are thoroughly investigated.

Three-dimensional free vibration analysis of rotating sandwich functionally graded truncated conical shells under various boundary conditions

In this work, the authors develop a Reissner’s mixed variational theorem (RMVT)-based weak formulation for the three-dimensional (3 D) free vibration analysis of sandwich functionally graded (FG) truncated conical shells under various boundary conditions rotating with a constant angular velocity with respect to the central axis of the shells. The material properties of the FG layers constituting the truncated conical shell are assumed to obey a power-law distribution of the volume fractions of the constituents through their thickness direction, for which the effective material properties are estimated using the rule of mixtures. Based on the weak formulation, a semi-analytical finite annular prism (FAP) method is developed for the current issue, where the effects of the initial hoop stress due to rotation and the centrifugal and Coriolis accelerations are considered. Implementation of the current FAP method indicates that its solutions converge rapidly, and the convergent solutions closely agree with accurate solutions available in the literature. The obtained 3 D frequency parameter solutions for rotating and non-rotating sandwich FG truncated conical shells can provide a standard by which to assess those obtained using assorted two-dimensional classical, advanced, and refined shell theories.

Shell finite element models with local kinematic refinements based on Reissner’s Mixed Variational Theorem with layer-wise descriptions

This work presents refined shell finite element (FE) models based on Reissner’s Mixed Variational Theorem (RMVT) with layer-wise (LW) Node-Dependent Kinematics (NDK) for the analysis of multi-layered composite structures. LW descriptions of displacements and transverse stresses employing hierarchical Legendre polynomials (HLP) are adopted. Developed in the framework of Carrera Unified Formulation (CUF), adaptable refinement through hierarchical shell elements with NDK is extended to the RMVT formulations. The numerical accuracy and efficiency are assessed by comparing the FE solutions for bending shells against references available in the literature or from modeling with commercial software.

Free vibration analysis of laminated composite and sandwich plates based on a mixed zigzag theory

In this article, an accurate mixed zigzag theory (HZZTM) is proposed for free vibration analysis of laminated composite and sandwich plates. In-plane displacements in the proposed zigzag theory nonlinearly distribute through the thickness and exhibit an abrupt discontinuity of slope at interfaces, which coincide with three-dimensional analysis. In order to obtain the accurate transverse shear stresses, a preprocessing approach based on the three-dimensional (3D) equilibrium equations and the Reissner mixed variational theorem (RMVT) is used. It is significant that the second-order derivatives of in-plane displacement variables have been removed from the transverse shear stress fields, such that the finite element implementation is greatly simplified. Thus, based on the proposed zigzag model, a computationally efficient C0-type three-node triangular plate element with linear interpolation function is proposed for free vibration analysis of thick laminated composite and sandwich plates. Moreover, the accurate transverse shear stresses are introduced in the dynamic equilibrium equation by employing the Hamilton’s principle, which can actively improve the accuracy of natural frequencies of the laminated composite plates with different thickness and material properties at each ply. Performance of the proposed model is assessed by comparing with several benchmark solutions. Agreement between the present results and the reference solutions is very good, and the proposed model only includes the seven displacement variables which can demonstrate the accuracy and effectiveness of the proposed model.

An accurate zigzag theory for bending and buckling analysis of thick laminated sandwich plates with soft core

An accurate and computationally attractive zigzag theory is developed for bending and buckling analysis of thick laminated soft core sandwich plates. The kinematic assumptions of the proposed zigzag theory are obtained by superimposing a nonlinear zigzag function on the first-order shear deformation theory. In order to obtain the accurate transverse shear stresses, a preprocessing approach based on the three-dimensional equilibrium equations and the Reissner mixed variational theorem is used. It is significant that the second-order derivatives of in-plane displacement variables have been removed from the transverse shear stresses, such that the finite element implementation is greatly simplified. Thus, based on the proposed zigzag model, a computationally efficient four-node C0 quadrilateral plate element with linear interpolation function is proposed for bending and buckling analysis of soft core sandwich plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Moreover, the accurate transverse shear stresses can be involved in the strain energy which can actively improve the accuracy of critical loads. Performance of the proposed model is assessed by comparing with several benchmark solutions. Agreement between the present results and the reference solutions is very good, and the proposed model only includes the seven displacement variables which can demonstrate the accuracy and effectiveness of the proposed model.

Layerwise mixed elements with node-dependent kinematics for global–local stress analysis of multilayered plates using high-order Legendre expansions

Carrera Unified Formulation (CUF) is taken a step further to render node-dependent kinematics (NDK) capabilities to new layerwise finite elements based on Reissner’s Mixed Variational Theorem (RMVT), especially suited for global–local stress analysis of multilayered plates, ensuring high numerical accuracy and computational efficiency, all together. In the framework of CUF, as introduced originally for multilayered structures, any degree of kinematic refinement can be considered in agreement with Equivalent Single-Layer (ESL) or Layer-Wise (LW) theories to develop advanced finite element models, whether based on the Principle of Virtual Displacements (PVD) or RMVT. The degree of kinematic refinement, which usually holds equally for the entire element, can be taken a step further, by being assigned locally to each of its nodes, making full use of CUF to render NDK capabilities to the elements. Besides, even though the elements can adopt any type of nodal shape functions, high p-order hierarchical Legendre expansions (HLE) can also be combined with NDK, achieving excellent convergence rates. These capabilities combined, explored first under the PVD, are for once integrated in the proposed elements under RMVT to further benefit accurate stress analysis. These elements can be applied throughout the entire mesh, adapting to local, transitional and global regions straightforwardly, providing high numerical accuracy, locally, with minimal computational efforts, globally. The numerical results focus on stress analysis of multilayered composite plates, including local effects, to demonstrate the predictive capabilities of the proposed RMVT-based LW elements with NDK and HLE combined, considering well-known benchmark three-dimensional exact solutions for assessment.

Thermoelastic Stress and Deformation Analyses of Functionally Graded Doubly Curved Shells

In this paper, the authors develop Reissner’s mixed variational theorem (RMVT)-based finite layer methods for the three-dimensional (3D) coupled thermoelastic analysis of simply supported, functionally graded, doubly curved (DC) shells with temperature-independent material properties. A two-phase composite material is considered to form the shell, and its material properties are assumed to obey a power–law distribution of the volume fractions of the constituents through the thickness direction of the shell. The effective material properties are estimated using the Mori–Tanaka scheme. The accuracy and convergence rate of these RMVT-based finite layer methods are validated by comparing their solutions with the quasi 3D and accurate two-dimensional solutions available in the literature.

Advanced models for nonlocal magneto-electro-elastic multilayered plates based on Reissner mixed variational theorem

In the present work, nonlocal layer-wise (LW) models for the analysis of magneto-electro-elastic multilayered plates are formulated. An Eringen non-local continuum behavior is assumed for the layers material; in particular, as usual in plate theories, partial in-plane nonlocality is assumed whereas local constitutive behavior is considered in the thickness direction. The proposed plate theories are obtained via the Reissner mixed variational theorem (RMVT), assuming the generalized displacements and generalized out-of-plane stresses as primary variables, and expressing them as through-the-thickness expansions of suitably selected functions, considering the expansion order as a free parameter. In the framework of the Carrera Unified Formulation, this allows the systematic generation of advanced high order plate theories via a layer-based assembly algorithm of the so-called fundamental nuclei associated with the variables expansion terms. The use of the LW approach and RMVT allows for (i) the explicit fulfillment of the transverse generalized stress interface equilibrium, which is crucial for a correct description of the plate fields, (ii) the straightforward analysis of plates with layers exhibiting different characteristic lengths in their nonlocal behavior. A Navier solution for the developed models has been implemented and tested for the static bending and free vibrations analyses of rectangular simply-supported plates. The obtained representative results favorably compare against available three-dimensional analytic results and demonstrate the features of the proposed theories.