Reissner-Mindlin Model Research Articles

Enhancing performance of sandwich panel with three-dimensional orthogonal accordion cores

This study introduces a novel three-dimensional orthogonal accordion structure (3D-OAS) as the cellular core of sandwich panels, achieving multi-directional zero Poisson’s ratio through the orthogonal combination of two-dimensional accordion structures. To analyze its static characteristics effectively, a two-dimensional equivalent Reissner–Mindlin model (2D-ERM) was established utilizing the variational asymptotic method (VAM). The accuracy of 2D-ERM was confirmed by conducting three-point bending tests on 3D-printed specimens and analyzing the in-plane and out-of-plane deformation results of the 3D finite element model (3D-FEM). The comparison of global displacement contours and path-displacement curves between 3D-FEM and 2D-ERM showed a high level of agreement in predicting static deformation. The equivalent stiffness of SP-3D-OAS steadily increased as the inclined angle deviates by 90-degree, irrespective of whether it pertains to the convex or concave angle. Evaluation of deformability in sandwich panels with different cellular core forms revealed superior comprehensive performance in 3D-OAS, followed by 3D-YRS and 3D-XYAS, with a reduction of 16.41% and 17.35% in specific stiffness, respectively. Compared to the 3D-FEM, 2D-ERM significantly reduces computation time without compromising engineering accuracy. The research results provide a useful reference for optimal design of sandwich panels with multi-directional ZPR cellular core.

Dynamic Characteristics of Composite Sandwich Panel with Triangular Chiral (Tri-Chi) Honeycomb under Random Vibration.

A full triangular chiral (Tri-Chi) honeycomb, combining a honeycomb structure with triangular chiral configuration, notably impacts the Poisson's ratio (PR) and stiffness. To assess the random vibration properties of a composite sandwich panel with a Tri-Chi honeycomb core (CSP-TCH), a two-dimensional equivalent Reissner-Mindlin model (2D-ERM) was created using the variational asymptotic method. The precision of the 2D-ERM in free and random vibration analysis was confirmed through numerical simulations employing 3D finite element analysis, encompassing PSD curves and RMS responses. Furthermore, the effects of selecting the model class were quantified through dynamic numerical examples. Modal analysis revealed that the relative error of the first eight natural frequencies predicted by the 2D-ERM consistently remained below 7%, with the modal cloud demonstrating high reliability. The PSD curves and their RMS values closely aligned with 3D finite element results under various boundary conditions, with a maximum error below 5%. Key factors influencing the vibration characteristics included the ligament-rib angle of the core layer and layup modes of the composite facesheets, while the rib-to-ligament thickness ratio and the aspect ratio exert minimal influence. The impact of the ligament-rib angle on the vibration properties primarily stems from the significant shift in the core layer's Poisson's ratio, transitioning from negative to positive. These findings offer a rapid and precise approach for optimizing the vibration design of CSP-TCH.

Pseudo-equivalent model for sandwich panels with egg-shaped honeycomb-grid core

The novel sandwich panel with egg-shaped honeycomb-grid core (SP-EHC), formed by chamfering the grid walls, can achieve a balance between lightweight and high-strength, as well as promote effective ventilation and drainage due to the semi-closed honeycomb design. To determine the panel’s equivalent stiffness properties, an asymptotic analysis of the energy functional stored within the periodic unit cell is conducted. On this base, a 2D pseudo-equivalent model (2D-PEM) in the form of the Reissner–Mindlin model was formulated. The pseudo-excitation method was employed to analyze the self-power spectral density function and root mean square response of the 2D-PEM under random vibration. The accuracy and effectiveness of the model were validated through comparisons of the three-point bending experimental results of the 3D-printed specimen, exhibiting relative errors of 2.95%. In addition, free and random vibration analyses were conducted on the hybrid SP-EHC to further validate the model. Furthermore, the influence of material and structural parameters on specific stiffness, natural frequencies and first velocity RMS was systematically investigated. The findings reveal that enhancing the grid wall’s chamfering ratio is beneficial in achieving the desired objectives of lightweight and high-strength. In addition, compared to sandwich panels with orthogonal and pyramid grid cores, this improvement positively impacts the vibration characteristics of the sandwich panel. These findings offer valuable insights for the future design and optimization of such sandwich panels.

VAM-based reduced plate model for composite sandwich folded plate (CSFP) with V-shaped folded cores

The composite sandwich folded plate (CSFP) not only has excellent mechanical properties but is multi-functional, owing to the internal transparent design. To investigate its effective performance, the original CSFP is reduced to constitutive modeling over the unit cell (providing the orthotropic elastic constants) and global analysis over the two-dimensional reduced-order plate model (2D-RPM) based on the variational asymptotic method, and transformed to a Reissner–Mindlin model for practical application. The accuracy and effectiveness of 2D-RPM is verified by comparison with the static and dynamic results of three-dimensional direct numerical simulation (3D-DNS) under different boundary conditions. Furthermore, a parameter study is carried out to investigate the effects of layup configurations and core structural parameters (opening angle, core height, wall thickness, and forms of folded core) on the effective performance of the CSFP. It is concluded that the CSFP has the best balanced performance with a layup of ±45°/0°/90°s and larger opening angle and wall thickness of the folded core, as well as a smaller Z-line length and height.

Optimum design of two-material bending plate compliant devices

PurposeThe purpose of this study is to explore the optimum design of bending plate compliant mechanisms subjected to pure mechanical excitations using topological-derivative-based topology optimization. The main objective is to design the reinforcement in a plate of base material.Design/methodology/approachThe optimum design is performed by means of a level-set representation method guided by topological derivatives. Kirchhoff and Reissner–Mindlin models are used to solve the linear bending plate problem. A qualitative comparison has been carried out between the optimal obtained topologies for each model.FindingsThe proposed methodology was able to design reinforcement in a plate of the base material. The obtained reinforcements notably improve the device’s behavior. The shape and topology of the reinforcements vary depending on the mechanical plate model considered. In fact, in the Reissner–Mindlin solutions, very thin flexo-torsional hinges connecting big zones of the reinforcement material are designed.Originality/valueUp to date, the synthesis of ortho-planar mechanisms by means of continuum topology optimization was only boarded within a multi-physics context. In this work, the optimal design of pure ortho-planar compliance actuators is addressed. The best performance is found by analyzing the results for two classical mechanical plate models.

The Bending-Gradient theory for flexural wave propagation in composite plates

This paper is concerned with the prediction of the propagation of flexural waves in anisotropic laminated plates with relatively high slenderness ratios by means of refined plate models. The study is conducted using the Bending-Gradient theory which is considered as an extension of the Reissner–Mindlin theory to multilayered plates. Two projections of the Bending-Gradient model on Reissner–Mindlin models are also explored. The relevance of the proposed models is tested by comparing them to well-known plate theories and to reference results obtained using the finite element method.

Generalized Reissner–Mindlin Model for Composite Plates with No Restrictions on Constituent Materials

Unlike previous works, the present work implements the variational-asymptotic method to develop a generalized Reissner–Mindlin model for composite plates made of inhomogeneous and anisotropic materials. From a mathematical perspective, an equivalent two-dimensional plate model is first constructed through a dimensional-reduction procedure that involves the asymptotically correct energy functional up to second order in small parameters and with no specific restrictions on constituent materials. Next, to obtain a generalized Reissner–Mindlin plate model, a hybrid energy transformation procedure is applied from an engineering perspective. During this procedure, the unnecessary mathematical complexity of partial-derivative terms in the energy functional derived herein and obscure physical interpretations of mechanical boundary conditions in the plate modeling are systematically resolved. To evaluate the accuracy and capability of these procedures, three-dimensional recovery relations are established to predict the three-dimensional fields of the original three-dimensional structure. Several examples from the literature are presented to demonstrate the consistency and efficiency of the proposed approach.

Shear-corrected Reissner-Mindlin plate model

A new shear-corrected Reissner-Mindlin model is presented. The method reduces the modeling error of the classical model without affecting the form of the classical plate equations. Therefore, implementation on an existing software for the Reissner-Mindlin plate model is simple. The principle of virtual work and an explicit set of kinematic and kinetic assumptions is used in derivation. There, displacement assumption of the classical model is enhanced by a warping part which is eliminated to end up with equations for the classical part only. The equations differ from the classical ones in shear correction factors, modification in the source term, and stress expressions.

Hybrid energy transformation to generalized Reissner–Mindlin model for laminated composite shells

By using the variational-asymptotic method, a two-dimensional mechanical model for laminated composite shells is established from a mathematical perspective, having the energy functional asymptotically correct up to the desired order in the small parameters. However, it is not in a practical form from an engineering perspective because of the appearance of partial derivative terms, which bring unnecessary mathematical complexity and obscure physical interpretation of mechanical boundary conditions in the shell modeling. Therefore, one more procedure is inevitably required – a so-called energy transformation procedure, which constructs a mathematical link between the energy functional derived herein and a simpler engineering model, such as a generalized Reissner–Mindlin model. In a different manner from previous works, this article introduces a hybrid energy transformation procedure composed of two successive steps: an equilibrium transformation via a linear algebraic approach, and an energy transformation via a perturbation approach. During this procedure, the first step is to transform the two-dimensional equilibrium equations from a hyper-static system into an isostatic system by augmenting them with the two-dimensional compatibility equations. Then, for obtaining a generalized Reissner–Mindlin model, the second step is to introduce initial curvature/twist as small parameters (in the sense of perturbations) into the constitutive law. The coupling stiffness terms between the transverse shear generalized strain measures and the remaining generalized Reissner–Mindlin strain measures are shown to be identically zero for laminated composite plates/shells (in contrast to the analogous terms of a similarly constructed generalized Timoshenko model for composite beams). Several examples are presented to demonstrate the capability and accuracy of this new approach.

Two-scale Reissner-Mindlin plate model

A two-scale plate model, in which the displacement assumption consists of the Reissner-Mindlin and warping parts, is presented. To reduce the modelling error of the classical Reissner-Mindlin model, the warping part is chosen so that the overall displacement satisfies the full 3D elasticity equations as well as possible. Pressure loaded isotropic homogeneous plate is used as an application example.

Shear stiffness prediction of Reissner–Mindlin plates with periodic microstructures

ABSTRACTIn this article, an intuitionistic interpretation of the new numerical implementation of asymptotic homogenization for effective bending stiffness of in-plane periodic plate structures is presented. Based on this interpretation, a two-step method of effective shear stiffness prediction for their Reissner–Mindlin model is developed. An equivalent displacement field of linear curvature is applied to a unit cell in the first step; in the second step, a new unit cell problem is constructed and solved, and the effective shear stiffness is obtained through energy equivalence. This method can be easily implemented in commercial software and several examples are given to demonstrate its validity.

The Bending–Gradient theory for the linear buckling of thick plates: Application to Cross Laminated Timber panels

In this paper, the linear buckling of a heterogeneous thick plate is studied using the Bending–Gradient theory which is an extension of the Reissner–Mindlin plate theory to the case of heterogeneous plates. Reference results are taken from a 3D numerical analysis using finite-elements and applied to Cross Laminated Timber panels which are thick and highly anisotropic laminates. First, it is shown that critical buckling loads are close to the material failure load which proves the necessity of a design model for the buckling of Cross Laminated Timber panels. Second, the soft simple support boundary condition is introduced as an opposition to the conventional hard simple support condition. It is shown that this distinction could be taken into account for designing timber structures depending on the accuracy needed. Third, it is observed that for varying plate geometries and arrangements, the Bending–Gradient theory predicts more precisely the critical load of CLT panels than classical lamination and first-order shear deformation theories. Finally, it is demonstrated that one of the suggested projections of the Bending–Gradient on a Reissner–Mindlin model gives very accurate results and could favorably allow the development of engineering recommendations for estimating properly transverse shear effects.

A variational asymptotic theory of composite laminated plates: Hybrid transformation to Reissner–Mindlin model

In previous work, a 2D plate model for composite laminates without invoking ad hoc kinematic assumptions is constructed by modifying the asymptotically correct energy functional up to the desired order. However, it is not possible in general to construct an asymptotically correct Reissner–Mindlin model for practical use. To obtain the energy functional that is of practical use, a different manner from the previous works is introduced in this paper. By performing a hybrid transformation procedure composed of modified equilibrium and compatibility equations, all partial derivative terms on the energy functional are systematically eliminated to obtain the 2D generalized and transverse stiffness matrices for the Reissner–Mindlin model. Unlike the previous works, these is no need to relax the established warping constraints and perform the optimization procedure as an unnecessary approximation. Several numerical examples are presented to compare with the classical laminated theory as well as 3D exact solutions, demonstrating the capability and accuracy of this new method.

Hybrid Transformation to a Generalized Reissner–Mindlin Theory for Composite Plates

An asymptotic theory of composite plates is constructed using the variational asymptotic method. To maximize simplicity and promote efficiency of the developed model, a transformation procedure is required to establish a mathematical link between an asymptotically correct energy functional derived herein and a simpler engineering model, such as a generalized Reissner–Mindlin model. Without relaxing the warping constraints and performing “smart minimization” or optimization procedures introduced in previous work, a different approach is suggested in this paper. To eliminate all partial derivatives of the 2D generalized strains in the asymptotically correct energy functional, a hybrid transformation procedure is systematically carried out by involving modified equilibrium and compatibility equations, and solving a system of linear algebraic equations via the pseudo-inverse method. Equivalent constitutive laws for the generalized Reissner–Mindlin plate model are then estimated. Several examples as a preliminary validation are used to demonstrate the capability and accuracy of this new model.

A multiphysics model for magneto-electro-elastic laminates

An efficient, geometrically nonlinear, multiphysics plate model for analyzing magneto-electro-elastic composite laminates is rigorously developed by applying the variational asymptotic method. By taking advantage of the inherent small parameter characterized by the ratio of the thickness to the in-plane deformation of the plate, we systematically reduced the original multiphysically coupled three-dimensional model to a series of two-dimensional plate models. A companion one-dimensional through-the-thickness analysis provides necessary constitutive models for the plate analysis. For practical uses, we also fit the asymptotically correct second-order electromagnetic enthalpy into a generalized Reissner–Mindlin model. Three-dimensional multiphysical components such as displacement/strain/stress fields as well as the electric/magnetic potentials and fluxes are obtained through a recovery process. Results for sample problems featuring electromagnetic and elastic coupling as well as characteristically different load/boundary conditions and material configurations are compared with exact solutions to systematically investigate the prediction capability of the present model.

Reissner–Mindlin plate model with uncertain input data

A Reissner–Mindlin model of a plate resting on unilateral rigid piers and a unilateral elastic foundation is considered. Since the material coefficients of the orthotropic plate, stiffness of the foundation, and the lateral loading are uncertain, a method of the worst scenario (anti-optimization) is employed to find maximal values of some quantity of interest.The state problem is formulated in terms of a variational inequality with a monotone operator. Using mixed-interpolated finite elements, approximations are proposed for the state problem and for the worst scenario problem. The solvability of the problems and a convergence of approximations is proved.

Analysis of the discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model

We analyze the discontinuous Petrov–Galerkin (DPG) method with optimal test functions when applied to solve the Reissner–Mindlin model of plate bending. We prove that the hybrid variational formulation underlying the DPG method is well-posed (stable) with a thickness-dependent constant in a norm encompassing the L2-norms of the bending moment, the shear force, the transverse deflection and the rotation vector. We then construct a numerical solution scheme based on quadrilateral scalar and vector finite elements of degree p. We show that for affine meshes the discretization inherits the stability of the continuous formulation provided that the optimal test functions are approximated by polynomials of degree p+3. We prove a theoretical error estimate in terms of the mesh size h and polynomial degree p and demonstrate numerical convergence on affine as well as non-affine mesh sequences.

Zeroth-order shear deformation micro-mechanical model for composite plates with in-plane heterogeneity

This article introduces a new model for investigating the mechanical behavior of heterogeneous plates, which are composed of periodically-repeated microstructures along the in-plane directions. We first formulate the original three-dimensional problem in an intrinsic form for implementation into a single unified formulation and application to geometrically nonlinear problem. Taking advantage of smallness of the plate thickness-to-length parameter and heterogeneity and performing homogenization along dimensional reduction simultaneously, the variational asymptotic method is used to rigorously construct an effective zeroth-order plate model, which is similar to a generalized Reissner–Mindlin model (the first-order shear deformation model) capable of capturing the transverse shear deformations, but still carries out the zeroth-order approximation which can maximize simplicity and promote efficiency. This present approach is incorporated into a commercial analysis package for the purpose of dealing with realistic and complex geometries and constituent materials at the microscopic level. A few examples available in literature are used to demonstrate the consistence and efficiency of this new model, especially for the structures, in which the effects of transverse shear deformations are significant.

Asymptotical construction of a Reissner-like model for multilayer functionally graded magneto-electro-elastic plates

A magneto-electro-elastic Reissner–Mindlin model is developed for heterogeneous multilayer laminates made of functionally graded magneto-electro-elastic material using the variational asymptotical method. This model is applicable to laminates without prescribed electric and magnetic potential though the thickness. Taking advantage of the smallness of the plate thickness, we asymptotically decouple the original 3D magneto-electro-elastic problem into a 1D through-the-thickness analysis and a 2D plate analysis, and both are fully-coupled magneto-electro-elastic analyses. Interface continuity conditions are explicitly derived and solved to obtain the multilayer solutions. Various examples have been used to demonstrate the accuracy and application of this model.

Asymptotical construction of a fully coupled, Reissner–Mindlin model for piezoelectric and piezomagnetic laminates

The variational asymptotic method is used to construct a fully coupled Reissner–Mindlin model for piezoelectric and piezomagnetic laminates with some surfaces parallel to the reference surface coated with electrodes and magnetism. Taking advantage of the smallness of the plate thickness, we asymptotically split the original 3D electromagneto-mechanical problem into a 1D through-the-thickness analysis and a 2D plate analysis, and both are fully-coupled multiphysics analyses. The through-the-thickness analysis serves as a link between the original 3D analysis and the plate analysis by providing a constitutive model for the plate analysis and recovering the 3D field variables in terms of global responses calculated by the plate analysis. The present theory is implemented into the computer program VAPAS (Variational Asymptotic Plate and Shell Analysis). A numerical example of three-layer sandwich plate has been used to validate the present model.