Reissner Mindlin Plate Model Research Articles

Stress-based dimensional reduction and dual-mixed hp finite elements for elastic plates

Starting from the linearized weak forms of the kinematic equation and the angular momentum balance equation of three-dimensional non-linear elasticity, a stressbased dimensional reduction procedure is presented for elastic plates. After expanding the three-dimensional non-symmetric stress tensor into power series with respect to the thickness coordinate, the translational equilibrium equations, written in terms of the expanded stress coefficients, are satisfied by introducing first-order stress functions. The symmetry of the stress field is satisfied in a weak sense by applying the material rotations as Lagrangian multipliers. The seven-field plate model developed in this way employs unmodified three-dimensional strain-stress relations. On the basis of the dimensionally reduced plate model derived, a new dual-mixed plate bending finite element model is developed and presented. The numerical performance of the hp-version plate elements is investigated through the solutions of standard plate bending problems. It is shown that the modeling error of the stress-based plate model in the energy norm is better than that of the displacement-based Kirchhoff- and Reissner-Mindlin plate models. The numerical solutions and their comparisons to reference solutions indicate that the dual-mixed hp elements are free from locking problems, in either the energy norm or the stress computations, both for h- and p-extensions, and the results obtained for the stresses are accurate and reliable even for extremely thin plates.

Locking alleviation technique for the peridynamic Reissner–Mindlin plate model: the developed reduced integration method

Locking alleviation technique for the peridynamic Reissner–Mindlin plate model: the developed reduced integration method

Galerkin neural network approximation of singularly-perturbed elliptic systems

We consider the neural network approximation of systems of partial differential equations exhibiting multiscale features such as the Reissner–Mindlin plate model which poses significant challenges due to the presence of boundary layers and numerical phenomena such as locking. This work builds on the basic Galerkin Neural Network approach established in Ainsworth and Dong (2021) for symmetric, positive-definite problems. The key contributions of this work are (1) the analysis and comparison of several new least squares-type variational formulations for the Reissner–Mindlin plate, and (2) their numerical approximation using the Galerkin Neural Network approach. Numerical examples are presented which demonstrate the ability of the approach to resolve multiscale phenomena such for the Reissner–Mindlin plate model for which we develop a new family of benchmark solutions which exhibit boundary layers.

A locking-free weak Galerkin finite element method for Reissner–Mindlin plate on polygonal meshes

A new weak Galerkin finite element method is introduced and analyzed for the Reissner–Mindlin plate model in the primary form (without introducing shear strain as an extra unknown), which results in a linear system with symmetric positive definite stiffness matrix. The proposed method achieves uniform convergence with respect to plate thickness (the so-called locking-free) without introducing any projection, reduced integration, etc. In addition, the new method can be applied to general polygonal meshes; in particular, we implement pentagonal and hexagonal meshes in our numerical tests. The numerical study confirms our theory.

Gradient-index phononic crystals for highly dense flexural energy harvesting

Gradient-index (GRIN) refers to a system where the refractive index changes spatially within a specific region. GRIN phononic crystals are capable of not only amplifying the magnitude of wave energies but also controlling the directional nature of the wave propagation, thus offering substantial benefits with regard to energy harvesting (EH) improvements. Here, we propose a systematic design method for GRIN phononic crystals which combine the two-dimensional Reissner–Mindlin plate model and a genetic algorithm for optimization. This design process allows us to design a GRIN phononic crystal with any arbitrary refractive index profile or complex shape of the unit cells. The experimentally verified focusing capability of the GRIN phononic crystals led to the realization of piezoelectric energy harvesting with a maximum areal power density value of up to 240.4 mW/m2, considerably outperforming the existing non-GRIN-based EH systems without direction controllability.

Anisotropic strain gradient thermoelasticity for cellular structures: Plate models, homogenization and isogeometric analysis

For three-dimensional cellular plate-like structures with a triangular (extruded lattice) microarchitecture, the article develops a pair of two-scale plate models relying on the anisotropic form of Mindlin’s strain gradient thermoelasticity theory. Accordingly, a computational homogenization method is proposed for determining the constitutive parameters of the related higher-order constitutive tensors. First, a Reissner–Mindlin plate model is derived by dimension reduction from a general framework of three-dimensional orthotropic strain gradient thermoelasticity and written as a variational formulation. An isogeometric conforming Galerkin method is formulated accordingly. Second, the plate model is modified in order to reduce the number of the constitutive strain gradient parameters. These steps are then repeated by following the kinematical assumptions of the Kirchhoff plate theory. Third, in order to see the cellular microarchitecture as a homogeneous three-dimensional material with classical modulae of transversal isotropy, classical computational homogenization is accomplished for determining the corresponding material parameters. Fourth, in order to see the cellular structures as two-dimensional plates, a non-classical homogenization procedure is proposed for the identification of the strain gradient modulae of the plate models. Finally, a set of numerical examples illustrates the reliability and efficiency of the resulting plate models in homogenizing cellular plate-like structures into strain gradient plate models capturing the bending size effects induced by the microarchitecture.

Modelling the radially polarised annular stator of a piezoelectric travelling wave ultrasonic motor based on the shear effect

This article deals with the mathematical–analytical model of a radially polarised stator, part of a piezoelectric travelling wave ultrasonic motor based on the shear effect. The stator is treated with a Reissner–Mindlin plate model containing piezoelectric terms. The so-obtained mathematical description of the disc stator takes into account its geometry, kinematics and characteristics that influence efficiency and torque. Rayleigh–Ritz discretisation is used to obtain eigenfrequencies and eigenmodes of the stator plate. In addition, there are often teeth over the contact surface of ring-shaped stators to minimise the friction losses during operation of the motor, and possible vibration modes are compared with respect to the deflexion of the contact points. In the laboratory, measured eigenfrequencies of the free vibrations of the plate corroborate the numerical method. Particularly, the generation of travelling waves requests the excitation of two degenerated vibration modes in a certain electrode configuration. A voltage inverter was designed for this purpose.

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.

Simple and extensible plate and shell finite element models through automatic code generation tools

A large number of advanced finite element shell formulations have been developed, but their adoption is hindered by complexities of transforming mathematical formulations into computer code. Furthermore, it is often not straightforward to adapt existing implementations to emerging frontier problems in thin structural mechanics including nonlinear material behaviour, complex microstructures, multi-physical couplings, or active materials. We show that by using a high-level mathematical modelling strategy and automatic code generation tools, a wide range of advanced plate and shell finite element models can be generated easily and efficiently, including: the linear and non-linear geometrically exact Naghdi shell models, the Marguerre-von Kármán shallow shell model, and the Reissner-Mindlin plate model. To solve shear and membrane-locking issues, we use: a novel re-interpretation of the Mixed Interpolation of Tensorial Component (MITC) procedure as a mixed-hybridisable finite element method, and a high polynomial order Partial Selective Reduced Integration (PSRI) method. The effectiveness of these approaches and the ease of writing solvers is illustrated through a large set of verification tests and demo codes, collected in an open-source library, FEniCS-Shells, that extends the FEniCS Project finite element problem solving environment.

A volume-averaged nodal projection method for the Reissner–Mindlin plate model

We introduce a novel meshfree Galerkin method for the solution of Reissner–Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and is devoid of shear-locking. The proposed approach uses linear maximum-entropy basis functions for field variables approximation and is built variationally on a two-field potential energy functional wherein the shear strain, written in terms of the primitive variables, is computed via a volume-averaged nodal projection operator that is constructed from the Kirchhoff constraint of the three-field mixed weak form. The meshfree approximation is constructed over a set of scattered nodes that are obtained from an integration mesh of three-node triangles on which the meshfree stiffness matrix and nodal force vector are numerically integrated. The stability of the method is rendered by adding bubble-like enrichment to the rotation degrees of freedom. Some benchmark problems are presented to demonstrate the accuracy and performance of the proposed method for a wide range of plate thicknesses.

An asymptotic Reissner–Mindlin plate model

A mathematical study via variational convergence of a periodic distribution of classical linearly elastic thin plates softly abutted together shows that it is not necessary to use a different continuum model nor to make constitutive symmetry hypothesis as starting points to deduce the Reissner–Mindlin plate model.

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.

Discrete and continuous models for static and modal analysis of out of plane loaded masonry

A critical review of analytical and numerical models for studying masonry out of plane behaviour is presented. One leaf historical masonry, composed by rigid blocks arranged regularly with dry or mortar joints, is considered. Discrete model with rigid blocks, Love-Kirchhoff and Reissner-Mindlin plate models and 3D heterogeneous FEM are adopted. An existing simple and effective discrete model is adopted and improved by applying matrix structural analysis techniques for static and modal analysis of masonry walls in the elastic field, but the formulation allows to account also for material nonlinearity. Elastic parameters of both plate models are based on an existing compatible identification between 3D discrete model and 2D plate models. Static and modal analysis of masonry walls with several boundary conditions are carried on, numerical tests account for in plane size of heterogeneity and structure thickness by means of in and out of plane scale factors. Results show that discrete model is simple and effective for representing masonry behaviour, especially when size of heterogeneity is smaller than overall panel size. Decreasing in plane scale factor, plate models converge to the discrete one, but the Reissner-Mindlin one shows a better convergence and also allows adopting a simple FE for performing numerical analysis.

Analysis of strain gradient Reissner–Mindlin plates using a C0 four-node quadrilateral element

The weak form of Reissner–Mindlin plate model in the context of the classical continuum theory requires C0-continuity of the lateral deflection as well as rotation fields. However, due to the presence of higher-order derivatives, the same plate model based on strain gradient elasticity demands C1-continuity of interpolation functions for a standard finite element (FE) formulation. On the other hand, it is known that C0 interpolation functions result in more stable and cost-effective elements. Accordingly, a four-node quadrilateral gradient-enhanced plate element using C0 interpolation of the field variables is introduced. The main idea is to start with an eight-node quadrilateral element on which any field variable and its spatial derivatives are independently interpolated via standard C0 shape functions. By introducing appropriate algebraic constraints, the degrees of freedom corresponding to the midside nodes of the eight-node element are then eliminated and a quadrilateral element with four nodes on its corners is constructed. Accuracy and performance of the proposed element in several examples and for a wide range of parameters is investigated. It is shown that the new element can successfully capture the size-dependent behaviour of plates at small scales, has a proper rank (contains no spurious zero-energy modes), passes the patch test for thin as well as thick plates in an arbitrary mesh, and is free of shear locking. Furthermore, the introduced element can reproduce the results of the Reissner–Mindlin plate model based on the classical continuum theory when the plate thickness is far greater than the material length scale parameter.

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.

Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model

This work presents new stabilised finite element methods for a bending moments formulation of the Reissner-Mindlin plate model. The introduction of the bending moment as an extra unknown leads to a new weak formulation, where the symmetry of this variable is imposed strongly in the space. This weak problem is proved to be well-posed, and stabilised Galerkin schemes for its discretisation are presented and analysed. The finite element methods are such that the bending moment tensor is sought in a finite element space constituted of piecewise linear continuos and symmetric tensors. Optimal error estimates are proved, and these findings are illustrated by representative numerical experiments.

Locking free quadrilateral continuous/discontinuous finite element methods for the Reissner–Mindlin plate

We develop a finite element method with continuous displacements and discontinuous rotations for the Reissner–Mindlin plate model on quadrilateral elements. To avoid shear locking, the rotations must have the same polynomial degree in the parametric reference plane as the parametric derivatives of the displacements, and obey the same transformation law to the physical plane as the gradient of displacements. We prove optimal convergence, uniformly in the plate thickness, and provide numerical results that confirm our estimates.

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.

An asymptotic strain gradient Reissner-Mindlin plate model

In this paper we derive a strain gradient plate model from the three-dimensional equations of strain gradient linearized elasticity. The deduction is based on the asymptotic analysis with respect of a small real parameter being the thickness of the elastic body we consider. The body is constituted by a second gradient isotropic linearly elastic material. The obtained model is recognized as a strain gradient Reissner-Mindlin plate model. We also provide a mathematical justification of the obtained plate model by means of a variational weak convergence result.