Reissner-Mindlin Plate Theory Research Articles

Thermal buckling adaptive multi-patch isogeometric analysis of arbitrary complex-shaped plates based on locally refined NURBS and Nitsche’s method

Plate structures suffering thermal environments are often encountered in practice. In this paper, thermal buckling behavior of complex-shaped plates by an adaptive multi-patch isogeometric analysis (IGA) based on locally refined NURBS and Nitsche’s method is studied. Kinematic equations are derived using Reissner–Mindlin plate theory, while complex geometries of plates are represented with multiple patches, where the connection between two adjacent patches is constructed through Nitsche’s method. To conduct adaptive local refinement, structural mesh refinement strategy is combined with a posterior error estimator, which is defined according to the recovered stresses of the first thermal buckling mode. Numerical examples of plates with simple and complex geometries are considered. The accuracy of critical buckling temperature rise obtained from this study is verified against reference solutions.

Numerical analysis on stability of functionally graded graphene platelets (GPLs) reinforced dielectric composite plate

The present study deals with the buckling and postbuckling performances of functionally graded graphene platelets reinforced composite (FG-GPLRC) plate under external electric field. The elastic and dielectric properties of GPLRC are evaluated by effective medium theory (EMT) while the Poisson's ratio is calculated by rule of mixture. The GPL distribution throughout the thickness of the structure is described by an average volume fraction and a slope factor describing functionally graded distribution. The equations governing the buckling behaviors of FG-GPLRC plate are derived on the basis of principle of virtual work and Mindlin-Reissner plate theory using von Kármán strains. Differential quadrature method (DQM) is then used to discretize the equations and direct iteration method is used to numerically solve the discrete differential equations. The results advise that the stability of FG-GPLRC plate is correlated to several factors, including the average volume fraction of GPLs, functionally graded distribution, slope factor and the parameters of the electric field. The stability of the FG-GPLRC plates could be artificially varied via adjusting parameters of external electric field. The current research is envisaged to offer supportive information to the design of smart GPLRC structures.

Assumed strain finite element for natural frequencies of bending plates

PurposeThis paper aims to describe the formulation of a new finite element by assuming the strain field rather than the displacement field and by using the Reissner–Mindlin plate theory for the free vibration analysis of bending plates. This quadrilateral element consists of four-nodes and twelve degrees of freedom. The suggested element is based on assumed functions of the strain field that satisfy the compatibility equation.Design/methodology/approachAfter the proposition of the new element, several numerical tests for plates with regular and distorted meshes are presented to assess the performance of the new element. In addition, a parametric study is carried out to analyze the effects of biaxial loads on the natural frequencies of square plates with various boundary conditions. Detailed discussions are proposed after each benchmark problem.FindingsThe formulated element has verified the shear locking test and passes the patch test. The obtained results from the developed element show an excellent accuracy and fast convergence, and the natural frequencies are in excellent agreement when compared with analytical and other available numerical solutions.Originality/valueThe present element is simple in its formulation and has been proven to be applicable to thin or thick plate situations with sufficient accuracy. This element with full integration is free from shear locking, however, the numerical results provided by the standard four-node plate element R4 element show locking phenomena in thin plates. In addition to these features, the imposition of the compatibility conditions and the rigid body modes allow obtaining a finite element with higher-order terms for displacements field, which can increase the performance of the finite elements.

Notable highlights on locking-free techniques of Reissner-Mindlin plate finite elements in elastostatics

In this paper, we discuss about the notable locking-free techniques of several simple plate bending finite elements for the Reissner-Mindlin plate bending theory. The brief background for Reissner-Mindlin plate theory is presented, in which stress and strain derivation are given along with one-field and two-field variational approaches. Afterwards, we classify several efficient robust techniques in a cluster of main categories to present sequentially, which are all able to overcome the locking phenomenon in thick plate bending problems. Only selective algorithms are programmed to conduct numerical simulations. The corresponding results are compared between these elements to show their performances.

Multi-patch local mesh refinement XIGA based on LR NURBS and Nitsche’s method for crack growth in complex cracked plates

The objective of this article is to simulate crack growth of complex Mindlin–Reissner plates by developing an adaptive multi-patch extended isogeometric analysis (XIGA). Nitsche’s method is used to treat continuity between multi-patches or the coupling of non-conforming meshes, exactly describing the geometry of complex plates. The computational meshes in XIGA are independent of the cracks by introducing some enrichment functions into the displacement approximation based on the partition of unity, thus it is convenient in modeling evolution of crack. The locally refined non-uniform rational B-splines (LR NURBS), which have the properties of local refinement and exact description of geometry, are used to construct geometric model and taken as the shape functions in XIGA. To implement the adaptive local refinement, a method based on recovery technique by Zienkiewicz and Zhu is proposed. Based on the Mindlin–Reissner plate theory, interaction integral method is employed to calculate stress intensity factors (SIFs) and the maximum circumferential stress criterion is used to determine the crack propagation direction. Several numerical examples are carried out to demonstrate the performance of the adaptive multi-patch XIGA for simulating crack growth in complex plates.

Elastic-plastic buckling analysis of stiffened panel subjected to global bending in forming process

A new method has been proposed in this study for the elastic-plastic buckling analysis of stiffened panels under global bending. In this method, a simplified model of the stiffened panels has been built with the application of two theories of plasticity, the incremental theory (IT) and the deformation theory (DT). The effect of transverse shear deformation through the stiffener thickness has been considered using the Mindlin-Reissner plate theory. The governing differential equations have been solved by the differential quadrature (DQ) method and an iteration process has been adopted due to the non-linearity of material properties in the elastic-plastic buckling analysis. Non-linear finite element (FE) modelling of elastic-plastic buckling analysis has been carried out, and the FE results are between those based on DT and IT in general. When the reciprocal of strain hardening exponent mt increases to 20, the FE results are in a good agreement with DT results. Based on the proposed method and FE simulations, the effect of geometric parameters of stiffened panels (stiffener thickness to height ratio, stiffened panel length to height ratio, width to height ratio, and skin thickness to stiffener thickness ratio) on buckling behaviour in the elastic-plastic region has been investigated and discussed. The proposed method provides an efficient way for parameter optimisation in the structure design of stiffened panels for the aerospace applications.

Development of an analytical framework for viscoelastic corrugated-core sandwich plates and validation against FEM

In this study, an analytical procedure for the bending problem of a viscoelastic sandwich plate with a corrugated core is presented. Reissner–Mindlin plate theory and N-termed Prony series are employed to define the elastic and time-dependent contributions of the governing equations, respectively. Three different corrugation patterns, i.e., rectangular, trapezoidal, and triangular, are examined. Moreover, the structure is analyzed under both simply support and clamp boundary conditions. The calibrated material parameters of polymethyl methacrylate (PMMA) for the Generalized Maxwell rheological model are employed to show the viscoelastic response of the structure. A 3D finite element simulation of the problem is also conducted to confirm the accuracy of the analytical formulation. The two well-known creep and stress relaxation phenomena of the viscoelastic materials are examined for the mentioned corrugation cores and both boundary conditions analytically and numerically. The time-dependent dimensionless deflection and resultant von Mises stress distributions are provided. Besides, the variation of the results with various rise-times and applied load are studied in detail. The von Mises stress contours of the upper surface of the structure at the end of the creep test are also presented. The finite element method outcomes verify the analytical results with excellent compatibility. The proposed analytical procedure can be used as an efficient tool to study the effects of various parameters such as material, geometrical constants, and corrugation pattern on bending of viscoelastic sandwich plates with corrugated core problems for design and optimization, which involves a high number of simulations.

Efficient Theoretical and Numerical Methods for Solving Free Vibrations and Transient Responses of a Circular Plate Coupled With Fluid Subjected to Impact Loadings

Abstract This paper presents an easy-to-use theoretical method and an efficient numerical method for solving free vibrations and transient responses of a circular plate coupled with fluid subjected to impact loadings and provides insights into various coupling cases with these developed methods. The Kirchhoff plate theory, Mindlin–Reissner plate theory, and the linear velocity potential function are used. The wet mode of the coupled system is described as the superposition of dry modes of the plate, which has been considered in few studies. The natural frequencies and corresponding mode shapes are solved using the orthogonality of dry modes. The transient responses of the plate are then solved using the superposition of the wet modes and the orthogonality of dry modes. To validate the theoretical results, an efficient and flexible finite element method is proposed and verified by comparing with commercial software. The four-node mixed interpolation of the tensorial component quadrilateral plate finite element (MITC4) and the eight-node acoustic pressure element are used to model the plate and the fluid, respectively. The theoretical and numerical methods provide reliable and accurate results. Parametric studies are performed to investigate the influence of geometric sizes, plate material properties, and fluid properties on the natural frequencies of the coupled system. A coupling parameter of fluid–structure interaction is proposed. The nondimensional added virtual mass incremental (NAVMI) factor decreases as the coupling parameter increases. Besides, the influence of fluid on wet modes of the plate decreases with the order.

Nonlinear strain gradient forced vibration analysis of shear deformable microplates via hermitian finite elements

Based on the modified strain gradient model (MSGM) and Mindlin-Reissner (M-R) plate theory, the geometrically nonlinear forced vibration of rectangular microplates via finite element (FE) method is investigated in this research. The nonlinear mathematical model of the problem is provided under the von Karman nonlinear kinematic relations within the M-R plate theory along with the MSGM. By the use of Hamilton’s principle and fully confirming rectangular hermitian finite elements, the stiffness and mass matrices, as well as the force vector, are obtained. Various numerical examples are represented to verify the efficiency and validity of the proposed model. By reducing the constitutive relations, the results for the modified couple stress model (MCSM) are also presented. Various numerical examples are given to investigate the impacts of length-scale parameter and geometrical parameters on the nonlinear forced vibration responses.

Modeling and Simulation Analysis of Rigid-flexible Coupling Dynamics of Ramming Mechanism Based on Reissner-Mindlin’s Medium-thick Plate Theory

In order to obtain the dynamic characteristics of a ramming mechanism, the multi-rigid-body and rigid-flexible coupling dynamic models were established in Recurdyn. Considering the structural characteristics of the link, the deformation motion of the link is described by using the Reissner-Mindlin plate theory, and the dynamic model of a bomb feeder is established considering the influence of non-ideal factors such as transmission clearance, friction and so on, the dynamic characteristics of a bullet-feeding machine are simulated and analyzed. Compared with the multi Rigid body dynamics model, the simulation results of the rigid flexible coupling model are more consistent with the experimental results, in addition to the polygon effect, other factors such as transmission gap and so on are also the causes of the fluctuation of the chain head speed of the ammunition conveyer. In addition, with the periodical movement of the Ammunition Conveyer, the wear between the pin holes of the connecting links will become more and more serious, and the closer you get to the end of the chain, the greater the trend

Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory

Nonlinear plate bending within Mindlin's strain gradient elasticity theory (SGT) is investigated by employing somewhat non-standard finite element methods. The main goal is to compare the bending results provided by the geometrically nonlinear three-dimensional (3D) theory and the geometrically nonlinear Reissner–Mindlin plate theory, i.e., the first-order shear deformation plate theory (FSDT), within the SGT. For the 3D theory, the nonlinear Green–Lagrange strain relations are adopted, while the von Kármán nonlinear strains are employed for the FSDT. The matrix-vector forms of the energy functionals are derived for both models. In order to perform the corresponding finite element discretizations, a quasi-C1-continuous 4-node tetrahedral solid element and a quasi-C1-continuous 6-node triangular plate element are employed for the 3D model and plate model, respectively. The first-order derivatives of the primal problem quantities are employed as additional nodal values to respond to the continuity requirements of class C1. A variety of computational results highlighting the differences between the 3D and FSDT models are given for two different plate geometries: a rectangular plate with a circular hole and an elliptical plate.

A multi-layer moving plate method for dynamic analysis of viscoelastically connected double-plate systems subjected to moving loads

This article aims to firstly introduce a computational approach, named multi-layer moving plate method (MMPM), to dynamic analysis of viscoelastically connected infinitely long double-plate systems subjected to moving loads. The Reissner-Mindlin plate theory is utilized to describe the displacement field through the thickness of each plate, whilst quadratic serendipity shape functions are employed to represent unknown fields in finite element analyses (FEAs). The governing equations of motion of connected double-plate system are established in a moving coordinate system attached to the moving load. As a consequence, the paradigm can absolutely eradicate the update process of force vector since the applied load is taken into account as “stationary” in its coordinate system. First, several numerical examples for static, free vibration and dynamic analyses are exhibited to verify the accuracy of the proposed MMPM. Then, the influences of various parameters such as load’s velocity, damping coefficient, stiffness coefficient, and plate thickness on the dynamic responses of double-plate system are examined in great detail.

Linear and Nonlinear Flutter of Supersonic Panels of Various Shapes

AbstractA fluid-structure model based on nonlinear Mindlin-Reissner plate theory and linearized piston theory is used to study the aeroelastic flutter of panels of arbitrary planforms at supersonic...

Isogeometric Collocation Method to solve the strong form equation of UI-RM Plate Theory

This work presents the formulation of the isogeometric collocation method to solve the strong form equation of a unified and integrated approach of Reissner Mindlin plate theory (UI-RM). In this plate theory model, the total displacement is expressed in terms of bending and shear displacements. Rotations, curvatures, and shear strains are represented as the first, the second, and the third derivatives of the bending displacement, respectively. The proposed formulation is free from shear locking in the Kirchhoff limit and is equally applicable to thin and thick plates. The displacement field is approximated using the B-splines functions, and the strong form equation of the fourth-order is solved using the collocation approach. The convergence properties and accuracy are demonstrated with square plate problems of thin and thick plates with different boundary conditions. Two approaches are used for convergence tests, e.g., increasing the polynomial degree (NELT = 1✕1 with p = 4, 5, 6, 7) and increasing the number of element (NELT = 1✕1, 2✕2, 3✕3, 4✕4 with p = 4) with the number of control variable (NCV) is used as a comparable equivalent variable. Compared with DKMQ element of a 64✕64 mesh as the reference for all L/h, the problem analysis with isogeometric collocation on UI-RM plate theory exhibits satisfying results.

Determination of Bending Failure Load of Hat-Type Folded Composite Plate Using Finite Element Method

Structural failure is initiated when a material is stressed beyond its strength limit. Determination of failure loads and failure location is one of the most important problems in design structures. This paper analyzed the failure of the hat-type laminated composite plate under a bending load. Based on the Reissner-Mindlin plate theory and isoparametric rectangular plate elements with five degrees of freedom per node, an algorithm and Matlab code were established to compute the stresses and bending failure load of the folded composite plate according to Tsai-Wu and Maximum stresses criteria. The numerical results are reliable when compared with the published results in the literature for some specific cases.

Development of High-Order Infinite Element Method for Bending Analysis of Mindlin–Reissner Plates

An approach is presented for solving plate bending problems using a high-order infinite element method (IEM) based on Mindlin–Reissner plate theory. In the proposed approach, the computational domain is partitioned into multiple layers of geometrically similar virtual elements which use only the data of the boundary nodes. Based on the similarity, a reduction process is developed to eliminate virtual elements and overcome the problem that the conventional reduction process cannot be directly applied. Several examples of plate bending problems with complicated geometries are reported to illustrate the applicability of the proposed approach and the results are compared with those obtained using ABAQUS software. Finally, the bending behavior of a rectangular plate with a central crack is analyzed to demonstrate that the stress intensity factor (SIF) obtained using the high-order PIEM converges faster and closer than low-order PIEM to the analytical solution.

Structural Modeling of Compliance-Based Camber Morphing Structures Under Transverse Shear Loading

A parametrically driven structural model based on Mindlin–Reissner plate theory is developed to capture the three-dimensional deflections of a compliance-based morphing trailing edge device with severe structural discontinuities. The model is used to study the Fish Bone Active Camber (FishBAC) device, which is represented as a discontinuous plate structure that captures the step changes in stiffness created by the concept’s geometrical configuration. Courant’s penalty method is implemented in the form of artificial penalty springs to account for stiffness discontinuities. A numerical validation is performed using finite element analysis, followed by experimental validation under actuation loads. This analytical model represents a robust, efficient, mesh-independent, and parameter-driven solution to modeling discontinuous plate structures. These traits make it useful for ongoing fluid–structure interaction analysis and optimization of the FishBAC concept and also for application to other complex composite structures.

Peridynamic modeling of composite laminates with material coupling and transverse shear deformation

This study presents a new state-based PeriDynamic (PD) model of a composite laminate with arbitrary laminate layup; it captures all types of material couplings in the presence of transverse shear deformation. It consists of normal and shear bonds instead of fiber and matrix bonds to allow arbitrary fiber orientations. Also, rotational degrees of freedom are included in the equilibrium equations in order to evaluate the transverse shear angle and curvature. Mindlin-Reissner plate theory, employed in this model, permits the use of existing composite constitutive law which leads to the coupling between different deformation modes. In addition, a quasi-local boundary condition applied only to the first row of material points avoids the approximation of nonlocal boundary values. The accuracy of this model is verified against benchmark solutions, and validated by comparison with experimental results.

An efficient three-node triangular Mindlin–Reissner flat shell element

Shell elements are extensively used by engineers for modeling the behavior of shell structures. Among common shell elements, triangular shell elements are not influenced by element warping. This paper proposes a new three-node triangular flat shell element with six degrees of freedom per each node, named TMRFS. The element is formed by assemblage of new bending and membrane elements. The bending element is formulated based on the hybrid displacement function element method and Mindlin–Reissner plate theory. In this element, an assumed displacement function is employed as the trial function. The membrane component is an unsymmetric triangular membrane element with drilling vertex rotations. The membrane element employs two different types of displacement fields as the test and trial functions. The test function is a displacement field which is the same as one used in well-known Allman triangular element. Meanwhile, instead of displacement field, the analytical stress field is considered as the trial function. Numerical tests show that the accuracy of the proposed flat shell element is reasonable in comparison with some popular triangular elements and its performance is insensitive to geometry, load and boundary conditions. Moreover, the proposed element preserves the advantages of its formulation including free of membrane locking, shear locking and stiffness matrix singularity problems.

On the improvement of analytical delamination model for drilling of laminated composites using Galerkin method

Despite the fact that fiber reinforced polymer composite laminates have high strength to density ratio, they have machinability limitations. Drilling operation is one of the mostly used machining processes on the laminated composites as it plays a significant role on assembly of composite parts. The most critical damage, affecting drilled composites, is delamination. Using Galerkin method, this paper, proposes a new model, removing most of the assumptions of previous models. The result of this study can be used to precisely and more reliably predict the critical thrust force-feed rate at which delamination could occur. The model uses the results of cutting theory to predict the distribution of the drilling load on delaminated area, a modified variety of Mindlin–Reissner plate theory in conjunction with the linear elastic fracture mechanic theory in mixed mode loading condition for the prediction of critical thrust force. Comparing with the previous proposed models, it has no assumptions for the load distribution and it considers mode I and mode II simultaneously for the growth of the delaminated zone, which makes it possible to consider non-uniform loading conditions of the drilling process. Using this model, the effect of the distribution of the drill force on the composite is investigated. The critical thrust force was studied for load-controlled and displacement-controlled loading conditions. For the experimental validation, the onset of the growth of the crack was detected using acoustic emission. The results of the experiment were compared with that of model.