Reissner Mindlin Research Articles

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.

Estimating the area of extreme inclusions in Reissner–Mindlin plates

We derive upper and lower estimates of the area of unknown defects in the form of either cavities or rigid inclusions in Mindlin–Reissner elastic plates in terms of the difference of the works exerted by boundary loads on the defected and on the reference plate. It turns out that the upper estimates depend linearly on , whereas the lower ones depend quadratically on . These results continue a line of research concerning size estimates of extreme inclusions in electric conductors, elastic bodies and plates.

Buckling of fiber metal laminated plates – numerical and experimental studies

PurposeThe purpose of this study is to study the buckling behavior of new aircraft material, i.e. glass fiber metal laminated (GFML) plates.Design/methodology/approachThe first-order Reissner–Mindlin theory is used in the present finite element formulation to determine the buckling loads of GFML plates. A program is developed in MATLAB for analyzing the effect of different parameters on buckling loads GFML plates. A set of experiments was performed to determine critical buckling loads of GFML plates using universal testing machine INSTRON 8862 and compared with predictions using the numerical model.FindingsThe effects of various parameters such as aspect ratio, side to thickness ratio, ply orientation and boundary conditions on buckling loads of GFMLs are examined. With the increase of aspect ratio, the reduction in buckling load is observed, while the increase inside to thickness ratio decreases the buckling load of GFML plates. There is a slight variation in buckling load with the increase of ply orientation. The buckling load is significantly influenced by boundary conditions because of restraint at the edges.Practical implicationsThese types of materials are used in lightweight structures such as aircraft, aerospace and military vehicles. The results reported in the present study can be used as design guidelines while designing fiber metal laminated (FML) plated structures.Originality/valueFor the first time, the authors have studied the buckling behavior of bidirectional woven FML plates using both numerical and experimental techniques.

A high-performance four-node flat shell element with drilling degrees of freedom

This paper presents a new four-node quadrilateral flat shell element, named QFSUQ, for analysis of shell structures. The element is formed by assemblage of a new membrane element and a plate-bending element. The membrane component is an unsymmetric quadrilateral element with drilling degrees of freedom. The trial functions of the membrane element are determined using the element stress fields formulated based on the analytical solutions of the Airy stress function in global Cartesian coordinate system. The corresponding test functions are obtained through the four-node isoparametric-based displacement fields which are enhanced by drilling rotations. The bending component is based on the Hellinger–Reissner variational principle for analysis of Reissner–Mindlin plates. To validate the performance of the proposed element several numerical benchmark problems are employed and the obtained results are compared with other shell elements. The results prove that the QFSUQ element preserves the advantages of the parent element formulation namely explicit stiffness matrix, free of membrane locking, shear locking and singularity problems and is also appropriate in analysis of shell structures with complex geometry, loading and boundary conditions.

Isogeometric analysis of thin Reissner–Mindlin shells: locking phenomena and B-bar method

We propose a local type of B-bar formulation, addressing locking in degenerated Reissner–Mindlin shell formulation in the context of isogeometric analysis. Parasitic strain components are projected onto the physical space locally, i.e. at the element level, using a least-squares approach. The formulation allows the flexible utilization of basis functions of different orders as the projection bases. The introduced formulation is much cheaper computationally than the classical $$\bar{B}$$ method. We show the numerical consistency of the scheme through numerical examples, moreover they show that the proposed formulation alleviates locking and yields good accuracy even for slenderness ratios of $$10^5$$ , and has the ability to capture deformations of thin shells using relatively coarse meshes. In addition it can be opined that the proposed method is less sensitive to locking with irregular meshes.

A unified solution for the vibration analysis of two-directional functionally graded axisymmetric Mindlin–Reissner plates with variable thickness

In this paper, an effective unified numerical approach is utilized to analyze the free and forced vibration response of two-directional functionally graded (2D-FG) thick circular and annular plates with variable thickness. The material properties are assumed to be graded continuously both in thickness and radial directions. The governing equations are converted to a set of ordinary differential equations (ODEs) based on the first order shear deformation theory. Obtained canonical equations are solved numerically by the Complementary Functions Method (CFM) combined with the Laplace transform for the first time. Using an effective and suitable inverse numerical Laplace transform method, the results are transferred back to time space. The damping model of Kelvin is used in the damped forced vibration analysis. The novelty of this paper is to infuse this efficient and accurate method to the dynamic analysis of a wide range of solid circular or annular plates, with linear or non-linear thickness profiles, radially Functionally Graded (FG), FG in the thickness direction or 2D-FG, without any restrictions. The influence of material variation exponents and thickness profiles on the considered problems are investigated. Several examples are presented and results are verified with those obtained by finite element method and available published literature. Excellent agreement is observed.

Free Vibration with Mindlin Plate Finite Element Based on the Strain Approach

This paper presents the development of a quadrilateral plate bending finite element for the computation of natural frequencies of plates with arbitrary geometry. This element is based on the Reissner–Mindlin plate theory using assumed strains rather than displacements and contains only the three physical degrees of freedom at each of the four corner nodes. Tests of convergence are first established for square plates with both simply supported and clamped on their edges where it is shown that a good rate of convergence is obtained using few elements. Other tests are then applied to rectangular, circular, skew and stepped plates, in addition to a rectangular plate with a central cutout as well. The numerical results obtained show that the present element has successfully passed patch tests and its results of the natural frequencies and the associated modes of vibration are in excellent agreement when compared with analytical and other available numerical solutions.

Shape-free polygonal hybrid displacement-function element method for analyses of Mindlin–Reissner plates

A high-performance shape-free polygonal hybrid displacement-function finite-element method is proposed for analyses of Mindlin–Reissner plates. The analytical solutions of displacement functions are employed to construct element resultant fields, and the three-node Timoshenko’s beam formulae are adopted to simulate the boundary displacements. Then, the element stiffness matrix is obtained by the modified principle of minimum complementary energy. With a simple division, the integration of all the necessary matrices can be performed within polygonal element region. Five new polygonal plate elements containing a mid-side node on each element edge are developed, in which element HDF-PE is for general case, while the other four, HDF-PE-SS1, HDF-PE-Free, IHDF-PE-SS1, and IHDF-PE-Free, are for the edge effects at different boundary types. Furthermore, the shapes of these new elements are quite free, i.e., there is almost no limitation on the element shape and the number of element sides. Numerical examples show that the new elements are insensitive to mesh distortions, possess excellent and much better performance and flexibility in dealing with challenging problems with edge effects, complicated loading, and material distributions.

Seamless integration of design and Kirchhoff–Love shell analysis using analysis-suitable unstructured T-splines

Analysis-suitable T-splines (ASTS) including both extraordinary points and T-junctions are used to solve Kirchhoff–Love shell problems. Extraordinary points are required to represent surfaces with arbitrary topological genus. T-junctions enable local refinement of regions where increased resolution is needed. The benefits of using ASTS to define shell geometries are at least two-fold: (1) The manual and time-consuming task of building a new mesh from scratch using the CAD geometry as an input is avoided and (2) C1 or higher inter-element continuity enables the discretization of shell formulations in primal form defined by fourth-order partial differential equations. A complete and state-of-the-art description of the development of ASTS, including extraordinary points and T-junctions, is presented. In particular, we improve the construction of C1-continuous non-negative spline basis functions near extraordinary points to obtain optimal convergence rates with respect to the square root of the number of degrees of freedom when solving linear elliptic problems. The applicability of the proposed technology to shell analysis is exemplified by performing geometrically nonlinear Kirchhoff–Love shell simulations of a pinched hemisphere, an oil sump of a car, a pipe junction, and a B-pillar of a car with 15 holes. Building ASTS for these examples involves using T-junctions and extraordinary points with valences 3, 5, and 6, which often suffice for the design of free-form surfaces. Our analysis results are compared with data from the literature using either a seven-parameter shell formulation or Kirchhoff–Love shells. We have also imported both finite element meshes and ASTS meshes into the commercial software LS-DYNA, used Reissner–Mindlin shells, and compared the result with our Kirchhoff–Love shell results. Excellent agreement is found in all cases. The complexity of the shell geometries considered in this paper shows that ASTS are applicable to real-world industrial problems.

Wedge and crack solutions for Mindlin–Reissner plates

Displacement potentials are used to define the complete set of displacement and stress-resultant fields in moderately thick plate bending problems. The asymptotic singular terms are used to define and solve the eigenproblems associated with all alternative homogeneous wedge boundary conditions, and extended to model bimaterial wedges. This solution is used to model edge cracks with a filler and thus simulate the effect of crack repair. Definitions for the stress intensity factors are given, as well as the constitutive relations used to model a repaired crack with a contact element.

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.

Equilibrium and displacement elements for the design of plates and shells

This paper reconsiders the finite-element modelling of the linear elastic behaviour of plates and shells as governed by the Reissner–Mindlin first-order shear deformation theory. Particular attention is given to the problems associated with the locking of thin forms of structure when modelled with isoparametric conforming elements. As a means of ameliorating or removing these problems, three recent alternative types of elements are studied. Two are displacement elements which include different approaches to the definition of assumed strains, and the third is based on a hybrid equilibrium formulation of a flat shell element. The purpose of the paper is to compare and explain their performances and outputs in the context of two benchmark problems: a trapezoidal plate and the Scordelis−Lo cylindrical shell. Numerical examples are used to illustrate the convergence of stress-resultant contours as well as global quantities such as strain energy. The main conclusion is that while all three alternative types of element overcome locking with regard to displacements, the hybrid models are generally more efficient at providing good quality stress resultants. This is particularly so for those which contribute little to the total strain energy but yet may be significant in design.

Coupled atomistic‐continuum simulation of the mechanical properties of single‐layered graphene sheets

AbstractThe purpose of this work is the multiscale modeling of a single‐layered graphene sheet. The model is divided into three parts. One is an atomistic domain which is simulated with the atomic‐scale finite element method (AFEM). Another is a continuum domain. In this domain, the mechanical properties are investigated by using a finite element based on a nonlocal continuum shell model with a high order strain gradient. To be exact, it is a 4‐node 60‐generalized degree of freedom (DOF) Mindlin–Reissner finite shell element with a second order strain gradient. In the third part, a new transitional finite element is developed for smoothing the transition between the atomistic domain and continuum domain.

Modelling of thin viscoelastic shell structures under Reissner–Mindlin kinematic assumption

In this paper, we study thin viscoelastic shell structures using a constitutive equation in hereditary integral form. An alternative mathematical formulations for several viscoelastic shell structures under the Reissner–Mindlin kinematical assumptions are obtained. The resulting equations are written as a Volterra equation of the second kind to allow further mathematical analysis. A locking-free finite element formulation, with selective reduced integration is used to approximate the equation. To perform numerical experiments we consider several situations suffering from locking in both cases dynamic and quasi-static. We show the good behavior of the model compared with other models from the literature.

Smoothing Technique Based Beta FEM (βFEM) for Static and Free Vibration Analyses of Reissner–Mindlin Plates

Recently, the edge-based and node-based smoothed finite element method (ES-FEM and NS-FEM) has been proposed for Reissner–Mindlin plate problems. In this work, in order to utilize the numerical advantages of both ES-FEM and NS-FEM for static and vibration analysis, a hybrid smoothing technique based beta FEM ([Formula: see text]FEM) is presented for Reissner–Mindlin plate problems. A tunable parameter [Formula: see text] is introduced to tune the proportion of smoothing domains calculated by ES-FEM or NS-FEM, which controlled the accuracy of the results. Numerical illustrations in both static and free vibration analysis are conducted. The shear locking free property, converge property and dynamic stability are carefully examined via several well-known benchmark examples. Moreover, an experimental test is carefully designed and conducted for validations, in which the mode values and shape of a rectangular steel plate is tested. Numerical examples demonstrate the advantages of [Formula: see text]FEM, in comparison with the standard FEM, ES-FEM and NS-FEM using the same meshes. The numerical and experimental results are in good agreement with each other and the [Formula: see text]FEM achieves the best accuracy among all the methods for the static or free vibration analysis of plates.

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.

A novel dynamics model for railway ballastless track with medium-thick slabs

Motivated by the requirements for elaborated slab ballastless track dynamics analysis in practical engineering application, a novel dynamic model for the railway ballastless tracks with medium-thick slabs is proposed in this work based on the Reissner–Mindlin plate theory, and it is implemented into the coupled dynamics analysis of a vehicle and the ballastless track. First, an efficient and easily programmable computational algorithm is adopted to solve the transverse deflection of the Reissner–Mindlin plate, in which the displacements and shear strains are chosen as the independent variables and subsequently constructed by spline functions, resulting in no shear-locking effect. The involved partial differential equations are transformed into ordinary ones by using the energy variation principle. Further, a mathematical model for the ballastless track dynamics analysis is established, which can consider the effects of the shear deformation and moment of inertia involved in the medium-thick track slab. Experimental verification and comparative analysis with other models demonstrate the accuracy and efficiency of the proposed model. Finally, a spatially coupled dynamics model of a vehicle and the ballastless track is developed, and it is efficiently solved by using the hybrid explicit-implicit time integration method. Compared with the widely used modelling the track slab by elastic thin plate, the reliability and advantages of the proposed vehicle-slab track coupled dynamics model are demonstrated.

Finite rotation meshfree formulation for geometrically nonlinear analysis of flat, curved and folded shells

Geometrically nonlinear analysis of flat, curved and folded shells under finite rotations is performed by enhanced six degrees of freedom (6-DOFs) meshfree formulation. Curvilinear surfaces are dealt with the concept of convected coordinates. Equilibrium equations are derived by total Lagrangian formulation with Green–Lagrange strain and Second Piola–Kirchhoff stress. Both shell geometry and its deformation are approximated by Reproducing Kernels (RKs). Transverse shear strains are considered by Mindlin–Reissner theory. Numerical integration of the stiffness matrix is estimated by using the Stabilized Conforming Nodal Integration (SCNI) method. To show accuracy and effectiveness of the proposed formulation and discretization, benchmark problems from the literatures are considered. Apart from reference solutions available in the literature, additional reference results based on finite element method (FEM) conducted by the present authors are also presented.

Some advances in high-performance finite element methods

PurposeThe purpose of this paper is to give a review on the newest developments of high-performance finite element methods (FEMs), and exhibit the recent contributions achieved by the authors’ group, especially showing some breakthroughs against inherent difficulties existing in the traditional FEM for a long time.Design/methodology/approachThree kinds of new FEMs are emphasized and introduced, including the hybrid stress-function element method, the hybrid displacement-function element method for Mindlin–Reissner plate and the improved unsymmetric FEM. The distinguished feature of these three methods is that they all apply the fundamental analytical solutions of elasticity expressed in different coordinates as their trial functions.FindingsThe new FEMs show advantages from both analytical and numerical approaches. All the models exhibit outstanding capacity for resisting various severe mesh distortions, and even perform well when other models cannot work. Some difficulties in the history of FEM are also broken through, such as the limitations defined by MacNeal’s theorem and the edge-effect problems of Mindlin–Reissner plate.Originality/valueThese contributions possess high value for solving the difficulties in engineering computations, and promote the progress of FEM.

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.