Two simple finite element methods for Reissner–Mindlin plates with clamped boundary condition

A stable extended/generalized finite element method with Lagrange multipliers and explicit damage update for distributed cracking in cohesive materials

A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems

Mortar methods with dual Lagrange multiplier bases provide a flexible, efficient and optimal way to couple different discretization schemes or nonmatching triangulations. Here, we generalize the concept of dual Lagrange multiplier bases by relaxing the condition that the trace space of the approximation space at the slave side with zero boundary condition on the interface and the Lagrange multiplier space have the same dimension. We provide a new theoretical framework within this relaxed setting, which opens a new and simpler way to construct dual Lagrange multiplier bases for higher order finite element spaces. As examples, we consider quadratic and cubic tetrahedral elements and quadratic serendipity hexahedral elements. Numerical results illustrate the performance of our approach.

Domain decomposition techniques provide a powerful tool for the numerical approxima- tion of partial differential equations. We focus on mortar finite element methods on non-matching triangulations. In particular, we discuss and analyze dual Lagrange multiplier spaces for lowest order finite elements. These non standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a consequence, stan- dard efficient iterative solvers as multigrid methods or domain decomposition techniques can be easily adapted to the nonconforming situation. Here, we introduce new dual Lagrange multiplier spaces. We concentrate on the construction of locally supported and continuous dual basis functions. The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach.

In this paper, an analysis of free vibration in functionally graded nanoplate is presented. Third-order shear deformation plate theory is used to reach more accuracy in results. Small-scale effects are investigated using Eringen`s nonlocal theory. The governing equations of motion are obtained by Hamilton`s principle. It is assumed that the properties of nanoplates vary through their thicknesses according to a volume fraction power law distribution. The finite element method (FEM) is presented to model the functionally graded nanoplate and solve mathematical equations accurately. The finite element formulation for HSDT nanoplate is also presented. Natural frequencies of FG nanoplate with various boundary conditions are compared with available results in the literature. At the end some numerical results are presented to evaluate the influence of different parameters, such as power law index, nonlocal parameter, aspect ratio and aspect of length to thickness of nanoplate. In addition, all combinations of simply supported and clamped boundary conditions are considered.

The objective of this investigation is to predict the natural frequency of a laminated carbon fiber reinforced thermoplastics (CF/PEEK) plate with a hole by using the finite element method. In the finite element formulation, a parabolic isoparametric plate bending element is used and the first-order shear deformation theory based on the Mindlin type is considered. The good agreement between the numerical and the experimental results of CF/PEEK unidirectional and cross-ply laminated plates with a hole shows that the natural frequencies and modes can be predicted by the present method. Futhermore, the changes of frequencies and modes of the antisymmetrically angle-ply and the cross-ply laminated plates with the size of the hole, the stacking sequence of lamina and the plate length-to-width ratio are examined by F. E. M. under simply supported and clamped boundary conditions.

Investigation of buckling behavior of laminated reinforced concrete plates with central rectangular hole using finite element method

This work performs the convergence analysis of the polytopal nodal discretisation of contact-mechanics (with Tresca friction) recently introduced in Droniou et al. (Comput. Methods Appl. Mech. Eng. 422 (2024) 116838) in the framework of poro-elastic models in fractured porous media. The scheme is based on a mixed formulation, using face-wise constant approximations of the Lagrange multipliers along the fracture network and a fully discrete first order nodal approximation of the displacement field. The displacement field is enriched with additional bubble degrees of freedom along the fractures to ensure the inf–sup stability with the Lagrange multiplier space. It is presented in a fully discrete formulation, which makes its study more straightforward, but also has a Virtual Element interpretation. The analysis establishes an abstract error estimate accounting for the fully discrete framework and the non-conformity of the discretisation. A first order error estimate is deduced for sufficiently smooth solutions both for the gradient of the displacement field and the Lagrange multiplier. A key difficulty of the numerical analysis is the proof of a discrete inf-sup condition, which is based on a non-standard H−1/2-norm (to deal with fracture networks) and involves the jump of the displacements, not their traces. The analysis also requires the proof of a discrete Korn inequality for the discrete displacement field which takes into account fracture networks. Numerical experiments based on analytical solutions confirm our theoretical findings.

Tire's are made up of a typical fiber–rubber composite structure. A finite element model of a tire with foundation contact is developed in this study. In this model, the rubber compounds are simulated by incompressible elements, which are treated using the Lagrangian multipliers method. The nonlinear mechanical properties of the elastomers are modeled by the Mooney–Rivlin model. The belts, carcass, and bead are modeled by an equivalent orthotropic material model in which the effective moduli are determined from the individual material properties of the rubber compound and cord based on the Halpin–Tsai equations. For composite elements consisting of multi-ply cord–rubber composites, three-dimensional effective elastic constants can be determined using the material model presented by Sun and Li. The contact constraint of the tire with a flat foundation and rigid rim is treated using the variable constraint method. For the large deformation description of the tire, the Lagrangian method is used here. The strain tensor and the stress tensor are selected as, respectively, the Green–Lagrangian strain tensor and the second Piola– Kirchhoff stress tensor. In finite element code, two kinds of three-dimensional elements are used – an eight-node brick isoparametric element (hexahedral element) and a six-node isoparametric element (pentahedral element). The present numerical results show that the reliability and convergence of the model are fairly good. Moreover, three groups of radial tires with different belt widths under inflation and static footprint loading are analyzed using the finite element method. Based on the detailed analysis for stress analysis parameters in the critical regions in the tires, the relative belt edge endurance is predicted.

A three-dimensional free vibration analysis of cross-ply laminated rectangular plates with clamped edges

A cubic Hermite finite element-continuation method for numerical solutions of the von Kármán equations

Nanoindentation experiments and simulations are carried out to estimate the elastic modulus of freely-suspended-multilayer graphene. However, due to the difficulty of clamping all layers of multilayer graphene in experiments, and to the ambiguity of imposing the clamped boundary conditions in numerical simulations, the estimated values of elastic modulus exhibit large variation. In particular, interlayer sliding can affect the estimation of elastic modulus. From a series of molecular dynamics simulations, we demonstrate that the estimated elastic modulus of multilayer graphene depends on the level of interlayer sliding involved in boundary conditions. Under fully clamped boundary conditions that prevent interlayer sliding, the elastic modulus is constant regardless of the number of layers. In contrast, under weakly clamped boundary conditions that involve interlayer sliding, the elastic modulus decreases with increasing number of layers. In the case of weakly clamped conditions, a few wrinkles form in the interlayer and then coalesce into a single large wrinkle due to interlayer sliding. Our findings provide an understanding of the variation of elastic modulus observed in other experimental and numerical studies.

In this paper, the large amplitude vibrations of clamped-clamped thin viscoelastic rectangular plates due to a concentrated transversal harmonic load are investigated both theoretically and experimentally. Clamped boundary condition on all edges and von Kármán nonlinear strain-displacement relationships are considered while rotary inertia, geometric imperfections, and shear deformation are neglected. In the theoretical study, the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model. In-plane loads applied during the assembly of the plate are taken into account and clamped boundary conditions are modelled using artificial rotational springs. The nonlinear ordinary differential equations for the considered Kelvin-Voigt model are obtained using the generalized energy approach. These equations contain quadratic and cubic nonlinear viscoelasticity terms in addition to quadratic and cubic stiffness terms. Non-dimensionalization of variables is carried out and each second order equation is converted into two first order equations. The resulting system of equations is solved using AUTO (software based on the arclength continuation method that allows bifurcation analysis), to get the frequency-response curves at various force levels. Moreover numerical time integration of equations was also performed using the fourth-order Runge-Kutta method to understand the time response of the structure. In the experimental study, two rubber plates with different material and thicknesses were considered; a silicone plate with 0.0015 m thickness and a neoprene plate with 0.003 m thickness. The plates were fixed on a heavy rectangular metal frame thereby ensuring the clamped boundary condition on all edges. Linear experimental modal analysis was carried out as a first step to estimate the mode shapes and natural frequencies. In the second step, the nonlinear vibration response of the plate around its first resonance was measured at various harmonic force levels. At each force level, the amplitude of the harmonic excitation was kept constant by LMS Data Acquisition System and Test.Lab Stepped Sine software module while slowly varying the frequency of excitation to get the frequency-response curves. Laser Doppler Vibrometry was used to measure the response from the plate as it eliminates the possible mass loading effect introduced by any contact type sensors. A maximum amplitude of more than three times the thickness of the plate was achieved. The nonlinear response curves showed a typical hardening type nonlinearity along with sudden jumps as expected for plates. Experimental frequency-response curves were compared with theoretical results and a good agreement was found. The influence of nonlinear viscoelastic damping terms was clearly noticed on the response curves of the plate. The retardation time, measured in seconds decreases with increasing excitation force and larger amplitude vibrations.

The present study deals with both experimental and numerical investigation on the free vibration behavior of laminated composite plates subjected to varying temperature and moisture. A simple laminated plate model is developed for the vibration of composite plates subjected to hygrothermal loading. A computer program based on finite element method in the MATLAB environment is developed to perform all necessary computations. The woven fiber glass/epoxy specimens were hygrothermally conditioned in a humidity cabinet where the conditions were maintained at temperatures of 300–425 K and relative humidity ranging from 0% to 1.0% for moisture concentrations. Quantitative results are presented to show the effects of geometry, material and lamination parameters of woven fiber laminate on the free vibration of composite plates for different temperature and moisture concentrations. The numerical and experimental results show that there is reduction in natural frequency in Hz with the increase in temperature and moisture concentration for laminates with simply supported and clamped boundary conditions with different aspect ratio and side-to-thickness ratios.

Residual stress variations and plastic zone growth behavior of steel fiber-reinforced aluminum metal–matrix laminated composite plates which are clamped and subjected to outof-plane loading are investigated for different hardening parameter (K) and strain hardening exponent (n). Laminated plates are composed of four orthotropic layers oriented with different angles in symmetric or antisymmetric manner. The plates are meshed into 64 elements and 289 nodes with clamped boundary condition. Laminates are subjected to a transverse uniform distributed load. The loading is gradually increased after yield point of the plate to the higher levels with 400, 500, and 600 loading steps. Numerical solution is performed using the finite element method and the first-order shear deformation theory. For solution, Lagrange elements with nine nodes are used.