Reissner Mindlin Plate Model Research Articles

Failure analysis of composite laminates under large deflection

This paper presents the results of the application of a procedure, developed by Reddy & Reddy [ Composites Science and Technology 1992, 44, 227–255], for the evaluation of the first ply-failure load in multilayered composite plates. The procedure, which is based on the use of the finite element method (FEM) and which is suitable for the analysis of generally loaded plates, uses the non-linear von Karman formulation and, therefore, allows comparison of the failure loads in both the linear and the geometrically non-linear behaviour. Nevertheless, the use of the Newton-Raphson technique in searching the non-linear equilibrium points restricts its application to the case of plates without limit-point behaviour. The displacement model adopted in the FEM formulation is the traditional firstorder Reissner-Mindlin plate model that takes the shear deformation effect into consideration. Concerning the failure criteria, the analysis is based on a tensor polynomial criterion to which all other polynomial and independent criteria are brought back as particular cases. The study refers to the failure analysis of thin and thick plates under a uniformly distributed transverse load. Furthermore, a comparison of the failure criteria when the shear stresses are evaluated by means of the constitutive equations and by means of the local equilibrium equations is carried out. Finally, adopting a very simple degradation model of the mechanical properties to account for the stiffness decrease consequent to the failure, the qualitative behaviour of plates after the first non-catastrophic failure is also presented.

Analysis of a Linear–Linear Finite Element for the Reissner–Mindlin Plate Model

An analysis is presented for a recently proposed finite element method for the Reissner–Mindlin plate problem. The method is based on the standard variational principle, uses nonconforming linear elements to approximate the rotations and conforming linear elements to approximate the transverse displacements, and avoids the usual "locking problem" by interpolating the shear stress into a rotated space of lowest order Raviart-Thomas elements. When the plate thickness t = O(h), it is proved that the method gives optimal order error estimates uniform in t. However, the analysis suggests and numerical calculations confirm that the method can produce poor approximations for moderate sized values of the plate thickness. Indeed, for t fixed, the method does not converge as the mesh size h tends to zero.

Interior estimates for a low order finite element method for the Reissner–Mindlin plate model

Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner–Mindlin plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the global approximability of the finite element method. As an application, we show that for the soft simply supported plate, the Arnold–Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal order convergence cannot hold globally due to the boundary layer. Numerical results are given which support our conclusion.

Preconditioning discrete approximations of the Reissner-Mindlin plate model

Nous considerons des methodes iteratives pour resoudre le systeme d'equations lineaires resultant de la discretisation par elements finis mixtes du modele de plaque de Reissner-Mindlin. Nous montrons comment construire une matrice de preconditionnement, symetrique, positive et diagonale par blocs, de sorte que le conditionnement du systeme lineaire associe soit independant de la taille du maillage h et aussi de l'epaisseur de la plaque t. Nous expliquons ensuite comment implementer ce preconditionneur et appliquons cette technique pour resoudre efficacement le systeme lineaire indefini associe au modele de plaques. Bien que la formulation mixte du probleme de Reissner-Mindlin ait une structure de point selle commune a ce type de problemes variationnels mixtes, la presence du petit parametre t et le fait que la matrice soit seulement positive semi-definie dans le coin superieur gauche de la partition, creent de nouvelles complications.

On Variational Approaches to Plate Modes

The three basic functionals of potential energy, complementary energy and Hellinger–Prange–Reissner are usedto obtain a rational derivation of Reissner–Mindlinplate models, starting from the three-dimensional theory.We show that the models so obtained are instances of the same plate theory; nevertheless, due to the different constitutive relations governing their response, they mimic the three-dimensional behaviour in three different manners.

Design principles and error analysis for reduced-shear plate-bending finite elements

In this paper, we consider the problem of designing plate-bending elements which are free of shear locking. This phenomenon is known to afflict several elements for the Reissner-Mindlin plate model when the thickness \(d\) of the plate is small, due to the inability of the approximating subspaces to satisfy the Kirchhoff constraint. To avoid locking, a “reduction operator” is often applied to the stress, to modify the variational formulation and reduce the effect of this constraint. We investigate the conditions required on such reduction operators to ensure that the approximability and consistency errors are of the right order. A set of sufficient conditions is presented, under which optimal errors can be obtained – these are derived directly, without transforming the problem via a Hemholtz decomposition, or considering it as a mixed method. Our analysis explicitly takes into account boundary layers and their resolution, and we prove, via an asymptotic analysis, that convergence of the finite element approximations will occur uniformly as \(d\rightarrow 0\), even on quasiuniform meshes. The analysis is carried out in the case of a free boundary, where the boundary layer is known to be strong. We also propose and analyze a simple post-processing scheme for the shear stress. Our general theory is used to analyze the well-known MITC elements for the Reissner-Mindlin plate. As we show, the theory makes it possible to analyze both straight and curved elements. We also analyze some other elements.

ON THE CONVERGENCE OF A TRIANGULAR MIXED FINITE ELEMENT METHOD FOR REISSNER–MINDLIN PLATES

We analyze the convergence of a mixed finite element method introduced by Zienkiewicz, Taylor, Papadopoulos and Oñate for the Reissner–Mindlin plate model. In order to do this, we compare it with a method which is known to be convergent with optimal order uniformly in the plate thickness. We show that the difference between the solutions of both methods is of higher order than the error. In particular the method does not present locking and is optimal order convergent. We also present several numerical experiments which confirm the similar behavior of both methods.

Asymptotic Analysis of the Boundary Layer for the Reissner–Mindlin Plate Model

We investigate the structure of the solution of the Reissner–Mindlin plate equations in its dependence on the plate thickness in the cases of soft and hard clamped, soft and hard simply supported, and traction free boundary conditions. For the transverse displacement, rotation, and shear stress, we develop asymptotic expansions in powers of the plate thickness. These expansions are uniform up to the boundary for the transverse displacement, but for the other variables there is a boundary layer, which is stronger for the soft simply supported and traction-free plate and weaker for the soft clamped plate than for the hard clamped and hard simply supported plate. We give rigorous error bounds for the errors in the expansions in Sobolev norms. As an application, we derive new regularity results for the solutions and new estimates for the difference between the Reissner–Mindlin solution and the solution to the corresponding biharmonic model.

Energy-release-rate evaluation for delamination growth prediction in a multi-plate model of a laminate composite

In this paper, numerical methods for the evaluation of the energy-release-rate along a delamination periphery under conditions of local buckling of the delaminate, as well as global buckling of the entire laminate, are presented. A multi-plate model, using independent Reissner-Mindlin plate models for each of the delaminated and undelaminated plies, with Reissner-Mindlin constraints for relating the degrees of freedom of the delaminated plates to those of the undelaminated plate at the crack front, is used to model the laminate with embedded delaminations. Explicit expressions, in terms of finite element nodal or Gauss-point variables, are derived for the pointwise energy release rate in terms of the J-integral and the Equivalent Domain Integral in the context of a typical multi-plate model for characterising the delamination growth. A finite element method with a 3-noded quasi-conforming shell element, and an automated post-buckling solution capability, is used for conducting the numerical analyses in this paper. Using these numerical results, mechanisms of multiple buckling modes and their effect on the propagation of embedded delaminations in plates, are studied.

A stable bilinear element for the Reissner-Mindlin plate model

In this paper, we introduce a new quadrilateral element using isoparametric bilinear basis functions for both components of the rotation vector and the deflection. The element is a stable modification of the MITC4 element. We report on calculations with this new element, the original MITC4 and also the bilinear element, with selective reduced integration. The numerical results are in accordance with the results of the numerical analysis and they show that (i) the method with reduced integration is highly unreliable and cannot be recommended; (ii) the MITC4 performs rather well, but its instability can lead to a decrease in accuracy for the deflection and especially to an inaccurate and oscillating shear force; (iii) the drawbacks of the MITC4 are not present in the modified method.

Uniform Interior Error Estimates for the Reissner-Mindlin Plate Model

Interior error estimates are derived for the solution of the Reissner-Mindlin plate model discretized by mixed-interpolated elements. Precisely, it is shown that the error in an interior domain can be estimated by the sum of two terms: the first has the best order of accuracy that is possible locally for the finite element spaces used, the second is a weak norm of the error on a slightly larger domain (this term measures the effects from outside of this domain). The analysis is based on some abstract properties enjoyed by the finite element spaces considered.

Uniform interior error estimates for the Reissner-Mindlin plate model

Interior error estimates are derived for the solution of the Reissner-Mindlin plate model discretized by mixed-interpolated elements. Precisely, it is shown that the error in an interior domain can be estimated by the sum of two terms: the first has the best order of accuracy that is possible locally for the finite element spaces used, the second is a weak norm of the error on a slightly larger domain (this term measures the effects from outside of this domain). The analysis is based on some abstract properties enjoyed by the finite element spaces considered.

On mixed finite element methods for the Reissner-Mindlin plate model

In this paper we analyze the convergence of mixed finite element approximations to the solution of the Reissner-Mindlin plate problem. We show that several known elements fall into our analysis, thus providing a unified approach. We also introduce a low-order triangular element which is optimal-order convergent uniformly in the plate thickness.

Accuracy of some plate models for clamped‐in boundary conditions

AbstractThe paper shows that the well‐known Reissner–Mindlin plate model for the clamped‐in boundary condition does not capture the boundary layer behaviour for the bending moments. This boundary layer is present in 3‐D formulation. In contrast the proposed (1,1,2) model shows this boundary layer. The strength of the boundary layer for the (1,1,2) model is analysed.

A Uniformly Accurate Finite Element Method for the Reissner–Mindlin Plate

A simple finite element method for the Reissner–Mindlin plate model in the prim-itive variables is presented and analyzed. The method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging. It is proved that the method converges with optimal order uniformly with respect to thickness.