Invariant 8-node hybrid-stress elements for thin and moderately thick plates

Eight-node hybrid-stress elements are developed for the analysis of plates ranging from arbitrarily thin to moderately thick. The displacement behaviour is characterized by a transverse displacement and independent cross-section rotations, which are interpolated using the 8-node Serendipity shape functions. All components of stress are included; alternative elements are developed which differe in the form of the inplane distribution of the stresses. Elements are sought for whic the stiffiness is invariant and of correct rank, and whic show on signs of deterioration in the thin-plate limit. A discussion of the prospects for developing a 4-node element with these characteristics is also presented. Example problems are used to compare the performance of the 8-node elements including convergence behaviour, intraelement stress distributions and optimal sampling locations, and range of applicability in terms of plate thickness ratio.     

Three-dimensional hybrid-stress isoparametric quadratic displacement elements

Two 20-node quadratic displacement three-dimensional isoparametric elements are developed based on the hybrid-stress model. The elements differ in the stress interpolation used. In one case, the stress polynomials are selected to correspond approximately to the strain polynomials obtained from the displacement field, and a 57β stress field results. In the other element, complete cubic polynomials are used which are forced to satisfy the equilibrium equations and stress compatibility equations, and a 69β stress field results. Both elements possess correct rank, but only the 69β element is invariant. Results obtained using these two elements, and the corresponding 20-node assumed-displacement element, are compared and the 69β element is shown to be the better element. The 2 × 2 × 2 Gauss stations are also verified to be the optimal sampling points for these elements.     

A Serendipity cubic-displacement hybrid-stress element for thin and moderately thick plates

A hybrid-stress element is developed for the analysis of thin and moderately thick plates. The independent transverse displacement and rotations are interpolated by the 12-node cubic Serendipity shape functions. All components of stress are included and 36β stress assumption is used. The element stiffness possesses correct rank and numerical results indicate that the element does not lock in the thin-plate limit. Results obtained using the present element are compared with those obtained using a 12-node assumed-displacement based Mindlin plate element with reduced integration; the present hybrid-stress element is shown to yield superior accuracy for all cases considered. In addition, the accuracy of the present element is compared against that of analogous 4-node and 8-node hybrid-stress Mindlin plate elements.     

Elastic-plastic analysis of plates by the hybrid-stress model and initial-stress approach

Two alternative hybrid-stress-based functionals are examined for the incremental elastic-plastic static analysis of single layer plates. Material nonlinear effects are incorporated via the initial-stress approach so that an equivalent nodal force vector is defined and the stiffness remains constant throughout the incremental loading. The alternative functionals differ in the incremental stress which is assumed to satisfy equilibrium; in the first, it is the actual stress increment, and in the second it is the elastic stress increment. Results are presented for two example problems, and comparisons of the alternative functionals and plausible iteration schemes are given. The effects of variation of pertinent solution parameters are also shown. A 4-node hybrid-stress plate element based on a Mindlin-type displacement field is used for most cases; however, limited results are also presented using an 8-node plate element, thus permitting comparisons of the relative efficiencies of the two elements.     

Improved hybrid-stress axisymmetric elements includng behaviour for nearly incompressible materials

A series of 4-node axisymmetric solid-of-revolution elements with quadrilateral cross-section are developed based on the hybrid-stress finite element model. The displacement interpolation is identical, and the elements differ in the stress field chosen; alternative schemes for selection of stress distribution are investigated. The performance of these elements is assessed and compared for the example problems of a thick cylinder and thick sphere under internal pressure. Of particular importance are convergence, intra-element stress distributions, and element performance as the incompressible state is approached.     

Hybrid stress reduced-Mindlin elements for thin multilayer plates

A hybrid-stress formulation of isoparametric elements for the analysis of thin multilayer laminated composite plates is presented. The element displacement behaviour is characterized by laminate reference surface inplane and transverse displacements and laminate non-normal cross-section rotations; as a result, simple C^o interpolation of displacement and rotation can be used, and the number of degrees-of-freedom is independent of the number of layers. All components of stress are included and are related to a set of laminate stress parameters, the number of which is independent of the number of layers. Attention is restricted here to thin laminates, for which it is shown that the contributions of transverse shear stress and transverse normal stress to the internal complementary energy can be neglected. As a result of this reduction, a modified stiffness-formation algorithm can be used which provides a significant improvement in computational efficiency. The formulation presented is used to develop an 8-node isoparametric thin multilayer plate element. The resulting element is naturally invariant, of correct rank, and non-locking in the thin plate limit.     

Modified and Trefftz unsymmetric finite element models

The unsymmetric finite element method employs compatible test functions but incompatible trial functions. The pertinent 8-node quadrilateral and 20-node hexahedron unsymmetric elements possess exceptional immunity to mesh distortion. It was noted later that they are not invariant and the proposed remedy is to formulate the element stiffness matrix in a local frame and then transform the matrix back to the global frame. In this paper, a more efficient approach will be proposed to secure the invariance. To our best knowledge, unsymmetric 4-node quadrilateral and 8-node hexahedron do not exist. They will be devised by using the Trefftz functions as the trial function. Numerical examples show that the two elements also possess exceptional immunity to mesh distortion with respect to other advanced elements of the same nodal configurations.     

Optimal stress sampling points of plane triangular elements for patch recovery of nodal stresses

The existence of optimal stress sampling points in finite elements was first observed by Barlow. Knowledge of optimal stress sampling points is important in stress‐recovery methods such as the superconvergent patch recovery (SPR). Recently, MacNeal observed that Barlow points and Gaussian quadrature points are the same for the linear and quadratic bar elements, and different for the cubic bar element. Prathap proposed the best‐fit approach to predict the optimal sampling points, and showed that the best‐fit points coincide with Gaussian quadrature points not only for the linear and quadratic bar elements but also for the cubic bar element. In this paper, the best‐fit approach for predicting the optimal sampling points is extended to the linear and quadratic plane triangular elements, and the effectiveness of Barlow points, Gaussian points and best‐fit points as candidates of sampling points for the patch recovery of nodal stresses with these triangular elements is investigated for typical problems. The numerical results suggest that Barlow points do not exist for all strain/stress components, and Gaussian quadrature points which are the same as or close to the best‐fit points are better candidates for patch recovery. Copyright © 2003 John Wiley & Sons, Ltd.     

Rational approach for assumed stress finite elements

A new method for the formulation of hybrid elements by the Hellinger-Reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element. The equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co-ordinates. The resulting element possesses all the ideal qualities, i.e. it is invariant, it is less sensitive to geometric distortion, it contains a minimum number of stress parameters and it provides accurate stress calculations. For the formulation of a 4-node plane stress element, a small perturbation method is used to determine the equilibrium constraint equations. The element has been proved to be always rank sufficient.     

Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

 Expressions for critical timesteps are provided for an explicit finite element method for plane elastodynamic problems in isotropic, linear elastic solids. Both 4-node and 8-node quadrilateral elements are considered. The method involves solving for the eigenvalues directly from the eigenvalue problem at the element level. The characteristic polynomial is of order 8 for 4-node elements and 16 for 8-node elements. Due to the complexity of these equations, direct solution of these polynomials had not been attempted previously. The commonly used critical time-step estimates in the literature were obtained by reducing the characteristic equation for 4-node elements to a second-order equation involving only the normal strain modes of deformation. Furthermore, the results appear to be valid only for lumped-mass 4-node elements. In this paper, the characteristic equations are solved directly for the eigenvalues using <ty>Mathematica<ty> and critical time-step estimates are provided for both lumped and consistent mass matrix formulations. For lumped-mass method, both full and reduced integration are considered. In each case, the natural modes of deformation are obtained and it is shown that when Poisson's ratio is below a certain transition value, either shear-mode or hourglass mode of deformation dominates depending on the formulation. And when Poisson's ratio is above the transition value, in all the cases, the uniform normal strain mode dominates. Consequently, depending on Poisson's ratio the critical time-step also assumes two different expressions. The approach used in this work also has a definite pedagogical merit as the same approach is used in obtaining time-step estimates for simpler problems such as rod and beam elements. Received: 8 January 2002 / Accepted: 12 July 2002 The support of NSF under grant number DMI-9820880 is gratefully acknowledged.     

A rational approach for choosing stress terms for hybrid finite element formulations

A new approach for choosing the stress terms for a hybrid stress element is based on the condition of vanishing of the virtual work along the element boundary due to the stress terms higher than constant and the additional incompatible displacement. Examples using 4-node plane stress elements have shown that when the incompatible displacements also satisfy the constant strain patch test the resulting elements will provide the most accurate solutions. Advantages of this approach for the formulation of an axisymmetric solid are also indicated.     

Finite element dispersion analysis for the three-dimensional second-order scalar wave equation

The dispersive properties of finite element semidiscretizations of the three-dimensional second-order scalar wave equation are examined for both plane and spherical waves. This analysis throws light on the performance and limitations of the finite element approximation over the entire spectrum of wavenumbers and provides guidance for optimal mesh discretization as well as mass representation. The 8-node trilinear element, 20-node serendipity element, 27-node triquadratic element and the linear and quadratic spherically symmetric elements are considered.     

Element by element a posteriori error estimation of the finite element analysis for three-dimensional elastic problems

An a posteriori error estimation method for finite element solutions for three-dimensional elastic problems is presented based on the theory developed by the authors for two-dimensional problems.¹ The error is estimated for the finite element solutions obtained using three-dimensional 8-node elements with a linear interpolation function in an arbitrary hexahedron. The method is successfully applied to three-dimensional elastic problems. In order to decrease computing time and memory use, the error is estimated element by element. The major difficulty in the element-wise error estimation technique is satisfying the self-equilibrium condition of applied forces, especially in three-dimensional problems. These forces are mainly due to traction discontinuity on the element boundaries. The difficulty is circumvented by employing an element-wise optimal procedure. It is also shown that a very accurate stress solution can be obtained by adding estimated error to the original finite element solutions.     

Plane strain finite element simulation of shear band formation during metal forming

A plane strain finite element formulation and solution procedure for shear band failure during the plane strain metal forming process are developed and presented. The large strain elastic-plastic formulation includes a 5-node 10-degree-of-freedom (d.o.f.) ‘crossed-triangle’ element, a 4-node 8-d.o.f. element with selective reduced integration, an 8-node 16-d.o.f. element and a 4-node 8-d.o.f. element with an embedded shear band. The formulation includes an elastic-plastic material model with a modified Gurson yield function and combined isotropic-kinematic hardening. The solution procedure is based on a Newton–Raphson incremental-iterative method with an orthogonal projection of zero or negative eigen-modes when required. Two different examples of plane strain tension test are studied with results compared with available numerical solutions to evaluate the present formulation and solution procedure of the four different elements. The results demonstrate that both types of the 4-node quadrilaterals are comparable to the 5-node crossed-triangle element as well as the 8-jiode element. To further validate and to demonstrate the predictive capability and practical applicability of the present development, two plane strain metal forming examples are investigated. The first application is a numerical simulation of a sheet-stretching test with results compared with experimental data for a commercially pure aluminium–magnesium 5182-O sheet. The load vs. extension history and the through-thickness strain are compared. The good agreement suggests that it is possible to numerically determine the parameters needed for the modified Gurson yield function. The second application is a numerical simulation of the formation of dead metal zones in the extrusion process. A plane strain extrusion of a short aluminium billet through straight-sided dies is presented and characteristic features of the formation of dead metal zone are observed.     

Four-node tetrahedral elements for gradient-elasticity analysis

Computational analyses of gradient-elasticity often require the trial solution to be C¹ yet constructing simple C¹ finite elements is not trivial. This article develops two 48-dof 4-node tetrahedral elements for 3D gradient-elasticity analyses by generalizing the discrete Kirchhoff method and a relaxed hybrid-stress method. Displacement and displacement-gradient are the only nodal dofs. Both methods start with the derivation of a C⁰ quadratic-complete displacement interpolation from which the strain is derived. In the first element, displacement-gradient at the mid-edge points are approximated and then interpolated together with those at the nodes whilst the strain-gradient is derived from the displacement-gradient interpolation. In the second element, the assumed constant double-stress modes are employed to enforce the continuity of the normal derivative of the displacement. The whole formulation can be viewed as if the strain-gradient matrix derived from the displacement interpolation matrix is refined by a constant matrix. Both elements are validated by the individual element patch test and other numerical benchmark tests. To the best knowledge of the authors, the proposed elements are probably the first nonmixed/penalty 3D elements which employ only displacement and displacement-gradient as the nodal dofs for gradient-elasticity analyses.     

A quadrilateral hybrid stress element for Mindlin plates based on incompatible displacements

An hybrid stress element formulation based on internal, incompatible displacements is used to develop efficient Mindlin plate elements. The 4-node quadrilateral Mindlin plate element is derived from a modified energy functional. Both displacements and stresses are defined in the natural co-ordinate interpolation system. The assumed stress field is obtained by tensor transformation and so chosen as to ensure that the element is co-ordinate invariant and stable. Shear locking is avoided through an appropriate identification of the internal, incompatible displacement field. The role played by incompatible displacements in the formulation of hybrid stress elements for thin and moderately thick plates is discussed. Numerical applications are presented to illustrate the accuracy and reliability of the suggested Mindlin plate element.     

New superconvergent points of the 8‐node serendipity plane element for patch recovery

The Gaussian quadrature points, which are generally observed to be the same as the Barlow points for lower order elements, have so far been used as the sampling points for the superconvergent patch recovery (SPR). Recent developments on the best‐fit method to calculate the optimal sampling points suggest that, for higher order elements, Barlow points need not be the optimal sampling points and also need not be the same as the Gaussian quadrature points. In this paper the best‐fit method is extended to predict the optimal points of the 8‐node serendipity rectangular element, and it is observed that best‐fit points do not exist. Next, a novel method is proposed, in which, the expressions for stress‐error based on the best‐fit are used in the least‐square fit of the patch recovery, and thereby the superconvergent points are obtained more directly. Application of this method to the 8‐node serendipity element reveals the existence of two sets of superconvergent points for patch recovery, one of which is the well‐known Gaussian points, ( ), and the other is the set of four points given by , the existence of which has not been known before. A detailed numerical study on the patch recovery of stresses for two demonstrative problems reveals that there indeed exist two sets of superconvergent points as predicted by the proposed method. The comparative performance of the two sets of points is tested for typical demonstrative problems and the results are discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

More on optimal stress points—reduced integration,element distortions and error estimation

The presented work addresses the relationships between optimal sampling points, reduced integration and geometric distortion with the objective of estimating errors in terms of those considerations. Isoparametric quadratic plane and solid elements are used as a vehicle for the study. Geometric distortion measures and evaluation conditions, based on convergence requirements, are first defined in terms of the polynomial orders of the geometry and applied strain. Using these, the concept of optimal stress sampling, already established for undistorted elements, is extended to distorted geometry and shown to be effective over a range of geometries and strains. Errors in the strain-displacement relationship and numerical integration of the strains are used to estimate the total response error and to rationalize the connection between optimal stress points and reduced integration. Enhanced convergence, by extension to the representation of linear strains in elements with quadratic geometry, is identified as the main advantage of reduced integration. The applicability of the proposed, and other, distortion parameters to vetting of element geometry and error prediction is discussed.     

On improved hybrid finite elements with rotational degrees of freedom

A set of three new hybrid elements with rotational degrees-of-freedom (d.o.f.'s) is introduced. The solid, 8-node, hexahedron element is developed for solving three-dimensional elasticity problems. This element has three translational and three rotational d.o.f.'s at each node and is based on a 42-parameter. three-dimensional stress field in the natural convected co-ordinate system. For two-dimensional, plane elasticity problems, an improved triangular hybrid element and a quadrilateral hybrid element are presented. These elements use two translational and one rotational d.o.f. at each node. Three different sets of five-parameter stress fields defined in a natural convected co-ordinate system for the entire element are used for the mixed triangular element. The mixed quadrilateral element is based on a nine-parameter complete linear stress field in natural space. The midside translational d.o.f.'s are expressed in terms of the corner nodal translations and rotations using appropriate transformations. The stiffness matrix is derived based on the Hellinger–Reissner variational principle. The elements pass the patch test and demonstrate an improved performance over the existing elements for prescribed test examples.     

A formulation for the 4-node quadrilateral element

A formulation for the plane 4-node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second-order Taylor series in the physical co-ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices. For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix. The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non-linear analysis. There, an application is given for the calculation of inelastic problems in physically non-linear elasticity. The element is efficient to implement and it is frame invariant. Locking effects and zero-energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.