Patch‐averaged assumed strain finite elements for stress analysis

A finite element model for linear‐elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain‐displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four‐node and linear hexahedral eight‐node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf‐sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

A reduced integration solid‐shell finite element based on the EAS and the ANS concept—Geometrically linear problems

In this paper a new reduced integration eight‐node solid‐shell finite element is presented. The enhanced assumed strain (EAS) concept based on the Hu–Washizu variational principle requires only one EAS degree‐of‐freedom to cure volumetric and Poisson thickness locking. One key point of the derivation is the Taylor expansion of the inverse Jacobian with respect to the element center, which closely approximates the element shape and allows us to implement the assumed natural strain (ANS) concept to eliminate the curvature thickness and the transverse shear locking. The second crucial point is a combined Taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking‐free hourglass stabilization without rank deficiency. Hence, the element requires only a single integration point in the shell plane and at least two integration points in thickness direction. The formulation fulfills both the membrane and the bending patch test exactly, which has, to the authors' knowledge, not yet been achieved for reduced integration eight‐node solid‐shell elements in the literature. Owing to the three‐dimensional modeling of the structure, fully three‐dimensional material models can be implemented without additional assumptions. Copyright © 2009 John Wiley & Sons, Ltd.     

A reduced integration solid‐shell finite element based on the EAS and the ANS concept—Large deformation problems

In this paper we address the extension of a recently proposed reduced integration eight‐node solid‐shell finite element to large deformations. The element requires only one integration point within the shell plane and at least two integration points over the thickness. The possibility to choose arbitrarily many Gauss points over the shell thickness enables a realistic and efficient modeling of the non‐linear material behavior. Only one enhanced degree‐of‐freedom is needed to avoid volumetric and Poisson thickness locking. One key point of the formulation is the Taylor expansion of the inverse Jacobian matrix with respect to the element center leading to a very accurate modeling of arbitrary element shapes. The transverse shear and curvature thickness locking are cured by means of the assumed natural strain concept. Further crucial points are the Taylor expansion of the compatible cartesian strain with respect to the center of the element as well as the Taylor expansion of the second Piola–Kirchhoff stress tensor with respect to the normal through the center of the element. Copyright © 2010 John Wiley & Sons, Ltd.     

A new reduced integration solid‐shell element based on EAS and ANS with hourglass stabilization

In this paper, a novel reduced integration eight‐node solid‐shell finite element formulation with hourglass stabilization is proposed. The enhanced assumed strain method is adopted to eliminate the well‐known volumetric and Poisson thickness locking phenomena with only one internal variable required. In order to alleviate the transverse shear and trapezoidal locking and correct rank deficiency simultaneously, the assumed natural strain method is implemented in conjunction with the Taylor expansion of the inverse Jacobian matrix. The projection of the hourglass strain‐displacement matrix and reconstruction of its transverse shear components are further employed to avoid excessive hourglass stiffness. The proposed solid‐shell element formulation successfully passes both the membrane and bending patch tests. Several typical examples are presented to demonstrate the excellent performance and extensive applicability of the proposed element. Copyright © 2015 John Wiley & Sons, Ltd.     

Curved quadratic triangular degenerated‐ and solid‐shell elements for geometric non‐linear analysis

Compared to the large number of curved quadrilateral degenerated‐ and solid‐shell elements, there are only a very few curved triangular degenerated‐ and solid‐shell elements. Based on the assumed natural strain sampling scheme previously developed for a quadratic degenerated‐shell element for linear analysis, this paper devises geometric non‐linear six‐node degenerated‐shell and twelve‐node solid‐shell elements. Both elements can be curved and are only equipped with the standard nodal d.o.f.s. Careful consideration has been exercised to circumvent various locking phenomena that plague degenerated‐ and solid‐shell elements. Numerical examples are presented to illustrate their efficacy. Copyright © 2003 John Wiley & Sons, Ltd.     

Locking‐free continuum displacement finite elements with nodal integration

An assumed‐strain finite element technique is presented for linear, elastic small‐deformation models. Weighted residual method (reminiscent of the strain–displacement functional) is used to weakly enforce the balance equation with the natural boundary condition and the kinematic equation (the strain–displacement relationship). A priori satisfaction of the kinematic weighted residual serves as a condition from which strain–displacement operators are derived via nodal integration. A variety of element shapes is treated: linear triangles, quadrilaterals, tetrahedra, hexahedra, and quadratic (six‐node) triangles and (27‐node) hexahedra. The degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials. A straightforward constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Furthermore, the numerical inf–sup test is applied in select problems and several variants of the proposed formulations (linear triangles, quadrilaterals, tetrahedra, hexahedra, and 27‐node hexahedra) pass the test. Examples are used to illustrate the performance with respect to sensitivity to shape distortion and the ability to resist volumetric locking. Copyright © 2008 John Wiley & Sons, Ltd.     

Assumed‐deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis

An assumed‐strain finite element technique for non‐linear finite deformation is presented. The weighted‐residual method enforces weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed‐deformation gradient. Assumed gradient operators are derived via nodal integration from the kinematic‐weighted residual. A variety of finite element shapes fits the derived framework: four‐node tetrahedra, eight‐, 27‐, and 64‐node hexahedra are presented here. Since the assumed‐deformation gradients are expressed entirely in terms of the nodal displacements, the degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials and no volumetric/deviatoric split is required. The consistent tangent operator is inexpensive and symmetric. Furthermore, the material update and the tangent moduli computation are carried out exactly as for classical displacement‐based models; the only deviation is the consistent use of the assumed‐deformation gradient in place of the displacement‐derived deformation gradient. Examples illustrate the performance with respect to the ability of the present technique to resist volumetric locking. A constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Copyright © 2008 John Wiley & Sons, Ltd.     

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

The degenerated approach for shell elements of Ahmad and co‐workers is revisited in this paper. To avoid transverse shear locking effects in four‐node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single‐element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear‐locking‐free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

Extension of the ‘solid‐shell’ concept for application to large elastic and large elastoplastic deformations

In the present contribution we extend a previously proposed so‐called solid–shell concept which incorporates only displacement degrees of freedom to the simulation of large elastic and large elastoplastic deformations of shells. Therefore, the modifications necessary for hyper‐elastic or elastoplastic material laws are discussed. These modifications concern the right Cauchy–Green tensor for large elastic deformations, respectively, the deformation gradient for elastoplasticity which then are consistent to the modified Green–Lagrange strains that are necessary for transverse shear and membrane locking free solid–shell element formulations. However, in addition to the locking mentioned above especially in the range of plasticity incompressibility locking becomes important. Thus, the second major aspect of this contribution is the discussion of several ways to avoid incompressibility locking also including the investigation of eigenmodes. Finally, a selective reduced integration scheme with reduced integration for the volumetric term is employed and described in detail, although it is limited to material laws which allow the decomposition into a volumetric and a deviatoric part. Some numerical examples show the range of application for the proposed elements. Copyright © 2000 John Wiley & Sons, Ltd.     

A new one‐point quadrature enhanced assumed strain (EAS) solid‐shell element with multiple integration points along thickness: Part I—geometrically linear applications

Accuracy and efficiency are the main features expected in finite element method. In the field of low‐order formulations, the treatment of locking phenomena is crucial to prevent poor results. For three‐dimensional analysis, the development of efficient and accurate eight‐node solid‐shell finite elements has been the principal goal of a number of recent published works. When modelling thin‐ and thick‐walled applications, the well‐known transverse shear and volumetric locking phenomena should be conveniently circumvented. In this work, the enhanced assumed strain method and a reduced in‐plane integration scheme are combined to produce a new eight‐node solid‐shell element, accommodating the use of any number of integration points along thickness direction. Furthermore, a physical stabilization procedure is employed in order to correct the element's rank deficiency. Several factors contribute to the high computational efficiency of the formulation, namely: (i) the use of only one internal variable per element for the enhanced part of the strain field; (ii) the reduced integration scheme; (iii) the prevention of using multiple elements' layers along thickness, which can be simply replaced by any number of integration points within a single element layer. Implementation guidelines and numerical results confirm the robustness and efficiency of the proposed approach when compared to conventional elements well‐established in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid stress ANS solid‐shell element and its generalization for smart structure modelling. Part I—solid‐shell element formulation

In the recent years, solid‐shell finite element models which possess no rotational degrees of freedom and applicable to thin plate/shell analyses have attracted considerable attention. Development of these elements are not straightforward. Shear, membrane, trapezoidal, thickness and dilatational lockings must been visioned. In this part of this paper, a novel eight‐node solid‐shell element is proposed. To resolve the shear and trapezoidal lockings, the assumed natural strain (ANS) method is resorted to. The hybrid‐stress formulation is employed to rectify the thickness and dilatational locking. The element is computationally more efficient than the conventional hybrid elements by adopting orthogonal‐assumed stress modes and enforcing admissible sparsity in the flexibility matrix. Popular benchmark tests are exercised to illustrate the efficacy of the elements. In Part II of the paper, the element will be generalized for smart structure modelling by including the piezoelectric effect. Copyright © 2000 John Wiley & Sons, Ltd.     

Collapse of thin shell structures—stress resultant plasticity modelling within a co‐rotated ANDES finite element formulation

Due to the very non‐linear behaviour of thin shells under collapse, numerical simulations are subject to challenges. Shell finite elements are attractive in these simulations. Rotational degrees of freedom do, however, complicate the solution. In the present study a co‐rotated formulation is employed. The deformation of the shell is decomposed in to a contribution from large rigid body rotation and a strain producing term. A triangular assumed strain shell finite element is used. Hence, a high performance elastic element is combined with the co‐rotated formulation. In the co‐rotated co‐ordinate system the plasticity is accounted for by a simplifyed Ilyushin stress resultant yield surface. The stress update is determined from the backward Euler difference, and a consistent geometrical and material tangent stiffness is derived. Comparison with other published analysis results show that the present formulation gives acceptable accuracy. Copyright © 1999 John Wiley & Sons, Ltd.     

A new one‐point quadrature enhanced assumed strain (EAS) solid‐shell element with multiple integration points along thickness—part II: nonlinear applications

In this work the recently proposed Reduced Enhanced Solid‐Shell (RESS) finite element, based on the enhanced assumed strain (EAS) method and a one‐point quadrature integration scheme, is extended in order to account for large deformation elastoplastic thin‐shell problems. One of the main features of this finite element consists in its minimal number of enhancing parameters (one), sufficient to circumvent the well‐known Poisson and volumetric locking phenomena, leading to a computationally efficient performance when compared to other 3D or solid‐shell enhanced strain elements. Furthermore, the employed numerical integration accounts for an arbitrary number of integration points through the thickness direction within a single layer of elements. The EAS formulation comprises an additive split of the Green–Lagrange material strain tensor, making the inclusion of nonlinear kinematics a straightforward task. A corotational coordinate system is used to integrate the constitutive law and to ensure incremental objectivity. A physical stabilization procedure is implemented in order to correct the element's rank deficiencies. A variety of shell‐type numerical benchmarks including plasticity, large deformations and contact are carried out, and good results are obtained when compared to well‐established formulations in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Development of geometrically exact new shell elements based on general curvilinear co‐ordinates

In the present study first‐order shear deformable shell finite elements based on general curvilinear co‐ordinates are proposed. For the development of the present shell elements, a partial mixed variational functional with independently assumed strains is provided in order to avoid the severe locking troubles known as transverse shear and membrane lockings. Bubble functions are included in the shape function of displacement to improve the performance of the developed element. The proposed assumed strain four‐ and nine‐node elements based on the general tensor shell theory provide an efficient linkage framework for shell surface modelling and finite element analysis. In the several benchmark problems, the present shell elements with exact geometric representations demonstrate their performance compared to previously reported results. Copyright © 2002 John Wiley & Sons, Ltd.     

A hybrid stress ANS solid‐shell element and its generalization for smart structure modelling. Part II—smart structure modelling

In Part I of the paper, a hybrid‐stress‐assumed natural strain eight‐node solid‐shell element immune to shear, membrane, trapezoidal, thickness and dilatational lockings has been developed. Moreover, the element computational cost is reduced by enforcing admissible sparsity in the flexibility matrix. In this part of the paper, the solid‐shell element is generalized to a piezoelectric solid‐shell element. Using the two solid‐shell elements, smart structures with segmented piezoelectric sensors and actuators can be conveniently modelled. A number of problems are studied and comparisons with other ad hoc element models for smart structure modelling are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

An assumed strain triangular curved solid shell element formulation for analysis of plates and shells undergoing finite rotations

A formulation for 36‐DOF assumed strain triangular solid shell element is developed for efficient analysis of plates and shells undergoing finite rotations. Higher order deformation modes described by the bubble function displacements are added to the assumed displacement field. The assumed strain field is carefully selected to alleviate locking effect. The resulting element shows little effect of membrane locking as well as shear locking, hence, it allows modelling of curved shell structures with curved elements. The kinematics of the present formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the present element is free of small angle assumptions, and thus it allows large load increments in the geometrically non‐linear analysis. Various numerical examples demonstrate the validity and effectiveness of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

Post‐buckling analysis of imperfect laminates using finite strips based on a higher‐order plate theory

A finite strip procedure has been developed for the post‐buckling analysis of composite laminates when subjected to progressive end shortening. The finite strips are developed based on a higher‐order shear deformation plate theory and there are nine variables at each nodal line. Initial imperfection expressed in the form of suitable trigonometric function is allowed. Examples including isotropic plates and laminates with arbitrary lay‐up arrangement are presented. Numerical results for laminates with and without initial imperfection are used to illustrate the effect of imperfection. Copyright © 2003 John Wiley & Sons, Ltd.