A novel versatile multilayer hybrid stress solid-shell element

This paper presents a versatile multilayer locking free hybrid stress solid-shell element that can be readily employed for a wide range of geometrically linear elastic structural analyses, i.e. from shell-like isotropic structures to multilayer anisotropic composites. This solid-shell element has eight nodes with only displacement degrees of freedom and a few internal parameters that provide the locking free behavior and accurate interlaminar stress resolution through the element thickness. These elements can be stacked on top of each other to model multilayer structures, fulfilling the interlaminar stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of composite laminates. The element formulation is based on the modified form of the well-known Fraeijs de Veubeke–Hu–Washizu (FHW) multifield variational principle with enhanced assumed strains (EAS formulation) and assumed natural strains (ANS formulation) to alleviate the different types of locking phenomena in solid-shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar stress field in multilayer structures, which is achieved by incorporating an assumed stress field in a standard EAS formulation based on the FHW principle. To assess the present formulation’s accuracy, a variety of popular numerical benchmark examples related to element patch tests, convergence, mesh distortion, shell and laminated composite analyses are investigated and the results are compared with those available in the literature. This assessment reveals that the proposed solid-shell formulation provides very accurate results for a wide range of structural analyses.     

EAS concept for higher‐order finite shell elements to eliminate volumetric locking

The deficiency of volumetric locking phenomena in finite elements using higher‐order shell element formulations based on Lagrangean polynomials and a linear finite shell kinematics cannot be avoided by the existent enhanced assumed strain (EAS) concept established for low‐order elements. In this paper a consistent modification of the EAS concept is proposed to extend its applicability to higher‐order shell elements. This modification, affecting the transversal normal strain for polynomial orders p>1, eliminates pathological modes caused by volumetric locking. The efficiency of the proposed extended EAS method is demonstrated by means of eigenvalue analyses and two representative numerical examples. Copyright © 2008 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.     

A systematic development of ‘solid-shell’ element formulations for linear and non-linear analyses employing only displacement degrees of freedom

In the present contribution we propose a so-called solid-shell concept which incorporates only displacement degrees of freedom. Thus, some major disadvantages of the usually used degenerated shell concept are overcome. These disadvantages are related to boundary conditions—the handling of soft and hard support, the need for special co-ordinate systems at boundaries, the connection with continuum elements—and, in geometrically non-linear analyses, to a complicated update of the rotation vector. First, the kinematics of the so-called solid-shell concept in analogy to the degenerated shell concept are introduced. Then several modifications of the solid-shell concept are proposed to obtain locking-free solid-shell elements, leading also to formulations which allow the use of general three-dimensional material laws and which are also able to represent the normal stresses and strains in thickness direction. Numerical analyses of geometrically linear and non-linear problems are finally performed using solely assumed natural shear strain elements with a linear approximation in in-plane direction. Although some considerations are needed to get comparable boundary conditions in the examples analysed, the solid-shell elements prove to work as good as the degenerated shell elements. The numerical examples show that neither thickness nor shear locking are present even for distorted element shapes. © 1998 John Wiley & Sons, Ltd.     

A new axi-symmetric element for thin walled structures

A new axi-symmetric finite element for thin walled structures is presented in this work. It uses the solid-shell element’s concept with only a single element and multiple integration points along the thickness direction. The cross-section of the element is composed of four nodes with two degrees of freedom each. The proposed formulation overcomes many locking pathologies including transverse shear locking, Poisson’s locking and volumetric locking. For transverse shear locking, the formulation uses the selective reduced integration technique, for Poisson’s locking it uses the enhanced assumed strain (EAS) method with only one enhancing variable. The B-bar approach is used to eliminate the isochoric deformations in the hourglass field while the EAS method is used to alleviate the volumetric locking in the constant part of the deformation tensor. Several examples are shown to demonstrate the performance and accuracy of the proposed element with special focus on the numerical simulations for the beverage can industry.     

An enhanced layerwise finite element using a robust solid-shell formulation approach

The application of layerwise theories to correctly model the displacement field of sandwich structures or laminates with high modulus ratios, usually employs plate or facet shell finite element formulations to compute the element stiffness and mass matrices for each layer. In this work, a different approach is proposed, using a high performance hexahedral finite element to represent the individual layer mass and stiffness. This 8-node hexahedral finite element is formulated based on the application of the enhanced assumed strain method (EAS) to resolve several locking pathologies coming from the high aspect ratios of the finite element and the usual incompressibility condition of the core materials. The solid-shell finite element formulation is introduced in the layerwise theory through the definition of a projection operator, which is based on the finite element variables transformation matrix. The new finite element is tested and the implemented numerical remedies are verified. The results for a soft core sandwich plate are hereby presented to demonstrate the proposed finite element applicability and robustness.     

On the implementation of the EAS and ANS concept into a reduced integration continuum shell element and applications to sheet forming

This contribution deals with the application of a new solid-shell finite element based on reduced integration with hourglass stabilization in the field of sheet metal forming. The formulation includes the enhanced assumed strain (EAS) concept getting by with a minimum of enhanced degrees-of-freedom to overcome the volumetric locking and Pois- son’s thickness locking. To circumvent further the well-known effects of curvature thickness locking and transverse shear locking present in standard eight-node hexahedral finite elements the assumed natural strain (ANS) concept is applied. The implementation of the latter key feature is not straight-forward in reduced integration solid-shells. The second crucial point is a combined Taylor expansion of the compatible Green-Lagrange strain tensor with respect to the center of the element and the normal through the element center leading to an efficient and locking-free hourglass stabilization. Due to the three-dimensional modeling of the structure fully three-dimensional materials can be implemented without additional assumptions. Furthermore simulations of double-side contact problems (e.g. sheet metal forming) benefit from an exact modeling of the sheet thickness.     

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.     

On the Assumed Natural Strain method to alleviate locking in solid-shell NURBS-based finite elements

In isogeometric analysis (IGA), the functions used to describe the CAD geometry (such as NURBS) are also employed, in an isoparametric fashion, for the approximation of the unknown fields, leading to an exact geometry representation. Since the introduction of IGA, it has been shown that the high regularity properties of the employed functions lead in many cases to superior accuracy per degree of freedom with respect to standard FEM. However, as in Lagrangian elements, NURBS-based formulations can be negatively affected by the appearance of non-physical phenomena that “lock” the solution when constrained problems are considered. In order to alleviate such locking behaviors, the Assumed Natural Strain (ANS) method proposed for Lagrangian formulations is extended to NURBS-based elements in the present work, within the context of solid-shell formulations. The performance of the proposed methodology is assessed by means of a set of numerical examples. The results allow to conclude that the employment of the ANS method to quadratic NURBS-based elements successfully alleviates non-physical phenomena such as shear and membrane locking, significantly improving the element performance.     

A solid-shell layerwise finite element for non-linear geometric and material analysis

The application of layerwise theories to correctly model the displacement field of sandwich structures or laminates with high modulus ratios usually employs plate or facet-shell finite element formulations to compute the element stiffness and mass matrices for each layer. In this work an alternative approach is proposed, using a high performance hexahedral finite element to represent the individual layer mass and stiffness. This eight-node hexahedral finite element is formulated based on the application of the enhanced assumed strain method (EAS) to solve several locking pathologies coming from the high aspect ratio of the finite element and the usual incompressibility condition of the core materials. The solid-shell finite element formulation is introduced in the layerwise theory through the definition of a projection operator, based on the finite element variables transformation matrix. The non-linear geometric and material capabilities are introduced into the finite element formulation, allowing for the representation of large displacements, large deformation and material non-linear behaviors. The developed formulation is numerically tested and benchmarked, being validated by using published experimental results obtained from sandwich specimens.     

Four‐node semi‐EAS element in six‐field nonlinear theory of shells

We propose a new four‐node C⁰ finite element for shell structures undergoing unlimited translations and rotations. The considerations concern the general six‐field theory of shells with asymmetric strain measures in geometrically nonlinear static problems. The shell kinematics is of the two‐dimensional Cosserat continuum type and is described by two independent fields: the vector field for translations and the proper orthogonal tensor field for rotations. All three rotational parameters are treated here as independent. Hence, as a consequence of the shell theory, the proposed element has naturally six engineering degrees of freedom at each node, with the so‐called drilling rotation. This property makes the element suitable for analysis of shell structures containing folds, branches or intersections. To avoid locking phenomena we use the enhanced assumed strain (EAS) concept. We derive and linearize the modified Hu–Washizu principle for six‐field theory of shells. What makes the present approach original is the combination of EAS method with asymmetric membrane strain measures. Based on literature, we propose new enhancing field and specify the transformation matrix that accounts for the lack of symmetry. To gain knowledge about the suitability of this field for asymmetric strain measures and to assess the performance of the element, we solve typical benchmark examples with smooth geometry and examples involving orthogonal intersections of shell branches. Copyright © 2006 John Wiley & Sons, Ltd.     

On the enhancement of low‐order mixed finite element methods for the large deformation analysis of diffusion in solids

A stabilized scheme is developed for mixed finite element methods for strongly coupled diffusion problems in solids capable of large deformations. Enhanced assumed strain techniques are employed to cure spurious oscillation patterns of low‐order displacement/pressure mixed formulations in the incompressible limit for quadrilateral elements and brick elements. A study is presented that shows how hourglass instabilities resulting from geometrically nonlinear enhanced assumed strain methods have to be distinguished from pressure oscillation patterns due to the violation of the inf‐sup condition. Moreover, an element formulation is proposed that provides stable results with respect to both types of instabilities. Comparisons are drawn between material models for incompressible solids of Mooney–Rivlin type and models for standard diffusion in solids with incompressible matrices such as polymeric gels. Representative numerical examples underline the ability of the proposed element formulation to cure instabilities of low‐order mixed formulations. Copyright © 2015 John Wiley & Sons, Ltd.     

An enhanced strain 3D element for large deformation elastoplastic thin-shell applications

In this work a previously proposed solid-shell finite element, entirely based on the Enhanced Assumed Strain (EAS) formulation, is extended in order to account for large deformation elastoplastic thin-shell problems. An optimal number of 12 enhanced (internal) variables is employed, leading to a computationally efficient performance when compared to other 3D or solid-shell enhanced elements. This low number of enhanced variables is sufficient to (directly) eliminate either volumetric and transverse shear lockings, the first one arising, for instance, in the fully plastic range, whilst the last appears for small thickness values. The enhanced formulation comprises an additive split of the Green-Lagrange material strain tensor, turning the inclusion of nonlinear kinematics a straightforward task. Finally, some shell-type numerical benchmarks are carried out with the present formulation, and good results are obtained, compared to well-established formulations in the literature.Funding by Ministério da Ciência e do Ensino Superior (FCT and FSE) (Portugal) under grant PRAXIS XXI/ BD/21662/99; as well as the funding by FEDER, under grant POCTI/EME/47289/2002, are gratefully acknowledged.     

An optimal versatile partial hybrid stress solid‐shell element for the analysis of multilayer composites

In the present contribution we propose an optimal low‐order versatile partial hybrid stress solid‐shell element that can be readily employed for a wide range of geometrically linear elastic structural analyses, that is, from shell‐like isotropic structures to multilayer anisotropic composites. This solid‐shell element has eight nodes with only displacement degrees of freedom and only a few internal parameters that provide the locking‐free behavior and accurate interlaminar shear stress resolution through the element thickness. These elements can be stacked on top of each other to model multilayer composite structures, fulfilling the interlaminar shear stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of composite laminates. The element formulation is based on the modified form of the well‐known Fraeijs de Veubeke–Hu–Washizu multifield variational principle with enhanced assumed strains formulation and assumed natural strains formulation to alleviate the different types of locking phenomena in solid‐shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar shear stress field in multilayer structures, which is achieved by the introduction of the assumed interlaminar shear stress field in a standard enhanced assumed strains formulation based on the Fraeijs de Veubeke–Hu–Washizu principle. The numerical testing of the present formulation, employing a variety of popular numerical benchmark examples related to element patch test, convergence, mesh distortion, shell and laminated composite analyses, proves its accuracy for a wide range of structural analyses.Copyright © 2012 John Wiley & Sons, Ltd.     

4-Node combined shell element with semi-EAS-ANS strain interpolations in 6-parameter shell theories with drilling degrees of freedom

A 4-node C ⁰ shell element with drilling degrees of freedom is presented in this paper. The element is developed within the nonlinear 6-field shell theory. Kinematics of the shell is described by two independent fields: the vector field for translations and the proper orthogonal tensor field for rotations. Within the theoretical formulation no restriction is applied on magnitudes of displacements and rotations. To avoid locking phenomena the proposed element combines two interpolation schemes: the assumed natural strain (ANS) for transverse shear strains and the enhanced assumed strain (EAS). The latter interpolation is used with asymmetric (in-plane) membrane strains. The performance of the element is evaluated by example of benchmark problems with special emphasis on shell structures containing orthogonal intersections.     

Extension of the enhanced assumed strain method based on the structure of polyconvex strain-energy functions

In this work, two well-known approaches for mixed finite elements are combined to render three novel classes of elements. First, the widely used enhanced assumed strain (EAS) method is considered. Its key idea is to enhance a compatible kinematic field with an incompatible part. The second concept is a framework for mixed elements inspired by polyconvex strain-energy functions, in which the deformation gradient, its cofactor and determinant are three principal kinematic fields. The key idea for the novel elements is to treat enhancement of those three fields separately. This approach leads to a plethora of novel enhancement strategies and promising mixed finite elements. Some key properties of the newly proposed mixed approaches are that they are based on a Hu-Washizu type variational functional, fulfill the patch test, are frame-invariant, can be constructed completely locking free and show no spurious hourglassing in elasticity. Furthermore, they give additional insight into the mechanisms of standard EAS elements. Extensive numerical investigations are performed to assess the elements' behavior in elastic and elasto-plastic simulations.     

A Simple Procedure to Develop Efficient & Stable Hybrid/Mixed Elements,and Voronoi Cell Finite Elements for Macro- & Micromechanics

A simple procedure to formulate efficient and stable hybrid/mixed finite elements is developed, for applications in macro- as well as micromechanics. In this method, the strain and displacement field are independently assumed. Instead of using two-field variational principles to enforce both equilibrium and compatibility conditions in a variational sense, the independently assumed element strains are related to the strains derived from the independently assumed element displacements, at a finite number of collocation points within the element. The element stiffness matrix is therefore derived, by simply using the principle of minimum potential energy. Taking the four-node plane isoparametric element as an example, different hybrid/mixed elements are derived, by adopting different element strain field assumptions, and using different collocation points. These elements are guaranteed to be stable. Moreover, the computational efficiency of these elements is far better than that for traditional hybrid/mixed elements, or even better than primal finite elements, because the strain field is expressed analytically as simple polynomials (whereas, in isoparametric displacement-based element, the strain field is far more complicated), with nodal displacements as unknowns. The essential idea is thereafter extended to Voronoi cell finite elements, for micromechanical analysis of materials. Neither these four-node hybrid/mixed elements nor the Vonroni cell finite elements need to satisfy the equilibrium conditions a priori, making them suitable for extension to geometrically nonlinear and dynamical analyses. Various numerical experiments are conducted using these new elements, and the results are compared to those obtained by using traditional hybrid/mixed elements and primal finite elements. Performances of the different elements are compared in terms of efficiency, stability, invariance, locking, sensitivity to mesh distortion, and convergence rates.     

Enhanced finite element formulations on the numerical simulation of tailor-welded hydroformed products

This work aims to assess the influence of different finite element formulations in the performance and quality of solution obtained by numerical simulation in the analysis of tailor-welded hydroformed tubular parts. Tube hydroforming represents a cost effective forming process for high-strength, low weight products on, as an example, automotive and airspace applications. On the other side, the use of tailor-welding in order to obtain custom-made combinations of thicknesses and materials - leading to a wide variety of user-defined products - can be introduced into conventional tubular hydroforming processes in order to further improve the applicability range of the later process. The main goal of the present work is to describe the state-of-the-art in the field, focusing on distinct finite element formulations and providing guidelines for the simulation of tubular hydroforming process combined with tailor-welded joining techniques. Hexahedral solid and solid-shell enhanced assumed strain elements, either with reduced and full numerical integration procedures, are analyzed in order to infer about the potentialities of the combined forming technology. Material characterization of the heat affected zone is included and the influence of finite element modeling on defects onset and prediction during forming is considered.     

An unsymmetric 8-node hexahedral solid-shell element with high distortion tolerance: Geometric nonlinear formulations

A recent distortion-tolerant unsymmetric 8-node hexahedral solid-shell element US-ATFHS8, which takes the analytical solutions of linear elasticity as the trial functions, is successfully extended to geometric nonlinear analysis. This extension is based on the corotational (CR) approach due to its simplicity and high efficiency, especially for geometric nonlinear analysis where the strain is still small. Based on the assumption that the analytical trial functions can properly work in each increment during the nonlinear analysis, the incremental corotational formulations of the nonlinear solid-shell element US-ATFHS8 are derived within the updated Lagrangian (UL) framework, in which an appropriate updated strategy for linear analytical trial functions is proposed. Numerical examples show that the present nonlinear element US-ATFHS8 possesses excellent performance for various rigorous tests no matter whether regular or distorted mesh is used. Especially, it even performs well in some situations that other conventional elements cannot work.     

土体三维弹塑性分析中的体积闭锁分析及单元选择

在材料不可压缩或胀/缩塑性流动情况下，传统低阶单元有体积闭锁问题。以摩尔－库仑模型为例，推导了塑性剪切应变和塑性体积应变的关系，揭示闭锁产生的原因。分析8节点等参元、Wilson非协调元、EAS单元和14节点单元的闭锁性态，表明8节点单元有严重闭锁性，Wilson非协调元也有闭锁性，EAS单元和采用降阶积分的14节点单元能克服闭锁。单元测试和方形基础的承载力计算两个数值算例证实了分析的结果，为土体三维分析中选择有效可靠的单元提供依据。