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 eight‐node hybrid‐stress solid‐shell element for geometric non‐linear analysis of elastic shells

This paper presents eight‐node solid‐shell elements for geometric non‐linear analysis of elastic shells. To subdue shear, trapezoidal and thickness locking, the assumed natural strain method and an ad hoc modified generalized laminate stiffness matrix are employed. A selectively reduced integrated element is formulated with its membrane and bending shear strain components taken to be constant and equal to the ones evaluated at the element centroid. With the generalized stresses arising from the modified generalized laminate stiffness matrix assumed to be independent from the ones obtained from the displacement, an extended Hellinger–Reissner functional can be derived. By choosing the assumed generalized stresses similar to the assumed stresses of a previous solid element, a hybrid‐stress solid‐shell element is formulated. Commonly employed geometric non‐linear homogeneous and laminated shell problems are attempted and our results are close to those of other state‐of‐the‐art elements. Moreover, the hybrid‐stress element converges more readily than the selectively reduced integrated element in all benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd.     

A mixed solid‐shell element for the analysis of laminated composites

This paper presents a versatile low order locking‐free mixed solid‐shell element that can be readily employed for a wide range of linear elastic structural analyses, that is, from thick isotropic structures to multilayer anisotropic composites. This solid‐shell element has eight nodes with only displacement degrees of freedom and few assumed stress parameters that provide very accurate interlaminar stress calculations 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 the laminate. The element formulation is based on the well‐known Fraeijs de Veubeke–Hu–Washizu mixed 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 stress field in multilayer structures, which is achieved by the introduction of a constraint equation on the interlaminar stresses in the Fraeijs de Veubeke–Hu–Washizu principle‐based enhanced assumed strains formulation. The intelligent computer coding of the present formulation makes the present element appropriate for a wide range of structural analyses. To assess the present formulation's accuracy, a variety of popular numerical benchmark examples related to element convergence, mesh distortion, and shell and laminated composite analyses are investigated and the results are compared with those available in the literature. These benchmark examples reveal that the proposed formulation provides very good results for the structural analysis of shells and multilayer composites. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

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.     

An improved solid‐shell element based on ANS and EAS concepts

This paper presents an eight‐node nonlinear solid‐shell element for static problems. The main goal of this work is to develop a solid‐shell formulation with improved membrane response compared with the previous solid‐shell element (MOS2013), presented in 1 . Assumed natural strain concept is implemented to account for the transverse shear and thickness strains to circumvent the curvature thickness and transverse shear locking problems. The enhanced assumed strain approach based on the Hu–Washizu variational principle with six enhanced assumed strain degrees of freedom is applied. Five extra degrees of freedom are applied on the in‐plane strains to improve the membrane response and one on the thickness strain to alleviate the volumetric and Poisson's thickness locking problems. The ensuing element performs well in both in‐plane and out‐of‐plane responses, besides the simplicity of implementation. The element formulation yields exact solutions for both the membrane and bending patch tests. The formulation is extended to the geometrically nonlinear regime using the corotational approach, explained in 2 . Numerical results from benchmarks show the robustness of the formulation in geometrically linear and nonlinear problems. Copyright © 2016 John Wiley & Sons, Ltd.     

An isogeometric locking‐free NURBS‐based solid‐shell element for geometrically nonlinear analysis

In this work, we develop an isogeometric non‐uniform rational B‐spline (NURBS)‐based solid‐shell element for the geometrically nonlinear static analysis of elastic shell structures. A single layer of continuous 3D elements through the thickness of the shell is considered, and the order of approximation in that direction is chosen to be equal to two. A complete 3D constitutive relation is assumed. The objective is to develop a highly accurate low‐order element for coarse meshes. We propose an extension of the mixed method of Bouclier et al. [11] to deal with locking in the context of large rotations and large displacements. The main idea is to modify the interpolation of the average through the thickness of the stress components. It is also necessary to stabilize the element in order to avoid the occurrence of spurious zero‐energy modes. This was achieved, for the quadratic version, through the adjunction of artificial elementary stabilization stiffnesses. The result is an element of order 2, which is at least as accurate as standard NURBS shell elements of order 4. Linear and nonlinear test calculations have been carried out along with comparisons with other published NURBS and classical techniques in order to assess the performance of the element. Copyright © 2014 John Wiley & Sons, Ltd.     

Mixed shell element for seven‐parameter formulation

A new mixed shell element is developed for a seven‐parameter formulation in this paper. The mixed shell element is constructed by assuming stress field and displacement field together. Assumed stress field and assumed displacement field can be combined by stress–strain relationship with Hu‐Washizu functional. The developed mixed shell element can provide more flexible stiffness than other commercial softwares. Additionally, seven‐parameter shell formulation is used instead of Reissner/Mindlin formulation, since it can provide the thickness change. Even though some commercial engineering software are not proper for very thick shell structure, the developed mixed shell element for seven‐parameter formulation can be used without distinction of thick shell and thin shell. An example of shell models with different thickness is provided with solid model. Static and modal analyses are also performed for verification. Copyright © 2005 John Wiley & Sons, Ltd.     

Efficient and accurate multilayer solid‐shell element: non‐linear materials at finite strain

We present in this paper an efficient and accurate low‐order solid‐shell element formulation for analyses of large deformable multilayer shell structures with non‐linear materials. The element has only displacement degrees of freedom (dofs), and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out‐of‐plane bending) and to remedy volumetric locking. Based on the mixed Fraeijs de Veubeke‐Hu‐Washizu (FHW) variational principle, the in‐plane and out‐of‐plane bending behaviours are improved and the locking associated with (nearly) incompressible materials is avoided via a new efficient enhancement of strain tensor. Shear locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. Two non‐linear 3‐D constitutive models (Mooney–Rivlin material and hyperelastoplastic material at finite strain) are applied directly without requiring the enforcement of the plane‐stress assumption. In particular, we give a simple derivation for the hyperelastoplastic model using spectral representations. In addition, the present element has a well‐defined lumped mass matrix, and provides double‐side contact surfaces for shell contact problems. With the dynamics referred to a fixed inertial frame, the present element can be used to analyse multilayer shell structures undergoing large overall motion. Numerical examples involving static analyses and implicit/explicit dynamic analyses of multilayer shell structures with both material and geometric non‐linearities are presented, and compared with existing results obtained from other shell elements and from a meshless method. It is shown that elements that did not pass the out‐of‐plane bending patch test could not provide accurate results, as compared to the present element formulation, which passed the out‐of‐plane bending patch test. The present element proves to be versatile and efficient in the modelling and analyses of general non‐linear composite multilayer shell structures. Copyright © 2005 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.     

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.     

An isogeometric solid‐like shell element for nonlinear analysis

An isogeometric solid‐like shell formulation is proposed in which B‐spline basis functions are used to construct the mid‐surface of the shell. In combination with a linear Lagrange shape function in the thickness direction, this yields a complete three‐dimensional representation of the shell. The proposed shell element is implemented in a standard finite element code using Bézier extraction. The formulation is verified using different benchmark tests. Copyright © 2013 John Wiley & Sons, Ltd.     

A new nine‐node solid‐shell finite element using complete 3D constitutive laws

The solid‐shell element presented in this paper has nine nodes: eight are classically located at the apexes and are fitted with three translational DOFs whereas the ninth is sited at the center and is endowed with only one DOF; a displacement along the ‘thickness’ direction. Indeed, to be used for modeling thin structures under bending effects, this kind of finite element has a favored direction where several integration points are distributed. Besides, there is solely one ‘in‐plane’ quadrature point to avoid locking phenomena and prohibitive CPU costs for large nonlinear computations. Because a reduced integration is not enough to completely prevent transverse shear locking, a shear–strain field is assumed. Compared with the other eight‐node ‘solid‐shell' bricks, the presence of a supplementary node has a main aim: getting a linear normal strain component which, along with a full three‐dimensional constitutive strain–stress behavior, allows to achieve similar results in bending cases as those obtained with the usual plane stress state hypothesis. For that, the ninth node DOF plays the role of an extra parameter essential for a quadratic interpolation of the displacement in the thickness direction. The advantage is that this DOF has a physical meaning and, for instance, a strength equivalent to a normal pressure can be prescribed. With a suitable nodal numbering, the band width is not significantly increased and meshes can easily be generated because the extra nodes are always located at element centers. To emphasize the peculiar features of such an element, a set of examples (linear and nonlinear) is carried out. Numerous comparisons with other elements show pretty good results in bending dominating problems while adding the event of a normal stress component in sheet metal forming simulations with double side contact. Copyright © 2012 John Wiley & Sons, Ltd.     

A large deformation solid‐shell concept based on reduced integration with hourglass stabilization

In this paper a new eight‐node (brick) solid‐shell finite element formulation based on the concept of reduced integration with hourglass stabilization is presented. The work focuses on static problems. The starting point of the derivation is the three‐field variational functional upon which meanwhile established 3D enhanced strain concepts are based. Important additional assumptions are made to transfer the approach into a powerful solid‐shell. First of all, a Taylor expansion of the first Piola–Kirchhoff stress tensor with respect to the normal through the centre of the element is carried out. In this way the stress becomes a linear function of the shell surface co‐ordinates whereas the dependence on the thickness co‐ordinate remains non‐linear. Secondly, the Jacobian matrix is replaced by its value in the centre of the element. These two assumptions lead to a computationally efficient shell element which requires only two Gauss points in the thickness direction (and one Gauss point in the plane of the shell element). Additionally three internal element degrees‐of‐freedom have to be determined to avoid thickness locking. One important advantage of the element is the fact that a fully three‐dimensional stress state can be modelled without any modification of the constitutive law. The formulation has only displacement degrees‐of‐freedom and the geometry in the thickness direction is correctly displayed. 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 solid‐like shell element allowing for arbitrary delaminations

In this contribution a new finite element is presented for the simulation of delamination growth in thin‐layered composite structures. The element is based on a solid‐like shell element: a volume element that can be used for very thin applications due to a higher‐order displacement field in the thickness direction. The delamination crack can occur at arbitrary locations and is incorporated in the element as a jump in the displacement field by using the partition of unity property of finite element shape functions. The kinematics of the element as well as the finite element formulation are described. The performance of the element is demonstrated by means of two examples. Copyright © 2003 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.     

Hybrid‐stress solid elements for shell structures based upon a modified variational functional

In this paper, we start with a modified generalized laminate stiffness matrix that serves as a remedy to resolve the thickness locking and some abnormalities encountered by solid‐shell elements in laminate analyses. A modified Hellinger–Reissner functional having displacement and a set of generalized stresses as independent fields is devised. Based upon the functional, eight‐node and 18‐node hybrid‐stress solid‐shell elements are proposed. A number of benchmark tests on homogenous and laminated plates/shells are conducted. The accuracy of the elements is promising. Copyright © 2002 John Wiley & Sons, Ltd.