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.     

A finite element formulation for piezoelectric shell structures considering geometrical and material non‐linearities

In this paper, we present a non‐linear finite element formulation for piezoelectric shell structures. Based on a mixed multi‐field variational formulation, an electro‐mechanical coupled shell element is developed considering geometrically and materially non‐linear behavior of ferroelectric ceramics. The mixed formulation includes the independent fields of displacements, electric potential, strains, electric field, stresses, and dielectric displacements. Besides the mechanical degrees of freedom, the shell counts only one electrical degree of freedom. This is the difference in the electric potential in the thickness direction of the shell. Incorporating non‐linear kinematic assumptions, structures with large deformations and stability problems can be analyzed. According to a Reissner–Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three‐dimensional transversal isotropic material law, thus the kinematic in the thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced by the independent resultant stress and the resultant dielectric displacement fields. Accounting for material non‐linearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperature‐dependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature‐dependent hysteresis curves. 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 mixed shell formulation accounting for thickness strains and finite strain 3d material models

A non‐linear quadrilateral shell element for the analysis of thin structures is presented. The Reissner–Mindlin theory with inextensible director vector is used to develop a three‐field variational formulation with independent displacements, stress resultants and shell strains. The interpolation of the independent shell strains consists of two parts. The first part corresponds to the interpolation of the stress resultants. Within the second part independent thickness strains are considered. This allows incorporation of arbitrary non‐linear 3d constitutive equations without further modifications. The developed mixed hybrid shell element possesses the correct rank and fulfills the in‐plane and bending patch test. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison with other element formulations. We present results for finite strain elasticity, inelasticity, bifurcation and post‐buckling problems. Copyright © 2007 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.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Energy and momentum conserving elastodynamics of a non‐linear brick‐type mixed finite shell element

The paper presents aspects of the finite element formulation of momentum and energy conserving algorithms for the non‐linear dynamic analysis of shell‐like structures. The key contribution is a detailed analysis of the implementation of a Simó–Tarnow‐type conservation scheme in a recently developed new mixed finite shell element. This continuum‐based shell element provides a well‐defined interface to strain‐driven constitutive stress updates algorithms. It is based on the classic brick‐type trilinear displacement element and is equipped with specific gradient‐type enhanced strain modes and shell‐typical assumed strain modifications. The excellent performance of the proposed dynamic shell formulation with respect to conservation properties and numerical stability behaviour is demonstrated by means of three representative numerical examples of elastodynamics which exhibit complex free motions of flexible structures undergoing large strains and large rigid‐body motions. Copyright © 2001 John Wiley & Sons, Ltd.     

Optimal solid shell element for large deformable composite structures with piezoelectric layers and active vibration control

In this paper, we present an optimal low‐order accurate piezoelectric solid‐shell element formulation to model active composite shell structures that can undergo large deformation and large overall motion. This element has only displacement and electric degrees of freedom (dofs), with no rotational dofs, and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out‐of‐plane bending). The combination of the present optimal piezoelectric solid‐shell element and the optimal solid‐shell element previously developed allows for efficient and accurate analyses of large deformable composite multilayer shell structures with piezoelectric layers. To make the 3‐D analysis of active composite shells containing discrete piezoelectric sensors and actuators even more efficient, the composite solid‐shell element is further developed here. Based on the mixed Fraeijs de Veubeke–Hu–Washizu (FHW) variational principle, the in‐plane and out‐of‐plane bending behaviours are improved via a new and efficient enhancement of the strain tensor. Shear‐locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. We also present an optimal‐control design for vibration suppression of a large deformable structure based on the general finite element approach. The linear‐quadratic regulator control scheme with output feedback is used as a control law on the basis of the state space model of the system. Numerical examples involving static analyses and dynamic analyses of active shell structures having a large range of element aspect ratios are presented. Active vibration control of a composite multilayer shell with distributed piezoelectric sensors and actuators is performed to test the present element and the control design procedure. Copyright © 2005 John Wiley & Sons, Ltd.     

A discontinuous Galerkin formulation of non‐linear Kirchhoff–Love shells

Discontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknown‐field derivatives and have particular appeal in problems involving high‐order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197 :2901–2929) to develop a formulation of linear Kirchhoff–Love shells considering only the membrane and bending responses. In this proposed one‐field method—the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG formulation. It is the purpose of the present paper to extend this formulation to finite deformations and non‐linear elastic behaviors. While the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple‐stress from the displacement field at the mid‐surface of the shell, the non‐linear formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reformulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple‐stress at the edges of the shells, the extension to non‐linear deformations is straightforward. Copyright © 2008 John Wiley & Sons, Ltd.     

A robust non‐linear mixed hybrid quadrilateral shell element

In the paper a non‐linear quadrilateral shell element for the analysis of thin structures is presented. The variational formulation is based on a Hu–Washizu functional with independent displacement, stress and strain fields. The interpolation matrices for the mid‐surface displacements and rotations as well as for the stress resultants and strains are specified. Restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived. The developed mixed hybrid shell element possesses the correct rank and fulfills the in‐plane and bending patch test. Using Newton's method the finite element approximation of the stationary condition is iteratively solved. Our formulation can accommodate arbitrary non‐linear material models for finite deformations. In the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post‐buckling response. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison to other element formulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Rotation‐free triangular shell element using node‐based smoothed finite element method

In this paper, a triangular thin flat shell element without rotation degrees of freedom is proposed. In the Kirchhoff hypothesis, the first derivative of the displacement must be continuous because there are second‐order differential terms of the displacement in the weak form of the governing equations. The displacement is expressed as a linear function and the nodal rotation is defined using node‐based smoothed finite element method. The rotation field is approximated using the nodal rotation and linear shape functions. This rotation field is linear in an element and continuous between elements. The curvature is defined by differentiating the rotation field, and the stiffness is calculated from the curvature. A hybrid stress triangular membrane element was used to construct the shell element. The penalty technique was used to apply the rotation boundary conditions. The proposed element was verified through several numerical examples.     

Multilayered shell finite element with interlaminar continuous shear stresses: a refinement of the Reissner–Mindlin formulation

A finite element formulation for refined linear analysis of multilayered shell structures of moderate thickness is presented. An underlying shell model is a direct extension of the first‐order shear‐deformation theory of Reissner–Mindlin type. A refined theory with seven unknown kinematic fields is developed: (i) by introducing an assumption of a zig‐zag (i.e. layer‐wise linear) variation of displacement field through the thickness, and (ii) by assuming an independent transverse shear stress fields in each layer in the framework of Reissner's mixed variational principle. The introduced transverse shear stress unknowns are eliminated on the cross‐section level. At this process, the interlaminar equilibrium conditions (i.e. the interlaminar shear stress continuity conditions) are imposed. As a result, the weak form of constitutive equations (the so‐called weak form of Hooke's law) is obtained for the transverse strains–transverse stress resultants relation. A finite element approximation is based on the four‐noded isoparametric element. To eliminate the shear locking effect, the assumed strain variational concept is used. Performance of the derived finite element is illustrated with some numerical examples. The results are compared with the exact three‐dimensional solutions, as well as with the analytical and numerical solutions obtained by the classical, the first‐order and some representative refined models. Copyright © 2000 John Wiley & Sons, Ltd.     

Adding transverse normal stresses to layered shell finite elements for the analysis of composite structures

This work presents a formulation developed to add capabilities for representing the through thickness distribution of the transverse normal stresses, σ_z, in first and higher order shear deformable shell elements within a finite element (FE) scheme. The formulation is developed within a displacement based shear deformation shell theory. Using the differential equilibrium equations for two-dimensional elasticity and the interlayer stress and strain continuity requirements, special treatment is developed for the transverse normal stresses, which are thus represented by a continuous piecewise cubic function. The implementation of this formulation requires only C⁰ continuity of the displacement functions regardless of whether it is added to a first or a higher order shell element. This makes the transverse normal stress treatment applicable to the most popular bilinear isoparametric 4-noded quadrilateral shell elements.

To assess the performance of the present approach it is included in the formulation of a recently developed third order shear deformable shell finite element. The element is added to the element library of the general nonlinear explicit dynamic FE code DYNA3D. Some illustrative problems are solved and results are presented and compared to other theoretical and numerical results.     

On the treatment of frictional contact in shell structures using variational inequalities

A new variational inequality‐based formulation is presented for the large deformation analysis of frictional contact in elastic shell structures. The formulations are based on a seven‐parameter continuum shell model, which account for the normal stress and strain through the shell thickness and accommodate double‐sided shell contact. The kinematic contact conditions are enforced accurately in terms of the physical contacting surfaces of the shell. Large deformations and rotations are accounted for by invoking the appropriate stress and strain measures. Lagrange multipliers are used to ensure that the kinematic contact constraints are accurately satisfied and that the solution is free from user‐defined parameters. Three numerical examples are used to verify the developed formulations and algorithms, and test the various aspects of the solution technique. Copyright © 1999 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.     

Sandwich shell finite element for dynamic explicit analysis

The contribution of this paper consists of new development of transverse shear stresses through the thickness and finding an expression for the critical time step for explicit time integration of layered shells. This work presents the finite element (FE) formulation and implementation of a higher‐order shear deformable shell element for dynamic explicit analysis of composite and sandwich shells. The formulation is developed using a displacement‐based third‐order shear deformation shell theory. Using the differential equilibrium equations and the interlayer requirements, special treatment is developed for the transverse shear, resulting in a continuous, piecewise quartic distribution of the transverse shear stresses through the shell thickness. Expressions are developed for the critical time step of the explicit time integration for orthotropic homogeneous and layered shells based on the developed third‐order formulation. To assess the performance of the present shell element, it is implemented in the general non‐linear explicit dynamic FE code DYNA3D. Several problems are solved and results are presented and compared to other theoretical and numerical results. Copyright © 2002 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.     

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 Continuum Based Thick Shell Element for Large Deformation Analysis of Layered Composites

A new continuum based thick shell model is presented for modeling orthotropic laminated shell structures undergoing large elastic deformations. An equivalent single-layer model involving seven nodal degrees of freedom is used. In that layered model, there are no restrictions on the number of layers, their thickness and their stacking sequence. The shell model accounts explicitly for the thickness change in the shell, as well as the normal stress and strain states through its thickness. Shear locking is avoided using an assumed natural strain formulation, while thickness locking is avoided using modified displacement interpolation functions. The performance of the layered shell element is tested using several linear and non-linear composite plate and shell problems involving anisotropic, angle and cross-ply laminates, cylindrical and spherical shells.     

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.