The Use of 9-Parameter Shell Theory for Development of Exact Geometry 12-Node Quadrilateral Piezoelectric Laminated Solid-Shell Elements

The present work focuses on the development of the exact geometry (EG) 12-node piezoelectric solid-shell element with three translational degrees of freedom per node. The term “EG” reflects the fact that coefficients of the first and second fundamental forms of the reference surface are taken exactly at each element node. The finite element formulation developed is based on the higher-order 9-parameter equivalent single-layer shell theory accounting for thickness stretching, which permits the use of 3D constitutive equations. In this theory, we introduce three sampling surfaces, namely, bottom, middle, and top, and choose nine displacements of these surfaces as basic shell unknowns. Such a way allows one to represent the EG piezoelectric solid-shell element formulation in a very compact form and to derive the strain-displacement equations, which describe exactly all rigid-body shell motions in any convected curvilinear coordinate system. The element matrices are evaluated through the use of 3D analytical integration by employing the extended ANS method. To avoid shear and membrane locking and have no spurious zero energy modes, the assumed displacement-independent strains and stress resultants fields are invoked.     

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.     

Assessment of nonlinear exact geometry sampling surfaces solid-shell elements and ANSYS solid elements for 3D stress analysis of piezoelectric shell structures

In this work, the finite rotation exact geometry four-node solid-shell element using the sampling surfaces (SaS) method is developed for the analysis of the second Piola-Kirchhoff stresses in laminated piezoelectric shells. The SaS method is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements and electric potentials of these surfaces as fundamental shell unknowns. The outer surfaces and interfaces are also included into a set of SaS. To circumvent shear and membrane locking, the hybrid-mixed solid-shell element on the basis of the Hu-Washizu variational principle is proposed. The tangent stiffness matrix is evaluated by 3D analytical integration throughout the finite element. This novelty provides a superior performance in the case of coarse meshes. A comparison with the SOLID226 element showed that the developed exact geometry SaS solid-shell element allows the use of load increments, which are much larger than possible with existing displacement-based finite elements. Thus, it can be recommended for the 3D stress analysis of doubly-curved laminated piezoelectric shells because the SaS formulation gives the opportunity to obtain the 3D solutions of electroelasticity with a prescribed accuracy.     

Analysis of the second Piola-Kirchhoff stress in nonlinear thick and thin structures using exact geometry SaS solid-shell elements

This paper presents the finite rotation exact geometry four-node solid-shell element using the sampling surfaces (SaS) method. The SaS formulation is based on choosing inside the shell N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains and stresses leads to a robust higher-order shell formulation. The SaS are located at only Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. The proposed hybrid-mixed four-node solid-shell element is based on the Hu-Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated through efficient 3D analytical integration and its explicit form is given. As a result, the proposed exact geometry solid-shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement-based solid-shell elements.     

Finite rotation exact geometry solid-shell element for laminated composite structures through extended SaS formulation and 3D analytical integration

A nonlinear exact geometry hybrid-mixed four-node solid-shell element using the sampling surfaces (SaS) formulation is developed for the analysis of the second Piola-Kirchhoff stress that extends the authors' finite element (Int J Numer Methods Eng. 2019;117:498-522) to laminated composite shells. The SaS formulation is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as basic shell unknowns. The external surfaces and interfaces are also included into a set of SaS. The proposed hybrid-mixed solid-shell element is based on the Hu-Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated by efficient three-dimensional (3D) analytical integration. As a result, the developed exact geometry solid-shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement-based solid-shell elements. It could be useful for the 3D stress analysis of thick and thin doubly curved laminated composite shells because the SaS formulation gives the possibility to obtain the 3D solutions with a prescribed accuracy.     

An Assumed Strain Triangular Solid Element for Efficient Analysis of Plates and Shells with Finite Rotation

A simple triangular solid shell element formulation is developed for efficient analysis of plates and shells undergoing finite rotations. The kinematics of the present solid shell element formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the kinematics of deformation is free of the limitation of small angle increments, and thus the formulation allows large load increments in the analysis of finite rotation. An assumed strain field is carefully selected to alleviate the locking effect without triggering undesirable spurious kinematic modes. In addition, the curved surface of shell structures is modeled with flat facet elements to obviate the membrane locking effect. Various numerical examples demonstrate the efficiency and accuracy of the present element formulation for the analysis of plates and shells undergoing finite rotation. The present formulation is attractive in that only three points are needed for numerical integration over an element.     

A general higher-order shell theory for compressible isotropic hyperelastic materials using orthonormal moving frame

The primary objective of this study is threefold: (1) to present a general higher-order shell theory to analyze large deformations of thin or thick shell structures made of general compressible hyperelastic materials; (2) to formulate an efficient shell theory using the orthonormal moving frame, and (3) to develop and apply the nonlinear weak-form Galerkin finite element model for the proposed shell theory. The displacement field of the line normal to the shell reference surface is approximated by the Taylor series/Legendre polynomials in the thickness coordinate of the shell. The use of an orthonormal moving frame makes it possible to represent kinematic quantities (e.g., the determinant of the deformation gradient) in a far more efficient manner compared with the nonorthogonal covariant bases. Kinematic quantities for the shell deformation are obtained in a novel way in the surface coordinate described in the appendix of this study with the help of exterior calculus. Furthermore, the governing equation of the shell deformation has been derived in the general surface coordinates. To obtain the nonlinear solution in the quasi-static cases, we develop the weak-form finite element model in which the reference surface of the shell is modeled exactly. The general invariant based compressible hyperelastic material model is considered. The formulation presented herein can be specialized for various other nonlinear compressible hyperelastic constitutive models, for example, in biomechanics and other soft-material problems (e.g., compressible neo-Hookean material, compressible Mooney–Rivlin material, Saint Venant–Kirchhoff model, and others). A number of numerical examples are presented to verify and validate the formulation presented in this study. The scope of potential extensions are outlined in the final section of this study.     

Exact geometry four-node solid-shell element for stress analysis of functionally graded shell structures via advanced SaS formulation

Abstract

The exact geometry four-node solid-shell element formulation using the sampling surfaces (SaS) method is developed. The SaS formulation is based on choosing inside the shell N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the use of Lagrange basis polynomials of degree N???1 in the through-thickness interpolations of displacements, strains, stresses and material properties leads to a very compact form of the SaS shell formulation. The SaS are located at Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. To implement efficient 3D analytical integration, the extended assumed natural strain method is employed. As a result, the proposed hybrid-mixed solid-shell element exhibits a superior performance in the case of coarse meshes. To circumvent shear and membrane locking, the assumed stress and strain approximations are utilized in the framework of the mixed Hu-Washizu variational formulation. It can be recommended for the 3D stress analysis of thick and thin doubly-curved functionally graded shells because the SaS formulation with only Chebyshev polynomial nodes allows the obtaining of numerical solutions, which asymptotically approach the 3D solutions of elasticity as the number of SaS tends to infinity.     

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 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.     

A FE2 shell model with periodic boundary conditions for thin and thick shells

In this article a FE2 shell model for thin and thick shells within a first order homogenization scheme is presented. A variational formulation for the two-scale boundary value problem and the associated finite element formulation is developed. Constraints with 5 or 9 Lagrange parameters are derived which eliminate both rigid body movements and dependencies of the shear stiffness on the size of the representative volume elements (RVEs). At the bottom and top surface of the RVEs which extend through the total thickness of the shell stress boundary conditions are present. The periodic boundary conditions at the lateral surfaces are applied in such a way that particular membrane, bending and shear modes are not restrained. This is shown by means of a homogeneous RVE. The first of all linear formulation is extended to finite strain problems introducing transformation relations for the stress resultants and the material matrix. The transformations are performed at the Gauss points on macro level. Several boundary value problems including large deformations, stability and inelasticity are computed and compared with 3D reference solutions.     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 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 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.     

A geometrically non‐linear piezoelectric solid shell element based on a mixed multi‐field variational formulation

This paper is concerned with a geometrically non‐linear solid shell element to analyse piezoelectric structures. The finite element formulation is based on a variational principle of the Hu–Washizu type and includes six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. The element has eight nodes with four nodal degrees of freedoms, three displacements and the electric potential. A bilinear distribution through the thickness of the independent electric field is assumed to fulfill the electric charge conservation law in bending dominated situations exactly. The presented finite shell element is able to model arbitrary curved shell structures and incorporates a 3D‐material law. A geometrically non‐linear theory allows large deformations and includes stability problems. Linear and non‐linear numerical examples demonstrate the ability of the proposed model to analyse piezoelectric devices. Copyright © 2005 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.     

NON-LINEAR FINITE ELEMENT ANALYSIS OF ISOTROPIC AND COMPOSITE SHELLS BY A TOTAL LAGRANGIAN DECOMPOSITION SCHEME

ABSTRACT

A total Lagrangian finite element scheme for arbitrarily large displacements and rotations is applied to a composite beam, an isotropic deep arch, and a section of an isotropic circular toroidal shell. The scheme decomposes the motion into stretches and rigid-body rotations, examining the deformed state with respect to an orthogonal, rigidly translated and rotated triad located at the deformed point of interest. The Jaumann stress and strain measures, which are resolved along the axes of this triad, are employed in the algorithm. Local and layer-wise thickness stretching and shear warping functions are used to model the three-dimensional behaviour of the shell. These functions are developed through the use of the constitutive equations, certain stress and displacement continuity requirements at ply interfaces and laminate surfaces, and the behaviour of the shell reference surface. Two finite elements are employed in the analyses: an eight-noded, 36 degree-of-freedom (DOF) element, and a four-noded, 44 DOF element. The 36 DOF element, which is not a compatible element with respect to the derivatives of in-plane deformations, proves adequate for moderate rotation problems, but fails in modelling very large rotation problems. The use of the 44 DOF element provides dramatically improved results in the large rotation problem.     

A FEM model for the active control of curved FGM shells using piezoelectric sensor/actuator layers

In this paper, a generic finite element formulation is developed for the static and dynamic control of FGM (functionally graded material) shells with piezoelectric sensor and actuator layers. The properties of the FGM shell are graded in the thickness direction according to a volume fraction power‐law distribution. The proposed finite element model is based on variational principle and linear piezoelectricity theory. A constant displacement and velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied in a closed‐loop system to provide feedback control of the integrated FGM shell structure. Both static and dynamic control of FGM shells are simulated to demonstrate the effectiveness of the proposed active control scheme within a framework of finite element discretization and piezoelectric integration. Copyright © 2002 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.     

Stress resultant geometrically non-linear shell theory with drilling rotations. Part III: Linearized kinematics

A consistent formulation of the geometrically linear shell theory with drilling rotations is obtained by the consistent linearization of the geometrically non-linear shell theory considered in Parts I and II of this work. It was also shown that the same formulation can be recovered by linearizing the governing variational principle for the three-dimensional geometrically non-linear continuum with independent rotation field. In the finite element implementation of the presented shell theory, relying on the modified method of incompatible modes, we were able to construct a four-node shell element which delivers a very high-level performance. In order to simplify finite element implementation, a shallow reference configuration is assumed over each shell finite element. This approach does not impair the element performance for the present four-node element. The results obtained herein match those obtained with the state-of-the-art implementations based on the classical shell theory, over the complete set of standard benchmark problems.