A hybrid/mixed model for non-linear shell analysis and its applications to large-rotation problems

Adopting an updated Lagrange approach, the general framework for the fully non-linear analysis of curved shells is developed using a simple quadrilateral C⁰ model (HMSH5). The governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified Hellinger/Reissner type with independent assumptions for displacement and strain fields. Emphasis is placed on devising effective solution procedures to deal with large rotations in space, finite stretches and generalized rate-type material models. In particular, a geometrically exact scheme for configuration update is developed by making use of the so-called exponential mapping algorithm, and the resulting element was shown to exhibit a quadratic rate of (asymptotic) convergence in solving practical shell problems with Newton–Raphson type iterative schemes. For the purpose of updating the spatial stress field of the element, an ‘objective’ generalized midpoint integration rule is utilized, which relies crucially on the concept of polar decomposition for the deformation gradient, and is in keeping with the underlying mixed method. Finally, the effectiveness and practical usefulness of the HMSH5 element are demonstrated through a number of test cases involving beams, plates and shells undergoing very large displacements and rotations.     

Finite element representation and pressure stiffness in shell stability analysis

In the light of recent contributions by Batoz¹ and Hibbitt,² two aspects of finite element formulations for shell stability analysis are examined. The first is the consistency of the shell strain-displacement equations employed; the second is the proper representation of ‘follower forces’—pressures that are always normal to the deforming surface. Numerical studies of an arch indicate that improper representation of either of these factors can have a significant effect on predicted buckling loads. Numerical studies of an arch indicate that improper representation of either of these factors can have a significant effect on predicted bukling loads.     

On the classical shell model underlying bilinear degenerated shell finite elements: general shell geometry

We study the shell models arising in the numerical modelling of shells by geometrically incompatible finite elements. We build a connection from the so‐called bilinear degenerated 3D FEM to the classical 2D shell theory of Reissner–Naghdi type showing how nearly equivalent finite element formulations can be constructed within the classical framework. The connection found here facilitates the mathematical error analysis of the bilinear elements based on the degenerated 3D approach. In particular, the connection reveals the ‘secrets’ that relate to the treatment of locking effects within this formulation. Copyright © 2002 John Wiley & Sons, Ltd.     

A triangular finite shell element based on a fully nonlinear shell formulation

This work presents a fully nonlinear six-parameter (3 displacements and 3 rotations) shell model for finite deformations together with a triangular shell finite element for the solution of the resulting static boundary value problem. Our approach defines energetically conjugated generalized cross-sectional stresses and strains, incorporating first-order shear deformations for an inextensible shell director (no thickness change). Finite rotations are treated by the Euler–Rodrigues formula in a very convenient way, and alternative parameterizations are also discussed herein. Condensation of the three-dimensional finite strain constitutive equations is performed by applying a mathematically consistent plane stress condition, which does not destroy the symmetry of the linearized weak form. The results are general and can be easily extended to inelastic shells once a stress integration scheme within a time step is at hand. A special displacement-based triangular shell element with 6 nodes is furthermore introduced. The element has a nonconforming linear rotation field and a compatible quadratic interpolation scheme for the displacements. Locking is not observed as the performance of the element is assessed by several numerical examples, which also illustrate the robustness of our formulation. We believe that the combination of reliable triangular shell elements with powerful mesh generators is an excellent tool for nonlinear finite element analysis.Fellowship funding from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and CNPq (Conselho Nacional de Pesquisa), together with the material support and stimulating discussions in IBNM (Institut für Baumechanik und Numerische Mechanik), are gratefully acknowledged in this work.     

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.     

Influence functions and goal‐oriented error estimation for finite element analysis of shell structures

In this paper, we first present a consistent procedure to establish influence functions for the finite element analysis of shell structures, where the influence function can be for any linear quantity of engineering interest. We then design some goal‐oriented error measures that take into account the cancellation effect of errors over the domain to overcome the issue of over‐estimation. These error measures include the error due to the approximation in the geometry of the shell structure. In the calculation of the influence functions we also consider the asymptotic behaviour of shells as the thickness approaches zero. Although our procedures are general and can be applied to any shell formulation, we focus on MITC finite element shell discretizations. In our numerical results, influence functions are shown for some shell test problems, and the proposed goal‐oriented error estimation procedure shows good effectivity indices. Copyright © 2005 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.     

Accelerated ‘relaxation’ or direct solution? Future prospects for fem

‘Dynamic’ or ‘viscous’ relaxation procedures have not gained much popularity in finite element analysis in which the direct (Gaussian elimination) solution dominates. Reasons for this are various—the most important being the rather slow convergence generally achieved for such procedures. However, it is possible to accelerate this quite dramatically and a method of doing so is shown in this paper. With the use of such acceleration and the inherent advantages of greatly reduced storage requirements and simplicity of programming, relaxation procedures promise an exciting possibility for the solution of large two- and three-dimensional problems in both linear and nonlinear ranges.     

The geometric stiffness of thick shell triangular finite elements for large rotations

This paper is concerned with the development of the geometric stiffness matrix of thick shell finite elements for geometrically nonlinear analysis of the Newton type. A linear shell element that is comprised of the constant stress triangular membrane element and the triangular discrete Kirchhoff Mindlin theory (DKMT) plate element is ‘upgraded’ to become a geometrically nonlinear thick shell finite element. Perturbation methods are used to derive the geometric stiffness matrix from the gradient, in global coordinates, of the nodal force vector when stresses are kept fixed. The present approach follows earlier works associated with trusses, space frames and thin shells. It has the advantage of explicitness and clear physical insight. A special procedure, tailored to triangular elements is used to isolate pure rotations to enable stress recovery via linear elastic constitutive relations. Several examples are solved. The results compare well with those available in the literature. Copyright © 2005 John Wiley & Sons, Ltd.     

Unique bending boundary layer behaviors in a composite non-circular cylindrical shell

A cylindrical composite shell with a non-circular, symmetric cross-section of flat sides and circular arc corners is analyzed using the abaqus finite element program. A trade study on the effects of various corner radii and shell lengths is performed. The response of the shell to constant internal pressurization is studied, with particular attention given to the bending boundary layer (BBL) near the ends. It is found that the extent of the BBL in the non-circular case is 2.5–4 times longer than that predicted by the classical equation, however, the ‘intensity’ of the bending boundary layer is reduced. An unusual ‘compounding’ effect in boundary layer response for short non-circular shells is described.     

A higher-order facet quadrilateral composite shell element

A C⁰ finite element formulation of flat faceted element based on a higher-order displacement model is presented for the analysis of general, thin-to-thick, fibre reinforced composite laminated plates and shells. This theory incorporates a realistic non-linear variation of displacements through the shell thickness, and eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with five and nine degrees of freedom per node. A comparison of results is also made with the 2-D thin classical and 3-D exact analytical results, and finite element solutions with 9-noded first-order element. © 1997 John Wiley & Sons, Ltd.     

Galerkin lumped parameter methods for transient problems

In this paper we propose a general methodology to obtain lumped parameter models for systems governed by parabolic partial differential equations which we call Galerkin lumped parameter methods. The idea consists of decomposing the computational domain into several subdomains connected through so‐called ports. Then a time‐independent adapted reduced basis is introduced by numerically solving several elliptic problems in each subdomain. The proposed lumped parameter model is the Galerkin approximation of the original problem in the space spanned by this basis. The relationship of these methods with classical lumped parameter models is analyzed. Numerical results are shown as well as a comparison of the solution obtained with the lumped model and the ‘exact’ one computed by standard finite element procedures. Copyright © 2011 John Wiley & Sons, Ltd.     

A shell problem ‘highly sensitive’ to thickness changes

In general, shell structural problems can be identified to fall into one of the categories of membrane‐dominated, bending‐dominated and mixed shell problems. The asymptotic behaviour with a well‐defined load‐scaling factor shows distinctly into which category a given shell problem falls. The objective of this paper is to present a shell problem and its solution for which there is no convergence to a well‐defined load‐scaling factor as the thickness of the shell decreases. Such shells are unduly sensitive in their behaviour because the ratio of membrane to bending energy stored changes significantly and indeed can fluctuate with changes in shell thickness. We briefly review the different asymptotic behaviours that shell problems can display, and then present the specific problem considered and its numerical solution using finite element analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

Derivation of geometrically nonlinear finite shell elements via tensor notation

To develop geometrically nonlinear, doubly curved finite shell elements the basic equations of nonlinear shell theories have to be transferred into the finite element model. As these equations in general are written in tensor notation, their implementation into the finite element matrix formulation requires considerable effort. The present paper will demonstrate how to derive the nonlinear element matrices directly from the incrementally formulated nonlinear shell equations using a tensor-oriented procedure. This enables the numerical realization of all structural responses, e.g. the calculation of pre- and post-buckling branches in snap-through analysis and especially in bifurcation analysis, including the detection of critical points and the consideration of geometric imperfections. To avoid loss of accuracy care is taken for a realistic computation of the geometric properties as well as of the external loads. Finally, the developed family of shell elements will be presented and its efficiency will be demonstrated by some applications to linear and geometrically nonlinear structural phenomena.     

Modelling the transient interaction of a thin elastic shell with an exterior acoustic field

Coupled finite and boundary element methods for solving transient fluid–structure interaction problems are developed. The finite element method is used to model the radiating structure, and the boundary element method (BEM) is used to determine the resulting acoustic field. The well‐known stability problems of time domain BEMs are avoided by using a Burton–Miller‐type integral equation. The stability, accuracy and efficiency of two alternative solution methods are compared using an exact solution for the case of a thin spherical elastic shell. The convergence properties of the preferred solution method are then investigated more thoroughly. Copyright © 2007 John Wiley & Sons, Ltd.     

Motorcyclist helmet composite outer shell characterisation and modelling

In this study a new finite element model of composite outer shell of motorcyclist helmet is proposed, by modelling each layer of the composite material that builds the laminated structure of the outer shell of the helmet. Elastic and rupture properties of the laminate are taken into account for developing the finite element (FE) model and are extracted experimentally. A coupled experimental–numerical method combined with experimental modal analysis on beam samples is used to obtain the elastic characteristics of each layer of the outer shell. The rupture properties for each layer are extracted by experimental impact tests. The FE model of the outer shell is then validated with experimental data for elastic and rupture behaviour.     

A fully non-linear axisymmetrical quasi-kirchhoff-type shell element for rubber-like materials

An axisymmetrical shell element for large deformations is developed by using Ogden's non-linear elastic material law. This constitutive equation, however, demands the neglect of transverse shear deformations in order to yield a consistent theory. Therefore, the theory can be applied to thin shells only. Eventually a ‘quasi-Kirchhoff-type theory’ emerges. Within this approach the computation of the deformed director vector d is a main assumption which is essential to describe the fully non-linear bending behaviour. Furthermore, special attention is paid to the linearization procedure in order to obtain quadratic convergence behaviour within Newton's method. Finally, the finite element formulation for a conical two-node element is given. Several examples show the applicability and performance of the proposed formulation.     

A continuum shell finite element model for impact simulation of woven fabrics

A new computational approach is developed to predict the impact behaviour of fabric panels based on the detailed response of the smallest repeating unit (unit cell) in the fabric. The unit cell is constructed and calibrated using measured geometrical (weave architecture, crimp, voids, etc.) and mechanical properties of the fabric. A pre-processor is developed to create a 3D finite element mesh of the unit cell using the measured fabric cross-sectional micro-images. To render an efficient method for simulation of multi-layer packs, these unit cells are replaced with orthotropic shell elements that have similar macroscopic (smeared) mechanical properties as the unit cell. The aim is to capture the essence of the response of a unit cell in a single representative shell element, which would replace the more complicated and numerically costly 3D solid model of the yarns in a crossover. The 3D finite element analysis of the unit cell is used to provide a baseline mechanical response for calibrating the constitutive model in the equivalent shell representation. This shell element takes advantage of a simple physics-based analytical relationship to predict the behaviour of the fabric's warp and weft yarns under general applied displacements in these directions. The analytical model is implemented in the commercial explicit finite element code, LS-DYNA, as a user material routine (UMAT) for shell elements. Layers of fabric constructed from these specialized elements are stacked together to create fabric targets that are then analysed under projectile impact. This approach provides an efficient numerical model for the dynamic analysis of multi-layer fabric structures while taking into account several geometrical and material attributes of the yarns and the fabric.     

A new shell formulation using complete 3D constitutive laws

This paper should not be only regarded as a presentation of new shell elements but rather as a methodology which can be applied to most classical shell elements and has two aims: Achieving the same results in bending cases while breaking from plane stress state hypothesis and adding a normal stress component for process simulations such as hydro‐forming, hemming, sheet metal forming with bottoming, flanging, incremental forming and so on. Owing to the non‐linear applications quoted before, only shell elements with one integration point on the mid‐plane are selected: Triangles that are naturally constant strain elements and reduced integration quadrilaterals. The method mainly consists of adding a central node at the center (of gravity for a triangle) with two degrees of freedom: Two translations normal to the mid‐surface for which one corresponds to the bottom surface (‘lower skin’ of the shell) and the other to the top surface (‘upper skin’ of the shell). Then a full 3D constitutive strain–stress behavior can be used. For triangles in bending state—either based on Kirchhoff's or on Mindlin's assumptions—, it is shown that the results are exactly the same as those given by the initial formulation of these elements using a plane stress hypothesis. For quadrilaterals, the results are slightly different but many numerical examples—including non‐linear computations—prove that those differences are not significant. Copyright © 2010 John Wiley & Sons, Ltd.