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SUMMARYStarting from continuum mechanics principles, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non-linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.

SUMMARYThe enhanced assumed strain (EAS) method, recently proposed by Simo and Rifai13, is used to develop new four-node membrane, plate and shell elements and eight-node solid elements. The equivalence of certain EAS-elements with Hellinger-Reissner (HR) elements is discussed. For instance, the seven-parameter element EAS-7 with 2 x 2 integration is identical to the HR-element of Pian and Sumihara'. Eight-node solid elements which are free of volumetric locking and four-node shell elements which have an improved membrane and bending behaviour, compared to the Bathe-Dvorkin shell element', are introduced. Numerical tests for linear elastic problems show an improved performance of the EAS-elements.

SUMMARYConventional shell formulations, such as 3-or 5-parameter theories or even 6-parameter theories including the thickness change as extra parameter, require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness. This means that the normal stress in thickness direction has to either vanish or be constant. In general, these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions.The main objective of the present study is to introduce directly a complete 3-D constitutive law without modification. Therefore, a 7-parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects. In order to preserve the basic features of a displacement formulation the extra strain term is incorporated via the enhanced assumed strain concept recently proposed by Simo and Rifai to improve the performance of finite elements. Since the extra strain parameter can be eliminated on the element level after discretization, the formulation preserves the formal structure of a 6-parameter shell theory. The resulting hybrid-mixed shell formulation is applied to large deformation problems with hyperelasticity, small and large strain plasticity. OBJECTIVEThe present study is restricted to a non-linear shell formulation with linear displacement variation across the thickness. It is a standard procedure in 3-parameter theories (with KirchhoffLove hypothesis) as well as 5-parameter theories (e.g. with Reissner-Mindlin kinematics) to condense the 3-D constitutive equations by employing the assumption of vanishing Cauchy stresses in thickness direction ~3~ = 0. Consequently, the thickness stretch is a dependent variable which for compressible materials can be determined by the zero normal stress assumption. The condensation may be directly incorporated, e.g. applying projection methods in classical small strain plasticity.'*2 For reinforced concrete shells it is even common to restrict the non-linear material law to a plane stress situation, i.e. the material influence of the transverse shear stress is neglected or at best roughly appr~ximated.~ A step back into the 3-D continuum is necessary if local 3-D effects become relevant in shell structures, i.e. if stresses and strains in thickness direction are of significant importance. As examples the concentrated loading in reinforced concrete shells or the delamination in composites are mentioned. This means that a shell theory has to be extended into a 6-parameter formulation including a constant normal stress in thickness direction. The result of this extension is in turn that the bending part of the constitutive law has to be condensed to invoke the constant stress assumption (e.g. Reference 4). Otherwise, even in a linear theory a relative error in the order CCC

Well-known finite element concepts like the Assumed Natural Strain (ANS) and the Enhanced Assumed Strain (EAS) techniques are combined to derive efficient and reliable finite elements for continuum based shell formulations. In the present study two aspects are covered:The first aspect focuses on the classical 5-parameter shell formulation with Reissner-Mindlin kinematics. The above-mentioned combinations, already discussed by Andelfinger and Ramm for the linear case of a four-node shell element, are extended to geometrical non-linearities. In addition a nine-node quadrilateral variant is presented. A geometrically non-linear version of the EAS-approach is applied which is based on the enhancement of the Green-Lagrange strains instead of the displacement gradient as originally proposed by Simo and Armero.In the second part elements are derived in a similar way for a higher order, so-called 7-parameter non-linear shell formulation which includes the thickness stretch of the shell (Bu¨chter and Ramm). In order to avoid artificial stiffening caused by the three dimensional displacement field and termed 'thickness locking', special provisions for the thickness stretch have to be introduced.

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The present study provides an overview of modeling and discretization aspects in finite element analysis of thin‐walled structures. Shell formulations based upon derivation from three‐dimensional continuum mechanics, the direct approach, and the degenerated solid concept are compared, highlighting conditions for their equivalence. Rather than individually describing the innumerable contributions to theories and finite element formulations for plates and shells, the essential decisions in modeling and discretization, along with their consequences, are discussed. It is hoped that this approach comprises a good amount of the existing literature by including most concepts in a generic format. The contribution focuses on nonlinear finite element formulations for large displacements and rotations in the context of elastostatics. Although application to dynamics and problems involving material nonlinearities is straightforward, these subjects are not taken into account explicitly.

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