http://hub.hku.hk/bitstream/10722/54269/2/92945.pdf?accept=1

A survey of recent shell finite elements

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.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/0470091355.ecm026

Models and finite elements for thin-walled structures

Outside of the classical microstructural detail-free estimates of effective moduli, micromechanical analyses of macroscopically uniform heterogeneous media may be grouped into two categories based on different geometric representations of material microstructure. Analysis of periodic materials is based on the repeating unit cell (RUC) concept and the associated periodic boundary conditions. This contrasts with analysis of statistically homogeneous materials based on the representative volume element (RVE) concept and the associated homogeneous boundary conditions. In this paper, using the above classification framework we provide a critical review of the various micromechanical approaches that had evolved along different paths, and outline recent emerging trends. We begin with the basic framework for the solution of micromechanics problems independent of microstructural representation, and then clarify the often confused RVE and RUC concepts. Next, we describe classical models, including the available RVE-based models, and critically examine their limitations. This is followed by discussion of models based on the concept of microstructural periodicity. In the final part, two recent unit cell-based models, which continue to evolve, are outlined. First, a homogenization technique called finite-volume direct averaging micromechanics theory is presented as a viable and easily implemented alternative to the mainstream finite-element based asymptotic homogenization of unit cells. The recent incorporation of parametric mapping into this approach has made it competitive with the finite-element method. Then, the latest work based on locally-exact solutions of unit cell problems is described. In this approach, the interior unit cell problem is solved exactly using the elasticity approach. The exterior problem is tackled with a new variational principle that successfully overcomes the non-separable nature of the overall unit cell problem.

Micromechanics of spatially uniform heterogeneous media: A critical review and emerging approaches

A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains

Though the basic equations of the linear theory of elasticity are well known, it is rather cumbersome to evaluate the exact state of stress and displacement for a multilayered medium. In this paper we discuss a formal solution based on integral transforms and matrix analysis. Essentially the problem is reduced to the evaluation of some basic matrices and thus the calculation takes place in a clearly arranged way. Here we restrict ourselves to plane elastic and isotropic layers and use cartesian coordinates.

Theory of elasticity of a multilayered medium

A non-linear shell theory, including transverse shear strains, with exact description of the kinematical fields is developed. The strain measures are derived via the polar decomposition theorem allowing for an explicit use of a three parametric rotation tensor. Thus in-plane rotations, also called drilling degrees of freedom, are included in a natural way. Various alternatives of the theory are derived. For a special version of the theory, with altogether six kinematical fields, different mixed variational principles are given. A hybrid finite element formulation, which does not exhibit locking phenomena, is developed. Numerical examples of shell deformation at finite rotations, with excellent element performance, are presented. Comparison with results reported in the literature demonstrates the features of the theory as well as the proposed finite element formulation.

An exact finite rotation shell theory, its mixed variational formulation and its finite element implementation

Remark on the paper Configuration dependent pressure potentials by Dawn Fisher

A series of problems in mechanics and physics are governed by the ordinary Poisson equation which demands linearity, isotropy, and material homogeneity. In this paper a generalization with respect to nonliearity, anisotropy, and inhomogeneity is made. Starting from the canonical basic equations in the primal and dual formulation respectively we derive systematically the corresponding generalized variational principles; under certain conditions they can be extended to so calle complementary extremum principles allowing for global bounds. For simplicity a restriction to two dimensional problems is made, including twice-connected domains.

Hans Bufler

Papers

Theory of elasticity of a multilayered medium

An exact finite rotation shell theory, its mixed variational formulation and its finite element implementation

Remark on the paper Configuration dependent pressure potentials by Dawn Fisher

The generalized (nonlinear) poisson problem: a dual variational approach