On FEM analysis of Cosserat-type stiffened shells: static and stability linear analysis

The present research investigates the theory and numerical analysis of shells stiffened with beams in the framework based on the geometrically exact theories of shells and beams. Shell’s and beam’s kinematics are described by the Cosserat surface and the Cosserat rod, respectively, which are consistent including deformation and strain measures. A FEM approximation of the virtual work principle leads to the conforming shell and beam FE with 6 DoFs (including the drilling rotation for shells) in each node. Examples of static and stability linear analyses are included. Novel design formulas for the stability of stiffened shells are included.

A New Method for Analysing Large Elasto-Plastic Deformations of a Thin Cosserat Shell

One of the best approaches for modelling the large deformation of shells is the Cosserat surface; however, the finite-element implementation of this model suffers from membrane and shear locking, especially for very thin shells. If the director vector is constrained to remain perpendicular to the mid-surface, during deformation, locking will be prevented. This constraint is in fact a limiting analysis of the Cosserat theory in which Kirchhoff's hypothesis is enforced. This has been considered for the first time. Simo's plastic approach is modified to implement the constrained director. This model includes both kinematic and isotropic hardening behaviours. A consistent elasto-plastic tangent modular matrix is extracted. Numerical solution is performed by interpolation of displacement on the whole domain, and a hierarchical finite-element scheme is developed. The principle of virtual work is used to obtain the weak form of the governing differential equations and the material and geometric stiffness matrices are derived through a linearization process. The validity and the accuracy of the method are illustrated by numerical examples.

Method of Virtual Power Applied to Cosserat Surfaces With Deformable Directors

Following a brief outline of the method of virtual power, the local equations of motion for a Cosserat surface with inextensible directors are derived by means of this method. The model obtained coincides with the results derived from three-dimensional theory by Simo and Fox. Subsequently the model is extended so as to account for deformable directors. Besides the linear-momentum and moment-of-momentum balance equations, one additional scalar equation is derived. This equation replaces the director-momentum balance equation of Naghdi and therefore eliminates the necessity of introducing constitutive restrictions. The equivalence between the model derived by the virtual-power method and the results from the direct method of Naghdi are finally noted.

On thermal effects in the theory of shells

This paper is concerned with thermomechanics of thin shells by a direct approach based on the theory of a Cosserat surface comprising a two-dimensional surface and a single director attached to every point of the surface. In almost all previous developments of the thermo-mechanical theory of shells by direct approach, only one temperature field has been admitted. This allows for the characterization of temperature changes along some reference surface, such as the middle surface, of the (three-dimensional) shell-like body, but not for temperature changes along the shell thickness. A main purpose of the present study is to incorporate the latter effect into the theory; and, in the context of the theory of a Cosserat surface, this is achieved by a recent approach to thermomechanics (Green & Naghdi 1977) which provides a natural way of introducing two (or more) temperature fields at each material point of the surface. Apart from full discussion of thermomechanics of shells and thermodynamical restrictions arising from the second law of thermodynamics for shells, attention is given to a discussion of symmetries (including material symmetries) and thermal effects in the nonlinear theory of elastic shells with detailed discussion of the linear theory of elastic plates.

Wave Propagation in the Linear Theory of Elastic Shells

The problem of wave propagation in elastic shells within the framework of a linear theory of a Cosserat surface is treated using the method of singular wave curves. The equations for determining the speeds of propagation and their associated wave mode shapes are obtained in a form involving the speeds of propagation in Cosserat plates and the curvature of the shell. A number of special cases in which the speeds and mode shapes simplify are considered. In particular, these special cases are shown to include as examples, certain systems of waves in elastic shells whose middle surfaces are the surface of revolution, the circular cylinder, the sphere, and the right helicoid.