An adaptive Strategy for the Multi-scale Analysis of Plate and Shell Structures with Elasto-plastic Material Behaviour

Modelling of structures on different scales has been a popular subject in the past. Within such a strategy the structural behaviour is modeled on a macro-level, describing the structure itself, whereas the material behaviour is modeled on a micro-level. Here typically RVEs are used. The proper choice of boundary conditions for the RVE is a difficult task in case of shell structures. It should be mentioned that the correct calculation of material parameters on the macro level is crucial for any associated nonlinear analysis. Here, results have been presented for homogeneous and layered structures for composite materials in Gruttmann and Wagner (2013). In the present paper we will discuss the influence of material nonlinear behaviour, here the elasto-plastic behaviour, within the above described setting. Typically these calculations are very time consuming, even if the FE-model is parallelized. Thus we will discuss possibilities to change material models on the structural model during the loading process, starting with elastic material models without a second scale and switching to a two-scale approach, where necessary

An advanced shell model for the analysis of geometrical and material nonlinear shells

In this paper layered shells subjected to static loading are considered. The displacements of the Reissner–Mindlin theory are enriched by a an additional part. These so-called fluctuation displacements include warping displacements and thickness changes. They lead to additional terms for the material deformation gradient and the Green–Lagrangian strain tensor. Within a nonlinear multi-field variational formulation the weak form of the boundary value problem accounts for the equilibrium of stress resultants and couple resultants, the local equilibrium of stresses, the geometrical field equations and the constitutive equations. For the independent shell strains an ansatz with quadratic shape functions is chosen. This leads to a significant improved convergence behaviour especially for distorted meshes. Elimination of a set of parameters on element level by static condensation yields an element stiffness matrix and residual vector of a quadrilateral shell element with the usual 5 or 6 nodal degrees of freedom. The developed model yields the complicated three-dimensional stress state in layered shells for elasticity and elasto-plasticity considering geometrical nonlinearity. In comparison with fully 3D solutions present 2D shell model requires only a fractional amount of computing time