A large‐strain elasto–plastic shell formulation using the Morley triangle

Abstract

SUMMARYThe paper describes a large-strain elasto-plastic formulation for shells, which is based on the faceted 'Morley triangle', with rotation variables only being provided at each mid-side in the form of a rotation about the side. A co-rotational formulation is adopted, with the local deformation gradient being obtained from a polar decomposition using F = RU. The material description is then based around 'log e U', with U being decomposed into principal directions. Plasticity is treated with the aid of the multiplicative F = FeFp. In contrast to many earlier formulations with the Morley triangle, the current formulation is invariant to the node numbering. In order to achieve this invariance, it is necessary to re-visit the origins of the Morley triangle and to re-cast the formulation as a special form of 'discreteKirchho approach'. The associated constraint is now applied in relation to the 'current conÿguration'.

A finite-strain quadrilateral shell element based on discrete Kirchhoff-Love constraints

“…Note that s i3 , s 33 and s 3i are not necessarily zero, but rather do not contribute to the internal work. As will be seen later, they are Lagrange multipliers for the KL constraints.…”

“…The director inextensibility, which is imposed by use of spherical co-ordinates as described later, eliminates the presence of s 33 (e.g. it is imposed by the use of a 'co-ordinate transformation technique').…”

“…Most previous work in shells with mid-side rotations was employed either triangles [24,27,32,33] or planar quadrilaterals [25,34], often under very restrictive assumptions (e.g. Reference [30]) such as moderate rotations and co-rotational formulations based on the super-position of a plane element with a plate bending element, valid for triangular elements only.…”

A 3-Node Co-Rotational Triangular Elasto-Plastic Shell Element Using Vectorial Rotational Variables

“…Existing 3-node triangular shell elements can be categorized into 4 types: Type 1 with only 3 displacement degrees-of-freedom (dofs) per node [14][15][16][17][18][19][20][21][22]; Type 2 with 3 displacement dofs and 2 rotational dofs per node [23][24][25]; Type 3 with 3 displacement dofs at the vertices and the rotational dofs at side nodes [26][27]; Type 4 with 3 displacement dofs and 3 rotational dofs per node [28][29][30][31][32][33]. Most existing 3-node rotation-free triangular shell elements are based on the Kirchhoff assumption, ignoring shear deformation, can therefore only be employed in modeling thin plate and shell problems [21].…”

“…In addition, flat shell elements with 5 degrees-of-freedom per node lack proper nodal degrees of freedom to model folded plate/shell structures, making the assembly of elements troublesome [35]. To avoid the singularity of the assembled global stiffness matrix, some researchers defined the displacement degrees-of-freedom at the vertices and the rotational degrees-of-freedom at side nodes of flat shell elements [26][27]. This type of elements cannot, however, be easily matched with other types of elements in modeling of complex structures.…”