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

Abstract

A 3-node co-rotational triangular elasto-plastic shell element is developed. The local coordinate system of the element employs a zero-'macro spin' framework at the macro element level, thus reducing the material spin over the element domain, and resulting in an invariance of the element tangent stiffness matrix to the order of the node numbering. The two smallest components of each nodal orientation vector are defined as rotational variables, achieving the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing co-rotational finite-element formulations, both element tangent stiffness matrices in the local and global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of strain energy with respect to the local nodal variables and, through chain differentiation with respect to the global nodal variables. For elasto-plastic analysis, the Maxwell-Huber-Hencky-von Mises yield criterion is employed together with the backward-Euler return-mapping method for the evaluation of the elasto-plastic stress state, where a consistent tangent modulus matrix is used. Assumed membrane strains and assumed shear strains---calculated respectively from the edge-member membrane strains and the edge-member transverse shear strains---are employed to overcome locking problems, and the residual bending flexibility is added to the transverse shear flexibility to improve further the accuracy of the element. The reliability and convergence of the proposed 3-node triangular shell element formulation are verified through two elastic plate patch tests as well as three elastic and three elasto-plastic plate/shell problems undergoing large displacements and large rotations.

A Co-Rotational Framework for Quadrilateral Shell Elements Based on the Pure Deformational Method

“…Similar to this simpler co-rotational framework, Izzuddin [19] defined a new local co-rotational system and adopted vectorial rotation for quadrilateral shell elements, through which not only the invariance to node ordering but also a symmetric tangent stiffness matrix can be achieved. Pertaining studies on shell elements with this co-rotational framework were also presented in references (Li and Vu-Quoc [20]; Li et al [21]; Li et al [22]; Li et al [23]; Li et al [24]; Izzuddin and Liang [25]; Li et al [26]). …”

A 4-Node Co-Rotational Quadrilateral Composite Shell Element

“…Accordingly, the transformation matrix is expressed as where the sub-matrices of T are the same as those in Ref. 39. The element tangent sti®ness matrix k TG in the global coordinate system can now be obtained as follows: …”

“…31,32 The versatile vectorial rotational variables had also been employed in a 4-node co-rotational°at quadrilateral shell element with hierarchic freedoms, 33 two co-rotational beam element formulations, 34,35 and co-rotational triangular and quadrilateral shell element formulations. [36][37][38][39] To overcome locking problems, the assumed membrane strains and transverse shear strains in the present 4-node co-rotational quadrilateral shell element are interpolated respectively by using ANS methods. 40,41 The outline of the paper is arranged as follows.…”

A nine‐node corotational curved quadrilateral shell element for smooth, folded, and multishell structures

“…At any node of a smooth shell or at a node away from the intersection of nonsmooth shells, the two smallest components of the midsurface normal vector in the global coordinate system are selected as vectorial rotational variables . Such vectorial rotational variables had been successfully employed in developing four‐node and nine‐node quadrilateral elements, three‐node, and six‐node triangular elements accommodating elastic or elastoplastic or composite behavior for smooth shells undergoing large displacement and large rotations . On the other hand, at a node on the intersection edge of nonsmooth shells, we introduce a novel treatment of rotation (without using an axial rotation vector) by using two smallest components of one vector and one smaller component of another vector of a triad oriented initially to three axes of the global coordinate system as vectorial rotational variables.…”

“…34 Such vectorial rotational variables had been successfully employed in developing four-node and nine-node quadrilateral elements, three-node, and six-node triangular elements accommodating elastic or elastoplastic or composite behavior for smooth shells undergoing large displacement and large rotations. [35][36][37][38][39][40] On the other hand, at a node on the intersection edge of nonsmooth shells, we introduce a novel treatment of rotation (without using an axial rotation vector) by using two smallest components of one vector and one smaller component of another vector of a triad oriented initially to three axes of the global coordinate system as vectorial rotational variables. These vectorial rotational variables could be different components of the same (polar/proper) vector(s) even at the same node in different incremental steps of a nonlinear incremental solution procedure.…”