Tangent stiffness matrix

About: Tangent stiffness matrix is a research topic. Over the lifetime, 1031 publications have been published within this topic receiving 21140 citations.

A four-node corotational quadrilateral elastoplastic shell element using vectorial rotational variables

Abstract: SUMMARY A four-node corotational quadrilateral elastoplastic shell element is presented. The local coordinate system of the element is defined by the two bisectors of the diagonal vectors generated from the four corner nodes and their cross product. This local coordinate system rotates rigidly with the element but does not deform with the element. As a result, the element rigid-body rotations are excluded in calculating the local nodal variables from the global nodal variables. The two smallest components of each nodal orientation vector are defined as rotational variables, leading to the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing corotational finite-element formulations, the resulting element tangent stiffness matrix is symmetric owing to the commutativity of the local nodal variables in calculating the second derivative of strains with respect to these variables. For elastoplastic analyses, the Maxwell–Huber–Hencky–von Mises yield criterion is employed, together with the backward-Euler return-mapping method, for the evaluation of the elastoplastic stress state; the consistent tangent modulus matrix is derived. To eliminate locking problems, we use the assumed strain method. Several elastic patch tests and elastoplastic plate/shell problems undergoing large deformation are solved to demonstrate the computational efficiency and accuracy of the proposed formulation. Copyright © 2013 John Wiley & Sons, Ltd.

A non‐linear numerical method for accurate determination of limit and bifurcation points

Abstract: This paper presents a numerical procedure for accurate determination of a limit or a bifurcation point. The method minimizes simultaneously the first and the second variations of an admissible functional or iterates to satisfy the equilibrium and the semi definite condition for the tangent stiffness matrix. It can be readily incorporated into a computer program for non-linear finite element analysis to improve its accuracy in the location of critical points.

Linearization and return mapping algorithms for elastoplasticity models

Abstract: The return mapping algorithm is one of the most efficient procedures to solve elasto-plastic problems. However, a criticism that may be lodged against this method is the difficulty of the practical computation of the consistent tangent matrix when the return is non-radial. Much research has been done to handle this matrix. In this paper, a unified approach is presented in such a way that a simple closed-form expression gives the consistent tangent matrix for the classical constitutive relations (von Mises, Tresca, Mohr–Coulomb, Drucker–Prager). The basic ideas are in the properties of eikonal equations appearing in several fields as image treatment, short time computation in elastic waves and others. The same kinds of ideas can be extended to non-classical models. Copyright © 2003 John Wiley & Sons, Ltd.

A mixed co-rotational 3d beam element formulation for arbitrarily large rotations

Abstract: A new 3-node co-rotational element formulation for 3D beam is presented. The present formulation differs from existing co-rotational formulations as follows: 1) vectorial rotational variables are used to replace traditional angular rotational variables, thus all nodal variables are additive in incremental solution procedure; 2) the Hellinger-Reissner functional is introduced to eliminate membrane and shear locking phenomena, with assumed membrane strains and shear strains employed to replace part of conforming strains; 3) all nodal variables are commutative in differentiating Hellinger-Reissner functional with respect to these variables, resulting in a symmetric element tangent stiffness matrix; 4) the total values of nodal variables are used to update the element tangent stiffness matrix, making it advantageous in solving dynamic problems. Several examples of elastic beams with large displacements and large rotations are analysed to verify the computational efficiency and reliability of the present beam element formulation.