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 Few Good Reasons to Consider a Beam Finite Element Formulation on the Lie Group SE(3)

Abstract: Based on an original interpolation method we develop a beam finite element formulation on the Lie group SE(3) which relies on a mathematically rigorous framework and provides compact notations. We work out the beam kinematics in the SE(3) context, the beam deformation measure and obtain the expression of the internal forces using the virtual work principle. The proposed formulation exhibits important features from both the theoretical and numerical points of view. The approach leads to a natural coupling of position and rotation variables and thus differs from classical Timoshenko/Cosserat formulations. We highlight several important properties such as a constant deformation measure over the element, an invariant tangent stiffness matrix under of rigid motions or the absence of shear locking.Copyright © 2013 by ASME

Inelastic and non-linear materials

Abstract: In Chapter 2 we presented a framework for solving general problems in solid mechanics. In this chapter we consider several classical models for describing the behaviour of engineering materials. Each model we describe is given in a strain-driven form in which a strain or strain increment obtained from each finite element solution step is used to compute the stress needed to evaluate the internal force, σ B T σ dΩ, as well as a tangent modulus matrix, or its approximation, for use in constructing the tangent stiffness matrix. Quite generally in the study of small deformation and inelastic materials (and indeed in some forms applied to large deformation) the strain (or strain rate) or the stress is assumed to split into an additive sum of parts. We can write this as e=e e +e i e = e e + e i (4.1) or σ=σ e +σ i σ = σ e + σ i (4.2) in which we shall generally assume that the elastic part is given by the linear model e e = D -1 σ e e = D - 1 σ (4.3) in which D is the matrix of elastic moduli.

Contribution à la Modélisation de Composites 2D/3D à l'Aide d'Eléments Finis Spéciaux.

Abstract: This doctoral thesis deals with the finite element formulation and evaluation of a modified Mindlin's discrete variational model for static, dynamic, linear and non-linear composite plates and shells analysis Including additional terms of zigzag type, in order to improve the accuracy of stress, the model has been reformulated to take into account the linear picewise of displacement variation Consequently, two finite plate and shell elements with four nodes, called DMQP and DMQS (Discrete Quadrilateral Mindlin Plates and Shells respectively), enhanced by quadratic field rotations, have been developed and validated under REFLEX and ABAQUS codes Both elements including the zigzag effect have been also developed in a second version, and validated through several static and dynamic test problems known from the literature, highlighting the independence towards the transverse shear correction and in particular the stress accuracy with respect to the initial model without the zigzag effect The satisfactory results of this model found through cases of linear isotropic shell tests, motivated us to extend this approach to the non-linear geometric applications An isoparametric curve element of shell has been developed for this purpose, where small elastic deformation assumptions of and large displacements and moderate rotations are adopted It is geometrically simple and has only four nodes at corners and 6 DOF/node The elementary calculation of the tangent stiffness matrix consists in combining the linear part of the curved shell element (DMQS) with that of the membrane Q4 non-linear part An Updated Lagrangian Formulation at each Iteration (ULFI) is used with Newton-Raphson resolution Method Some standard tests of nonlinear geometrical shell structures are presented; they show a very good convergence and global behavior better than such elements

Branch Switching for Pitchfork Bifurcation in Nonlinear Finite Element Method

Abstract: A general branch-switching approach is proposed, based on pseudo arc-length continuation in the nonlinear finite element method (FEM), to trace the branch solution curve at the pitchfork bifurcation point in parameter space. By this approach, branch direction can be determined without derivatives of tangent stiffness matrix. This approach is proved mathematically and is inserted into a general FEM code. Stability of equilibrium corresponding to each solution curve is estimated by the Lagrange-Dirichlet criteria. As an example, the whole process of the elastic mode jumping for rectangular thin plates, in which there are many pitchfork bifurcations in parameter space in loading and unloading, is simulated by the improved FEM code.The numerical results are identical with that of an experiment done by previous researchers.