M. A. Crisfield

Energy‐conserving and decaying Algorithms in non‐linear structural dynamics

A generalized formulation of the Energy-Momentum Methodwill be developed within the framework of the GeneralizedMethodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response. Furthermore, the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the Constraint Energy-Momentum Algorithm. The goal of this general approach of implicit energyconserving and decaying time integration schemes is, to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties. Numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non-linear structural dynamics. Among the methods considered will be the Newmark Method, the classical -methods, the Energy-Momentum Methodwith and without numerical dissipation, the Constraint EnergyMomentum Algorithm and the Constraint Energy Method. Copyright ? 1999 John Wiley & Sons, Ltd.

Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics

Geometrically exact 3D beam theory has been used as a basis for development of a variety of finite element formulations. It has recently become apparent that the important requirement of objectivity of adopted strain measures, although provided by the theory itself, does not automatically extend to a finite element formulation. In this paper we present a new finite element formulation of the geometrically exact 3D beam theory, specifically designed to preserve the objectivity of the adopted strain measures. In order to do so the current local rotations are interpolated in a manner similar to that adopted in co-rotational approaches. However, no approximations typical for co-rotational approaches are introduced into the procedure, so in contrast to co-rotational formulations, the present formulation fully preserves the geometric exactness of the theory. A range of numerical examples serves to illustrate the problem and to assess the formulation.

OBJECTIVITY OF STRAIN MEASURES IN THE GEOMETRICALLY EXACT THREE-DIMENSIONAL BEAM THEORY AND ITS FINITE-ELEMENT IMPLEMENTATION

The paper discusses the issue of discretization of the strain–configuration relationships in the geometrically exact theory of three–dimensional (3D) beams, which has been at the heart of most recent nonlinear finite–element formulations. It is demonstrated that the usual discretization procedures for implementing these strain measures within a finite–element framework violate the important property of objectivity: invariance to rigid body rotations. A method is proposed for overcoming this limitation, which paves the way for an objective finite–element formulation of the geometrically exact 3D beam theory. For a two–noded element, this method involves obtaining the relative rotation matrix that rotates one nodal triad onto the other and then interpolating the resulting rotation vector.

An interface-element formulation for the simulation of delamination with buckling

The paper describes a simple corotational formulation applied to one-dimensional interface elements which embed a fracturing procedure for mixed-mode delaminations. Having thereby introduced geometric non-linearity, the technique can be applied to situations involving a combination of buckling and delamination. Detailed comparisons are made with experimental results for such a problem.

INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS

The formulation of dynamic procedures for three-dimensional (3-D) beams requires extensive use of the algebra pertaining to the non-linear character of the rotation group in space. The corresponding extraction procedure to obtain the rotations that span a time step has certain limitations, which can have a detrimental eect on the overall stability of a time-integration scheme. The paper describes two algorithms for the dynamics of 3-D beams, which dier in their manifestation of the above limitation. The rst has already been described in the literature and involves the interpolation of iterative rotations, while an alternative formulation, which eliminates the above eect by design, requires interpolation of incremental rotations. Theoretical arguments are backed by numerical results. Similarities between the conventional and new formulation are pointed out and are shown to be big enough to enable easy transformation of the conventional formulation into the new one. ? 1998 John Wiley & Sons, Ltd.

Re-Visiting the Contact Patch Test

The paper uses the contact patch test to assess both linear and quadratic elements in relation to straight and curved contact surfaces. A new contact formulation is proposed which uses a combination of linear and quadratic shape functions. The potential advantages of the new formulation are demonstrated. Copyright

Dynamic analysis of 3D beams with joints in presence of large rotations

Abstract In this paper we present a way to extend the earlier static master–slave formulation for beams and joints [1] to dynamic problems. The dynamic master–slave approach is capable of (i) handling the problems of linear elasticity in a geometrically non-linear environment, (ii) accounting for the non-linear kinematics of arbitrary types of joints and (iii) performing the numerical time integration while preserving some of the important constants of motion like total energy and the total momenta for Hamiltonian systems in the absence of external loading. The performance of the formulation is demonstrated with the aid of two representative numerical examples.

Enhanced lower-order element formulations for large strains

The paper describes a range of lower-order element formulations that can be applied to both elastic and elasto-plastic large-strain elements. For plane-strain analysis, this process involves four-noded quadrilaterals while the enhancements involve incompatible modes or enhanced strains. One particular new formulation can be considered as either a “co-rotational approach” or a modified form of “Biot stress procedure”.

Some aspects of the non-linear finite element method

Abstract The paper discusses four separate aspects of the non-linear finite element method: (i) An alternative formulation for the static co-rotational technique in conjunction with a simple faceted shell idealisation. (ii) Solution procedures for non-linear dynamics with emphasis on energy conserving techniques (iii) The use of interface elements and fracture energy related softening “stress-strain” curves for modelling mixed mode delamination in “composites” (iv) Hybrid static/dynamic solution procedures. While the topics are separate, there are links and these are explored.

Non-linear ‘master-slave’ relationships for joints in 3-D beams with large rotations

A novel approach is presented for the analysis of spatial beam elements with end releases, in which, for each joint in the structure, an additional (slave) set of kinematic variables is introduced, which is directly related to the existing (master) set of variables at that node. This relationship is established according to the nature of the joint to be modelled and takes into account that the sliding/rotation takes place along/around the axis that is rigidly attached to the structural node and is thus not fixed in space. In this way, the most interesting releases such as revolute, spherical, prismatic and cylindrical joints can be analysed accurately and efficiently. The main concepts are applicable to any spatial beam finite element, provided that a displacement vector and a rotation matrix are defined at both end nodes. The numerical examples, in which a highly deformable space frame with different joints is analysed, demonstrates the accuracy of the proposed procedure and its advantages compared to the ‘penalty’ technique.