GEOMETRICAL AND PHYSICAL NONLINEARITIES SOME DEVELOPMENTS IN THE NETHERLANDS

By finite element elastic-plastic analysis it has been possible to obtain detailed information on the deformation process in the vicinity of a growing crack in sheet metal. It is found that the crack grows when the value of the crack tip opening angle exceeds some critical value. Particular attention is given to the closure behaviour if the loading forces are reduced. The results obtained are compared with experimental data. A flat triangular element with initial curvature can be looked upon as a doubly curved shell element. Thus a shell element is developed with nodal displacements in the three corner points and in the three midside points and with two parameters for rotations about each side. Element properties can be derived from quadratic displacement functions, linearly varying bending moments and linearly varying membrane deformations, the latter as an interpolation between the values in the corners. These corner values are determined by the appropriate nonlinear expressions in the displacements. On the basis of shell theory and the principle of minimum potential energy the merits of this particular finite element were thoroughly investigated. In view of its accuracy in linear problems, and since rigid body motions are fully represented, the element will also be very suitable for geometrically nonlinear problems. Though for the derivation of the features and properties of the finite elements the continuum theory will be used, in the analysis of a structure finite elements may be looked upon as models of mechanical behaviour in their own right. This approach proved to be particularly useful in the kinematic and dynamic analysis of mechanisms. Examples will be given of the kinematic description of some special elements, such as gears. Also for stability analysis the discrete approach was found to be fruitful of results. With properly chosen deformation parameters as generalized strains the buckling and postbuckling behaviour, as well as the nonlinear behaviour in general with small but finite rotations, can be analysed for struts, plates and shells, whilst the physical linearity on the level of the finite elements is retained. An effective elastic analysis of nonlinear problems is obtained in terms of perturbation equations for the participation factors of eigenvectors. These equations, limited in number, can be solved by a special method that was devised for the computation of critical equilibrium states of nonlinear elastic systems. In this method difficulties due to matrix singularity in the incremental equations are overcome by the introduction of the path length as independent variable. © 1979.