Localization phenomena & FEM

J. Wackerfuß

In the field of material research, the term 'localization' represents processes which are characterized by a concentration of inelastic deformations in narrow zones within solids; beyond these zones, pure elastic behavior can normally be observed. The appearance of this phenomenon depends primarily on the specific characteristics of the material. Under experimental conditions, the following kinds of localizations can be observed: shear bands in ductile materials, slip surfaces in granular materials and cracks in brittle materials.

A continuum mechanical description based on standard stress-strain constitutive equations including strain-softening is not suitable to describe localization phenomena, since the type of the partial differential equations governing the problem changes when the localization appears; this finally leads to an ill-posed boundary value problem. In this case a finite-element calculation reveals that there is a pathological dependency of the results on the spatial discretization.

In this research project different models for the regularization of this problem are presented. In addition to extended continuum theories, which are primarily based on the consideration of a characteristic length in the material model, discontinuous models have also been investigated. They interpret the localization zone as a singular surface inside the body which contains certain kinematic discontinuities. With the help of numerical tests, a regularizing effect of the models which were tested could be demonstrated. The results did not reveal the presence of any pathological dependency on the chosen spatial discretization.

When describing structural-mechanical localization phenomena with discontinuous models, the displacement field is decomposed into a continuous and a discontinuous part. For the approximation of the latter, special approach functions are required within the framework of a finite element formulation. The choice of these approach functions is responsible for the character of the respective method. In general, a distinction is made between the embedded discontinuities method and the extended finite element method (XFEM). While the additional unknowns can be condensed out at the element level in the former method, they must be considered as global nodal values in the latter. A suitable finite element formulation must basically lead to a stable calculation in which the numerical results converge towards a fixed solution with increasing discretization density - independent of the selected orientation of the element mesh.

In experimental investigations on metallic bodies, so-called shear bands (Lüders bands) are observed when a critical stress state is exceeded. The figure on the right shows the results of a numerical simulation of a vertically compressed metal disk in animated form. Within the finite element calculation, different spatial discretizations were investigated. Both the number and the orientation of the elements were varied. As a representative example, the deformed meshes for different loading conditions are shown here for a structured (left) and an unstructured (right) element mesh. The corresponding vertical component of the displacement field is indicated in color. Regardless of the element mesh selected, the diagonally developing shear band can be clearly seen, which suddenly appears when a certain stress is exceeded.

Continuous models:

In experimental investigations of brittle material behavior, the formation and growth of microcracks are observed after a limit stress is exceeded. When using continuum mechanical models to describe localization phenomena on a macroscopic level, such cracks are not described discretely, but continuously. Continuous models are essentially based on the assumption of an extended classical continuum theory characterized by an extended set of independent variables, which ultimately introduces a measure of the thickness of the localization zone.

• The use of rate-dependent material models offers one possibility for regularization. In this project, the free energy function describing a damaging (rate-independent) material behavior was extended by a viscous (rate-dependent) term, which is assigned only the task of numerical stabilization or regularization. To avoid unwanted overstresses, it is necessary to increase the stress not linearly but in a staircase manner within the framework of an incremental load increase. The numerical simulation of a one-dimensional tensile test is used to explain how the method works. The (global) load-displacement curves shown in the following figures show that the rate-independent model (based on classical continuum mechanics) exhibits a strong mesh dependence in the post-critical region for different element discretizations (left), while the relaxed solution of the stabilized model (viscous stabilization) reproduces the exact solution, independent of the chosen discretization. The relaxed solution is characterized by a state of complete decay of the (viscous) overstresses, which must be achieved after each load increase - by a sufficiently long relaxation time. It should be noted that the regularizing effect of the process is very much dependent on the choice of viscosity parameters.

• An alternative possibility of regularization is given by the extension of the classical continuum model in the sense of a micropolar continuum. While in the framework of a classical continuum theory (Boltzmann continuum) three translational degrees of freedom are assigned to each material point, in the micropolar continuum theory (Cosserat continuum) 3 rotational degrees of freedom are additionally considered. The following diagrams compare the results of different finite element simulations for a simple shear test on a metallic disk. The left figure shows the global load-displacement curves for a plasticity model with linear isotropic hardening for different discretizations. It can be seen that as the discretization density increases, the results converge towards a fixed solution; this is true for both the Boltzmann and Cosserat models. A fundamentally different behavior is obtained for the plasticity model with linear isotropic softening. While the results of the Cosserat model converge towards a fixed solution with increasing discretization density, this is not the case for the Boltzmann model; here one obtains a strong dependence of the numerical results on the chosen spatial discretization. In this case, the calculation becomes numerically unstable and sometimes breaks off after a few load steps. On the basis of further numerical investigations, it was found that the regularizing effect of the model is very strongly dependent on the choice of the additional (Cosserat) material parameters, which in turn have a strong influence on the elastic or pre-critical load-bearing behavior.

Wackerfuß, J.: Numerische Beschreibung von Lokalisierungsphänomenen unter Berücksichtigung von diskontinuierlichen Verschiebungen, In N. Gebbeken, K.-U. Bletzinger, H. Rothert (editor): Aktuelle Beiträge aus Baustatik und Computational Mechanics, pp. 109–122, Universität der Bundeswehr München, Berichte aus dem Konstruktiven Ingenieurbau [03/3], 2003

Wackerfuß, J.: Theoretische und numerische Beiträge zur Beschreibung von Lokalisierungsphänomenen in der Strukturmechanik, Dissertation, Shaker-Verlag, Juni 2005