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 behaviour 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 divided into a continuous and a discontinuous part. To approximate 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 discontinuity method and the extended finite element method (XFEM). While the additional unknowns can be condensed out at element level in the former method, they must be taken into account as global node values in the latter method. A suitable finite element formulation must always lead to a stable calculation in which the numerical results converge to a fixed solution with increasing discretization density - regardless 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 adjacent figure shows the results of a numerical simulation of a vertically compressed metal disk in animated form. Different spatial discretizations were investigated as part of the finite element calculation. Both the number and the orientation of the elements were varied. The deformed meshes for different stress states are shown here for a structured (left) and an unstructured (right) element mesh. The corresponding vertical component of the displacement field is shown in color. Regardless of the selected element mesh, the diagonally developing shear band that suddenly appears when a certain stress is exceeded can be clearly seen.

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 for the thickness of the localization zone.

• One possibility of regularization is the use of rate-dependent material models. Within the scope of the project, the free energy function for the description of a damaging (rate-independent) material behaviour was extended by a viscous (rate-dependent) term, which is only assigned the task of numerical stabilization or regularization. To avoid undesired overstressing, it is necessary to increase the stress in the context of an incremental load increase not linearly, but in a stepped manner. The mode of operation of the method is explained using the numerical simulation of a one-dimensional tensile test. 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, irrespective of the chosen discretization. The relaxed solution is characterized by a state of complete decay of the (viscous) overstresses, which has to be achieved after each load increase - by a sufficiently long relaxation time. It should be noted that the regularizing effect of the process is highly 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 degrees of freedom of translation are assigned to each material point, in the micropolar continuum theory (Cosserat continuum) three additional degrees of freedom of rotation are taken into account. 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 with increasing discretization density the results converge towards a fixed solution; this is true for both the Boltzmann and the Cosserat model. 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 selected spatial discretization. In this case, the calculation becomes numerically unstable and sometimes breaks off after a few load steps. Further numerical investigations have shown that the regularizing effect of the model is 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 behaviour.

Wackerfuß, J.: Numerical description of localization phenomena considering discontinuous displacements, In N. Gebbeken, K.-U. Bletzinger, H. Rothert (editors): 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, June 2005