Modeling, analysis and simulation of 2D dynamic crack propagation

Abstract

The goal is to investigate mathematically the behavior of a linear elastic, isotropic, homogeneous and finite body with a running crack under the influence of a wave. Here we concentrate on bounded crack velocities. Reducing the 3D wave model given by Navier-Lame equation system to a 2D one we get an in-plane model for plane elastic waves and an out-of-plane model for shear waves. The main points are in both cases: the description of the behavior of the elastic fields near the running crack tip and the derivation of the equations of motion of the crack tip. The method: Analysis of the transformed problem. A well-tried method in solid mechanics is to transform the current configuration (a non-cylindrical space-time domain). For this purpose it will be assumed there is a family of mapping which maps the reference configuration (cylindrical domain) into the current configuration. Additionally a further configuration, where the isotropic Laplacian appears, is considered. Roughly speaking, $h(t)$ describes the motion of the crack tip. Performing the above change of variables we get elastic wave equations with time-dependent coefficients and lower order terms in the reference domain. Using functional analysis arguments the solvability of the transformed initial boundary value problem is studied. Derivation of the equation of motion: Starting from the rate of the total energy in the current configuration we have to derive an equation of motion of crack tip, that means, a nonlinear ordinary differential equation for $h(t)$. In order to calculate the rate of the contributing energies we transform again the integrals into the reference configuration and use the results of the first item. Most of the studies on the dynamic crack propagation from the viewpoint of the fracture mechanics postulate the body is infinite in a thickness direction because the mathematical treatment is simple. However this postulate is not pertinent to most practical cases. We report here an analytical study on the dynamic crack propagation in a finite configuration. For both cases, the out of- and in-plane, the analytical solution is determined in the vicinity of a moving crack located in an isotropic medium. As specially, we are concerned with the variation of stress intensity factors and displacement fields near the crack tip. In order to construct the asymptotic of the singular solutions for in-plane fracture case, the matching procedure will be used once more. In contrast to before, the method is not directly applicable. For getting these fields we introduce auxiliary potentials related to the in-plane motion of the crack which are separable into Mode I (opening mode) and Mode II (in-plane sliding mode) crack problems. Once such asymptotic fields near the tip of a propagating crack are determined, other important quantities of relevance in dynamic fracture mechanics, such as dynamic stress intensity factors and dynamic energy release rate can be determinate.

Die mathematische Modellierung, Analysis und numerische Berechnung des Bruchwachstums in unterschiedlichen Materialien und unter verschiedenen Belastung ist nach wie vor ein aktuelles, umfangreiches und anspruchsvolles Forschungsthema. Dieser Arbeit beschäftigt sich mit makroskopischen Modellen, die das Verhalten eines laufenden Risses unter Einfluss dynamischer Belastungen beschreiben. Dabei geht sie von einem energetischen Bruchkonzept aus, das besagt, dass stets ein Energiegleichgewichtszustand angenommen wird. Dies kann als Verallgemeinerung des Griffith'schen Bruchkriteriums des quasistatischen Falls auf den dynamischen Fall gesehen werden, wobei die kinetische Energie zusätzlich auftritt. Betrachtet man das ebene Problem eines geradlinig laufenden Risses, dann kann durch Kopplung der elastischen Wellengleichung mit dem Bruchkriterium eine gewöhnliche Differentialgleichung zur Bestimmung des Risspfades und der Rissgeschwindigkeit hergeleitet werden. Die analytische Herleitung dieser Bewegungsgleichungen für die Rissspitze ist ein anspruchsvolles Problem. Die schwierigen numerischen Berechnungen beruhen auf einem iterativen Verfahren und wurden mit dem FEM-Packet PDE2D durchgeführt. In dieser Dissertation wird unterschieden zwischen der out-of-plane und der in-plane Situation für die lineare Elastizität.