Numerical 3D-bifurcation analysis of star-shaped crack patterns using the energy method

Abstract

The present research deals with a three-dimensional (3D) Finite Element Method (FEM) bifurcation analysis based on the global mechanical potential that can be used to find the parameters at which a crack pattern changes. In our case we want to analyze at which point the star-shaped shrinkage cracks in an aqueous colloidal suspension filled in a glass cylinder change from four to three or two cracks growing. The driving force for the crack growth is shrinkage caused by diffusion controlled drying. The 3D crack front geometry is described efficiently by using a Fourier series approach. Based on steady-state crack growth, the Fourier coefficients are determined in a first step using an optimization algorithm. As a result, the time dependent crack growth can be determined. In a second step, the bifurcation point is determined by an eigenvalue analysis of the second order derivatives of the potential energy of the system. If the lowest eigenvalue reaches zero the fundamental solution becomes unstable and a transition will occur. Our analysis shows that the transition from four to two cracks is preferred over the transition from four to three cracks