FEM-bifurcation analysis for 3D crack patterns

Abstract

During the shrinkage of volcanic rock, columnar joints evolve with an almost regular hexagonal pattern formation and a very distinct column diameter defining the crack spacing. This phenomenon is caused by an inhomogeneous, constantly moving temperature field. The constant column growth is presumably always close to a bifurcation point at which the ideal regular pattern of hexagons could branch off to a pattern with a different symmetry due to slightly changing cooling conditions. This paper presents an energy-based fracture mechanical bifurcation analysis to determine this point and its stability by evaluating first and second order derivatives of the free Helmholtz energy with respect to a limited number of coefficients of a truncated Fourier series expansion of the three-dimensional crack front contour. The energy is obtained by linear elastic simulations on three-dimensional Finite Element models of the periodically repeatable patterns before and after bifurcation. While providing an understanding of the column growth procedure of volcanic rock, the characteristics of the approach suggest applications for various complex three-dimensional crack growth systems in engineering.