3D crack tracking algorithms based on weight function approximation schemes

Minimum theorems for Linear Elastic Fracture Mechanics (LEFM) are derived in [1] in terms of crack front quasi-static velocity, moving from the analogy between LEFM and plasticity theory. They have been recently rephrased in terms of weight functions [2]. Weight functions, introduced by Rice [3] for three dimensional problems, are displacements analytical solutions, in a distributional sense, of the linear elastic boundary value problem. Their closed form is available for a very limited set of geometries of the crack front. The need to supply a high quality approximation of weight functions in all cases for which they are not known in closed form obstructed the way toward an effective implementation of crack tracking algorithms. A general method to approximate weight functions is here handled by means of an algorithm based on the definition of weight function itself, thus allowing an implicit and effective 3D crack tracking algorithm. This latter is based on a Newton-Raphson numerical strategy for the Griffith-Maximum Energy Release Rate (MERR) condition, which is endowed with a variational formulation at every iteration [4]. Owing to the convexity of the MERR safe equilibrium domain, the algorithm provides a finite elongation at each point of the crack front based upon the increment of the external loads as the minimum of a constrained quadratic functional. The constraint is computationally handled by means of the penalty method. Numerical experiments show that the proposed algorithm is more accurate, and numerically stable than explicit algorithms in time, which have been formulated moving from the map of velocities of crack elongation along the crack front. References [1] Salvadori A., Fantoni F., “Minimum theorems in 3D incremental linear elastic fracture mechanics”, International Journal of Fracture, 184(1): 57-74, 2013 [2] Salvadori A., Fantoni F., “Weight function theory and variational formulations for three dimensional plane elastic crack advancing”, International Journal of Solids and Structures, 51(5): 1030-1045, 2014 [3]Rice J., “ Weight function theory for three-dimensional elastic crack analysis”, in R.P. Wei, R.P. Gangloff (Eds.), Fracture Mechanics: perspectives and directions (20 th symposium) vol. 2, ASTM STP 1020, American Society for Testing and Materials, Philadelphia: 29-57, 1989 [4]Salvadori A., Fantoni F., “On a 3D crack tracking algorithm and its variational nature”, Journal of the European Ceramic Society 34: 2807-2821, 2014