Fracture Mechanics in Smoothed Particle Hydrodynamics: An algorithm to calculate the J-Integral

The stress intensity factors or strain energy release rate are typically usedto characterise the stress ?eld in the vicinity of the crack in the fracturemechanics experiments. One way to obtain the strain energy release ratein elastic-plastic fracture mechanics is from the stress and deformation ?eldaround the crack tip and calculation of the J integral. The J-integral is con-tour independent, although the contour must start and end from a traction-free surface, such as the crack surface. Using Green's theorem, the J-integralcan be formulated as a surface or area integral, which makes it convenientfor implementation in ?nite element and the SPH analysis codes. More im-portantly, the J-integral calculation is insensitive to uncertainty of the exactcrack tip location, can be applied for linear elastic analysis with small scaleyielding and in an improved formulation for elastic plastic fracture.The aim of the work presented in this paper is to develop an algorithm for cal-culation of the J integral and implement it into an SPH explicit code. Theimplementation is based on a new de?nition of the weighting function q1,as appropriately normalised kernel function, which inherently satisfy all thespeci?c requirements. The function is su?ciently smooth in the J-integral area, is equal to unit inside contour path of the integral and zero outsideof the path. In the current implementation, the gradient of this functionwas evaluated analytically. The veri?cation and validation of developed al-gorithm was based on simulation of the standard single edge notch tensiontest (SENT) under the plain strain conditions. The SPH results were com-pared to the FEM results for stress and displacement ?elds in the vicinityof the crack tip, as well as the J integral solutions, including the compar-ison of the J integral results with existing exact analytical solution. TheSPH results demonstrated convergence and were within 2% of the convergedFEM solutions, which makes the implementation validated. The simulationprogram also included the sensitivity analysis of the SPH results to the sizeand discretisation of the area used for calculation of the J-integral, includingthe rate of the convergence of the results for the ?rst few contours. Theimplementation is currently developed for linear elastic fracture mechanicsapplications, but its generalisation and application to the elastic plastic frac-ture mechanics, including the combination with elastic plastic constitutivemodels is straightforward.