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

The stress intensity factors or strain energy release rate are typically used to characterise the stress field in the vicinity of a crack in fracture mechanics. One way to obtain the strain energy release rate in elastic–plastic fracture mechanics is from the stress and deformation field around the crack tip through the calculation of the J integral. The J-integral is contour independent, although the contour must start and end from a traction-free surface, such as the crack surface. Using Green’s theorem, the J-integral can be formulated as a surface or area integral, which makes it convenient for implementation in finite element method (FEM). More importantly, the J-integral calculation is insensitive to uncertainty of the exact crack tip location, can be applied for linear elastic analysis with small scale yielding and in an improved formulation for elastic plastic fracture. In short, the J-integral is an indispensable tool in the study of fracture mechanics. Despite the J-integral being widely used in FEM, including availability in most commercial FEM codes, there is currently no algorithm to calculate the J-integral in the Smoothed Particle Hydrodynamics (SPH) method. This is somewhat surprising since the SPH method, due to its meshless nature, has inherent advantages in dealing with cracks compared to mesh based methods such as FEM. In this paper we will therefore address this deficiency and develop an algorithm for calculation of the J integral in the SPH method. The implementation of his new alghorithm is based on a new definition of the weighting function q 1 , as appropriately normalised kernel function, which inherently satisfies all the specific requirements on q 1 : The function is sufficiently smooth in the J-integral area, it is equal to unit inside contour path of the integral and zero outside of the path. A further element of novelty is that in the current implementation, the gradient of this function is evaluated analytically rather than through a numerical approximation. The verification and validation of developed algorithm is based on simulation of the standard single edge notch tension test (SENT) under the plain strain conditions. The SPH results are compared to the FEM results for stress and displacement fields in the vicinity of the crack tip, as well as the J integral solutions. The SPH results demonstrated convergence and were within 2% of the converged FEM solutions. The validation also allows for the definition of simple guidelines for the definition of the J-integral area to achieve accurate results. The implementation is currently developed for linear elastic fracture mechanics applications, but its generalisation and application to elastic plastic fracture mechanics, including the combination with elastic plastic constitutive models is straightforward.