Abstract
The stress intensity factors or strain energy release rate are typically used
to characterise the stress ?eld in the vicinity of the crack in the fracture
mechanics experiments. One way to obtain the strain energy release rate
in elastic-plastic fracture mechanics is from the stress and deformation ?eld
around 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-integral
can be formulated as a surface or area integral, which makes it convenient
for implementation in ?nite element and the SPH analysis codes. More im-
portantly, 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.
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. The
implementation is based on a new de?nition of the weighting function q1,
as appropriately normalised kernel function, which inherently satisfy all the
speci?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 outside
of the path. In the current implementation, the gradient of this function
was evaluated analytically. The veri?cation and validation of developed al-
gorithm was based on simulation of the standard single edge notch tension
test (SENT) under the plain strain conditions. The SPH results were com-
pared to the FEM results for stress and displacement ?elds in the vicinity
of the crack tip, as well as the J integral solutions, including the compar-
ison of the J integral results with existing exact analytical solution. The
SPH results demonstrated convergence and were within 2% of the converged
FEM solutions, which makes the implementation validated. The simulation
program also included the sensitivity analysis of the SPH results to the size
and discretisation of the area used for calculation of the J-integral, including
the rate of the convergence of the results for the ?rst few contours. The
implementation is currently developed for linear elastic fracture mechanics
applications, but its generalisation and application to the elastic plastic frac-
ture mechanics, including the combination with elastic plastic constitutive
models is straightforward.
to characterise the stress ?eld in the vicinity of the crack in the fracture
mechanics experiments. One way to obtain the strain energy release rate
in elastic-plastic fracture mechanics is from the stress and deformation ?eld
around 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-integral
can be formulated as a surface or area integral, which makes it convenient
for implementation in ?nite element and the SPH analysis codes. More im-
portantly, 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.
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. The
implementation is based on a new de?nition of the weighting function q1,
as appropriately normalised kernel function, which inherently satisfy all the
speci?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 outside
of the path. In the current implementation, the gradient of this function
was evaluated analytically. The veri?cation and validation of developed al-
gorithm was based on simulation of the standard single edge notch tension
test (SENT) under the plain strain conditions. The SPH results were com-
pared to the FEM results for stress and displacement ?elds in the vicinity
of the crack tip, as well as the J integral solutions, including the compar-
ison of the J integral results with existing exact analytical solution. The
SPH results demonstrated convergence and were within 2% of the converged
FEM solutions, which makes the implementation validated. The simulation
program also included the sensitivity analysis of the SPH results to the size
and discretisation of the area used for calculation of the J-integral, including
the rate of the convergence of the results for the ?rst few contours. The
implementation is currently developed for linear elastic fracture mechanics
applications, but its generalisation and application to the elastic plastic frac-
ture mechanics, including the combination with elastic plastic constitutive
models is straightforward.
| Original language | English |
|---|---|
| Journal | Computer Methods in Applied Mechanics and Engineering |
| Publication status | Submitted - 3 Jul 2025 |