Summary: We use the Newton-Raphson method to analyse elasto-plastic fracture problems. The stress and strain fields at the crack tip as well as the J- integral have been calculated (J can be thought of as the amplitude of the crack tip singularity fields). As Poisson’s ratio \(\mu\) approaches 0.5, regular finite element methods become inapplicable. In order to overcome this difficulty, we transform the stiffness matrix into \(K_ 1+\alpha K_ 2\) (as \(\mu\to 0.5\), \(\alpha\to \infty)\), and then, use the regular Gauss integral for \(K_ 1\) and the reduced integral for \(K_ 2\). This method can be applied to two and three-dimensional problems.