Suitability of nonlinear root-solvers whose convergence rates are better than the quadratic Newton–Raphson method and that do not require higher derivatives is examined for solving nonlinear equations encountered in the implementation of constitutive models. First, the performance of six cubic convergent methods is demonstrated by means of examples. These cubic methods are used in place of the Newton–Raphson method to solve the nonlinear equations in the J2 plasticity and Gurson plasticity constitutive models. Few cubic methods are found to be computationally efficient and relatively insensitive to the initial guess when compared to the Newton–Raphson method for the considered models.