We have investigated the influence of the excluded volume on the configurational properties of simple sampling off-lattice polymer chains. This was done by direct Monte Carlo estimates of chain configurations supplemented by extrapolation of physicochemical parameters to infinitely long walks. In addition, the effects of a local chain rigidity parameter on the chain attrition phenomena, mean-square end-to-end distance and radius of gyration as a function of the chain number of monomers and space dimensions (2, 3, 4 d) have been determined. The results suggest that although scaling concepts defined for lattice models are also applicable to continuum models, the introduction of a chain rigidity parameter to take into account torsional rotations about the bonds of the backbone and short range excluded volume effects results in a significant difference in the qualitative and quantitative description of the chain attrition phenomena. Results also suggest that by increasing the chain rigidity and space dimension, the emergence of asymptotic domains in which the characteristic chain lengths are described by scaling laws occurs at larger chain sizes.