In spite of a vast background on cohesive constitutive law and its use for various analyses such as delamination in composite laminates, some numerical aspects of that have been less explored and reported in the literature. The aim of this paper is to study the phenomenon of spurious stress oscillation and also dominant process zone (where damage has its most significant evolution) in delamination modeling. For this purpose, distribution of normal stress and damage parameter of different DCB specimen are analyzed. Distribution of stress around the process zone indicates spurious oscillation just ahead of the delamination tip where damage parameter is zero and no effect of this phenomenon is seen on results of applied load versus opening displacement. Additionally, it is shown that larger values of penalty stiffness lead in smaller length of dominant process zone and very large values of penalty stiffness pushes the distribution of damage parameter to become step-like function. Authors believe that this effect is in fact the main reason of un-converged solution of models with too large penalty values.