The convergence problem during the cohesive zone modelling of hydrogen embrittlement in constant displacement scenario is attributed to the numerical instability which is studied analytically in the present work. The property of numerical stability is directly associated with the number of solutions for the controlling equations from the failure initiation point. It is shown that all the cases with a non-unique solution are numerically unstable thereby having convergence problem. Linear elastic and elasto-plastic material models are considered in the derivation, and the convergence properties for both models are proved essentially the same. The viscous regularization proposed by Gao and Bower  proves effective in solving the convergence problem with good accuracy under constant displacement, provided that the viscosity is small enough. This is further supported by a pipeline engineering case study where the viscosity regularized cohesive zone approach is applied to the hydrogen embrittlement simulation. The stabilizing mechanism of the viscous regularization is attributed to its capacity to enforce a single solution by modifying the controlling equations. The influence of viscous regularization on symmetry modelling is also discussed.