Voicing or vocal folds auto-oscillation results when the expiratory fluid flow coming from the lungs interacts with the deformable vocal folds tissues in the glottis. Consequently, the outcome of this fluid-structure interaction is determined by fluid flow properties, the glottal geometry as well as structural tissue properties. Changes to the vocal fold structure often happens in voice pathologies attributed to structural vocal fold tissue abnormalities. These pathologies affect the general population and in particular voice professionals such as teachers, singers, etc. Whereas an extensive literature attests to the medical, social and economic relevance of these voice pathologies, systematic physical studies are few. Therefore, in this paper, we present an experimental study of the influence of a local rigidification on the auto-oscillation of the vocal folds. Four-layer silicone molded mechanical vocal folds replicas are used. The layers composition and elasticity properties reflect the multi-layer physiological representation of a human vocal fold. Besides this normal reference replica, replicas with an abnormal vocal fold structure are obtained by adding a stiff rectangular inclusion during the molding process so that these vocal folds replicas exhibit a local rigidification. The elastic Young's modulus is at least four times greater than 65 kPa, which is generally the upper range associated with a normal vocal fold. The position and orientation of the inclusion are varied in a way that the replicas elasticity can be determined. For each replica, the auto-oscillation characteristics are experimentally studied by placing the replica in a suitable setup allowing to measure pertinent physical quantities such as upstream pressure, vocal folds displacement and acoustics. Changes to these auto-oscillation characteristics are outlined and discussed. The implications for pathological voice modeling are then discussed.