Adding fillers in elastomers is known to increase the elastic modulus and the wear resistance of elastomers, but also to increase nonlinear dissipation, a phenomenon known as the Payne effect. Indeed, when submitted to deformations of the order of a few per cents or more, the elastic modulus can decrease down to values much smaller than the initial one. On the other hand, when submitted to large amplitude oscillatory shear at a frequency omega, frequency analysis shows that the contribution of higher harmonics 3 omega, 5 omega, .., to the response is quite small. This might appear somehow as a paradox since the nonlinear behavior of filled elastomers can be strongly marked. We discuss here in detail a possible physical origin of these various features. We do it by comparing experimental results performed on model elastomers to the prediction of a model proposed recently, based on the presence of glassy bridges linking neighboring particles. We show that the kinetics of rupture and rebirth of these glassy bridges can explain these effects.