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In this paper several damage equations are analysed with respect to their properties at damage initialization. This is particularly important for soft biological tissues since two different loading regimes have to be clearly distinguished: the physiological domain where no damage evolution should be considered and the supra-physiological domain where damage evolves. At the transition between these two domains the behaviour of different damage models may influence the convergence of the Newton iteration when solving, for example, nonlinear finite element problems. It is shown that the model proposed by Balzani et al. (Comput Meth Appl Mech Eng 2012; 213-216: 139-151) a priori ensures smooth tangent moduli. In addition to that, a new damage function is proposed able to describe a slow damage evolution at damage initialization also providing smooth tangent moduli. Using this new damage function the approach given by Balzani et al. (Acta Biomater 2006; 2(6): 609-618) can also be modified such that smooth tangent moduli are guaranteed. Numerical analyses of a circumferentially overstretched artery are performed and show that no convergence problems are observed at the transition from the undamaged to the damaged domain, even when a model is used that has non-smooth tangent moduli. © The Author(s) 2013.