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A new material model is proposed for the description of stress-softening observed in cyclic tension tests performed on soft biological tissues. The modeling framework is based on the concept of internal variables introducing a scalar-valued variable for the representation of fiber damage. Remanent strains in fiber direction can be represented as a result of microscopic damage of the fiber crosslinks. Particular internal variables are defined able to capture the nature of soft biological tissues that no damage occurs in the physiological loading domain. A specific model is adjusted to experimental data taking into account the supra-physiological loading regime. For the description of the physiological domain polyconvex functions are used which also take into account fiber dispersion in a phenomenological approach. The applicability of the model in numerical simulations is shown by a representative example where the damage distribution in an arterial cross-section is analyzed.