Damage-induced strain softening is of vital importance for the modeling of localized failure in frictional-cohesive materials. This paper addresses strain localization of damaging solids and the resulting consistent frictional-cohesive crack models. As a supplement to the framework recently established for stress-based continuum material models in rate form (Wu and Cervera 2015, 2016), several classical strain-based damage models, expressed usually in total and secant format, are considered. Upon strain localization of such damaging solids, Maxwell's kinematics of a strong (or regularized) discontinuity has to be reproduced by the inelastic damage strains, which are defined by a bounded characteristic tensor and an unbounded scalar related to the damage variable. This kinematic constraint yields a set of nonlinear equations from which the discontinuity orientation and damage-type localized cohesive relations can be derived. It is found that for the

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