Continuum elastoplastic damage mbdels employing irreversible thermodynamics and internal state variables are developed within two alternative dual frameworks. In a strain [stress]-based formulation, damage is characterized through the effective stress [strain] concept together with the hypothesis of strain [stress] equivalence, and plastic flow is introduced by means of an additive split of the stress [strain] tensor. In a strain-based formulation we redefine the equivalent strain, usually defined as the J2-norm of the strain tensor, as the (undamaged) energy norm of the strain tensor. In a stress-based approach we employ the complementary energy norm of the stress tensor. These thermodynamically motivated definitions result, for ductile damage, in symmetric elastic-damage moduli. For brittle damage, a simple strain-based anisotropic characterization of damage is proposed that can predict crack development parallel to the axis of loading (splitting mode). The strain- and stress-based frameworks lead to dual but not equivalent formulations, neither physically nor computationally. A viscous regularization of strain-based, rate-independent damage models is also developed, with a structure analogous to viscoplasticity of the Perzyna type, which produces retardation of microcrack growth at higher strain rates. This regularization leads to well-posed initial value problems. Application is made to the cap model with an isotropic strain-based damage mechanism. Comparisons with experimental results and numerical simulations are undertaken in Part II of this work.