Delayed long-term strains of concrete caused by creep are a known problem leading to, \eg the loss of pre-stress and additional microcracking in concrete structures. In order to improve predictions of the creep strains and damage state of cementitious materials, a coupled experimental and numerical study of the creep/microcracking interactions was designed. Compressive creep tests on cement paste and mortar were carried out to analyze the influence of the material heterogeneity and stress level on the creep rate. The obtained data were used for the calibration of a creep constitutive model and as benchmark for predictions of the creep/damage interactions. The creep model was supplemented with separate damage models for the bulk matrix and matrix-aggregate interfaces. The resulting viscodamage models were applied on artificial microstructures of mortar to simulate, using the finite element method, damage effects on the effective mortar creep behavior. This numerical model was able to reproduce the compressive creep behavior of mortar at low stresses and predicted a tertiary creep stage in tension. Nonlinear creep at higher stresses could only be partially reproduced by taking into account these damage mechanisms, pointing toward nonlinear creep phenomena at the microscale.