Prediction of failure mechanisms in concrete is a fairly complex task due to heterogeneous concrete microstructure, localization process triggered by cracks, multiple crack interactions during their growth and coalescence, and different dissipative mechanisms in a fracture process zone prior to localized failure and in a localization zone during the failure. None of the currently used phenomenological models can represent the full set of 3D failure modes. This work presents an attempt to solve this with the 3D meso-scale model based on discrete lattice approach. In particular we show that we can capture such complexities at the meso-scale, which is able to represent microcracks in fracture process zone along with the localized failure represented in terms of embedded strong discontinuity and accompanied with softening constitutive law. The model can also successfully simulate salient features of concrete response, such as order of magnitude of reduction in strength in uniaxial tension versus compression, strength increase in biaxial loading or hydrostatic tension. Moreover, macro-scale representation of failure surfaces obtained with presented model for different loading programs confirms the need for failure concrete criterion of multi-surface kind. Part I of this work presents the proposed meso-scale based on extensive number of numerical simulations with multiple realizations of different concrete specimens, along as the optimal deterministic fit for several common concrete failure models. The ultimate interest of the work is to provide detailed data set for different failure modes which can be used for identification of probability distribution of material parameters for different criteria. Such task is carried in Part II of this work.