Analytical and numerical studies on continuum models for the elastic-plastic behavior of uniformly periodic lattice materials under multi-axial loading are presented in this paper. This study firstly investigates the basic topology of unit cell structures for three different lattice materials with cubic symmetry. By homogenizing the mechanical properties of these materials within the unit volume space, the equivalent continuum models are obtained with the internal variables which result in the mechanical and geometrical characteristics of discrete truss members at the micro-scale such as structural packing, axial stiffness, and material density. Therefore, in this study, the strain hardening was applied to the material model of individual truss members in a valuable effort to explain the plastic behavior of the homogenized lattice material. The expansion of pressure-dependent stress surface at the macro-scale level is estimated by analytical predictions, which are derived from the equivalent continuum models. Analytical predictions show good agreements with existing results obtained by finite element (FE) analyses.