The mechanical behaviors of lithium iron phosphate battery cells are investigated by conducting in-plane and out-of-plane compression tests of representative volume element (RVE) specimens of dry cells. The test results of cell RVE specimens under in-plane constrained compression indicate that the load carrying behavior of the cell RVE specimens is characterized by the buckling, the kink and shear band formation, and the final densification of the cell components. The SEM images of the active materials on electrodes and the test results of the cell RVE specimens under out-of-plane compression suggest that the porosity in the components and the macroscopic gaps between the components is up to 40%. The test results suggest that the lithium-ion battery cells can be modeled as anisotropic foams or cellular materials. The elastic buckling analyses for a beam with lateral constraints indicate that the higher order buckling modes and the critical buckling stresses in general agree with those observed in experiments. The elastic buckling analyses also justify the length selection of the cell RVE specimens. Finally, an idealized kinematic model is presented to explain the physical mechanisms of the kink and shear band formation in the cell RVE specimens under in-plane constrained compression.