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We develop a numerical methodology for the calculation of mode-I R-curves of brittle and elastoplastic lattice materials, and unveil the impact of lattice topology, relative density and constituent material behavior on the toughening response of 2D isotropic lattices. The approach is based on finite element calculations of the J-integral on a single-edge-notch-bend (SENB) specimen, with individual bars modeled as beams having a linear elastic or a power-law elasto-plastic constitutive behavior and a maximum strain-based damage model. Results for three 2D isotropic lattice topologies (triangular, hexagonal and kagome) and two constituent materials (representative of a brittle ceramic (silicon carbide) and a strain hardening elasto-plastic metal (titanium alloy)) are presented. We extract initial fracture toughness and R-curves for all lattices and show that (i) elastic brittle triangular lattices exhibit toughening (rising R-curve), and (ii) elasto-plastic triangular lattices display significant toughening, while elasto-plastic hexagonal lattices fail in a brittle manner. We show that the difference in such failure behavior can be explained by the size of the plastic zone that grows upon crack propagation, and conclude that the nature of crack propagation in lattices (brittle vs ductile) depends both on the constituent material and the lattice architecture. While results are presented for 2D truss-lattices, the proposed approach can be easily applied to 3D truss and shell-lattices, as long as the crack tip lies within the empty space of a unit cell.