Lattice materials are used as the core of sandwich panels to construct light and strong structures. This thesis focuses on metallic sandwich structures and has two main objectives: (i) explore how a surface treatment can improve the strength of a lattice material and (ii) investigate the collapse response of two competing prismatic sandwich cores employed in ship hulls. First, the finite element method is used to examine the effect of carburisation and strain hardening upon the compressive response of a pyramidal lattice made from hollow tubes or solid struts. The carburisation surface treatment increases the yield strength of the material, but its effects on pyramidal lattices are not known. Here, it is demonstrated that carburisation increases the plastic buckling strength of the lattice and reduces the slenderness ratio at which the transition from plastic to elastic buckling occurs. The predictions also showed that strain hardening increases the compressive strength of stocky lattices with a slenderness ratio inferior to ten, but without affecting the collapse mode of the lattice. Second, the quasi-static three-point bending responses of simply supported and clamped sandwich beams with a corrugated core or a Y-frame core are compared via experiments and finite element simulations. The role of the face-sheets is assessed by considering beams with (i) front-and-back faces present and (ii) front face present, but back face absent. These two beam designs are used to represent single hull and double hull ship structures, and they are compared on an equal mass basis by doubling the thickness of the front face when the back face is absent. Beams with a corrugated core are found to be slightly stronger than those with a Y-frame core, and two collapse mechanisms are identified depending upon beam span. Short beams collapse by indentation and for this collapse mechanism, beams without a back face outperform those with front-and back faces present. In contrast, long beams fail by Brazier plastic buckling and for this collapse mechanism, the presence of a back face strengthens the beam. Third, drop weight tests with an impact velocity of 5 m/s are performed on simply supported and clamped sandwich beams with a corrugated core or a Y-frame core. These tests are conducted to mimic the response of a sandwich hull in a ship collision. The responses measured at 5 m/s are found to be slightly stronger than those measured quasi-statically. The measurements are in reasonable agreement with finite element predictions. In addition, the finite element method is used to investigate whether the collapse mechanism at 5 m/s is different from the one obtained quasi-statically. The predictions indicate that sandwich beams that collapse quasi-statically by indentation also fail by indentation at 5 m/s. In contrast, the simulations for beams that fail quasi-statically by Brazier plastic buckling show that they collapse by indentation at 5 m/s. Finally, the dynamic indentation response of sandwich panels with a corrugated core or a Y-frame core is simulated using the finite element method. The panels are indented at a constant velocity ranging from quasi-static loading to 100 m/s, and two indenters are considered: a flat-bottomed indenter and a cylindrical roller. For indentation velocities representative of a ship collision, i.e. below 10 m/s, the predictions indicate that the force applied to the front face of the panel is approximately equal to the force transmitted to the back face. Even at such low indentation velocities, inertia stabilisation effects increase the dynamic initial peak load above its quasi-static value. This strengthening effect is more important for the corrugated core than for the Y-frame core. For velocities greater than 10 m/s, the force applied to the front face exceeds the force transmitted to the back face due to wave propagation effects. The results are also found to be very sensitive to the size of the flat-bottomed indenter; increasing its width enhances both inertia stabilisation and wave propagation effects. In contrast, increasing the roller diameter has a smaller effect on the dynamic indentation response. Lastly, it is demonstrated that material strain-rate sensitivity has a small effect on the dynamic indentation response of both corrugated and Y-frame sandwich panels.