In this work, we studied the relaxation parameters, h_e-p and h_res, of three composite structures Cu/LiF, Cu/MgO, and Cu/Si, which have different types of a chemical bond between the substrates (ionic (LiF), ionic-covalent (MgO), and covalent (Si)) and differ in hardness (H_Cu = 0.6 GPa, H_LiF, H_MgO and H_Si are 1.2, 7.5 and 8.2 GPa, respectively). For each type of substrates, coated systems (CSs) were fabricated with a following Cu film thickness: t₁ = 85; t₂ = 470 and t₃ = 1000 nm. The behavior of relaxation parameters was examined over a wide range of loads, P = 2–900 mN, during nanoindentation. The elastic-plastic parameters were shown to depend on the CS type, as well as on the film thickness and the magnitude of the applied load.