The indentation problem of an elastic-perfectly plastic film by a rigid plane indenter of flat end was solved to the zeroth order of the ratio of the film thickness to the contact width for frictionless contact and the contact width much larger than the film thickness. Explicit relation between the indentation force and the indentation depth was obtained in the plastic range. The finite element results supported this relationship for the ratio of the film thickness to the contact width less than or equal to 0.4. The ratio of the elastic energy to the plastic energy in an indentation cycle was found to be proportional to the film thickness and inversely proportional to the plastic indentation depth. For the elastic indentation of a multilayer structure by a rigid plane indenter of flat end with frictionless contact, the inverse of the effective contact modulus of the multilayer structure linearly depends on the thickness ratio of each layer to the structure thickness and the inverse of the contact modulus of each layer. For the indentation of a bilayer structure with non-slip contact between the structure and a rigid substrate, the finite element results showed that both the effective contact modulus and the indentation load for the occurrence of plastic deformation increase with decreasing the layer thickness, demonstrating the constraint-induced enhancement of the mechanical strength of materials.