Simple qualitative considerations suggest that the inherent mechanical properties-equilibrium structure and elastic constants-of thin epilayers are influenced by proximity effects. The computational effort to calculate these properties, using any form of atomic interaction, becomes enormous when epilayer and substrate are discommensurate and the repeat period is large-an important feature of misfit strain relief in epilayers. A procedure is proposed by which this problem can be overcome for an epimonolayer, and can be extended to multilayers. The procedure involves the following assumptions : (i) the mechanical behavior of the monolayer (ML) is governed by the principle of minimum energy, the average energy per ML atom being minimized in this case, (ii) the field of interaction emanating from the substrate is periodic (with the periodicity of the substrate surface) within the plane of the ML, (iii) the substrate field can be mapped by calculation involving the translation of the ML-as if "rigid", having registry dimensions and allowing for height equilibration-on the substrate surface, (iv) the ML may be constrained to its average (constant) equilibrium height with negligible discrepancies in the energy. The procedure is demonstrated, and numerically justified, by its application to {111} MLs of Ni and Cu in Nishiyama-Wassermann orientation on W {110}, using embedded-atom-method potentials. The calculations produce convincing evidence to substantiate the validity of the procedure, showing that the contribution of the substrate to the embedding energy of ML atoms can be fairly accurately described in terms of its average electron density in the ML plane, the effect of the periodic oscillations in the electron density being negligible. A similar procedure is valid for the embedding of substrate atoms. The application to Ni and Cu on W shows that proximity effects are drastic : the in-plane elastic constants of a supported ML, an ML in the crystal interior and a free standing ML are respectively and crudely in the ratio 1:1.5:2.5. Proximity effects are likewise important in anharmonicity.