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A systematic theoretical analysis is presented of the combined effects of substrate compliance and film compositional grading on the relaxation of strain due to lattice mismatch in layer-by-layer semiconductor heteroepitaxy. The analysis is based on a combination of continuum elasticity theory and a novel atomistic simulation approach for modeling structural and compositional relaxation in layer-by-layer heteroepitaxial systems. Results are presented for InAs epitaxy on GaAs(1 1 1)A compliant substrates with some marginal film compositional grading that consists of one monolayer of In0.50Ga0.50As grown on the substrate surface prior to InAs growth. A parametric study is carried out over a wide range of substrate thicknesses. Interfacial stability with respect to misfit dislocation formation, the dependence on substrate thickness of a thermodynamic critical film thickness, and the completion of the coherent-to-semicoherent interfacial transition are examined in detail. In addition, the structural characteristics and compositional distribution of the corresponding semicoherent interfaces, the associated strain fields, as well as the film surface morphological characteristics are analyzed. Most importantly, the role of segregation at defects of a semicoherent interface in the thermodynamics of layer-by-layer heteroepitaxial growth is demonstrated. Our study shows that systematic combination of the mechanical behavior of thin compliant substrates with grading of the epitaxial film composition provides a very promising engineering strategy for strain relaxation in heteroepitaxy. [Copyright &y& Elsevier]