The problems of non-slipping contacts between dissimilar elastic solids are studied under the conditions of plane strain. When two dissimilar solids are incrementally pressed into contact, a relative tangential displacement along the contact interface emerges because of the material property mismatch. The spatial derivative of the relative tangential displacement is referred to as the interface mismatch eigenstrain, which is separately investigated subject to the non-adhesive and adhesive surface conditions. The explicit solutions of the interface mismatch eigenstrain for non-slipping contacts with symmetrical indenter profiles are obtained. These results provide the foundation for improved precision contact deformation modelling at different scales.