Strained epitaxial material clusters or islands grown on a relatively thick substrate are considered in the context of continuum mechanics. The islands and the substrate are modeled as isotropic elastic solids with similar material properties. The geometry is assumed to be two-dimensional. Under the assumption that the total free energy of the system consists of the energy of the free surface and the strain energy, minimum energy profiles of dislocation-free islands are calculated. It is shown that even though the surface energy contribution to the free energy favors a flat film morphology, an islanded morphology is the preferred growth mode of a strained film with a sufficiently large lattice mismatch. The coalescence of two islands which are relatively close to each other is also considered. It is demonstrated that the free energy of the system is reduced through the coalescence of islands and that coalescence may occur spontaneously. Minimum energy profiles of islands containing interface misfit dislocations are also calculated. Sufficiently large dislocated islands are shown to have a smaller height-to-width aspect ratio and a lower surface chemical potential at equilibrium than dislocation-free equilibrium islands of the same size. This result is used to explain qualitatively how the nucleation of a dislocation can affect the growth characteristics of an island.