The strong metal--support interaction (SMSI) is the encapsulation of a supported metal particle by an oxide layer that diffuses from the substrate. This process is usually described as being driven by a reduction in the surface energy of the metal particle and has a significant influence on the catalytic activity of the metal. Here, epitaxial Pd nanocrystals grown in ultrahigh vacuum on $\mathrm{SrTi}{\mathrm{O}}_{3}(001)$ and anatase $\mathrm{Ti}{\mathrm{O}}_{2}(001)$ substrates are studied by scanning tunneling microscopy. At annealing temperatures above \ensuremath{\sim}600 \textordmasculine{}C, the Pd crystals can become encapsulated by a $\mathrm{Ti}{\mathrm{O}}_{x}$ monolayer originating from the substrates. For both bare and encapsulated Pd crystals, their height-to-width ratio increases with the crystal height as a mechanism to partially release their interfacial misfit strain with the substrate. However, the rate of this increase is lower for encapsulated crystals, indicating that during the SMSI process the interface between the particle and the oxide is modified to form a lower energy interfacial structure which also results in less strain in the encapsulated particle. The SMSI is found to preferentially occur on larger crystals, driven by the reduction in their elastic strain energy, which scales with the crystal volume. Compared with the traditional view of SMSI our results provide a more complete description of the encapsulation process.