We investigate the buildup of strain in InP quantum dots with the addition of shells of the lower-lattice constant materials ZnSe and ZnS by Raman spectroscopy. Both materials induce compressive strain in the core, which increases with increasing shell volume. We observe a difference in the shell behavior between the two materials: the thickness-dependence points toward an influence of the material stiffness. ZnS has a larger Young's modulus and requires less material to develop stress on the InP lattice at the interface, while ZnSe requires several layers to form a stress-inducing lattice at the interface. This hints at the material stiffness being an additional parameter of relevance for designing strained core/shell quantum dots.