We describe a facile technique based on polymer encapsulation to apply several percent (>5%) controllable strains to monolayer and few-layer Transition Metal Dichalcogenides (TMDs). We use this technique to study the lattice response to strain via polarized Raman spectroscopy in monolayer WSe2 and WS2. The application of strain causes mode-dependent redshifts, with larger shift rates observed for in-plane modes. We observe a splitting of the degeneracy of the in-plane modes in both materials and measure the Grüneisen parameters. At large strain, we observe that the reduction of crystal symmetry can lead to a change in the polarization response of the mode in WS2. While both WSe2 and WS2 exhibit similar qualitative changes in the phonon structure with strain, we observe much larger changes in mode positions and intensities with strain in WS2. These differences can be explained simply by the degree of iconicity of the metal-chalcogen bond.