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Strain engineering has been extended recently to the ultrafast timescales, driving metal-insulator phase transitions and the propagation of ultrasonic demagnetization fronts. However, the non-linear lattice dynamics underpinning interfacial optoelectronic phase switching have not yet been addressed. Here we focus on the lattice dynamics initiated by impulsive resonant excitation of polar lattice vibrations in LaAlO$_3$ single crystals, one of the most widely utilized substrates for oxide electronics. We show that ionic Raman scattering drives coherent oxygen octahedra rotations around a high-symmetry crystal axis and we identify, by means of DFT calculations, the underlying phonon-phonon coupling channel. Resonant lattice excitation is shown to generate longitudinal and transverse acoustic wavepackets, enabled by anisotropic optically-induced strain in and out of equilibrium. Importantly, shear strain wavepackets are found to be generated with extraordinary efficiency at the phonon resonance, being comparable in amplitude to the more conventional longitudinal acoustic waves, opening exciting perspectives for ultrafast material control.