Coda Wave Interferometry (CWI) is a highly sensitive monitoring technique built on the sensitivity of elastic coda waves to small changes in a diffusive medium. However, a clear connection between the physical processes involved in the evolution of the medium and the time changes observed by CWI has not been clearly described yet. Here, we quantify the impact of elastic deformation on CWI measurements at laboratory scales. We compare experimental results from wave scattering measurements during a uniaxial compression test to those of a numerical approach based on the combination of two codes (SPECFEM2D and Code_Aster), which allows us to model wave propagation in complex diffusive media during its elastic deformation. In both approaches, the reversible time delays measured between waveforms increase with the elastic deformation of the sample. From the numerical modeling, we gain insight to the relative contributions of different physical effects on the CWI measurement: local density changes from volumetric strain, the deformation of scatterers, and acoustoelastic effects. Our results suggest that acoustoelastics effects related to nonlinear elasticity are dominant.