Modeling velocity changes in response to stress changes plays an important part in understanding seismic responses from the subsurface. One branch of such modeling consists of treating an assemblage of grains as an effective medium and using established grain contact theories to determine the elastic moduli. Such models are commonly limited to hydrostatic or uniaxial strain scenarios, not capable of capturing the anisotropy induced by a general triaxial stress state. A new set of expressions is developed by extending an existing effective medium theory to a general stress state in which the radial (horizontal) stress is different from the axial (vertical) stress. The theory is valid at the limits of slip and no slip. Novel functions to combine the no-slip and slip limits are implemented to match the model to observed laboratory data on glass beads and sand grain assemblages loaded along different stress paths. The new expressions provide a good agreement between the modeled and measured stress and stress path dependence of the compressional wave (P-wave) and shear wave velocities and associated P-wave anisotropy. This stress dependence is of particular interest in rock-physics modeling workflows evaluating time-lapse feasibility for shallow or unconsolidated sand reservoirs or for characterizing burial history.