A computer-implemented method of estimating mechanical properties and stresses of rocks around a borehole. In situ stresses and dynamic and static moduli of a rock are jointly inferred from acoustic radial profiles measured with a borehole sonic tool. Inversion is performed on equations that govern the near-borehole distributions of the compressional wave slowness, the fast shear wave slowness, the slow shear wave slowness, and the shear wave modulus in the plane perpendicular to the borehole axis for in situ stresses, the dynamic shear modulus, the dynamic Lamé parameter, λ, the static drained Young's modulus, and the static drained Poisson's ratio. Third-order nonlinear dynamic moduli are also inferred by the procedure. Material properties are retrieved in an isotropically stressed reference state. The input data used by the inversion is appropriately prescribed. Methods for constraining the solution in the event of noisy or limited data are demonstrated.