Tomography of small residual stresses

We study the inverse problem of determining the residual stress in Man's model using tomographic data. Theoretically, the tomographic data are obtained at the zeroth approximation of geometrical optics for Man's residual stress model. For compressional waves, the inverse problem is equivalent to the problem of inverting the longitudinal ray transform of a symmetric tensor field. For shear waves, the inverse problem, after the linearization, leads to another integral geometry operator which is called the mixed ray transform. Under some restrictions on coefficients, we are able to prove the uniqueness results in these two cases.