how to determine stiffness matrix of transversely isotropic material

Dear all, I am an applied mathematician and I  have been stucked by the this general question for a while, i.e.,  to determine the stiffness matrix of transversely isotropic material from known strain and stress tensors.  The definition of transversely isotropic tells us that the stiffness matrix can be defined by 5 constants, see attached figure. C44 is easy to determine, so we can focus on the remaining 4 constants. I attempted to establish 4 linear equations for these 4 constants with the known strain and stress components. However, the coefficient for this 4 by 4 linear system turns out to be singular, hence not sovable. I would want to know how these constants are determined in the communities of solid mechanics - I did not find answer in all references I can find. Also I would want to know why my method described above fails. Thank you very much!  - yzhou