Computing Young Modulus from Elastic Constants?

Is the material truly homogenous with no dislocations what so ever? Otherwise you would be trying to measure the modulus of a sub nanomaterial.

My material is single crystal and have two phases. Material is after 7steps homogenisation annealing and 2steps ageing (Ni base superalloy). The effect of dislocation can be neglected in my case. I need the modulus behaviour for FEM measurements.

Youngs Modulus is delta stress / delta strain for initial linear portion of the curve. I suspect you may be interested in the total area under the curve though. Ideally you should determine modulus by testing in at least three locations between. Assuming your initial directions 001= 0 degrees and 011= 90 degrees, test at positions 22.5/45/68 degrees then plot in three dimensional coordinates.

Modulus of elasticity E is the slope of straight line in elastic portion of stress-strain curve, E = Δ Stress / Δ Strain.

It sounds like you are trying (or need to ) plot Mohr's Circle. Many on-line resources exist, such as http://www.engapplets.vt.edu/Mohr/java/nsfapplets/MohrCircles2-3D/mohrcircles2-3d.htm . Be aware that you cannot simply evaluate components as you might when resolving triangles.