If you use flexural mode for testing and determining toughness, you will most likely use a SENB (Single Edge Notched Beam Configuration).  The size of the defect/crack is known (notch depth-usually 50% of total sample thickness) and the results for K1C will be more controlled and accurate. The crack will actually be subject to tensile opening during bending. In a simple compression test, the flaw size and distribution is unknown.  The areas under the stress strain curve can be used on a comparative basis measure fracture energies.  The results from the two techniques will not be comparable.

I agree with Dr Rajendra U. Vaidya. If you apply pure tension into a certain cross section of sample in bending and pure compression into sufficient size of sample without end effect etc., results gathered from the two methods can be related eachother with a satisfactory regression curve. Plain concrete works to compression rather than tension because compression rehabilitate crack development in concrete during the test up to a certain point.  Concrete is too fragile to fracture in flexural test. thus, there are sensitive test procedures with regard to loading rate, satisfactory sample cross section etc.

I think that dépends on the nature of the material ''Elastic, hyperelastic, plastic ...etc'' you might find some corrélation or no.