I am looking for the relationship between the elastic modulus of concrete measured from concrete cylinder in compression versus the one determined indirectly from the bending test of single notch concrete prism.
As you know, the standard for determination of elastic modulus of concrete is to test a cylinder in compression.
There are some models such as Effecttive Crack Model (ECM) for detrmination of fracture toughness of concrete, which allows to INDIRECTLY determine the elastic modulus of concrete based on an arbitrary load level and its corresponding deflection in the initial (linear) portion of the load-deflection curve of the notched beam specimen. In ECM, the concrete prism having a single notch is tested in 3 point bending.
A comment on your question instead of an answer. Conceptually, the elastic modulus is a property of the material, that means it does not depend on the specimen type, size or loading configuration. If you consider a sample or component under loading, the ratio of load and deflection is called compliance. Compliance is not a material property, it depends on the elastic modulus of the material, but also on the specimen type, size and loading configuration. So the best you can do is to look for a relationship between the compliance of a notched bar of given size, notch depth, and loading configuration (e.g. three-point bending, four-point bending, clamped bar) and the elastic modulus of the material. You should find this kind of relationship in mechanical engineering handbooks. I understand that you wish to calculate the elastic modulus of the material from the compliance of a notched bar in bending. While this is possible in principle, it is dangerous, because the deformation in the stress field of the notch may not be purely elastic. I hope this will help you to get a clear view at the problem and the clear view at the problem will help you to find the solution.
Firstly thanks for Professor Bergner's comments, very clear and inspiring.
I also have a short comment here:
As you may understand, three point bending test of a notched beam is traditionally used for obtaining the parameters describing fracture properties (e.g. fracture energy Gf), which should be determined AFTER knowing elastic modulus "E".
I think one problem is these parameters (referring to fracture properties) of concrete generally vary in a big range (quasi-brittle material), not as stable as E. If you want to back calculate the E from them, you could introduce "disturbance", not a very good idea.
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