Dear Khaled Megahed ,

I did 4-point-bending-tests (deformation controlled) with steel-fibre-concrete to measure

force-displacement and normalized the measured values to stress-strain.

Please find here an abstract concerning my nonlinear simulation in Ansys: https://tu-dresden.de/zih/hochleistungsrechnen/hpc-monat?set_language=en

or here the pre-print:

Preprint Modeling the stiffness degradation of concrete based on phys...

What do you exactly mean by "effective" ?

As damage progresses, the loaded section area of the material will include damaged and undamaged area. The effective stress or undamaged stress is the force over the undamaged area. The effective stress = true stress/(1-D), where D is the damage density (=Damaged area/ total area of the loaded section).

Effective stress is non-physical. In plasticity-damage formulation it is product of hardening of yield surface and nominal stress is then calculated as you described by damage variable. Various hardening/damage variable combinations can produce same nominal stress. See:

https://dragozarkovic.blogspot.com/p/concrete.html?m=1

Drago Žarković

In the attached link, does effective strain curve has an empirical formulation? Also, should it be ascending (no softening part)?

Effective stress tell us how high would stress be if the material is not damaged for that strain level. So it can not be measured.

That stress is not something you can measure.

It is acending because plasticity part of this constitutive model is defined with hardening.

I think that unloading process could provide me with the elastic modulus, and from it I could compare it with the initial elastic modulus at the start of loading and extract damage parameter. Is it correct?

No. Look at the attached picture sig-eps experiment. As you said

effective stress = true stress/(1-D)

so

D = 1 - effective stress/true stress.

Damage correlates to stress and not modulus of elasticity. Since effective stress is non-physical, not measurable, damage is also not measurable.

You could fit experimental data with some theoretical plasticity-damage model and determine D (of that model) when modeling with it. See picture sig-eps experiment-numeric.

Thanks Drago Žarković

Stiffness = force / displacement

Normalized stiffness = stress / strain (flexural modulus) (Material parameter)

Stiffness Change = Load Rate ​​/ Strain Rate = Force Change / Deformation Change (Material parameter) (Damage rate)

You can measure the stiffness change (damage rate) in experiment:

delta stiffness = delta F / delta u

I think that thi could also be useful to yo. See, Chapter 3.

Book Viscoelasticity and Rate-Dependent Continuum Damage Models

Thank you Miguel Cervera