I measured the softening behaviour of concrete in a deformation controlled 4-point-bending-test. Within the experiment the load speed will be reduced to reach a constant deformation speed. This reduction of the load speed represents the stiffness degradation of the concrete / the damage of the concrete.

More information see my project: https://www.researchgate.net/project/CC-Technology-Conserving-resources-with-light-weighted-concrete-constructions-for-slabs-and-walls

The presentation at the conference in Meisdorf should be interesting for you - sorry, up to now only in german language but I will upload an english version of my text soon.

Jürgen Ries please elaborate “This reduction of the load speed represents the stiffness degradation of the concrete / the damage of the concrete.”

Damage is associated with stress, not with stiffness.

Dear Drago Žarković,

in my presentation "Meisdorf" (project CC) you will find at slide 25:

Change of stiffness / damage:

∆ S = ∆F / ∆u = ∆σ / ∆ε

∆F load speed

∆u deformation speed

∆S ~ remaining energy / total energy

𝜎 = (𝐹 ∗ 𝑙) / (𝑏 ∗ ℎ^2)

𝜎 ~ 𝐹

𝜀 = (ℎ∗𝑤𝑚𝑖𝑡𝑡𝑒∗28,1) / (6∗𝑙^2)

𝜀 ~ 𝑤𝑚𝑖𝑡𝑡𝑒

F=Force l=length h=height b=width wmitte=displacement (u)

The effective stress is the true stress associated with the undamaged part of a material point. As a result, as long as damage is not significant, the true and effective stresses are the same. You can therefore use the experimental stress-strain curve from a tension test to identify the material parameters associated with elasticity and plasticity as long as it is legitimate to consider damage is negligible (e.g. in the absence of softening). You can then adjust the damage-related parameters of your constitutive model to get a correct description of the softening.

Thank you Charles Mareau

How could I adjust the damage-related parameters?

Do you have an example or reference?

For the softening / stiffness degradation:

- use a theory (e.g. exponential curve for concrete)

- if you want to know the truth: measure softening by experiment like I did

Softening at specimen and structural levels is very much related to size effect.

This applies to elasto-plastic behaviour, see for instance Article Size effect and localization in J2 plasticity

as well as damage behaviour, see also

Article Phase-field regularized cohesive zone model (CZM) and size e...

Depending on the material of your interest, different strategies can be proposed. For instance, you can use an inverse method that consists in simulating the tension test and iteratively adjusts damage parameters until the agreement between the numerical and experimental force-displacement curves is reasonable. An other strategy would consists in evaluating the evolution of the damage variable by performing an incremental tension test with successive loading/unloading. Once the evolution of the damage variable is known, the identification of damage related parameters is straight forward.

For a good overview on the application of damage mechanics, you can read J. Lemaitre, A course on damage mechanics, Springer, 2nd edition (1996)

Thanks Charles Mareau and Miguel Cervera

Hey, there a way to relation the initial properties of material (E, 𝜎e, 𝜎r) with the 𝜎i that the damage occurs? What the stress that capable to generate damage in material?

Luan Bezerra Silva It is possible to build a model where damage occurs at a certain stress level. For instance, for a 1D problem (the generalization to 3D is straight forward), you can consider only the elasticity/damage coupling so that the corresponding state equations are:

s= (1-D) E0 ee

-y= 1/2 E0 ee2 = 1/2 s2/E02/(1-D)2

where y is the driving force for damage (i.e. the elastic energy restitution rate), s is the stress state, ee is the elastic strain tensor, E0 is the initial Young's modulus and D is the damage variable.

You can then use an evolution rule for the damage variable which has the following form:

dD/dt= 0 if y=yc

In this case, damage will grow only if y>=yc. yc therefore represents a threshold for damage development. Since y can be calculated from s, it is easy to express yc from a critical stress value (say sc), which corresponds to the stress level needed for damage development. The damage growth rate can be adjusted with the S and M parameters.

Khaled Megahed Khaled, there is much to it then stated here. To answer your question, yes, it should not have softening part. By your terminology, it seems you are dealing with concrete or some similar material, since for metals, true stress is the one without softening. True stress could be expressed mathematically (usually without perfect matching), and damage also can be expressed through some sort of function. But it depends on many more parameters, firstly material and type of degradation that is causing damage, than triaxility, Lode parameter, than strain rate etc. Then, you probably need this for numerical analysis, and by using Abaqus or similar software, you will need to make a choice about yield law, hardening law, damage type and law (if any), or simply use damage as a multi-linear for some points you choose. This choice is highly influenced by the experiment you did, for example, is the strain rate important and varied etc, or did you used extensometers so you have only global softening, or you used DIC so you already know the true stress-strain relation for your material. If the latter is not the case, I suggest you follow the advice Charles gave, and iteratively obtain satisfactory P-delta through numerical analysis. Some papers that might help would be:

An Experimental-Numerical Combined Method to Determine the True Constitutive Relation of Tensile Specimens after Necking, You-de Wang, Shan-hua Xu, Song-bo Ren, and Hao Wang

True Stress-True Strain Models for Structural Steel Elements P. Arasaratnam K. S. Sivakumaran and M. J. Tait

Bolted shear connectors vs. headed studs behaviour in push-out tests, Marko Pavlović, Zlatko Marković , Milan Veljković , Dragan Buđevac

ELse, if you used Aramis, then maybe take a look at:

Strain and stress relation for non-linear finite element simulations, Sohren Ehlers , Petri Varsta