Good afternoon.

If you are exploring the properties of the same material that changes over time, or considering a combination of hardened and unhardened parts, then it would be logical to redefine the tangential modulus as well. When studying the properties of defective pipes, I took into account local changes in the properties of the material, including the tangential modulus, and interesting results came to light.

Good afternoon Dmitry.

If the tubes are manufactured by different -different amounts of work hardening (during the forming process of the tube), then I think the tangent modulus is different for different work-hardened tubes, and we should consider different tangent moduli for tubes (Although the material is same but the amount of work hardening is different for different tube).

This is true, but this is provable only in the presence of tests, and is relevant if there is a significant difference in the readings. In mass production, such changes can be neglected.

Dear Shyam Kishor Sharma,

With respect to the description given by you in the question, let us consider that the material is made out of a single constant manufacturing process, and the stress-strain curve obtained from experiments is fixed.

Let us further consider that during service, there are no external physics involved such as temperature-changes, moisture absorption etc. and the load applied onto the component is purely mechanical.

Therefore, with the above considered points, the stress-strain curve would remain fixed just like a material-property or material-behaviour.

Now, let us come to the question of Tangent modulus.

I would try to explain from a generalized 3D concept of purely isotropic hardening.

Kindly have a look into the attached file for understanding the description below.

Case-I : Let us assume that the stress-strain response is 'Linear Isotropic Hardening'. In this case, the hardening modulus ( H ) remains constant, just like the Young's modulus.

So, in the generalized 3D case, the expression of consistent tangent modulus matrix ( D-ep ) would be as shown in the attached file.

Here, we need to note that the tangent modulus matrix is dependent on a constant hardening modulus ( H ).

The response between equivalent total stress and equivalent total strain can be plotted as shown in the document with a linear behaviour.

But case-I is often not experimentally found. So let us move to case-II

Case-II: The stress-strain response follows Ramberg-Osgood hardening.

In this case, the tangent modulus matrix becomes a function of Ramberg-Osgood constant parameters (k , n) and also a function of varying values of equivalent plastic strain (epsilon-p) as shown in the attached document.

Therefore the expression of consistent tangent modulus matrix ( D-ep ) now incorporates a varying hardening modulus depending on the variation of equivalent plastic strain.

The response between equivalent total stress and equivalent total strain can be plotted as shown in the document with a non-linear behaviour.

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Now let us come to the actual point of concern here.

In case-I, if simulation is performed, the iterative updating of variables would not incorporate any requirement of updating the Hardening modulus (since, it is constant in this case).

But, in case-II, if simulation is performed, the iterative updating of variables would compulsorily incorporate updating the Hardening modulus with every iteration (since, it is dependent on equivalent plastic strain in this case). This situation often leads to higher number of iterations required for achieving convergence, or sometimes even leads to non-convergence (if the stress-strain curve is too much non-linear).

Due to the above complexity in case-II, the computational effort is higher in there.

As such, many computational researchers assume that the hardening modulus is constant even though it shows a curved behaviour. In other words, they prefer an idealization of Case-II through Case-I. This often solves problems of non-convergence and also leads to faster convergence and less computational effort.

So, the fact which you are saying about 'some authors assuming approximate hardening modulus' is probably due to this reason.

Moreover, those authors might have probably done a little more tweaking. In order to accurately idealize the curved stress-strain response, they might have assumed a range of constant values of hardening modulus instead of one single approximate value.

This approach seems to have been by those authors mainly for faster simulations. In that process, they might have analysed the effect of the range of constant hardening values in their papers.

However, this does not mean the material-response or stress-strain curve is changing.

For any particularly manufactured fixed material, without any external influence of temperature, moisture etc., the actual stress-strain curved response and the dependence of hardening modulus / tangent modulus matrix on plastic strain would remain fixed.

Work hardening and strain hardening during service would not change it.

Only, if there is some alteration in the manufacturing process of the material, or if the material during service is subjected to high-temperature or other aggressive environments, the stress-strain curve and corresponding hardening modulus / tangent modulus would change.