Implementing hydrogen effects in constitutive modeling of metal plasticity

[PerennialII]

I've been working on/off with problems of environmentally assisted cracking, primarily in the fracture mechanics side of things and on occation some damage mechanics work. As a continuation for this I started looking for plasticity models which introduce the effects of hydrogen directly into typical models of incremental plasticity, and so far the one I've available as a complete implementation is given by Sofronis et al, in which the hydrogen concentration is introduced to stress-strain behavior as :

$$\sigma_y(\epsilon_p,c)=\sigma_0[(\gamma-1)c+1](1+\frac{\epsilon_p}{\epsilon_0})^{1/n}$$

(local hydrogen enhances dislocation mobility -> flow stress decreases), where the yield strength $$\sigma_y$$ is dependent on the concentration $$c$$, effective plastic strain $$\epsilon_p$$, $$\gamma$$ is a softening parameter, $$\epsilon_0$$ the yield strain and $$\sigma_0$$ the yield strength for $$c = 0$$. The model has a number of other parameter ranging from dislocation density to specification to lattice traps etc.

So the question I'm thinking about now is what else is out there, anyone got any experience on what material models are out there and might be usable considering linkage to damage / fracture mechanics ?

 

[Astronuc]

Perennial, what references do you have for Sofronis? He has done a lot of work on a lot of different materials. He has co-authored several papers with Birnbaum, and both have done a fair amount of work with Zr (hcp) alloys and effects of hydrogen.

Zr alloys are interesting since they form Zr hydride, ZrH_x where x~1.6. Locally though the x~2. The hydrides dissolve at relatively low temp, and hydrogen solubility is another factor. Then to complicate matters, the dissolution temperature is different than the precipitation temperature.

In general the plasticity is very localized, such that macroscopically, the material shows little strain and effectively behaves in a brittle manner.

I believe this behavior is quite different for steels and Ni-alloys.

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A major issue in Zr-alloys is the waterside corrosion and the hydrogen which is produced. The hydrogen which diffuses into the metal will migrate under thermal gradients and certainly will prefer to precipitate near defects, especially where tensile fields are present. This is somewhat different from the environmental hydrogen-assisted cracking in which the hydrogen forms at the crack tip (usually by a chemical or galvanic reaction) and diffuses from that point up the stress gradient.

I found a TOC from a conference - Hydrogen Effects in Materials (pdf file)

 

[PerennialII]

Thanks for a great reply as always !

Perennial, what references do you have for Sofronis? He has done a lot of work on a lot of different materials. He has co-authored several papers with Birnbaum, and both have done a fair amount of work with Zr (hcp) alloys and effects of hydrogen.

The prime reference in this respect where he presents a basis for hydrogen concentration modified von Mises type plasticity model is : Sofronis, P. Liang, Y. and Aravas, N. "Hydrogen induced shear localization of plastic flow in metals and alloys", European Journal of Mechanics A/Solids, 20(6), 857-872, 2001. I have some other works where he addresses hydrogen dislocation interaction, overall really elegant work.

Zr alloys are interesting since they form Zr hydride, ZrH_x where x~1.6. Locally though the x~2. The hydrides dissolve at relatively low temp, and hydrogen solubility is another factor. Then to complicate matters, the dissolution temperature is different than the precipitation temperature.

In general the plasticity is very localized, such that macroscopically, the material shows little strain and effectively behaves in a brittle manner.

I believe this behavior is quite different for steels and Ni-alloys.

Yeah, the underlying behavior of hydrogen makes the problem a really complex one (and extremely interesting) and material dependent, it appears that in the model I'm testing they've eliminated localization effects by characterizing hydrogen concentration to affect yield, but retain hardening from thereon. On the other hand, the ultimate goal of our work is to connect plasticity and damage (and cracking), which pretty much dictates that the problem needs to be approached in a non-local sense (the other thread a while back got linked here), in order to capture the softening and damage in any reasonable manner (I'm doing finite element analyses of fracture mechanics specimen with the material models, using SCC crack propagation & hydrogen charged toughness data). Getting the whole chain to work qualitatively is something that'll need quite a bit of pushing, quantitative is years away (treatment of localization issues ... usually ).

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A major issue in Zr-alloys is the waterside corrosion and the hydrogen which is produced. The hydrogen which diffuses into the metal will migrate under thermal gradients and certainly will prefer to precipitate near defects, especially where tensile fields are present. This is somewhat different from the environmental hydrogen-assisted cracking in which the hydrogen forms at the crack tip (usually by a chemical or galvanic reaction) and diffuses from that point up the stress gradient.

I found a TOC from a conference - Hydrogen Effects in Materials (pdf file)

Considering the ties of this research to e.g. nuclear side of things the behavior of Zr alloys is of high relevance. And yes, the differences in EAC and traditional HE (of welds for example) are certainly there, I think the EAC is actually easier to approach due to the predefined nature of the defects and extensive studies of hydrogen formation and transport near cracks in SCC, in HE you're always lost with initial defects, initiation, incubation and so forth. Thanks for the TOC ... I noticed a few guys from our institute attended so I'll hit our library first thing Monday morning and should be able to find the whole proceedings.

 

[PerennialII]

... a few years ago listed the couplings what we wanted to investigate in relation to the whole hydrogen - damage problem, went somewhere along these lines :

1. Hydrogen diffusion problem, feed from a thermal problem and complete coupling to stress analysis PDE problem (typically residual stress PDE problem e.g. from simulation of welding etc.)
   
   The question arising in this area are primarily related to accurate modeling of hydrogen diffusion (not a problem really), when adding it to a realistic cracking scenario (like welding or welds overall, necessitating in principle a complete analysis of the welding residual stresses and thermal history numerically; complex but not really much more than lots of work), but the problematic area is really the coupling of hydrogen diffusion with elastoplasticity. And when in 2) below we apply damage and fracture mechanics, how do we properly account for the interaction to hydrogen enhanced plastic deformation. The questions I'm wondering, in addition to the softening constitutive equation concept overall, are related to accounting for the hydrogen concentration itself correctly. Meaning, that hydrogen diffusion is relatively easy to model such that stress gradient effects are included, but connecting it to where the hydrogen resides is somewhat of a puzzle (like assessing hydrogen residing in interstial lattice sites and reversible trap sites such that the whole coupling thing remains coherent).
   
   
2. Coupling between the elastic-plastic crack or crack initiation problem and the hydrogen diffusion problem, using damage mechanics
   
   The definition of a suitable damage concept is one of the problems here, and in particular, how to do it such that the constitutive framework is complete and the interaction with plasticity is addressed properly. Since material damage is present in any event, localization needs to be addressed ... we're currently doing this using non-local methods and wouldn't expect that to be a huge issues. What is, however, is whether to derive a constitutive framework where the evolution equations for damage variables have hydrogen concentration etc. as a parameter, or to state that all hydrogen effects are contained within plasticity and from thereon failure is evaluated e.g. using cohesive theories. Using damage theories the softening could be input directly into many existing damage laws, if the hydrogen effects are to be input into cohesive models, it is likely that the models themselves will need to contain concentration terms introducing the "softening" there as well. The latter approach is feasible modeling wise, like cohesive theories seem to be in general, but a damage theory might be able to handle the damage evolution itself better. Remains to be seen, anyways, going to continue with the research.
   
   Building links to more macroscopic issues can be done quite simply using fracture mechanics, i.e. evaluating contour integrals over problems modeled using a framework such as above is a good way to derive "trend" curves etc. methods.
   
   

 

[sanchin]

Hydrogen + constitutive behavior + fracture

If you only want to model what happens to the bulk material upon hydrogen absorption, you may be very well off changing only the bulk constitutive behavior. For instance, Sofronis, Liang, and Aravas obtained in that way the point of structural instability derived from such a bulk behavior. But you will not be able to simulate crack propagation. If you actually want to track the evolution of cracks, you need more than that. Cohesive models are very useful for this purpose, as used by Serebrinsky, Carter and Ortiz, "A quantum-mechanically informed continuum model of hydrogen embrittlement", Journal of the Mechanics and Physics of Solids 52 (2004) 2403-2430, and also Liang and Sofronis, "Toward a phenomenological description of hydrogen-induced decohesion at particle/matrix interfaces", Journal of the Mechanics and Physics of Solids 51 (2003) 1509-1531. If you decide to use cohesive models to let the crack/s evolve, then you face the issue of how to derive the appropriate constitutive behavior of the cohesive zone. And on this point there is a large literature too.