In Hall-Petch formula there is relation between Vickers hardness and grain size. In fact, there are 2 constants in this relation. How can I obtain them for pure Copper? I think they should be derived from the initial specimen conditions i.e. initial grain size and hardness, but how?
H=H0+k/d1/2, d - grain size, k - koeff. of H-P. for Cu k=5700 MPa/nm -1/2
For example: H - is hardness of HPT (high pressure torsion) Cu, H0 - is hardness of annealed HPT Cu, d - is grain size of HPT Cu (is about 200 nm)
1. C. Suryuanarayana, D. Mukhopadhyay, S.N. Patankar, F.H. Froes. J. Mater. Res.7, 2114 (1992).
2. G.W. Nieman, J.R. Weertman, R.W. Siegel. Nanostruct. Mater.1,185(1992).
Dear Rinat,
Thanks for your answer. These 2 articles mention constants for noncrystalline materials, while in our case we face micro-size grains. Would you tell me, whether these constants true for micro scale or not?
Dear Mehdi. Unfortunately I can't say- is it possible this equation to use for your case or not. In our case we were just trying to evaluate hardness depending on grain size . Could you write more about the data of your experiment?
Hi Mehidi, you can get the constants experimentally using the classic method. The factor k is the slope of the straight line that is obtained when is plotted H vs. D^(-1/2) and H_0 is the intersection with the ordinate in this plot. Take a look in the Dieter´s book, "Mechanical Metallurgy"
Dear Rinat,
Our experimental data is based on below article:
http://www.sciencedirect.com/science/article/pii/S0927025609001177
As you can see, we have just data of hardness and we want to estimate grain size.
Dear Dr. Garcia-Sanchez,
As I mentioned, we have just data of hardness and we cannot plot H vs. d^(-1/2). Indeed, in a reverse way, by using of hardness, H0 and k we want to estimate d (grain size). But unfortunately we don't have H0 and k data for pure copper.
It is not that difficult. You need a few samples with different grain sizes .. :
http://www.sciencedirect.com/science/article/pii/S0167663614001835
But it is not that easy to interpret the grain size by knowing the hardness value! There are constants in the Hall-Petch equation that depend on the strain and strain rate values etc.
Also be careful of the size effect, as mentioned in the above link.
similar to many experimental relationships, use of a set of empirically calibrated H-P coefficients has its limitations. One has to make sure that the reported/published coefficients match well your experimental conditions. One key condition is "the deformation path/process" and homogeneity of the deformation. Overlooking these, you could easily end up using the non-representative coefficients..
Dear Ehsan and Shahin,
Thank you very much for your answers. Estimating grain size by hardness seems very complex and not precise.
Dear all
There is a new work that tries to combine both the Hall-Petch and its inverse in one model through presenting a multiscale model that enables description of both the Hall-Petch relation and its inverse in one equation without the need of prior knowledge of the grain size distribution, for more details, please see the following link:
https://www.researchgate.net/publication/324507369_A_new_multiscale_model_to_describe_a_modified_Hall-Petch_relation_at_different_scales_for_Nano_and_micro_materials
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