My guess as follows: For very small grain sizes the  new mode of plastic deformation enters into the scenario, which is nothing but the grain boundary sliding and grain rotation, which  becomes energetically dominant  mechanism for deformation, the way that material responses to external  tractions and body forces to give desired micro- distortion and the shape changes. That what happens in real life during extrusion, forging  and rolling of macro-bodies in blacksmith shops once upon a time!.. Where GB region acts as a viscous media to reduce the energy of dissipation due to the frictional nonconservative contact forces acting between the  rolling and sliding grains.

Hardness of the material mainly depends on the crystallize size and interaction between the grain boundaries. When the  crystallize size is small then material become more harder and brittle. But the interaction between grain boundaries plays a counter role to reduce its strength. 

 Dear Tarik ,

is your meaning Hall-petch reverse below 10 nm in Nano crystalline film ?

Dear Abbas, Yes for small size of  grain structure the  plastic deformation by grain sliding even rotation becomes dominant mechanism of deformation. If you plot Yield stress as function of  inverse  SQR of grain size, you may get two intersecting lines, which has an  intersection at  d= 10 nm.   The straight  line segment on the left side has positive slope which is called dislocation sliding region. On the right side of the intersection the line segment has negative slope, which correspond to grain sliding mode, and it is also reverse Hall-Petch  mode. Don't forget in this type of piloting, the grain size  increases from right to left direction on the abscissa. 

Best Luck

Tarık

Dear Tarik ,

very very thanks due to valuable answers.

Best Regards,A

l gave very detailed reasoning  and two equations previously to explain this behavior at this platform :

1) The first one operates above the threshold grain boundary size, 10 nm, which is also called Hall-Petch connection  defined by YieldSigma=  Sigmao+K/SQWR(d), and its closely related to the dislocation pile -up against the  GB barier, and the back stress created by dislocation pile-up at the Franck-Read Source. This region is called also as the dislocation sliding region.  Because of the stress multiplication  at the leading dislocation line (SigmaP= n Sigmad) that may be high enough to break down the grain boundary barrier,  and dislocations  proceed in sliding into  the next grains. 

2) For small grain size region, which is also called reverse Hall-Petch domain,  where deformation takes place by grain sliding and grain rotation.   Yield Sigma= SigmaR -  KR/SQR(d). where the grain boundary region acts as a viscous media, which lowers the frictional energy loses. during plastic deformation process. 

Dear Tarik ,

 thanks , i enjoy from your answer.

now please tell me Why the hardness can be further enhanced in the nanocomposite coating design even though the Nano crystalline size is below 10 nm ?

Dear Abbas; when I was in your age, or  iet say in the second year of graduate studies I could easily answer this question!  Probably,  I had had exceptionally good lecturers at Stanford in those years.

Now Let me guess the answer to your question. If could remember the slip band formation mechanism, where the state of the surface namely whether it is protected by an oxide layer having not enough rigidly   or not  plays significant role  for the reflection of dislocation lines, which are continuously generated by FR sources.

a) If the surface state doesn't  have rigid protective layer then the  inelastic dislocation reflection takes place by creating a new dislocation with an opposite sign and plus the slip step at the surface.  This way kinetics energy of the incoming dislocation transfer to the formation energy of a new one: According Frank (1948) theory of dynamic multiplication because of the high kinetic energy of the  dislocations arriving to the edge of specimen they go beyond the original surface dragging a few further rows with them. These  few rows slip one atomic distance further then the rest producing a slipped plane bounded by a dislocation having opposite sign and moves in opposite direction under the same resolved shear stress and hits opposite edge of the specimen.  This dynamic process is repeated again again till FR  is exhausted.  What one  sees at the surface of the specimen ^deformation slip bands each have signatures of a  few thousands  of dislocations.  

b)  If you have stiff boundary such as alumina over the aluminum specimen, or to have protecting coating as proposed in this question this multiplication doesn't take place and the dislocation pile-up  occurs at the edges of the specimen till the back stress stop the operation of the FR.   Phenomena exactly the same as the GB barrier problem treated by Hall and Petch. The yield stress becomes inversely proportional with some 'fractional power' of  the specimen  dimension over which the slip planes are active under the applied stress system.

Dear Prof.Tarik ,

Thanks.

All the best,

Abbas

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