Hall-Petch relation is connecting grain size and tensile properties. With smaller grain you usually get better mechanical properties.

@David Bombač, Smaller grains results in more grain boundary fraction in the material and thus result in improved strength. But my question is , will it not result in loss of ductility at the same time?

I wouldn't think so. If you have smaller grain and same density of faults (dislocations, SF, ...) there is smaller amount of them in each grain and also distance to grain boundary where they can relive is shorter. But still it would depend a lot what material are you using, especially if there are some pinning particles.

And another question, are we talking about room temperature ductility or not?

@David Bombač :Yes we are talking about room temperature ductility. My understanding is that when we have a large fraction of grain boundaries in the material, the dislocations can be easily arrested at these grain boundaries and thus leading to increased strength and loss of ductility. Though we have still same defect density , their mobility is severely restricted because of smaller grain size , then how the ductility can remain the same?

Thank You @Hamed Mirzadeh

By decreasing the grain size, there is a possibility to activation of new deformation mechanisms that are different from slip and twining, such as grain boundary sliding and grain rotation. these new mechanisms can increase the ductility in materials even by reduction of grain sizes, there is a very popular paper about this apposite response which is written by valiev et.al also I attached a photo you can see the unique behavior of some nano materials

Dear all

To answer the question, we need to understand the mechanism of the Hall-Petch relation and its inverse. 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

It is often stated that “Grain boundary strengthening has the advantage that the ductility of the material does not decrease with decreasing grain size and increasing strength.” (Rösler, Harders, Bäker, Mechanical Behavior of Engineering Materials). This is an exception to the generally observed inverse relation between strength and toughness.

In this context it is important to differentiate between toughness, which requires a pre-existing crack or notch and ductility, which is the ability of the material to plastically deform and is usually determined in a tensile test, e.g. by the fracture strain.

Fine-grained structures usually have smaller potential flaws, which increases the stress to fracture.

Grain boundaries can also act as barriers to crack propagation, the different crystallographic orientations cause the crack change direction, and cracks can be bridged in fibrous grain structures.

For fracture to occur in a tensile test of a semi-brittle material, slip bands have to nucleate at the yield stress, these have to nucleate micro-cracks, and these micro-cracks have to propagate.

Cottrell (A.H. Cottrell, Theory of brittle fracture in steel and similar metals, Trans. AIME 212 (1958) 192-203.) developed a model in which the stress for propagating a microcrack that was initiated at the intersection of two slip planes can be calculated to $\sigma_{f} \approx \frac{4 G \gamma_{m}}{k_{y}} d^{-1 / 2}$

with $\sigma_{f}=$ fracture stress $G=$ shear modulus $\gamma_{m}=$ plastic work done around a crack as it moves through the crystal $\begin{aligned} k_{y} &=\text { dislocation locking term from Hall-Petch relation \\ d &=\text { grain size } \end{aligned}$.

Which is similar to the Hall-Patch equation.

For more, see e.g. chap. 7.2.2. of Hertzberg, Vinci, Hertzberg, Deformation and Fracture Mechanics of Engineering Materials or p.325 of Haasen, Physical Metallurgy.

Hope this helps,

Erik