Dear Muhammad,

In my opinion it is no so simple, because your material probably has a complex microstructure and the involved strain energy is not simply related to the increase of the dislocations density (so the energy of the dislocation per unit of length multiplied by the number of dislocations). This simply relationship will work only for the deformation of a pure metal in which the strain hardening would be the only operating mechanism.

When dislocations interact with other defects (point defects, precipitates, grain boundaries) your strain energy will exhibit a more complex dependence on several factors.

One way to follow experimentally the plastic strain energy as a function of cycling is to follow the evolution of the stress-strain cycle, measuring the area of the internal loop normalised to the maximum applied energy (which represent the energy disipated per unit of volume) and follow its evolution as a function of cycling. The you can correlate this magnitude with the density of dislocations measured by TEM.

I hope this could be useful.

Kind regards

Dear Prof. Juan,

Thank you for the explanation.

Dear  Muhammad,

please see the paper in RG and Figures 7-10 (might help you):- 

https://www.researchgate.net/publication/255945588_Temperature_effect_on_the_low-cycle_fatigue_behavior_of_type_316L_stainless_steel_Cyclic_non-stabilization_and_an_invariable_fatigue_parameter

Best wishes

Article Temperature effect on the low-cycle fatigue behavior of type...

Dear Dr. Phaniraj,

Thank you for your answer. I will definitely read this article.

Dear Dear Muhammad,

I hope you have got some good feel of the issue. I can echo Prof Jose San Juan's comment. This is not a solvable problem by simple calculation. One concept which is at mistake to me is the statement that "Energy given to the material in form of deformation during cycling loading is stored in the material as dislocations". The energy is not all stored, but "dissipated". Some part is stored within the materials (not all as dislocations) and some is converted into other forms, like heat and acoustic emissions. For example, when a dislocation moves, heat is generated and dissipated into the environment, consuming the external mechanical energy provided to the material, yet no new dislocation length is created. That means there does not exist a quantitative correlation between the net energy lost and the net total length of dislocations generated during mechanical deformation, and deformation cycling. 

Hope this is useful.

Best regards,

Yinong Liu