Recently, there have been many strain gradient theories that are used for the interpretation of size effect at the micron and submicron length scales. The basic idea of these theories is the introduction of a first, or second (or both) gradients of strain or any internal state variable in the governing equations of classical theories.

 I have been working for the last 5 years on the development of several strain gradient plasticity and damage theories that are based on dislocation mechanics and laws of thermodynamics. Please check my recent publication in IJSS that will give you a comprehensive review of the recent developments of scale-dependent theories and will give you a systematic framework for development of any gradient plasticity theory: 

Abu Al-Rub, R.K., Voyiadjis, G.Z., and Bammann, D.J., “A thermodynamic based higher-order gradient theory for size dependent plasticity,” International Journal of Solids and Structures, Vol. 44, No. 9, pp. 2888–2923, 2007. 

Moreover, if you need to know more information about the physical interpretation of the gradients in terms of geometrically necessary dislocations, please check this reference 

Abu Al-Rub, R.K., Voyiadjis, G.Z. “A physically based gradient plasticity theory,” International Journal of Plasticity, Vol. 22, No. 4, pp. 654-684, 2006. 

One of the most crucial aspects of the strain gradient theories is the determination of the material length scale which scales the gradient effect. For a systematic study on the determination of this length scale from micro and nanoindentation tests, check these papers: 

Abu Al-Rub, R.K. “Prediction of micro- and nano indentation size effect from conical or pyramidal indentation,” Mechanics of Materials, Vol. 39, No. 8, pp. 787-802. 

Abu Al-Rub, R.K., Voyiadjis, G.Z. “Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments,” International Journal of Plasticity, Vol. 20, No. 6, pp. 1139-1182, 2004.

Finally, I know that the finite element implementation of the strain gradient theories is not an easy task and cannot be directly implemented in the available commercial Finite Element software like ABAQUS. Therefore, I have developed a very simple and accurate computational framework that allows you to implement these theories in ABAQUS directly by just developing a user material subroutine (either UMAT in ABAQUS/Standard or VUMAT in ABAQUS/Explicit). Check this paper for details about this computational framework:

Abu Al-Rub, R.K., Voyiadjis, G.Z. “A Direct finite element implementation of the gradient plasticity theory,” International Journal for Numerical Methods in Engineering, Vol. 63, No. 4, pp. 603–629, 2005.

I hope that this will help.

Rashid K. Abu Al-Rub, Ph.D.

Catholic University of America