Non Local Model Vs Implicit Gradient Model

Kapil,

There are several nonlocal theories in the literature: (a) Integral-type nonlocal theories; (b) explicit gradient theories, which are weak forms of the integral-type nonlocal theories; (c) implicit gradient theories, which are strong nonlocal theories and are equivalent to Integral-type nonlocal theories. Moreover, the last three nonlocal theories can be considered either higher-order or lower-order gradient-dependent theories, depending on the introduction of higher-order stresses and additional non-classical boundary conditions (i.e. higher-order). As far as the numerical implementation, the lower-order explicit gradient-dependent theories are the easiest to implement in numerical codes. One can implement those either directly through the stress update material subroutine (for example using least square fitting to get the gradient value) or through introducing additional degrees of freedom at selected nodal points in a finite element context. The numerical implementation of the nonlocal integral-type theories is not attractive and is not an easy task. This is attributed to the unknown limits of the integral at which the weighting function vanishes. Therefore, imaginary nodal points outside the physical boundaries need to be introduced to be able to calculate the integral term, which is a non-physical procedure.

For more details about the above issues, you can read the following articles of mine:

Voyiadjis, G.Z., Abu Al-Rub, R.K., Palazotto, A.N. “Thermodynamic formulations for non-local coupling of viscoplasticity and anisotropic viscodamage for dynamic localization problems using gradient theory,” International Journal of Plasticity, Vol. 20, No. 6, pp. 981-1038, 2004.

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.

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. 

Rashid K. Abu Al-Rub, Ph.D. Texas A&M University