An implicit gradient model by a reproducing kernel strain regularization in strain localization problems Original Research Article

A reproducing kernel strain regularization (RKSR) as a mathematical generalization of gradient theory and non-local theory for strain localization problems is presented. RKSR introduces a correction of the weight function in the non-local strain by imposition of gradient reproducing conditions. Both continuum and discrete forms of RKSR are presented, and they lead to an implicit representation of gradient models. As such, RKSR provides a gradient type regularization to the localization problem without increasing the order of differentiation in the governing equations. Hence no additional boundary conditions are required, and the need for higher order continuity for the approximation of unknowns in the governing equations is no longer an issue. A von Neumann spectral analysis is employed to study the spectral properties of RKSR of various orders in one dimension. It is shown that RKSR almost duplicates the spectral properties of second and fourth order gradient theories. In summary, RKSR reproduces the regularization properties of gradient methods without dealing with additional boundary conditions or higher order continuity issues.