Computational modeling issues of gradient-extended viscoplasticity

Crystal (visco)plasticity is the accepted model framework for incorporating microstructural in-formation in continuum theory with application to crystalline metals where dislocations constitute thephysical mechanism behind inelastic deformation. In order to account for the size effects due to the ex-istence of grain boundaries in a polycrystal, it is convenient to include some sort of gradient-extensionof the flow properties along the slip directions, either in the dragstress or backstress (from GND, whichare generally of two types: edge and screw dislocations). Various explicit models based on this con-ceptual background have been proposed, not the least by Gurtin and coworkers 1 ; however, severalmodeling issues still await its resolution. An elegant way of unifying gradient theory for differentapplication models was presented by Miehe 2 .In this contribution we focus on issues related to the theoretical as well as the computationalformat, while (for the sake of clarity) restricting to gradient-extended viscoplasticity for a standardcontinuum. Thereby, we avoid the additional complications associated with the proper version ofcrystal (visco) plasticity, such as higher order boundary conditions. The so-called “primal” formatexploits the internal variables as the primary unknown field together with the displacement field. Analternative format is coined the “semi-dual format”, which exploits (in addition) the microstresses,thereby defining a mixed variational problem. We note that a mixed method that bears resemblancewith the semi-dual format has been used extensively in our research group in recent years 3 ; however,without possessing a well-defined variational structure.We compare the primal and semi dual variational formats in terms of pros and cons from variousaspects. We also discuss the pertinent FE-spaces that appear as the natural/possible choices. In partic-ular, for the semi-dual format we investigate the possibility to use a minimal degree of regularity thathas so far not been discussed in the literature