A Comparison of Computational Formats of Gradient-Extended Crystal Viscoplasticity in the Context of Selective Homogenization

Paper in proceeding, 2016

Crystal (visco)plasticity is the accepted model framework for incorporating microstructural information in continuum theory with application to crystalline metals, where dislocations constitute the physical mechanism behind inelastic deformation. In order to account for the size effects due to the existence of grain boundaries in a polycrystal, it is convenient to include some sort of gradient-extension of the flow properties along the slip directions, either in the dragstress or backstress (from GND’s). Various explicit models based on this conceptual background have been proposed, e.g. Gurtin et al.[1] , Gottschalk et al.[2]; however, several modeling issues still await its resolution. A comprehensive unifying account of gradient theory for a variety of application models was presented by Miehe[3] . When applied to a polycrystal, it is desirable that the homogenization strategy will result in a standard continuum format on the macroscale, whereas micro-stresses are confined to the mesoscale and and automatically "suppressed" during the procedure of (selective) homogenization. This can be achieved within a fairly general setting of variationally consistent homogenization. In this contribution we focus on issues related to the computational format of gradient-extended crystal viscoplasticity that constitutes the RVE-problem. A few different variational formats are thereby investigated. The so-called “primal” format exploits the slip on each slip system together with the displacement field as the unknown global fields. An alternative format is coined the “semi-dual format”, in which the slip variables are replaced by the microstresses as global fields, thereby defining a mixed variational problem. For both the primal and semi-dual formulations, we establish variational principles for the time incremental FE-problems which ensure symmetry of the corresponding tangent problems. We note that a mixed method that bears strong resemblance with the semi-dual format has been used extensively in our research group in recent years, e.g. Bargmann et al.[4] ; however, without possessing a well-defined variational structure. We compare the primal and semi-dual variational formats in terms of pros and cons from various aspects. We also discuss the pertinent FE-spaces that appear as the natural/possible choices and assess the computational efficiency of the FE-approximations with the aid of numerical examples pertaining to a single crystal as well as to a polycrystal in the homogenization context. References: [1] M. E. Gurtin, L. Anand, S. P. Lele, Journal of the Mechanics and Physics of Solids, 55, 1853, (2007) [2] D. Gottschalk, A. McBride, B.D. Reddy, A. Javili, P. Wriggers, C.B. Hirschberger, Computational Material Science, 111, 443, (2016) [3] C. Miehe, J. Mech. Phys. Solids, 59 898, (2011) [4] S. Bargmann, M. Ekh, K. Runesson, B. Svendsen, Philosophical Magazine, 90, 1263, (2010)