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

• Crystal (visco)plasticity is the accepted model framework for incorporating microstructural information incontinuum theory with application to crystalline metals, where dislocations constitute the physicalmechanism behind inelastic deformation. In order to account for the size effects due to the existence ofgrain boundaries in a polycrystal, it is convenient to include some sort of gradient-extension of the flowproperties along the slip directions, either in the dragstress or backstress (from GND’s). Various explicitmodels 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 gradienttheory for a variety of application models was presented by Miehe[3]. When applied to a polycrystal, it isdesirable 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 theprocedure of (selective) homogenization. This can be achieved within a fairly general setting of variationallyconsistent homogenization. In this contribution we focus on issues related to the computational format ofgradient-extended crystal viscoplasticity that constitutes the RVE-problem. A few different variationalformats are thereby investigated. The so-called “primal” format exploits the slip on each slip system togetherwith the displacement field as the unknown global fields. An alternative format is coined the “semi-dualformat”, in which the slip variables are replaced by the microstresses as global fields, thereby defining amixed variational problem. For both the primal and semi-dual formulations, we establish variationalprinciples for the time incremental FE-problems which ensure symmetry of the corresponding tangentproblems. We note that a mixed method that bears strong resemblance with the semi-dual format has beenused extensively in our research group in recent years, e.g. Bargmann et al.[4]; however, without possessing awell-defined variational structure. We compare the primal and semi-dual variational formats in terms of prosand cons from various aspects. We also discuss the pertinent FE-spaces that appear as the natural/possiblechoices and assess the computational efficiency of the FE-approximations with the aid of numericalexamples 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, ComputationalMaterial 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)

Subject headings

TEKNIK OCH TEKNOLOGIER  -- Materialteknik -- Metallurgi och metalliska material (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Materials Engineering -- Metallurgy and Metallic Materials (hsv//eng)

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