Professor Marc Geers, Technical University of Eindhoven

Miniaturization of metallic systems: size effects and continuum modelling approaches
When Apr 22, 2013
from 02:00 PM to 03:00 PM
Where LR8
Contact Name
Contact Phone 01865 283302
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In the past decade, industry is increasingly focused on the behaviour of materials in micro- and nanosystems. At the level of many microsystems, metallic structures and films are used ranging from sizes of a few microns to hundreds of microns. The scientific community has given a lot of attention to this subject, in particular in the range where size effects have a dominating contribution.


This presentation focuses on a full classification and computational modelling of different size effects, related to different underlying physical mechanisms in the crystalline microstructure. To this purpose, size effects in the plastic response of thin Al structures and films have been measured and modelled accordingly. Within this context, the following aspects will be addressed:


• first-order size effects, highlighting the influence of the ratio between the size of grains and the characteristic dimensions of micro-components [1, 2]:

    – processing-induced size effects, studied on the basis of mechanical cutting and laser cutting of small scale specimens [3]

    – grain boundary induced size effects

    – surface-induced size effects, related to the presence of an outer free surface

    – statistical size effects, related to the statistical variation of the number of grains in miniaturized specimens [4]

• second-order size effects, studied by a strain gradient crystal plasticity approach, which accounts for essential short-range dislocation interactions [5].


The computational description of the FCC behaviour relies on a recently developed strain gradient dependent crystal plasticity approach, which incorporates an intrinsic scale dependence [6, 7, 8, 9, 10, 11]. The heterogeneous deformation-induced evolution and distribution of geometrically- necessary dislocations (GND’s) are incorporated into a physically based continuum theory of crystal plasticity, which is briefly presented. Additional boundary conditions are formulated at the grain boundaries, obstructing the slip at the slip system level in the direction perpendicular to those boundaries. At the free (external) surfaces, the GND densities are prescribed to be zero, which allows to capture an intrinsic size dependence upon varying grain size and/or sample size.


Comments on the physical justification and interpretation of the higher-order terms will be presented. An idealized dislocation pile-up configuration is considered, for which a sharp comparison between discrete and continuum solutions can be made. A rigorous connection has been established with the work of Groma et al. [12], whereby the resulting higher-order theory has been demonstrated to correlate well with discrete dislocation simulations [13]. In the present contribution we demonstrate how a virtually identical theory can be formulated on a purely deterministic basis thus providing additional insight into the origin of the nonstandard terms in crystal plasticity [14]. Continuum interaction terms are derived by considering all interactions in a collection of discrete dislocations on a single slip system. The dislocations are arranged in an idealized configuration of infinite walls. Based on this discrete configuration a continuum expression is derived for the dislocation interactions by assuming that each wall interacts with many walls. The resulting interaction stresses are incorporated in equations governing the transport of dislocation densities along slip planes. The resulting continuum model is compared with discrete dislocation simulations. This analysis shows the importance of including all short-range interactions in the model, especially in the case where positive and negative dislocations are equally present. The additional role of dislocation climb is briefly addressed through a DDD simulation [15].




[1] P.J.M. Janssen, Th.H. de Keijser, and M.G.D. Geers. An experimental assessment of grain size effects in the uniaxial straining of thin Al sheet with a few grains across the thickness. Materials Science and Engineering: A, 419(1-2):238–248, 2006.

[2] C. Bayley, W.A.M. Brekelmans, and M.G.D. Geers. A three-dimensional dislocation field crystal plasticity approach applied to miniaturized structures. Philosophical Magazine, 87(8-9):1361–1378, 2007.

[3] P.J.M. Janssen, J.P.M. Hoegnagels, Th.H. de Keijser, and M.G.D. Geers. Processing induced size effects in plastic yielding upon miniaturisation. Journal of the Mechanics and Physics of Solids, 56:2687–2706, 2008.

[4] T. Fulop, W.A.M. Brekelmans, and M.G.D. Geers. Size effects from grain statistics in ultra-thin metal sheets. Journal of Materials Processing Technology, 174(1-3):233–238, 2006.

[5] M.G.D. Geers, W.A.M. Brekelmans, and P.J.M. Janssen. Size effects in miniaturized polycrystalline fcc samples: Strengthening versus weakening. International Journal of Solids and Structures, 43(24):7304–7321, 2006.

[6] L.P. Evers, W.A.M. Brekelmans, and M.G.D. Geers. Non-local crystal plasticity model with intrinsic ssd and gnd effects. Journal of the Mechanics and Physics of Solids, 52(10):2379–2401, 2004.

[7] L.P. Evers, W.A.M. Brekelmans, and M.G.D. Geers. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures, 41(18-19):5209–5230, 2004.

[8] C. Bayley, W.A.M. Brekelmans, and M.G.D. Geers. A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. International Journal of Solids and Structures, 43(24):7268–7286, 2006.

[9] M.G.D. Geers, W.A.M. Brekelmans, and C. Bayley. Second-order crystal plasticity: internal stress effects and cyclic loading. Modelling and Simulation in Materials Science and Engineering, 15(1):S133–S145, 2007.

[10] I. Erturk, J.A.W. van Dommelen, and M.G.D. Geers. Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories. Journal of the Mechanics and Physics of Solids, 57:1801–1814, 2009.

[11] R.H.J. Peerlings, L.H. Poh, and M.G.D. Geers. An implicit gradient plasticity-damage theory for predicting size effects in hardening and softening. Engineering Fracture Mechanics, 95:2–12, 2012.

[12] I. Groma, F.F. Csikor, and M. Zaiser. Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics. Acta Materialia, 51(5):1271–1281, 2003.

[13] A. Roy, R.H.J. Peerlings, M.G.D. Geers, and Y. Kasyanyuk. Continuum modeling of dislocation interactions: Why discreteness matters? Materials Science and Engineering: A, 486:653–661, 2008.

[14] M.G.D. Geers, R.H.J. Peerlings, J.P.M. Hoefnagels, and Y. Kasyanyuk. On a proper account of first- and second-order size effects in crystal plasticity. Advanced Engineering Materials, 11(3):143–147, 2009.

[15] C. Ayas, V.S. Deshpande, and M.G.D. Geers. Tensile response of passivated films with climb-assisted dislocation glide. Journal of the Mechanics and Physics of Solids, 60(9):1626–1643, 2012.