Crystal size effect in two dimensions – Influence of size and shape R. Maaß, a,⇑ C.A. Volkert a and P.M. Derlet b a Institute for Materials Physics, University of Go ¨ ttingen, Friedrich-HundPlatz 1, 37077 Go ¨ ttingen, Germany b Paul Scherrer Institut, Condensed Matter Theory, 5232 Villigen PSI, Switzerland Received 1 January 2015; revised 30 January 2015; accepted 1 February 2015 Available online 19 February 2015 Based on the statistics of both the stress and strain of a plastic event, the well known size-effect in strength can be linked to a crystal’s critical stress distribution and the universal scaling exponent of intermittent plasticity. We successfully test these hypotheses with small-scale deformation experi- ments as a function of diameter and aspect ratio, and find that the latter affects the material’s strength in a way that gives direct insight into the underlying critical stress distribution of the deforming volume. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Size effect; Plasticity; Single crystals; Stress scaling Classical laws on strength of crystalline materials did consider strength to be independent of external dimen- sions of the tested material. About 10 years ago it was con- vincingly demonstrated that there is an extrinsic size-strength scaling in crystal plasticity at small scales that is well described by a power-law: r / d n , where r is the flow strength and d is some measure of crystal size. Today’s consolidated picture of this strength-size scaling shows that n is strongly dependent on the initial microstruc- ture of the material [1–3]. While no size-scaling exists at the macroscopic scale, bulk crystals have in common with their small counterparts that their elastic to plastic transition is also very much dependent on the details of the underlying microstructure. More specifically, it is the initial defect structure that deter- mines strength across all length scales. The mobile part of the dislocation network will facilitate plastic deformation by collectively moving and multiplying within the crystal. Such plastic activity – may it be in a bulk or small-scale crystal – can be characterized by the external stress at which it occurs and thus the statistical properties of the early stages of a deformation sequence should be strongly influ- enced by a distribution P(r c ) of such critical stresses. P(r c ) therefore characterizes the underlying dislocation net- work microstructure in terms of the stress scale at which plastic evolution occurs. In addition to the strength determining initial defect dis- tribution, crystals at all length scales deform plastically in a discrete fashion in terms of both stress and plastic strain [4]. At the bulk scale this is hard to observe in a regular straining experiment, with the exception of some cases of ultra high strain resolution [5]. Reducing sample size, the intermittency of the discretely evolving dislocation struc- ture can be readily observed with nano-scale extensometry, which reinvigorated numerous efforts investigating scale-free slip events [6,7] as well as the temporal dynamics of avalanches [8,9] occurring in critically evolving disloca- tion systems. The general conclusion became that a plasti- cally evolving dislocation structure exhibits power-law slip-size (plastic strain) distributions, P(S) / S s , where s is a critical exponent, suggesting that dislocation based plasticity belongs to a class of phenomenon that is funda- mentally scale free and universal [4,10]. Such universality in the statistics of plastic strain implies an insensitivity to the microscopic details, while on the other hand the stress scale at which such plastic events occur does appear to be very dependent on the microstruc- tural details. The emerging questions are how these two very different aspects of deformation can be linked, and how this may result in the “smaller is stronger” paradigm observed in micro-compression experiments? One recent statistical approach which relates these aspects in a unified way gives the simple expression n =(s + 1)/(a + 1) [11], for the size-effect exponent, where a is the leading order expo- nent of the low critical stress side of P(r c ) and s is the afore- mentioned critical exponent for scale-free plastic activity. This expression relates material strength (as a function of deforming volume) directly with the universal behavior of intermittent plastic strain activity and the statistics of the critical stresses at which it occurs. As will be subsequently discussed, the developed connection also infers a broader class of scaling laws in which both size and shape can affect material strength. In particular the aspect ratio of a http://dx.doi.org/10.1016/j.scriptamat.2015.02.006 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. ⇑ Corresponding author; e-mail: robert.maass@ingenieur.de Available online at www.sciencedirect.com ScienceDirect Scripta Materialia 102 (2015) 27–30 www.elsevier.com/locate/scriptamat