Geometrical Critical Thickness Theory for the Size Effect at the Initiation of Plasticity
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1185-II05-02
ABSTRACT Recently, size effects in the initiation of plasticity have been clearly observed and reported in different geometries; e.g., bending (Ehrler et al. Phil. Mag. 2008), twisting (Ehrler et al., MRS, Spring Meeting 2009) and indentation (Zhu et al. J. Mech. Phys. Sol. 56, 1170, 2008). Strain gradient plasticity theory is the principal approach for explaining size effects during plastic deformation in these geometries. However, it fails to account for any size effect at the initial yield. Geometrical critical thickness theory was proposed to explain the yield size effect in bending and torsion (Dunstan and Bushby, Proc. Roy. Soc. A460, 2781, 2004). The theory shows that the initial yield strength is scaled with the inverse square root of the characteristic length scale without requiring any free fitting parameters. Here, we extend the theory to describe the yield size effect in indentation. The theory agrees fairly well with experimental observations in micro-torsion (Ehrler et al., MRS, Spring Meeting 2009) and nanoindentation (Zhu et al., J. Mech. Phys. Solid, 2008). 1. INTRODUCTION It has been known for several decades that materials display strong size effects at the micron or sub-micron scale, in which the strength is enhanced when the size of the structure or of the stressed volume is decreased. Generally, size effects can be categorized as intrinsic and extrinsic size effect [1]. Intrinsic size effects are due to microstructural constraints of materials. Microstructural size effects include those due to grain boundaries [2] and particle reinforcement [3]. The extrinsic size effects have been presented for uniform deformation (without introducing plastic strain gradient) and non-uniform deformation. Uniform size effects are observed in, e.g., compression of pillars [4] and tension of whiskers [5]. Non-uniform or strain-gradient size effects are seen in torsion of wires [6, 7], bending of foils [8, 9], indentation [10-12], etc. Yield is always an important phenomenon but difficult to measure in materials science and engineering. Recently, size effects in the initiation of plasticity have been clearly observed and reported in different geometries, e.g, twisting [7] and indentation [12]. In all these different geometries, the yield strength clearly increases as the dimensional length scale is decreased. Various theories account for the size effects. Their application is often controversial, and each may account for the size effect in a different situation, i.e., different strain regime (flow strength or yield strength, gradient or uniform). Strain-gradient plasticity theory is well-known, in which the size effect is attributed to hardening due to geometrically-necessary dislocations (GNDs) [13, 14]. Since GNDs can only exist in the non-uniform plastic deformation (Ashby, 1970), strain gradient plasticity theory is not able to explain the initial yield size effect. Slipdistance theory is based on the ideas of Conrad et al. [15], and has been more widely applied recently [16, 17]; in this theory
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