The Strain Gradient Effects in Micro-Indentation Hardness Experiments

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where

If' 2 (e)±

,

(1)

is a reference stress in uniaxial tension such that the uniaxial stress-strain law can be written as a =--freff(e), 17 is the effective strain gradient, and 1 is identified as the internal constitutive length in strain gradient plasticity, given by Urref

2

S= 18a' •lUref

Yb.

(2)

53 Mat. Res. Soc. Symp. Proc. Vol. 578 © 2000 Materials Research Society

Here u is the shear modulus, b is the Burgers vector, and a is an empirical constant in the Taylor relation12 and is on the order of one 13. For micro-indentation hardness experiments, the fundamental law (1) predicts a linear dependence of the square of the hardness, H 2 , on the inverse of indentation depth, 1/ h, which agrees remarkably well with the hardness data for single crystal and cold worked polycrystalline copper 7 , as well as with single crystal silver5 and polycrystalline copper 6 Motivated by this remarkable agreement, Gao, Huang, Nix and Hutchinson 14' 15 have developed a theory of mechanism-based strain gradient (MSG) plasticity based on the Taylor model in dislocation theory. The theory is established from a multi-scale, hierarchical framework to link the micro-scale (e.g., 10-lOOnm), at which dislocation interactions are governed by the Taylor model, to the meso-scale (e.g., 1-10 microns), at which the plasticity theory is formulated. On the micro-scale, the strain gradient term is treated as a measure of the density of geometrically necessary dislocations16 whose accumulation increases the flow stress strictly following the Taylor model.

On the meso-scale, the constitutive equations are obtained

by averaging micro-scale plasticity laws over a representative cell. As the characteristic length associated with deformation becomes much larger than microns, MSG plasticity degenerates naturally to the classical plasticity theories. The theory of MSG plasticity14" 5 and the associated finite element method17 are used in this paper to investigate the micro-indentation experiments. The MSG plasticity theory is presented next. The hardness predicted by MSG plasticity is then compared with the micro-indentation hardness experiments in order to valid this continuum plasticity theory at the micron and submicron scales. MECHANISM-BASED STRAIN GRADIENT (MSG) PLASTICITY The theory of MSG plasticity"'",4" 5 is briefly described in this paper. Its starting point is the Taylor relation12 between the shear strength z and the total dislocation density Pi' in a material,

r = pbo=aublp, +PG,

(3)

where p is the shear modulus, b is the Burgers vector, a is an empirical constant on the order of one in the Taylor relation 12, PG is the density of geometrically necessary dislocations and is 16 17 related to the effective strain gradient 7qas ' (4)

PG = 27l/b,

and Ps is the density of statistically stored dislocations, which can be determined from the uniaxial stress-strain law a = areff(e) since there are no strain gradients in uniaxial tension and therefore no geometrically necessary dislocations'" 7 , Uref f(e) = a = 3r = 3a/b

js.

(

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