A model for the indentation size effect in polycrystalline alloys coupling intrinsic and extrinsic length scales
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A model for the indentation size effect in polycrystalline alloys coupling intrinsic and extrinsic length scales Simon P.A. Gill1,a) Christopher J. Campbell1 1
Department of Engineering, University of Leicester, Leicester LE1 7RH, U.K. Address all correspondence to this author. e-mail: [email protected]
a)
Received: 11 January 2019; accepted: 4 March 2019
The measured hardness of a metal crystal depends on a variety of length scales. Microstructural features, such as grain size and precipitate spacing, determine the intrinsic material length scale. Extrinsic (test) length scales, such as the indentation depth, lead to the indentation size effect (ISE), whereby it is typically found that smaller is stronger. Nix and Gao [J. Mech. Phys. Solids 46, 411 (1998)] developed a widely used model for interpreting the ISE based on forest hardening in single crystalline pure metals. This work extends that model to consider the hardness of polycrystals and alloys, as well as introducing a finite limit to the hardness at very small extrinsic length scales. The resulting expressions are validated against data from the literature. It is shown that a reasonable estimate of the intrinsic material length scale can be extracted from a suite of hardness tests conducted across a range of indentation depths using spherical indenters of various radii.
Introduction The measured hardness of a metal crystal depends on both the intrinsic and extrinsic length scales concerned [1, 2, 3]. The intrinsic length scale of a material is governed by the size of the features within its microstructure. Extrinsic length scales arise from the specific nature of the conditions used for the hardness test. In cantilever beam tests and pillar compression tests, the extrinsic length scale is typically the dimensions of the specimen itself. In hardness measurements, the extrinsic length scale is typically the indentation depth or the radius of the indenter. For flat punch indentations, it is a mixture of both [4]. The indentation size effect (ISE) is an experimentally observed phenomenon, in which the measured hardness of a material is found to increase as the extrinsic length scale decreases. Nix and Gao [5] first proposed a model to explain this behavior based on forest hardening due to the presence of geometrically necessary dislocations (GNDs). These are dislocations that are required to accommodate the plastic strain gradients within a crystal. The shear yield stress is assumed to be
parameter, which represents the strengthening contribution due to dislocation hardening. The total dislocation density, q 5 qS 1 qG, is assumed to be the sum of a constant statistically stored dislocation (SSD) population of density qS, and a locally evolving population of GNDs, qG, derived from the plastic strain gradient. The size-dependent hardness of the material is then expressed as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi q ð2Þ H ¼ 3 3sY ¼ H0 1 þ G ; qS
ð1Þ
pffiffiffi pffiffiffiffiffi where H0 ¼ 3 3bD Gb qS is the macroscopic hardness. GNDs are required to form the numerous surface steps of
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