Bainite tip radius prediction by analogy with indentation O. Bouaziz a, * , P. Maugis a , J.D. Embury b a Arcelor Research, Voie Romaine-BP30320, 57283 Maizie ` res-le `s-Metz Cedex, France b Department of Materials Science and Engineering, McMaster University, Hamilton, Ont., Canada Received 16 September 2005; received in revised form 20 December 2005; accepted 22 December 2005 Available online 25 January 2006 Abstract Through an analogy between plastic indentation and phase transformation with a curved front an approach is proposed which is suit- able to determine the tip radius of the growing phase in the case where plastic dissipation occurs. Including a scaling effect in the mechan- ical dissipation and assuming a maximum of the dissipated energy rate the modelling provides a prediction which is in good agreement with experimental tendencies especially for bainite. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Phase transformation; Bainite; Size effect; Modelling 1. Introduction The transformation of austenite on cooling can occur by a variety of mechanisms including the formation of bainite. The bainitic transformation occurs in a range between purely diffusional transformation to ferrite or pearlite and low temperature transformation to martensite by a displa- cive mechanism. Thus the bainite transformation exhibits features of both diffusional and displacive transformations and has given rise to a large amount of research activity (see Refs. [1,2] for recent reviews). A major part of the research has concerned modelling of the kinetics of the transformation. However, the prediction of the scale and morphology of the bainitic lath structure are not well understood despite their key role in understanding the mechanical properties of bainite [3,4]. In the current work the tip radius of the growing phase is considered as analo- gous to a spherical indenter, in order to produce a simple analysis to include a scaling effect in the mechanical dissi- pation during the transformation. 2. The proposed approach The plasticity under a spherical indenter of radius r can be described by a density of geometrically necessary dislo- cations [5,6] q g ¼ 2 b  r ; ð1Þ where b is the Burgers vector and the volume of the plastic zone is V p ¼ 2 3  p  r 3 . ð2Þ Neglecting the contribution of the statistically stored dislo- cations density the flow stress of the indented material is s c ¼ s 0 þ a  l  b  ffiffiffiffi q g p ð3Þ with s 0 a friction stress due to the lattice and solid solution hardening, a a constant, l the shear modulus. Using Eq. (1) the flow stress can be expressed as s c ¼ s 0 þ a  l  ffiffiffiffiffi 2b p ffiffi r p . ð4Þ As illustrated by the schematic diagram in Fig. 1 the expressions for the flow stress and volume of plastic zone 1359-6462/$ - see front matter Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2005.12.041 * Corresponding author. Tel.: +33 3 87 70 47 81; fax: +33 3 87 70 47 12. E-mail address: olivier.bouaziz@arcelor.com (O. Bouaziz). www.actamat-journals.com Scripta Materialia 54 (2006) 1527–1529