Phase structural ordering kinetics of second-phase formation in the vicinity of a crack

Abstract

The formation of a second phase in presence of a crack in a crystalline material is modelled and studied for different prevailing conditions in order to predict and a posteriori prevent failure, e.g. by delayed hydride cracking. To this end, the phase field formulation of Ginzburg-Landau is selected to describe the phase transformation, and simulations using the finite volume method are performed for a wide range of material properties. A sixth order Landau potential for a single structural order parameter is adopted because it allows the modeling of both first and second order transitions and its corresponding phase diagram can be outlined analytically. The elastic stress field induced by the crack is found to cause a space-dependent shift in the transition temperature, which promotes a second-phase precipitation in vicinity of the crack tip. The spatiotemporal evolution during nucleation and growth is successfully monitored for different combinations of material properties and applied loads. Results for the second-phase shape and size evolution are presented and discussed for a number of selected characteristic cases. The numerical results at steady state are compared to mean-field equilibrium solutions and a good agreement is achieved. For materials applicable to the model, the results can be used to predict the evolution of an eventual second-phase formation through a dimen-

Phase-field modelling: effect of an interface crack on precipitation kinetics in a multi-phase microstructure

“…s 0 = 0.0, the geometry of the second-phase region, growing self-similarly and symmetrically with respect to the x-axis, is, by formulation, directly related to the geometry of the isostress contours, described by the applied hydrostatic stress. This has also been shown in Nigro et al (2018) for a specific set of parameters. The difference between the isostress contour geometry and the left-hand side of the precipitate can be imputed to the presence of the interfacial energy, which tends to reduce the interfaces, through the gradient free energy term.…”

“…The term F int includes a coupling between the applied stress and the phase transformation-induced dilatation effect. The variation of the interaction free energy through a change of the applied stress corresponds to a modification of the terminal solid solubility (Nigro et al 2018). The total strains ε i j are expressed as…”

“…This simplification implies that the value of the phase field components in a material point remains unaffected by the surrounding points. By omitting this term, singularities, discontinuities and sharp transitions/interfaces can appear as noted in Nigro et al (2018). Thus, this analysis boils down to the examination of the Landau potential functional with respect to η 1 while being modified by the variation of s and the applied stress.…”

“…A number of models related to second-phase formation at a flaw tip in various crystalline materials have been developed in the past years (Varias and Massih 2002;Deschamps and Bréchet 1998;Gómez-Ramírez and Pound 1973;Boulbitch and Korzhenevskii 2016b;Léonard and Desai 1998;Hin et al 2008;Massih 2011;Bjerkén and Massih 2014;Jernkvist and Massih 2014;Jernkvist 2014;Nigro et al 2018). The phasefield method (Provatas and Elder 2010), based on the Ginzburg-Landau theory and employed in some of the cited works, has been not only extensively used within the magnetic field (Cyrot 1973;Berger 2005;Barba-Ortega et al 2009;Cao et al 2013;Gonçalves et al 2014) but also and particularly to predict microstructure evolutions in material.…”

“…For instance, to study a quasi-static phase transformation in the process zone of a propagating crack, Boulbitch and Korzhenevskii (2016a) employ a non-conserved order parameter, which interacts with the crack-induced stress field. A parametric study of second-phase formation in presence of a crack based on a similar set-up was carried out in Nigro et al (2018).…”