Benoît Appolaire

Combining phase field approach and homogenization methods for modelling phase transformation in elastoplastic media

A general constitutive framework is proposed to incorporate linear and nonlinear mechanical behaviour laws into a standard phase field model. In the diffuse interface region where both phases coexist, two mixture rules for strain and stress are introduced, which are based on the Voigt/Taylor and Reuss/Sachs well-known homogenization schemes and compared to the commonly used mixture rules in phase field models. Finite element calculations have been performed considering an elastoplastic precipitate growing in an elastic matrix in order to investigate the plastic accommodation processes.

Modeling of the effect of the β phase deformation on the α phase precipitation in near-β titanium alloys

Abstract A model has been developed in order to describe the influence of a prior plastic deformation of the β phase above the transus on the precipitation of the α phase occurring during subsequent cooling. The model relies on the calculation of the nucleation and growth rates of the α precipitates at the grain boundaries, based on a model presented formerly. Two major modifications have been made: first, the geometrical representation of the β microstructure accounts for subgrains resulting from the deformation process; second, the calculation of the nucleation rate is dependent on the conditions of the plastic deformation. A careful analysis of the main parameters of the model has led to a distinction between several assumptions: Widmanstatten colonies are likely to cross the subgrains during their growth; and the critical width of the transition from allotriomorphs to Widmanstatten plates is likely to decrease when the misorientation angle of the grain boundary decreases. Calculations performed to assess the influence of the strain and strain rate on the transformation kinetics are in good agreement with previous measurements.

Boundary layer correlation for dendrite tip growth with fluid flow

Abstract A correlation is presented for the calculation of the size of a stagnant film located in the liquid ahead of a paraboloidal dendrite, which grows steadily into an undercooled melt in the presence of fluid flow. The coefficients of the correlation are deduced from a fitting exercise of the supersaturation versus dimensionless numbers to the analytical solution developed for the Stokes flow approximation. It is shown that the correlation presented in this contribution offers several advantages compared with the Wang and Beckermann correlation previously published in the literature (Metall. Mater. Trans. 27A (1996) 2754). Not only the fit is improved, but it can also be extended to account for the angle between the dendrite growth direction and the fluid flow direction. Applications are given for an Al–7 wt.% Si alloy.

Free growth of equiaxed crystals settling in undercooled NH4Cl–H2O melts

Abstract Recently theoretical works concerning the effect of convection on the growth of isolated dendrites have been compared with experiments on NH 4 Cl settling equiaxed crystals. It was inferred that more accurate theories were still needed to describe properly the equiaxed crystal growth in the presence of convection. Some new results have been obtained using the following experimental set-up: in a tube containing an undercooled solution of NH 4 Cl–H 2 O, settling NH 4 Cl equiaxed crystals have been filmed with a video camera so as to determine the evolution with time of their size and of their settling velocity. After a careful comparison of the experimental results with some calculations involving the choice of a stability constant, no major discrepancy has been found to prevent the application of the theories in question to moving equiaxed crystals.

Phase field modelling of grain boundary motion driven by curvature and stored energy gradients. Part I: theory and numerical implementation

During thermo-mechanical processing, the strain energy stored in the microstucture of an FCC polycrystalline aggregate is generally reduced by physical phenomena controlled, at least partially, by mechanisms involving dislocation cell or grain boundary motion such as recovery, recrystallisation and grain growth. This work presents a novel coupled phase field-single crystal constitutive framework capable of describing the microstructural evolution driven by grain boundary curvature and/or stored energy during recrystallisation and grain growth. Thus, the minimisation of stored and grain boundary energies provides the driving force for grain boundary motion. To describe interface motion, a phase field model taking into account the stored energy distribution is formulated and implemented within a continuum mechanics framework. The single crystal constitutive behaviour is described using a dislocation mechanics-based crystallographic formulation. The coupling between the grain boundary kinematics and the crystal plasticity formulation is made through the dislocation densities and the grain orientations. Furthermore, the free energy parameters are calibrated from existing Read–Shockley boundary energy data and those describing grain boundary mobilities from published experimental data.

Phase field modelling of grain boundary motion driven by curvature and stored energy gradients. Part II: Application to recrystallisation

In this work, the coupled phase field–crystal plasticity constitutive framework proposed in a companion publication [G. Abrivard, E.P. Busso, S. Forest and B. Apolaire, Phil. Mag. (2012) (this issue)] is applied to study the microstructural evolution driven by grain boundary curvature and/or stored energy. Different microstructures involving bicrystals and polycrystals of pure Al are studied and the results compared against experimental data and known analytical solutions. First, the study of a bicrystal with only curvature as the driving force for boundary migration enables the model to reproduce the different mobilities between low and high angle grain boundaries in the absence of Σ-type boundaries, and to identify the threshold misorientation below which the mobility is negligible. The growth of a small dislocation-free grain embedded within a highly deformed one is considered having both curvature and stored energy as the competing driving forces. A parametric study enabled the effect of the initial size of the nucleus on the minimum level of stored energy required for grain migration to be quantified. Finally, a study of recrystallisation and grain growth phenomena on a representative polycrystal aggregate revealed that grains with the lowest stored energy are dominant at the end of the recrystallisation process. The predicted recrystallised material volume fraction evolution and the kinetics of recrystallisation and grain growth were found to have the same dependence on deformation levels and temperature as those reported in the literature. Several outstanding modelling issues are identified and suggestions for further developments are discussed.

Phase field modeling of elasto-plastic deformation induced by diffusion controlled growth of a misfitting spherical precipitate

A phase field model accounting for plasticity has been developed using an homogenization scheme for interpolating the constitutive laws within the diffuse interface. The influence of plasticity on the growth of a misfitting spherical precipitate, controlled by solute diffusion has been investigated: plasticity in the matrix slows down the transformation. Moreover, an excellent agreement with the corresponding analytical sharp interface solutions has been achieved.

Precipitation in a near Beta Titanium Alloy on Ageing: Influence of Heating Rate and Chemical Composition of the Beta-Metastable Phase

In the present study we focus on the precipitation processes during heating and ageing of β-metastable phase in the near β Ti-5553 alloy. Transformation processes have been studied using continuous high energy X-Ray Diffraction (XRD) and electrical resistivity for two different states of the β-metastable phase. Microstructures have been observed by electron microscopy. Different transformation sequences are highlighted depending on both heating rate and chemical composition of the β-metastable phase. At low temperatures and low heating rates, the hexagonal ωiso phase is first formed as generally mentioned in the literature. Increasing the temperature, XRD evidences the formation of an orthorhombic phase (α’’), which evolves toward the hexagonal pseudo compact α phase. For higher heating rates or for richer composition in β-stabilizing elements of the β-metastable phase, ω phase may not form and α’’ forms directly and again transforms into α phase. A direct transformation from β-metastable to a phase is observed for the highest heating rate. The formation of the metastable ωiso and α’’ phases clearly influences the final morphology of α.

A phase field model for dislocation climb

We propose a phase field method to model consistently dislocation climb by vacancy absorption or emission. It automatically incorporates the exact balance between the vacancy flux and the phase field associated with the dislocation evolution, enforced by the conserved character of the total population of vacancies. One of its major advantage is the natural introduction of a dynamic coefficient controlling the kinetics of vacancy emission/absorption by the dislocation. We also derived a closed-form expression of the climb rate valid from the diffusion-limited to the attachment-limited regimes.

Multiscale Theory of Dislocation Climb

Dislocation climb is a ubiquitous mechanism playing a major role in the plastic deformation of crystals at high temperature. We propose a multiscale approach to model quantitatively this mechanism at mesoscopic length and time scales. First, we analyze climb at a nanoscopic scale and derive an analytical expression of the climb rate of a jogged dislocation. Next, we deduce from this expression the activation energy of the process, bringing valuable insights to experimental studies. Finally, we show how to rigorously upscale the climb rate to a mesoscopic phase-field model of dislocation climb. This upscaling procedure opens the way to large scale simulations where climb processes are quantitatively reproduced even though the mesoscopic length scale of the simulation is orders of magnitude larger than the atomic one.