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Insights into the role of mechanics on diffusion-controlled phase transformations using phase field models

By Benoit Appolaire (Laboratoire d'étude des Microstructures, Cnrs/Onera)
Co-authors: Yann Le Bouar (Laboratoire d'étude des Microstructures, Cnrs/Onera)
Alphonse Finel (Laboratoire d'étude des Microstructures, Cnrs/Onera)
Maeva Cottura (University of Berkeley)
Samuel Forest (Centre des Matériaux, Mines ParisTech/Cnrs)
Kais Ammar (Centre des Matériaux, Mines ParisTech/Cnrs)
Elisabeth Aeby-Gautier (Institut Jean Lamour, Université de Lorraine/Cnrs)

The role played by the microstructures ensuing from phase transformations on the mechanical properties is now very well documented and thoroughly studied, in particular in metallic alloys. Although there is also a large number of works devoted to the reverse, i.e. the influence of mechanics on the microstructure formation and evolution, a general picture is still lacking, in particular when the phase transformations are diffusion-controlled. Indeed, drawing such a general picture requires to address at the same time the issues related to both mechanical behavior and phase transformations, as well as to address new issues arising from the tight coupling between evolving interfaces and evolving strain/stress fields. Besides the well-known modification of two-phase thermodynamic equilibrium by elasticity, as explored by LarchE` and Cahn (e.g. [10]) and Voorhees and Johnson (e.g. [9]), the trends are less obvious concerning the morphological evolutions when the chemical and mechanical driving forces are competing to reduce the overall free energy of the materials. In this contribution, an attempt will be made to draw some trends by examining several situations in different types of alloys where the different contributions that mechanics encompasses are decoupled. For that purpose, we will resort to extensive calculations with phase field models that have been specifically developed [1,7]. First we will discuss the role of elasticity on the shape selection of precipitates, beyond the classical results of hard precipitates in a soft matrix against soft precipitates in a hard matrix, that are relevant only for isotropic elasticity. Indeed, we will show how the anisotropy of the elastic energy arising from either the moduli or the eigenstrain is crucial for the shape selection, even for diffusion-controlled transformation at high temperatures where it is usually believed that elasticity is totally relaxed by plasticity. The examples supporting our analysis will concern cuboidal ordered precipitates in Ni-base superalloys [6] and acicular structures in alloys exhibiting allotropic transformation such as Ti alloys, steels or brass [3]. In a second part, we will show that in many cases, plasticity does not relax totally the elastic strain associated with the phase transformations [4]. As a direct consequence, plasticity may not change qualitatively the shape evolutions driven by elasticity, as it will be illustrated on the rafting of the ordered precipitates in Ni-base superalloys [5] and on the acicular structures [4], although it can change the kinetics of the processes. However, we will show that in some cases, plasticity may induce shape bifurcations [5,8] that are difficult to infer with simplified qualitative arguments, as usually done in the literature on diffusion-controlled phase transformations. We will conclude with a few open questions that we have been able to identify thanks to our phase field calculations, such as the inheritance of plastic strain by the growing phases [2]. [1] K. Ammar, B. Appolaire, G. Cailletaud, and S. Forest. Combining phase field approach and homogenization methods for modelling phase transformation in elastoplastic media. European Journal of Computational Mechanics, 18(5- 6):485-523, 2009. [2] K. Ammar, B. Appolaire, S. Forest, M. Cottura, Y. Le Bouar, and A. Finel. Modelling inheritance of plastic deformation during migration of phase boundaries using a phase field method. Meccanica, 49:2699-2717, 2014. [3] M. Cottura, B. Appolaire, A. Finel, and Y. Le Bouar. Phase field study of acicular growth: Role of elasticity in Widmansta?R tten structure. Acta Mate- rialia, 72:200-210, 2014. [4] M. Cottura, B. Appolaire, A. Finel, and Y. Le Bouar. Plastic relaxation dur- ing diffusion-controlled growth of Widmansta?R tten plates. Scripta Materi- alia, 108:117-121, 2015. [5] M. Cottura, B. Appolaire, A. Finel, and Y. Le Bouar. Coupling the phase field method for diffusive transformations with dislocation density-based crystal plasticity: Application to Ni-based superalloys. Journal of the Mechanics and Physics of Solids, 94:473-489, 2016. [6] M. Cottura, Y. Le Bouar, B. Appolaire, and A. Finel. Role of elastic inhomogeneity in the development of cuboidal microstructures in Ni-based su- peralloys. Acta Materialia, 94:15-25, 2015. [7] M. Cottura, Y. Le Bouar, A. Finel, B. Appolaire, K. Ammar, and S. Forest. A phase field model incorporating strain gradient viscoplasticity: Application to rafting in Ni-base superalloys. Journal of the Mechanics and Physics of Solids, 60:1243-1256, 2012. [8] V. de Rancourt, K. Ammar, B. Appolaire, and S. Forest. Homogenization of viscoplastic constitutive laws within a phase field approach. Journal of the Mechanics and Physics of Solids, 88:291-319, 2016. [9] W.C. Johnson and P.W. Voorhees. Interfacial stress, interfacial energy, and phase equilibria in binary alloys. Journal of Statistical Physics, 95(5-6):1281- 1309, 1999. [10] F. Larche and J.W. Cahn. Thermochemical equilibrium of multiphase solids under stress. Acta Metallurgica, 26(10):1579-1589, 1978.

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