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Cosserat phase field modelling and simulation of viscoplasticity induced grain boundary migration and recrystallisation in metallic polycrystals

Periodic Reporting for period 1 - MIGRATE (Cosserat phase field modelling and simulation of viscoplasticity induced grain boundary migration and recrystallisation in metallic polycrystals)

Reporting period: 2016-11-01 to 2018-10-31

Metals and metallic alloys are so important to humans that there are even epochs in history named after them (Iron Age, Bronze Age). Long before modern powerful microscopy, skilled metal workers knew how to manipulate the microstructure and thereby the behavior of metals. Imagine the blacksmith, alternately heating the sword in the fire, deforming it and then quenching it in a barrel of water. These principles for manufacturing largely persist to this day.

We now know that the atoms in metals arrange themselves in regular, crystalline structures.During solidification a microstructure is formed of highly ordered crystalline grains, separated by grain boundaries that are a few layers of atoms thick. The macoscopic behavior of a metal can be manipulated by changing the microstructure and the microstructure can be changed by deforming, heating and cooling the material, like the blacksmith did. This is referred to as thermomechanical processing and usually consists of at least two steps. heavily deform the metal beyond the elastic limit (the deformation is permanent even on release of load) and then heat it. During the deformation, energy is stored in the microstructure through the production and accumulation of atomistic defects. When the metal is heated, new grains free of defects nucleate and grow and after a certain time the old microstructure may be entirely replaced.

In this digital age, numerical computations is an important tool which complements and sometimes replaces experimental investigations. Simulations may for instance allow one to study what is going on inside a material or crash hundreds of digital car prototypes in rather than actually building them and crashing them in the lab. In order for the simulation models to be useful, they need to be robust and efficient and of course, most importantly, must faithfully represent the real phenomena they are intended to model. In the case of metals, the underlying physics is very rich. Much of the macroscopic behavior can be explained by the presence of atomistic defects in the crystal lattice. On the other hand, metals are often used in technological applications on a large scale (turbine blades, car, boat hull…) It is therefore not surprising that metals are studied on many scales and in many different research fields. Formicrostructure evolution during manufacturing, the scale of interest is the microscopic (micrometer or possibly nanometer) scale. Although information from the atomistic scale is useful and should be incorporated, atomistic simulations require too much in terms of computational resources to be feasible for modeling microstructure evolution in a polycrystal. Instead, the collective behavior of a large number of individual defects can be used to describe how energy is stored in the grain structure.

Models that deal with microstructure evolution in metals and metallic alloys undergoing thermomechanical processing is a topic of extensive research. Existing approaches tend to combine different models for the three main phenomena occuring : deformation, nucleation and grain growth. The separate models may often require using separate numerical methods, including discretization, for computations. In addition, an overall thermodynamic framework is not applied. With this project, the aim is to establish a unified model, formulated to be thermodynamically consistent, which can take into consideration both deformation and grain boundary migration. In addition, such a model should contain the possibility to account for nucleation without needing to resort to artificially introduce nuclei (new grains) in the structure.

The activities and outcome of the project can be summarized in the following research objectives:

1. Formulate a modeling framework combining, in a unified manner, methods that describe microstructure evolution through deformation and grain boundary migration due to stored energy in the structure
2. A numerical platform enabling simulation o
The research objectives have largely been achieved during the project. A unified model has been formulated for deformation and grain boundary migration in metallic polycrystals. The deformation part of the model is based on a Cosserat single crystal plasticity model. This is a mechanical model for viscoplastic deformation on the mesoscopic level which is non-local (it takes into consideration size effects). In a standard mechanical model the deformation is described by the displacement of material points, represented by the displacements in the three spatial directions. In a Cosserat model, each point can also rotate which provides three additional degrees of freedom. This allows for a representation of orientation in space of the crystal lattice which can be linked to the overall deformation. The mechanical model is strongly coupled with a so called phase-field model, which is a mathematical approach to represent distinct regions in space separated by boundaries through an order parameter. This order parameter is endowed with an evolution equation which allows the microstructure to evolve.

The model has been formulated in a thermodynamically consistent framework, taking into account conservation and dissipation of energy. The calibration process of the model, which is complex, has been completely established and applied to copper. In order to use the model for numerical simulations, it has been discretized using a numerical method called finite elements and implemented in a software called Z-set which is co-developed by the hosting institution MINES ParisTech. The behavior of grain boundaries in the presence of mechanical loading has been studied and reported on. The developed model is promising for future research and represents a true progress beyond the state-of-the art in the field.

In the course of the project, three journal publications have been submitted and accepted for publication and the researcher has participated in 9 scientific conferences with 8 oral presentations and one poster presentation.
In this work, a unified and thermodynamically consistent simulation model has been developed which allows studying simultaneous deformation and grain boundary migration in metallic polycrystals. The model can also be used to study such processes sequentially in a seamless fashion and without need for passing information between steps through algorithmic procedures. This represents a true progress beyond the state-of-the-art as previously existing approaches combined different methods and did not rely on strong coupling. It is a significant step towards better simulation models for thermomechanical processing of metals. The model is based on important descriptors of metallic microstructures (such as density of dislocation defects) and is formulated in a thermodynamically consistent way. The framework is quite general and leaves a lot of freedom for constituive choices. It is thus not restricted to study deformation and grain growth but can also, with suitable modifications, be used to study other phenomena such as phase transformations, intergranular damage and grain-boundary/dislocation interactions.
Geometry of the triple junction problem and relation between angles and grain boundaries
Simulation of the triple junction problem - grain boundary migration