Tag Archives: Modeling

New paper on grain boundary texture stability in UO2

A paper was recently published in Journal of Nuclear Materials on the stability of certain grain boundary configurations in UO2 (Uranium Dioxide) during grain growth. The paper is titled Stability of grain boundary texture during isothermal grain growth in UO2 considering anisotropic grain boundary properties, and can be found here or at the at Science Direct.

HallbergZhu2015_jnm_cover

Nucleation of Recrystallization

As a metallic material is plastically deformed, energy will be accumulated in the material through generation and rearrangement of defects, mainly dislocations. As dislocation networks develop they will tend to form entanglements that subsequently form dislocation cells which, in turn, eventually leads to the formation of subgrains in the grain interiors. These subgrains will be the birthplaces of the recrystallization nuclei. In order for the subgrains to become viable nuclei, however, two criteria need to be met locally in the microstructure. The first is a kinematic criterion requiring mobile high-angle grain boundaries to be formed by the nucleation event. The second is a thermodynamic criterion requiring a stored energy gradient to be present across the interface that is sufficient to provide enough positive driving pressure for grain growth to occur. These criteria are usually found to be met at grain boundary triple junctions and along grain boundaries. Such sites provide enough lattice curvature and sufficient stored energy differences for providing possible nucleation sites. Nucleation may also, but less common, take place in the grain interiors, for example along shear bands and near particle inclusions. The latter is often referred to as particle-stimulated nucleation. As nucleation preferentially takes place along pre-existing grain boundaries, it is common to observe so-called necklace patterns of recrystallized material along the boundaries.

The process of subgrain formation is in mesoscale models usually captured by considering an incubation time t_{\mathrm{c}} or a threshold dislocation density \rho_{\mathrm{c}}, i.e. a threshold stored energy, that needs to be reached before nucleation is initiated. In macroscale models, the same process is usually formulated in terms of a critical strain \varepsilon_{\mathrm{c}} that has to be achieved in order for nucleation to take place. Also the notion of a critical stress \sigma_{\mathrm{c}} has been proposed to identify the onset of recrystallization.

During the formation of the subgrains, recovery processes very rapidly reduce the dislocation content in the subgrain interiors by annihilation of dislocations and by accumulation of dislocations at the subgrain boundaries. This increases the subgrain misorientation with respect to neighboring regions and eventually provide high-angle, mobile, boundaries. The same processes also provide subgrains with very low internal stored energy.

The crystallographic orientation of the nuclei will influence both the progression of recrystallization and also the evolution of any recrystallization texture in the material. Recrystallization nuclei are products of the cold worked parent material but they do not necessarily inherit their orientation from the parent grains although it is not uncommon that clear evidence of the texture present in the initial, cold worked, microstructure can be observed also in the recrystallized material. High-angle boundaries with respect to the surrounding material are also required in order to permit growth of the nuclei, causing differences to develop between the initial and recrystallized textures. It can be noted that the development of recrystallization texture is usually attributed to either orientated nucleation or oriented growth or possibly a combination of these processes.

Having defined the initiation of recrystallization, the next question is at what frequency new nucleation events take place. This issue has been approached in a number of ways in different studies. The two main trends is to consider either of  site-saturated nucleation where all nuclei are assumed to be present at the start of the simulation and no new nuclei are added over time or, alternatively, continuous nucleation where new nuclei are continuously added according to some expression of the rate of nucleation. In the latter scenario, both constant and non-constant rates of nucleation have been considered.

Although constant nucleation rates are often assumed, there are experimental studies indicating nucleation to be non-constant and continuous process. However, it is inherently difficult to experimentally determine the rate of nucleation as it is not easy to define through experimental observations if a region in the microstructure is a subgrain, a nucleus or an expanding grain and the exact time at which one changes into the other.

Continuum scale modeling of phase transformation

Taking a continuum-mechanical perspective, the isothermal model in Hallberg et al. (2007) introduces the volume fraction of martensite as an internal variable. Along with a transformation condition, dependent on the state of deformation and on temperature, this allows the evolution of the martensitic phase to be traced. The presence of a transformation condition allows establishment of a transformation potential surface, much like the yield condition and yield surface found in plasticity theory. The transformation surface is illustrated in deviatoric stress space and in the meridian plane below.

Transformation surface in the deviatoric and in the meridian plane, respectively.
Transformation surface in the deviatoric and in the meridian plane, respectively.

Depending on which one is active, the yield and transformation conditions determine the response of the material. The relative influence of austenite and martensite on mechanical material properties is considered through a homogenization procedure, based on the phase fractions.

The above isothermal model is further elaborated in Hallberg et. al (2010b), where full thermo-mechanical coupling is considered. These models are suitable for large-scale simulations of metal forming processes involving materials exposed to martensitic phase transformation. The application to sheet metal forming is illustrated below by images from simulations of a deep-drawing process.

Volume fraction of martensite in a stainless steel sheet during deep-drawing at different temperatures. Note that three drawing stages are shown at each temperature. a) T=213K, b) T=233K, c) T=293K and d) T=313K.
Volume fraction of martensite in a stainless steel sheet during deep-drawing at different temperatures. Note that three drawing stages are shown at each temperature. a) T=213K, b) T=233K, c) T=293K and d) T=313K.

Phase transformations

Phase transformations in metallic materials have a major impact on vital engineering aspects of the material behavior such as ductility, strength and formability. Some phase transformations, such as the formation of pearlite and bainite, occur through diffusion-based processes where the constituents in the microstructure are redistributed. Being based on diffusion, these kinds of phase transformations tend to be relatively slow. On the other hand, phase transformations can also proceed by pure displacements in the crystal lattice structure. This is typical for the very rapid and diffusionless formation of martensite in austenitic steels.

Distribution of martensite (blue is austenite, red is martensite) in an austenitic metal sheet at three stages during a deep-drawing process at 213K.
Distribution of martensite (blue is austenite, red is martensite) in an austenitic metal sheet at three stages during a deep-drawing process at 213K.

Specifically, the latter kind of materials, undergoing microstructural changes in terms of austenite-martensite transformation, have in recent years gained increasing attention in relation to shape memory alloys (SMAs) and alloys exhibiting transformation-induced plasticity (TRIP steels).

Description of phase transformations is further involved due to the strong temperature-dependence of the process. Combined with significant differences in mechanical properties between the phases and the volumetric deformations accompanying e.g. martensitic phase transformations, strongly thermo-mechanically coupled phenomena arise.

The presence of martensite also changes the fracture behavior of a material since the martensite is considerably harder than the more ductile austenite parent phase. This influences e.g. initiation and propagation of crack and may become detrimental to metal forming and forging processes.

Continuum scale modeling of recrystallization

Adopting a continuum mechanical approach, recrystallization can be modeled using an internal variable representation of the pertaining quantities, such as the average grain size and the dislocation density. The macro-scale material behavior will in this way be based on parameters related to the evolving microstructure.

An example of continuum-scale modeling and simulation of ECAP-processing of Aluminum is given in Hallberg et al. (2010). Some results on the distributions of grain size and dislocation density in the work piece are shown in the figures below.

Results from simulations of ECAP-processing. The top figure shows the distribution of average grain size after and two ECAP-passes, respectively. The bottom figure shows the distribution of normalized dislocation density, also after one and two ECAP-passes, respectively. Note that after two passes, both grain size and dislocation density remain at relatively constant levels all through the specimen along the indicated lines.

Results from simulations of ECAP-processing. The top figure shows the distribution of average grain size after and two ECAP-passes, respectively. The bottom figure shows the distribution of normalized dislocation density, also after one and two ECAP-passes, respectively. Note that after two passes, both grain size and dislocation density remain at relatively constant levels all through the specimen along the indicated lines.
Results from simulations of ECAP-processing. The top figure shows the distribution of average grain size after and two ECAP-passes, respectively. The bottom figure shows the distribution of normalized dislocation density, also after one and two ECAP-passes, respectively. Note that after two passes, both grain size and dislocation density remain at relatively constant levels all through the specimen along the indicated lines.

 

Animation showing a 2D level set simulation of Dynamic Discontinuous Recrystallization (DDRX)

A simple 2D simulation of dynamic discontinuous recrystallization (DDRX) in pure Cu at an elevated temperature. The simulation is based on level sets in a finite element setting. Adaptive remeshing is performed in each step. The animation speed is increased compared to actual time.