# Paper published on evolution of grain boundary character distribution during grain growth

A new paper, titled "Influence of anisotropic grain boundary properties on the evolution of grain boundary character distribution during grain growth - A 2D level set study", was recently published in Modelling and Simulation in Materials Science and Engineering and can be found at the publisher's site following this link or here. # Production of very fine grained materials

Since recrystallization is driven by reduction of stored energy, the amount of cold work influences the grain size evolution. Materials with grain sizes down to the nanometer scale can be obtained by exposing the material to severe plastic deformation (SPD). Such SPD processes include Equal Channel Angular Pressing/Extrusion (ECAP/ECAE), Asymmetric Rolling (ASR), Multi-Directional Forging (MDF), High-Pressure Torsion (HPT) and Twist Extrusion (TE). Some of these processes are discussed further below.

During ECAP, studied in Hallberg et al. (2010), the material specimen is pressed through a channel die as illustrated below.

The amount of effective plastic deformation, denoted by $\displaystyle \varepsilon_{\mathrm{eff}}^{\mathrm{p}}$, that is imposed onto the specimen in each pass can be estimated from knowledge of the die geometry according to $\displaystyle \varepsilon_{\mathrm{eff}}^{\mathrm{p}}=\frac{N_{\mathrm{pass}}}{\sqrt{3}}\left[2\mathrm{cot}\left(\frac{\Phi}{2}+\frac{\Psi}{2}\right)+\Psi\mathrm{cosec}\left(\frac{\Phi}{2}+\frac{\Psi}{2}\right)\right]$

Varying the channel geometry thus allows control of the amount of plastic deformation that is exerted onto the specimen in each pass through the die. If the specimen is rotated between each pass through the die, different standardized processing routes can be obtained. 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.

Another SPD process is asymmetric rolling, discussed in Hallberg (2013), where a conventional rolling process can be made asymmetric by different methods.

The asymmetry of the process can be induced by having different radii $r$ of the rolls, by different roller velocities, i.e. $\omega_{1}\ne\omega_{2}$ or by different friction/lubrication conditions at each side of the sheet. The asymmetry increases the shear deformation of the rolled sheet and hence the total amount of effective plastic deformation. Distribution of dislocation density obtained from a symmetric rolling operation with 40% thickness reduction per pass and a rolling friction of 0.25. The top figure shows the results after the first pass and the bottom figure after the second pass.

# Metal forming and materials processing

Beside casting and machining, forming is one of the main processes for manufacturing of components from metallic materials. Forming processes include sheet metal forming as well as bulk metal forging. Metal forming generally involves significant plastic deformations, elevated temperatures, high deformation velocities and an increased risk for initiation of cracks in the material. These macroscopic process conditions are intimately connected to the microstructure evolution inside the material.

In Hallberg et al. (2007) and Hallberg et al. (2010), deep-drawing of stainless steel is used as application examples for constitutive models where martensitic phase transformation is considered, see the illustration below. As the martensite phase is much harder than the parent austenitic phase, the material properties may change dramatically in the presence of this kind of diffusionless - and thus rapid - phase transformation. Distribution of martensite (blue is austenite, red is martensite) in an austenitic metal sheet at three stages during a deep-drawing process at 213K.

The influence of deformation rate and material pre-processing in metal forging is studied in Hallberg et al. 2009. Different behavior of a 100Cr6 steel, due to previous tempering or annealing, was studied in high strain rate axisymmetric compression, experimentally as well as through numerical simulations. Axisymmetric compression of a cylindrical specimen. Due to friction at the top and bottom surfaces, the deformed specimen gets a typical "barrel" shape.

The illustration below is taken from Hallberg et al. 2009. Note the development of a "shear cross" (white lines) in the tempered - and much harder - material. This localized deformation is absent in the annealed specimen. Another example of metal forming is rolling, for example discussed in Hallberg (2013). A conventional rolling process can be made asymmetric by different methods in order to increase the deformation imposed onto the sheet.

The asymmetry of the process can be induced by having different radii $r$ of the rolls, by different roller velocities, i.e. $\omega_{1}\ne\omega_{2}$ or by different friction/lubrication conditions at each side of the sheet. The asymmetry increases the shear deformation of the rolled sheet and hence the total amount of effective plastic deformation. This is utilized in severe plastic deformation (SPD) processes for production of very fine-grained metals.

# Dislocation and grain boundary interaction

Mesoscale models of microstructure evolution allow studying of heterogeneous dislocation density distributions and related gradient effects. This is closely connected to the well-renowned Hall-Petch effect, stating a proportionality between the yield stress $\sigma_{\mathrm{y}}$ and the inverse of the square-root of the average grain size $d$ according to $\sigma_{\mathrm{y}}\propto\frac{1}{\sqrt{d}}$

The interaction between dislocation motion and grain boundaries can be modeled by different approaches. Some examples are given below.

In Hallberg and Ristinmaa (2013) (also discussed in this conference presentation), a hybrid finite difference/cellular automaton model is established where dislocation density gradient are modeled in a reaction-diffusion system. This approach results in the expected Hall-Petch behavior of the macroscopic flow stress – without including an explicit dependence on the grain size – in addition to providing a physically sound dislocation density distribution, indicated in the figure below. Simulated distribution of total dislocation density in an artificial polycrystal. Due to the reaction-diffusion modeling of dislocation density evolution and the presence of grain boundaries, dislocation pile-ups are formed along the grain boundaries and particularly at triple junctions.

By this modeling approach, the dislocation density will be concentrated at grain boundaries and particularly at triple junctions, directly providing the sites for nucleation of recrystallization. This is in contrast to the common modeling approach where nuclei are placed manually at appropriate sites in the microstructure.

Another approach to modeling of dislocation and grain boundary interaction is taken in Hallberg (2013) where a level set formulation is used to model polycrystal grain structures. Dislocation density distribution influenced by the presence of grain boundaries. The microstructure model is based on level sets, representing the individual grains. The extent of dislocation accumulation at grain boundaries can be controlled by the parameter w, as illustrated in images a-d.

# Grain boundaries

Many important aspects of the macroscopic material behavior in metals are controlled by the microlevel grain structure. Strength and ductility depends largely on the size and distribution of the grains. Especially the presence of grain boundaries plays an important role, not least in the macroscopic deformation hardening of the material. Grain boundaries pose obstacles to slip deformation by preventing dislocation motion, resulting in localized dislocation storage and heterogeneous deformation fields within the grains. This is closely related to the classical Hall-Petch relation, where the macroscopic flow stress is porportional to the inverse of the square root of the average grain size. More fine-grained materials will thus exhibit a higher flow stress than more coarse-grained materials. Schematic illustration of a dislocation pileup at a grain boundary. The individual dislocations glide along the same slip plane under an external resolved shear stress . The grain boundary blocks the first dislocation and the other dislocations pileup behind it. The first dislocation will experience a stress due to the pileup corresponding to . The pileup also gives rise to a backstress that interferes with the generation of additional dislocation loops. This is an important mechanism in work hardening.

The Hall-Petch effect will, however, tend to reverse as very small grains are considered. At a grain sizes of approximately 10 nm or below, other mechanisms such as grain boundary sliding usually cause a softening effect instead of the common Hall-Petch hardening.

Grain boundaries also contribute to the material behavior in many other aspects. They are often sites for initiation of defects such as cracks, they influence the material's susceptibility to chemical attack such as through corrosion and they influence the electrical conductivity of the material. In addition, migrating grain boundaries tend to interact with second-phase particles and the presence and arrangement of grain boundaries influence both the electrical conductivity and the diffusion properties of the material.

# Martensitic phase transformation and fracture

As the relatively ductile austenite phase is transformed in to harder and more brittle martensite in the vicinity stress-concentrations, the material conditions change and also the conditions for crack formation and propagation. Fatigue fracture can be considerably influenced by this kind of diffusionless phase transformation due to the higher fracture strength of martensite, compared to that of austenite. In Hallberg et al. (2012) the influence of martensite formation on fracture behavior and crack tip conditions is investigated, as illustrated below. Transformed zone at the tip of a stationary crack. The crack tip is located at coordinates (0,0). The contour lines in each figure correspond to different load levels. a) T=213K and b) T=233K.

# 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.

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.

# 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.

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.

# Impediment of grain boundary migration due to particle drag

Metallic materials used in engineering applications frequently contain some volume fraction of particles. These may be impurity particles or particles that are deliberately added to influence the microstructure behavior.

As grain boundaries migrate through the microstructure, driven by boundary curvature and/or stored energy gradients, any particles present will impede - or even prevent - the movement by exerting drag forces on the passing boundaries. Classically this effect is termed Zener drag or Zener pinning. This pinning pressure is usually written as $p=-z_{1}\gamma\frac{f_{\mathrm{V}}^{z_{2}}}{r_{\mathrm{p}}}$

where the negative sign indicates a retarding pressure and where $\gamma$ is the boundary energy while $f_{\mathrm{V}}$ and $r_{\mathrm{p}}$ are the volume fraction of particles and the average particle size, respectively. $z_{1}$ and $z_{2}$ are parameters.

In the original model by Zener (published by C.S. Smith in 1948), the formulation is based on the assumption of rigid grain boundaries between the particles, providing $z_{i}=\left\{3/2,1\right\}$. Later studies have tried to incorporate the tendency of the grain boundary to bow out between particles, arriving at other values of the parameters $z_{1}$ and $z_{2}$.

A mesoscale RVE model of dynamic discontinuous recrystallization, considering particle drag, is established in Hallberg et al. (2014). The model is based on a 3D cellular automaton formulation. See illustration below. Influence on recrystallization kinetics from a small volume fraction of homogeneously distributed particles of size (in microns). The recystallized volume fraction is shown on the vertical axis and time on the horizontal axis. Results are shown at two different temperatures. It is obvious that an increased presence of particles retard the progression of recrystallization. Simulation results from a 3D cellular automata model.