Optimization of Microstructure-Property Relationship in Materials
By Balasubramaniam (Rad) Radhakrishnan, Gorti Sarma, and Thomas Zacharia

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By modeling the nonuniform deformation of metals using a parallel supercomputer, Balasubramaniam Radhakrishnan (left), Gorti Sarma, and Thomas Zacharia can obtain information on how to optimize processing conditions to obtain desired microstructures and properties for materials such as aluminum used to make beverage cans. Photograph by Tom Cerniglio.

The production of aluminum beverage cans involves a number of thermal and mechanical processing steps, starting with the cast ingot. The reduction in thickness from the ingot to the final sheet is achieved by a series of rolling steps that reduce the aluminum ingot from a thickness of about 300 millimeters (mm) to 0.3 mm—the sheet thickness required for can making. The can is made by deep drawing, in which the sheet is held between a punch and a die and then drawn into the die by stretching. The success of the can-making operation depends upon the properties of the sheet, especially the orientations of the grains, referred to as texture. The texture control in the sheet depends on the proper choice of the processing variables during hot- and cold-rolling operations. Recrystallization is an important metallurgical phenomenon that influences the texture in the sheet during hot rolling.

ORNL researchers are using the Intel Paragon supercomputer to model the microstructural evolution of aluminum during thermomechanical processing. The models capture the rotation and elongation of the grains, the hardening of the material through the generation of dislocations by crystallographic slip during cold rolling, and the nucleation and growth of strain-free grains during annealing. The ORNL model can be readily used to develop the optimum processing window for producing controlled texture in aluminum to minimize scrap produced during can making.


Metals and alloys are made of crystalline grains whose characteristics and arrangements, which are visible in an electron microscope, can be altered by the application of heat (e.g., annealing) or mechanical action (e.g., compression). The microstructural features that determine the properties of a material include the size, shape, and orientation of the grains and their distribution in the microstructure. Optimizing the material microstructure for performance by designing and applying appropriate thermomechanical processing steps has been the prime role of metallurgists for decades. Until recently the study of processing-microstructure relationships has largely remained within the purview of experimental metallurgists because the mechanisms that contribute to these microstructural changes are very complex, and the changes occur either simultaneously or successively to varying degrees, depending on location within the material. The development of a computational model for predicting the overall response of the material to such a complex collection and sequence of metallurgical phenomena is extremely difficult. However, recent advances in high-performance computing have led to considerable progress in addressing this challenge.

Deformation Mechanisms

One of the basic deformation mechanisms in metallic crystals is crystallographic slip. Crystallographic slip involves the movement of dislocations in certain planes and directions in each crystal. A dislocation is an imperfection in the crystal structure of a metal resulting from an absence of atoms in one or more layers of a crystal. For a given crystal structure there are well-defined combinations of planes and directions in which slip occurs. For example, in the case of face centered cubic materials, slip occurs on the {111} type of planes and in the <110> directions. There are four different kinds of {111} planes and three <110> type directions in each of those planes, giving a total of 12 slip systems.

The occurrence of crystallographic slip during plastic deformation gives rise to two other phenomena that are of technological importance. The first is the increase in dislocation density during deformation due to the presence of locks within the crystal that arrest further movement of dislocations. These locks are produced by the interaction of dislocations moving in different slip systems. Because the resistance to dislocation movement increases in the presence of the locks, the crystal requires additional stress in the slip system to move further dislocations, and the crystal is said to “work-harden.” Work hardening is one of the mechanisms by which the deformation energy is stored within the crystal. The microstructure of the work-hardened material consists of a large number of cells, or subgrains, whose boundaries are essentially composed of a dense wall of dislocations. The second phenomenon that occurs is the rotation of the crystal during deformation. The crystal rotation is required for accommodating an arbitrary deformation because deformation by slip is possible only along a small number of slip systems. The crystal rotation accompanying arbitrary deformation is the underlying reason for the development of textures, or preferred orientations in materials, which results in anisotropic mechanical properties. The presence of texture is exploited in many industrial operations such as can making and other sheet-forming operations.

So far we have discussed the deformation of single crystals. However, a material microstructure consists of several crystals, or grains, connected together. Each of the grains may have a random crystallographic orientation, or the grains may have a preferred orientation, depending on the prior processing history. The surfaces along which the grains are connected remain intact during deformation. Hence, the deformation of individual grains in a material microstructure should satisfy the condition that the displacements are continuous across grain surfaces. This results in the operation of different slip systems in the grain interior as opposed to regions in the vicinity of grain boundaries. As the extent of deformation increases, a single grain breaks up into smaller “grains” of different orientations. Hence, formation of texture or preferred orientations during the deformation of a polycrystalline material depends not only on the orientations to which the grains rotate, but also on the orientation spread within each grain as different parts of the grain rotate to different orientations, as described above. Because the deformation of the grains is heterogeneous, the stored energy of deformation is also heterogeneous at the microstructural level. Both the heterogeneity in the stored energy of deformation and the orientation spread in the deformed microstructure are important variables that determine the further evolution of texture during thermal processing such as annealing.

Microstructural Events During Annealing

The microstructural events that occur during annealing of a deformed microstructure lead to a reduction in the stored energy of deformation by decreasing the dislocation density in the material. The two competing events that occur are recovery and recrystallization. Recovery is the process by which the stored energy of the material decreases continuously throughout the microstructure by the rearrangement of the dislocations into certain low-energy configurations. On the other hand, during recrystallization, there is a discontinuous change in the dislocation density because of the sweeping of the microstructure by certain surfaces that separate regions of high dislocation density and extremely low dislocation density. The resulting grain structure and texture depend on the orientation and the spatial distribution of the recrystallized nuclei that provide these surfaces. The relationship between the deformed microstructure and nucleation during recrystallization is a highly debated subject, and there is significant interest among researchers from both a fundamental and a technological viewpoint. Although several models have been proposed for the formation of nuclei during recrystallization, the extent to which these models apply to a real microstructure has not been quantified because of the lack of quantitative information on the deformed microstructure. The effect of simultaneous recovery during recrystallization is to diminish the driving force for the migration of the high-energy surfaces between recrystallized and deformed regions, with the result that the interface velocity decreases continuously and may even vanish before the recrystallization is complete.

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Beverage cans (left) are stamped out in the Coors Container Plant (right) in Golden, Colorado. ORNL is modeling the microstructure of rolled and heated aluminum before it is formed into cans.

Research at ORNL presented here shows how the use of massively parallel computing makes it possible to quantify the complex metallurgical phenomena that occur in aluminum during thermomechanical processing. For the first time, it is possible to model the deformation of a three-dimensional grain structure and quantitatively predict the development of not only the bulk texture but also the orientation gradients that exist in the deformed microstructure. In addition, the simulations also predict the heterogeneous distribution of the stored energy of deformation from grain to grain, as well as the intragranular variation of stored energy between sites at the interior of the grains and those sites in the vicinity of grain boundaries. The deformation simulation results have been mapped to a three-dimensional grid of regularly spaced points and used to predict the nucleation and growth of recrystallized grains, as well as the simultaneous recovery that occurs during subsequent thermal processing.

ORNL’s Methodology

The simulations for predicting microstructural changes during thermomechanical processing of metals address two processes—deformation and recrystallization. Simulations of the plastic deformation of the material (aluminum, in this case) require computing the motion of the workpiece under applied boundary conditions. Balance laws for conservation of momentum, mass, and energy form the basis for solving the resulting initial-boundary value problem. In addition to the balance laws, it is necessary to prescribe the constitutive behavior of the material, which relates the applied stress to the rate of deformation and the microstructural state. The plastic deformation is modeled in incremental fashion: the displacement or velocity of the workpiece is determined keeping the material state fixed, and then the state is updated before moving to the next increment.

As discussed earlier, the deformation of the metal is assumed to occur by the movement of dislocations along specific directions in specific planes. The rate of shear on these planes is related to the shear stress through a constitutive relation, typically a power law. The rate of deformation of the grain is given by a linear combination of the shear rates on all the slip systems, while the shear stress on the slip plane is the projection of the stress applied to the grain. A constitutive law between the rate of deformation and the stress tensors for the grain can be derived by eliminating the slip system shear rates from the above-mentioned relations. This constitutive law is nonlinear in the stress and requires an iterative method to compute the stress for a given rate of deformation.

The governing laws lead to differential equations that must be integrated to obtain the motion or deformation of the workpiece. The complex nature of the problems makes it necessary to use numerical methods for solving them, and the finite element method provides a suitable framework for this purpose. The domain of interest is discretized into several elements, and the balance laws are satisfied in an average sense over all the elements. The approach using finite elements for computing the motion of the workpiece consists of two main parts. The first task involves computing the stiffness for each element in the discretization, which represents a measure of the material response in that element. A stiffness matrix is set up based on the current material state by integrating the equations associated with the element. These stiffness matrices are then assembled in the second step to compute the discretized velocity or displacement field of the workpiece. The complexity of the two steps depends on the nature of the constitutive response and the size of the discretization.

The nature of the constitutive model based on crystal plasticity makes the stiffness computations extremely time-consuming. For a given rate of deformation with five independent components, a system of nonlinear equations must be solved to get the stress components. However, these computations can be carried out independently for each element, making them quite efficient on a parallel computer. It is this feature of the formulation which enables the treatment of fairly large-sized problems in a reasonable time frame. Although the stiffness computations scale fairly well with increasing numbers of processors, the assembly and solver routines are not so scalable. These parts of the program require communication between processors, which is an overhead that increases with the number of processors.

A finite element code capable of simulating the deformation of metals has been developed for the Intel Paragon at ORNL using High-Performance Fortran. A certain amount of optimization was achieved by making use of native NX message-passing calls. The code has been used to simulate the deformation of face-centered cubic metals in plane strain compression, and the resulting data on the distributions of stored energy and orientations have been used in modeling the subsequent recrystallization process.

The simulation of microstructural evolution during annealing is carried out using a Monte Carlo technique. The stored energy of deformation and the orientations of the elements in the deformed mesh are first mapped to a grid of regularly spaced points. As described previously, microstructural phenomena such as recovery, nucleation, and growth of recrystallized grains occur to varying degrees depending on the local deformation microstructure. In the Monte Carlo technique, each site in the grid is visited in a random fashion and each event is implemented on the basis of its probability of occurrence. The nucleation step is modeled by the growth of subgrains at each Monte Carlo site. In those sites where the stored energy of deformation is high and where there is also a monotonic increase in the misorientation of the subgrains during growth, the probability of forming a recrystallization nucleus is high. The quantitative description of the deformed microstructure allows, for the first time, the incorporation of a nucleation model based on subgrain growth at the mesoscopic scale which represents a collection of statistically significant number of grains. Hence, both the spatial distribution and the orientations of the nuclei can be obtained from first principles. The movement of a boundary separating a deformed region from the recrystallized region is driven by the elimination of the stored energy of deformation as the boundary advances. However, the driving force is constantly decreasing because of the recovery process. Also, the advancement of the boundary requires additional energy needed to create the new boundaries. The net energy change resulting from these processes is calculated, and the advancement of the boundary is allowed only if the net change in energy is negative.

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Fig. 1. (a) Initial microstructure with finite element mesh, (b) deformed mesh, and (c) deformed microstructure.
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Fig. 2. (a) Deformed microstructure, (b) stored energy distribution, and (c) nucleation parameter for a section normal to the y-axis.

Simulation Results

Figure l(a) shows the initial microstructure and its finite element discretization. The different colors in Fig. l(a) indicate different grains in the microstructure with a specific crystallographic orientation for each grain. The deformed mesh after a 50% reduction in height by plane strain compression is shown in Fig. l(b). The nonuniform deformation of the mesh is seen clearly. Figure l(c) shows the deformed microstructure without the mesh. A comparison of the grain structures in Figs. l(a) and l(c) shows that the grain deformation is also accompanied by a reorientation, the extent of which varies from grain to grain. In addition to an overall reorientation, there is also an orientation spread within each grain, which again varies from grain to grain. It is this orientation spread which gives rise to selective nucleation of recrystallized grains during annealing as discussed later. The deformation energy stored in the microstructure seen in Fig. 2(a) is shown in Fig. 2(b). Observe that the magnitude of the stored energy varies from grain to grain as well as within the grains. The stored energy is high at certain intragranular locations where there is an abrupt change in the grain orientation. These regions correspond to transition bands that are experimentally observed in deformed microstructures. The stored energy in the vicinity of grain boundaries as opposed grain interiors depends on the orientations of the surrounding grains. Hence, situations can be seen where the stored energy at the grain boundary is both lower and higher than in the grain interior.

Figure 2(c) shows the variation of a quantity called the nucleation parameter, which is the product of the stored energy and the misorientation of an element with respect to its neighbors, within the deformed microstructure. The nucleation parameter is an indicator of the probability of nucleation of the recrystallized grains on subsequent annealing. In regions where the stored energy is high, the dislocation sub-structure consists of a large number of small subgrains whose misorientation with the neighboring subgrains is fairly large. On the other hand, when the stored energy is smaller, the subgrain size is larger and the misorientation between subgrains smaller. Hence, the kinetics of subgrain growth is faster at those locations where the stored energy is high. However, a high value of stored energy alone is not sufficient for nucleation to occur. It is also necessary for the misorientation between subgrains to increase monotonically and to exceed the critical angle of 15° to form high-angle boundaries. This effect occurs in those regions where there is a significant local spread in the orientations, so that a given element has a large misorientation with respect to its surroundings. Hence, a site with a high value for the nucleation parameter also has a high probability for nucleation during annealing. The nucleation parameter is seen to have high values along certain grain boundaries and at triple points, as well as at certain intragranular locations where transition bands occur. It is important to note that the spatial distribution of the nucleation parameter is nonrandom.

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Fig. 3. Temporal evolution of recrystallized microstructure during annealing. “MCS” refers to time in Monte Carlo steps.
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Fig. 4. <111> pole figures showing (a) the deformed texture, (b) the orientations of nuclei, and (c) the recrystallized texture.

Figure 3 shows the temporal evolution of the recrystallized microstructure on annealing. The locations where the recrystallized nuclei form coincide with those where the nucleation parameter [Fig. 2(c)] is high. The nonuniform spatial distribution of the nuclei results in early impingement of the adjacent recrystallization fronts, resulting in an overall decrease in the rate of recrystallization. This effect is accentuated by the occurrence of simultaneous recovery in the deformed locations. Hence, the material undergoes only partial recrystallization, with islands of recovered areas in the microstructure. This result has a significant influence on the mechanical properties of the annealed material. Figure 4 shows the overall texture in the material in the deformed and recrystallized conditions. Also shown is the orientation distribution of the nuclei. Although recrystallization does not lead to the production of new orientations in the current simulations, it significantly redistributes the intensities of the texture components in the cold-worked material.

Conclusions

The work presented here (sponsored by DOE’s Division of Material Sciences) is the first of its kind in the materials literature in which a quantitative description of the deformed microstructure has been used to predict the evolution of texture and microstructure during subsequent annealing. Although the present work is specifically concerned with face-centered cubic crystal structure, the modeling can be easily extended to handle body-centered cubic and hexagonal close-packed crystal structures. The availability of quantitative information on the cold-worked state has allowed, for the first time, the prediction of both the orientation and spatial distribution of recrystallized nuclei, thus enabling a more realistic simulation of the microstructure and texture evolution during annealing.


BIOGRAPHICAL SKETCHES

BALASUBRAMANIAM (RAD) RADHAKRISHNAN received an M.S. degree in metallurgy from the Indian Institute of Technology, Madras, India, and a Ph.D. degree in materials science and engineering from the University of Alabama at Birmingham. Before pursuing his doctoral studies in the United States in 1986, he worked as a scientist at the National Aeronautical Laboratory in Bangalore, India. He joined the Modeling and Simulation Group in ORNL’s Metals and Ceramics Division as a postdoctoral fellow in 1993 and became a staff member in 1994. His research interests involve the modeling of microstructural evolution during materials processing. He has developed and validated Monte Carlo–based simulation techniques for studying microstructural evolution during grain growth, recrystallization, Oswald Ripening, and solidification.

GORTI SARMA received his B.S. degree in mechanical engineering from the Indian Institute of Technology, Madras, and his Ph.D. degree in mechanical engineering from Cornell University in Ithaca, New York. He joined the Modeling and Simulation Group in ORNL’s Metals and Ceramics Division as a postdoctoral research associate in September 1995. His main research interests are development of models to predict microstructural evolution during deformation processing, and application of the models to study problems in metal forming.

THOMAS ZACHARIA is director of ORNL’s Computer Science and Mathematics Division and formerly leader of the Modeling and Simulation Group in the Metals and Ceramics Division. (A more extensive biographical sketch is in "Analysis of Material Performance in Automotive Applications".)

 

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