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Modeling sintering without constitutive equations

Schematics of  linking of macroscopic and mesoscopic structure levels in the direct multi-scale modeling approach, which eliminates the use of constitutive equations. Credit: JACerS; Wiley.

Sintering is among the most fundamental and important processes applied to ceramic materials.

In its essentials, it is simple: It’s densification by heating. The details, however, are complicated, and modeling has proven to be a very useful tool for understanding the driving forces and mechanisms that govern sintering processes.

One challenge with modeling is the issue of scale. Different mathematical models are needed to describe physical processes depending on the scale of the object of interest. It is difficult and not necessarily accurate to derive meaning from the output of a model in one size regime that is applied to a different size regime. For example, the mathematics that describes the atom-level process of solidification of a metal is very different from the mathematics that describes the casting shrinkage of an engine block, even though the former determines the latter.

A new paper in the Journal of the American Ceramic Society describes a new approach to modeling the sintering process that gets around the scale problem. The paper has a deceptively simple title, “Direct Multi-Scale Modeling of Sintering,” and is available via Early View. (One of the authors is ACerS Fellow Eugene Olevsky, from San Diego State University. The other three authors are from institutes in the Ukraine and Russia.)

The paper addresses “continuum sintering theory,” which models “sintering of macroscopically inhomogeneous porous specimens.” The macroscopic inhomogeneity arises from presintering processes such as powder pressing or forming. It can also arise from boundary constraints or from structural constraints within the specimen itself, such as with multilayer composites.

Advances in continuum theory, the paper says, are owing largely to “the use of comparatively simple special constitutive equations of sintered porous materials.”

Constitutive equations are mathematical descriptions of the relationship between stress, strain, strain rate and microstructural features like grain size. The coefficients in the equations are functions of internal material parameters. Sintering kinetics are known to depend on grain size, pore size distribution, pore coarsening, pore and grain morphology and other parameters. A limitation of previous models has been an inability to expand the models to include more than one parameter.

The question is how to solve this limitation and expand the models to accommodate multiple parameters. The paper list three current barriers: experimentally determining the interrelationships between multiple parameters is difficult and time consuming; a multi-parameter approximation of the experimental data in the constitutive equations is necessary; and the number and identification parameters to include in the model is not known.

However, the new approach does not rely on constitutive equations. Instead, “the results of modeling at the mesoscopic level are directly used for the prediction of the macroscopic behavior, ” hence the “direct” part of the paper’s title.

Mesoscopic modeling defines and analyzes the “evolution of a set of unit cells of a powder material during sintering.” In this paper, a unit cell is a “representative mesoscopic volume of the material,” and the mesoscopic scale correlates to the structure level of the powder particles. The macroscopic level is the specimen being sintered. The “multi-scale” aspect refers to a simultaneous numerical analysis of the sintering processes at the powder particle (mesoscopic level) and specimen (macroscopic level).

The model starts by setting a control point, and then several finite elements around the point comprise the mesostructure/unit cell. Using numerical analysis, macroscopic material parameters like viscosity and sintering stress are found in what amounts to a virtual testing of the unit cell independent of constitutive equations. Values of properties are fed into a macroscopic finite element model, which allows macroscopic effects like distortion and shrinkage to be determined.

The paper demonstrates the modeling method for two cases: diffusion sintering of ceramic composites and viscous sintering. They are able to show that the “evolution of the unit cells is connected with the macroscopic level shape distortion,” based on assumptions about the similarity of the macroscopic and mesoscopic strain rates.

Besides Olevsky, the paper (doi:10.1111/j.1551-2916.2012.05083.x) is authored by Andrey Maximenko, Andrey Kuzmov, and Evgeny Grigoryev. Maximenko and Kuzmov are affiliated with the Frantsevich Institute for Problems of Materials Science, Kiev, Ukraine. Maximenko is also affiliated with the Key Laboratory for Electromagnetic Field Assisted Processing of Novel Materials of the Moscow Engineering Physics Institute, as is Grigoryev and Olevsky.