Mathematical Modelling in Application to Plasticizers

Mathematical models are very valuable because they permit the use of empirical data for calculation of other useful quantities and prediction of complex variables. Mathematical models usually explain reasons for observed behavior by giving the relationships and data used in development and validation of mathematical models. Accumulation of knowledge and data is a usual prerequisite to formulation of a mathematical model. In this sense, existence of a mathematical model usually indicates that sufficient experimental work was conducted to interpret data in a fundamental way. Below, some of these existing relationships, which help to use data on plasticizers, are discussed. 

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This paper describes application of mathematical modeling to three specific problems warpage of layered composite panels, stress relaxation during a post-forming cooling, and buckling of a plastic column. Information provided here is focused on identification of basic physical mechanisms and their incorporation into the models. Mathematical details and systematic analysis of these models can be found in references to the paper. 

The physical context concept in the conceptual model is extended to describe the behavior of plastics in the form of pellets through the class solid state condition which encapsulates properties such as pellet type. This part of the implementation model concerns the mathematical modeling of some of the properties of polymers, which correspond to their djmamic or flow behavior. A class for a concrete mathematical model not only holds declarative information such as the list of parameters, but also provides a method for calculating the value of the property modeled. This method requires an implementation which is usually different from the one for another mathematical model. Therefore, mathematical models are organized in this application through further classification. 

Also, no more mechanistical models were proposed, but a series of mathematical models appeared, most of which are based on the flee volume theory. These models try to correlate the Tg of the plasticized systems with Tg. and Tg2 of polymer and plasticizer, respectively. The extent to which a plasticizer reduces the glass trarrsition temperatrrre of polymer is irsed as a measure of plasticizer efficiency. Maiuitz et al did an exterrsive sirrvey of these earlier models. These models are applicable in a certain range of experimental conditiorrs and they contain parameters which are diffrcrrlt to obtain experimentally, and thus they are not very useful for predictive purposes. 

Mendelson, A., Plasticity Theory and Applications. MacMillan, New York, 1970. Robertson, R.E. and Stiff, H.A., An improved mathematical model for relating shear stress to shear rate in drilling fluids and cement slurries. Soc. Pet. Eng. J., (February), 31-36 (1976). 

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