Nonlinear models of mechanical properties reduce rubber recipe development time.
Recent increases of environmental legislation and energy labeling of tires (e.g., in Japan, South Korea and EU) have added one more dimension to tire development: All compounds must also be optimized to decrease rolling resistance and improve fuel economy. Hence, the development work is even more complex than before, and appropriate tools to shorten time-to-market are worth their weight in gold.
The rubber mix before vulcanization needs to have viscosity within certain limits. After vulcanization, hardness, tensile modulus, tensile strength, tear strength, elongation at break, dynamic mechanical properties, wear resistance, etc., are of interest. To achieve a desired combination of several of these properties is no easy task. Trial and error experiments are often carried out in large numbers for developing rubber recipes which lead to a desired combination of several of these properties.
Mathematical modeling can be performed in various ways, and different ways are suitable in different situations. Mathematical models represent knowledge of quantitative effects of relevant variables in a concise and precise form. They can be used instead of experimentation if they are reliable enough. Mathematical models also permit the user to carry out various kinds of calculations, like determining suitable values of variables which will result in desired material properties or product characteristics.
Physical or phenomenological modeling is not particularly effective for predicting material behavior. Physical modeling usually requires assumptions and simplifications. Empirical and semi-empirical modeling, on the other hand, does not need any major assumptions or simplifications. Empirical models simply describe the observed behavior of a system. Empirical modeling is feasible when the relevant variables are measurable.
Conventional techniques of empirical modeling, however, are linear statistical techniques. These tend to have serious limitations because nothing in nature is very linear, and particularly so in materials science. It therefore makes sense to use better techniques of empirical and semi-empirical modeling which take nonlinearities into account.
There is hardly any material behavior which is absolutely linear. It is therefore wise to treat the nonlinearities rather than ignore them. The proponents of linear techniques draw on their simplicity and the possibility of adding nonlinear terms in linear regression. Often, this is not done, and is not efficient even if it is done. Nature does not follow the simplicities that we try to fit it in, using common linear techniques.
Nonlinear modeling is empirical or semi-empirical modeling which takes at least some nonlinearities into account. Nonlinear modeling can be carried out with a variety of methods. The older techniques include polynomial regression, linear regression with nonlinear terms and nonlinear regression. These techniques have several disadvantages compared to the new techniques of nonlinear modeling based on free-form nonlinearities.
The newer methods include feed-forward neural networks, series of basis functions, kernel regression, multivariate splines, etc., which do not require a prior knowledge of the nonlinearities in the relations. Among these new techniques, feed-forward neural networks have turned out to be particularly valuable in materials science and chemical engineering (ref. 1). Feed-forward neural networks have several features which make them better tools for nonlinear empirical modeling. Besides their universal approximation capability (ref. 2), it is usually possible to produce nonlinear models with some extrapolation capabilities with feed-forward neural networks.
There are many different types of neural networks, and some of them have practical uses in process industries. Neural networks have been in use in process industries for about 20 years (ref. 3). The multilayer perception, a kind of a feed-forward neural network, is the most common one. Most neural network applications in industries are based on them (refs. 4-18).
Feed-forward neural networks have the universal approximation capability, which makes them particularly suitable for most function approximation tasks we come across in materials science and process engineering. The user does not need to know the type and severity of nonlinearities while developing the models. In other words, we have free-form nonlinearities in feed-forward neural network models.
There have been several earlier attempts in the recent past to develop neural network models of rubber properties (refs. 4-8). For example, Wang, Ma and Wu (ref. 4) have reported developing a neural network model for abrasion rate. Xiang, Xiang and Wu (ref. 5) have reported a neural network model for fatigue life prediction. Padmavathi et al. (ref. 6) report modeling of Mooney viscosity using neural networks. There are also reports (refs. 7 and 8) of neural network models of polymerization and vulcanization processes. Karaagac, Inal and Deniz (ref. 7) describe optimizing the curing time using a neural network model. Martinez Delfa, Olivieri and Boschetti (ref. 8) report optimization of emulsion polymerization of styrene butadiene rubber using neural network models. All these people have used feed-forward neural networks, but the quality of their models seems to be modest. They have only one rubber material under consideration, unlike this work which has three rubber materials in the recipes besides silica, and has been taken into industrial use.
Feed-forward neural networks resemble structurally and, to a smaller extent functionally, the networks of neurons in biological systems. Like the networks of neurons in the brain, artificial neural networks also consist of neurons in layers directionally connected to others in the adjacent layers (figure 1).
In a feed-forward neural network of the kind shown in figure 1, the output of each neuron i in the feed-forward neural network is usually given by:
[z.sub.i] = [sigma]([N.summation over (j=0)] [w.sub.ij][x.sub.j])
where [sigma] is called the activation function, and is usually the logistic sigmoid, given by
[sigma](a) = 1/1 + [e.sup.-a]
The incoming signals to the neuron are [x.sub.j], and [w.sub.ij] are the weights for each connection from the incoming signals to the ith neuron. The [w.sub.i0] terms are called biases. This results in a set of algebraic equations which relate the input variables to the output variables. Thus, for each observation (a set of input and output variables), the outputs can be predicted from these equations based on a given set of weights. The training procedure aims at determining the weights which result in the smallest sum of squares of prediction errors.
A variety of training methods are in use today. Back-propagation used to be the most common training method. Today, most people use good optimization methods (ref. 19) instead.
Nonlinear modeling in materials science
Nonlinear modeling has been utilized successfully for various materials, including plastics (refs. 9-11), metals (refs. 12-15), concrete (ref. 16), semiconductors (ref. 17), mineral wools (ref. 18), glass, ceramics, paper, etc. Different materials have different characteristics; different raw materials, different compositions, and are produced by different batch, continuous or fed-batch processes. However, some things are common to modeling of various kinds of material behavior. Material properties or product properties depend on composition variables, process variables and dimension variables, as summarized in figure 2.
One would like to determine the best values of composition variables (or feed characteristics), process variables and/or dimension variables such that the resulting material properties or product properties will be within desired limits. Sometimes, the composition variables or feed characteristics might be constants. In more common situations, the process variables may be constant or dependent variables, and the only degrees of freedom in materials development may be the composition of the feed, the amounts of raw materials and possibly dimension variables.
The problem looks somewhat similar from the modeling point of view for a wide variety of materials. Nonlinear models of the kind shown in figure 2 make materials development more efficient by reducing expensive experimentation and by helping achieve better combinations of material properties, sometimes optimized for cost.
Sometimes, the process variables do not matter, or are not independent variables. In some cases, dimension variables like particle sizes may be constant. In such cases, the problem is essentially of recipe development like this case, in which only composition variables matter and determine material properties.
As mentioned earlier, nonlinear modeling needs either experimental or production data. In the recent work of Nokian Tyres, an experimental approach was preferred since it is not very expensive to carry out experiments. The original experimental plan had only 28 experiments. A total of 43 experiments, however, were carried out with different recipes. Results from 38 of these experiments were used for the model development work. A much smaller number would have been sufficient for this work. The experiments were planned such that sufficient information on nonlinearities can be extracted from the experimental data. Several material properties were measured from the samples produced in the experiments.
From the raw data set, it was possible to see the effects of certain composition variables. Figure 3 shows the effect of silica content in phr on tensile moduli at three different elongations. The fractions of natural rubber, styrene butadiene rubber and butadiene rubber are the same for all these points on the plot. The effects of silica on tensile moduli are not very linear, but some of the effects of natural rubber content on some properties are not even monotonic.
The raw experimental data were analyzed and preprocessed, after which nonlinear models were developed and tested for several material properties over a period of a few months using the NLS 020 software. The experimental data taken into use were consistent and of very good quality; and as a consequence, excellent nonlinear models could be developed. The correlation coefficients of all the models were well above 90%, and for some properties like viscosity, they were above 99%. It is natural that the nonlinear models perform very well, since the effects are not very linear, while the linear models will not hesitate to predict even negative values of material properties.
Nonlinear models of material properties
Nonlinear models were developed for Mooney viscosity before vulcanization and for several material properties of the compounded rubbers after vulcanization. A large number of models in the form of feed-forward neural networks were attempted with different configurations. One or more of the weights in those models was constrained. Since the quality of the experimental data was very good, the resulting models were also of very good quality.
For durometer hardness, two observations were excluded from the data set because of obvious inconsistencies. The nonlinear model which was finally taken into use had the following statistical characteristics.
* Output variable 1: Durometer hardness A
* Training set: 36 observations
* rms err of output variable 1: 1.176599
* mean [absolute value of err] of output variable 1: 0.8747641
* rms % err of output variable 1: 1.8579
* max [absolute value of err] of output variable 1: 3.345741
* Correlation of output variable 1: 0.9812
The root mean square (rms) error, roughly speaking the standard deviation of prediction errors, was 1.18, which is close to the repeatability of the experiments and measurements. This amounts to a correlation coefficient of 98.12%. Figure 4 shows a plot of measured values of hardness against the values predicted by the nonlinear model on the vertical axis. As can be seen from the plot, the predictions are fairly close to the measured values.
For tensile modulus, no observations had to be excluded from the data set. The nonlinear model which was finally taken into use had the following statistical characteristics:
* Output variable 2: Tensile modulus at 100% elongation (MPa)
* Training set: 38 observations
* rms err of output variable 2: 0.1639614
* mean [absolute value of err] of output variable 2: 0.125768
* rms % err of output variable 2: 6.899
* max [absolute value of err] of output variable 2: 0.5173544
* Correlation of output variable 2: 0.9914
The rms error was 0.16 MPa, which is close to the repeatability of the experiments and measurements. This amounts to a correlation coefficient of 99.14%. Figure 5 shows a plot of measured values of tensile modulus against the values predicted by the nonlinear model on the vertical axis. As can be seen from the plot, the predictions are close to the measured values. Unlike linear regression models, with nonlinear models we can ensure that they never predict negative values.
Similarly, the nonlinear models for elongation at break, loss factor tan [delta] at 0[degrees]C and Mooney viscosity had the following statistical characteristics:
* Output variable 3: Elongation at break (%)
* Training set: 38 observations
* rms err of output variable 3: 33.31148
* mean [absolute value of err] of output variable 3: 25.5586
* rms % err of output variable 3: 9.5733
* max [absolute value of err] of output variable 3: 76.20193
* Correlation of output variable 3: 0.9333
* Output variable 4: Tan 8 at 0[degrees]C
* Training set: 38 observations
* rms err of output variable 4: 7.986302E-03
* mean [absolute value of err] of output variable 4: 6.445058E-03
* rms % err of output variable 4: 3.1368
* max [absolute value of err] of output variable 4: 2.090886E-02
* Correlation of output variable 4: 0.9404
* Output variable 5: Mooney viscosity
* Training set: 38 observations
* rms err of output variable 5:1.16931
* mean [absolute value of err] of output variable 5: 0.8254648
* rms % err of output variable 5: 2.4329
* max [absolute value of err] of output variable 5: 3.124168
* Correlation of output variable 5: 0.9969
As can be seen from the numbers above, the models had high correlation coefficients. Nonlinear models were developed for a few other material properties also, but not everything can be reported here.
Using the nonlinear models
Once the models are ready, they can be used for several purposes with appropriate software like Lumet systems. Besides being able to predict the values of the material properties, one would like to see the effects of different ingredients on the properties. In figure 6, hardness is plotted against sulfur content for different amounts of natural rubber. Figure 7 shows the effect of silica content on tensile modulus at 100% elongation for different amounts of sulfur. Figure 8 shows the effect of silica on elongation at break for different amounts of sulfur. Figure 9 shows the effect of styrene butadiene rubber on loss factor at 0[degrees]C for different amounts of sulfur.
The rubber recipes considered here are essentially a ternary system, since it has three main components which are natural rubber, styrene butadiene rubber and butadiene rubber. Therefore, the material properties of all different compositions can be plotted on ternary diagrams like the one shown in figure 10, with fixed amounts of other ingredients like sulfur and vulcanization accelerator.
Figure 10 shows 35 contours of viscosity on the ternary plot from 17 to 51 at intervals of one in different colors. The color scale shows the color for each value of viscosity. Thus, the highest values near 51 are near the top vertex of the triangle, which represents 100% SBR, while the lowest values near 17 are near the lower right vertex of the triangle, which represents 100% natural rubber. The fourth contour with a viscosity of 20 seems to pass through the center of the triangle, which represents 33.3% natural rubber, 33.3% SBR and 33.3% butadiene rubber.
The objective of materials development or development of new recipes is to determine compositions which result in desired combinations of material properties. Lumet systems also have mathematical tools that allow the user to calculate suitable recipes or compositions by specifying the constraints on any or several of the variables in the nonlinear models. Such calculations can take a few seconds or a minute or two. This helps speed up materials development work by a large margin for a variety of materials.
The objective of materials development for automobile tires is usually to achieve a desired combination of mechanical and other properties by proper compounding of rubbers and additives, and by suitable conditions for mixing, extrusion and vulcanization. These composition variables and process variables affect the material properties in a complicated manner, and people with even decades of experience cannot predict the quantitative effects of the relevant variables. Recently increased requirements to improve fuel economy of automobiles by reducing the rolling resistance of tires significantly have made the work even more complex. Reliable mathematical models are therefore highly useful in this situation. The materials development manager of Nokian Tyres stated that he expects these nonlinear models to reduce the recipe development time to less than half.
Materials development work is often done by trial and error experimentation. It can be done much more efficiently if the experiments are planned properly, followed by development of nonlinear models, and the necessary mathematics used to determine suitable values of the variables which will result in the desired combination of material properties. This has been demonstrated with various kinds of materials.
by Abhay Bulsari, Nonlinear Solutions Oy, and Jouko Ilomaki, Mika Lahtinen and Risto Perkio, Nokian Tyres PLC, Finland
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Figure 10--contours of viscosity on a ternary plot Contours at the following values Viscosity 17 27 37 18 28 38 19 29 39 20 30 40 21 31 41 22 32 42 23 33 43 24 34 44 25 35 -- 26 36 51
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|Author:||Bulsari, Abhay; Ilomaki, Jouko; Lahtinen, Mika; Perkio, Risto|
|Date:||Sep 1, 2015|
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