# Radial Basis Function Neural Network-Based Modeling of the Dynamic Thermo-Mechanical Response and Damping Behavior of Thermoplastic Elastomer Systems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Thermoplastic Elastomers

#### 1.2. Thermoplastic Polyurethanes

#### 1.3. Viscoelastic Behavior of Thermoplastic Polyurethanes

#### 1.4. Dynamic meChanical Analysis of Polymers

^{−1}. Information obtained through the strategic use of the three parameters of applied force with a very small amplitude, temperature, and frequency (or time) provides the basis for predicting polymer performance in real-world applications. DMA measurements usually result in temperature and/or frequency dependencies of the storage (elastic) modulus E′, loss (viscous) modulus E″ (as a real and imaginary component of complex dynamic modulus E

^{*}, respectively) and loss tangent tanδ. They make it possible to quantify the amount of elastic energy stored in the material, the amount of energy dissipated into heat under dynamic mechanical load and the mechanical damping or the ratio of loss and storage modulus, respectively. Due to the viscoelastic nature of the polymers, the sinusoidal strain lags the applied stress by a phase angle δ, which is a measure of its viscoelastic damping or internal molecular friction [14].

_{g}or T

_{α}, crystalline melt temperature (or melting point) T

_{m}and flow temperature T

_{f}[12] are characteristic of semi-crystalline thermoplastics, non-crosslinked elastomers and thermoplastic elastomers, while the crosslinked polymers up to the temperature of degradation T

_{d}only show the glass transition temperature. At very low temperatures in glassy stage, secondary gamma transition can also be observed at temperature T

_{γ}and beta transition at temperature T

_{β}due to localized bond movements (bending and stretching) and side chain movements, as well as by the whole side chains and localized groups of 4–8 backbone atoms movements, respectively. The beta transition at temperature T

_{β}often represents the T

_{g}of a secondary component in a blend or a specific block in the block copolymers such as TPUs [15]. The secondary delta transition at temperature T

_{δ}is related to very small motions within macromolecule at temperatures under T

_{γ}. Above the glass transition temperature, crystalline and semi-crystalline polymers may pass through T

^{*}and T

_{II}transition temperatures, indicating a slippage of the crystallites past each other and a movement of coordinated segments in the amorphous phase at higher temperatures that relate to reduced viscosity [16].

#### 1.5. Artificial Neural Networks Modeling

#### 1.6. Radial Basis Function Artificial Neural Network

## 2. Materials and Methods

#### 2.1. Sample Preparation

^{−1}and a force of 100 ± 2 kN. Ten rectangular shaped samples with the dimensions of 30 ± 0.3 mm (length) × 6 ± 0.06 mm (width) × 1.5 ± 0.02 mm (thickness) were cut off from the pressed TPU using a Gravograph LS100 40W CO

_{2}laser cutter (Gravograph, Paris, France).

#### 2.2. DMA Testing

^{−1}. The strain measurements were done by a linear variable differential transformer of the DMA equipment with a span of 2 mm and a mean resolution of 2 nm. Low temperatures in the DMA measuring chamber were provided by liquid nitrogen from a 50 K Dewar flask (Perkin Elmer, Baesweiler, Germany), without direct contact of the refrigerant with the measured sample. The measurements uncertainty was about 2%. The average values of storage modulus E′(T, f), loss modulus E″(T, f), and loss tangent tanδ (T, f) were computed.

#### 2.3. RBF-ANN Modeling

_{j}, σ

_{j}and φ

_{j}(x

_{i}) denote its well-pointed center (centroid), spread width (stretch constant) and the response of the j-th hidden neuron corresponding to its input x

_{i}, respectively. The term

_{i}and the corresponding centroid of Gaussian μ

_{j}(Euclidean norm of RBF).

_{i}is the number of temperature samples T

_{i}for each frequency.

_{i}of the input signal p both input variables is multiplied by the respective element b

_{j}of the hidden layer bias vector b

^{1}with the length S

^{1}corresponding to the number of hidden neurons. Each bias b

_{j}of hidden neuron j represents a constant external input with a value of 1 multiplied by its weight w

_{ji}, which is the appropriate element of the input weight matrix of hidden layer IW

^{1,1}with the size S

^{1}xR, and is the primary activity level of the hidden neuron j. The vector n

^{1}with the length S

^{1}, whose elements n

_{j}represent the distances between the multiplied inputs p

_{i}and their related weights w

_{ji}, serves as the net input n

^{1}of the hidden layer, or as the activation potentials [24] of its RBF neurons.

^{1}with the length S

^{1}. The weight w

_{ji}represents the centroid of the j-th hidden RBF neuron. The total output of nonlinear hidden layer 1

_{j}and biases b

_{k}, so

_{kj}are the corresponding elements of the output layer weight matrix LW

^{2,1}with the size S

^{2}× S

^{1}, b

_{k}are the elements of the vector of biases of the output layer b

^{2}with the length S

^{2}and n

_{k}elements of the vector n

^{2}with the length S

^{2}, which serves as the net input n

^{2}of the output layer 2, or as the activation potential of its linear neurons.

_{k}of the linear transfer function of k-th output layer neuron with the input n

_{k}for each input variable of the model is given by the expression

_{k}is the element of the output layer 2 output vector [38]

^{2}, which is also the overall output of RBF-ANN y, and

^{®}Version 9.0.0.341360 R2016a 64-bit software package (MathWorks, Natic, MA, USA) that provides number of built-in functions with sufficiently powerful and user-friendly RBF-ANNs training and simulation algorithms.

^{®}development environment, a built-in newrb function was used. The algorithm of this function adds one by one neuron with the specified width of the Gaussian RBF function spread to the hidden layer until their predetermined maximum number MN is reached, or the predetermined smallest value of mean squared error (MSE) goal. The iterative algorithm of simultaneous networking and training starts from an empty hidden layer. After adding a new hidden neuron, a network simulation is performed in each iteration step; the input vector with the greatest network’s MSE is found; and Gaussian RBF (radbas) neuron is added to the hidden layer with weights corresponding to this vector and with constant bias; the linear layer weights and biases (purelin) are redesignated to minimize the network’s MSE. This procedure is repeated until the network’s MSE falls below goal or until the maximum number of neurons MN is reached [39]. The ANN’s error criterion MSE for each ANN output variable is calculated as the mean squared of the errors

_{i}are target values E′(T, f), E″(T, f) and tanδ(T, f), y

^{m}

_{i}represent the network response to its T

_{i}and f

_{i}input values, and N is the number of data points in the ANN patterns. With fixed MN and goal, the best-fit function of RBF-ANN targets for both input variables T and f was searched by trial and error method for different spread parameter values for the training set of data points.

**y**and simulated outputs

**y**for validation, as well as testing data

^{m}_{p}and s

_{y}are their standard deviations, respectively [39].

**z**represents the original data before normalization,

**z**and

_{min}**z**are their minimum and maximum values before normalization, and

_{max}**z**are data after normalization. With the applied Gaussian RBF activation functions of the hidden layer, such an adjustment of the input and target data to the interval [0, 1] is able to ensure fast and accurate convergence of network output data to the values of the analyzed experimental data. The reverse of network outputs normalized to original data after RBF-ANN simulation with training, validation and testing data was realized according to the relationship [27]

_{norm}## 3. Results and Discussion

#### 3.1. Results of Dynamic Mechanical Analysis

_{α}, T

_{β}and T

_{γ}, respectively, which are particularly pronounced on the curves E″(T, f). The temperature T

_{γ}corresponds to the glass transition of crystalline soft segments from stiff glassy stage to compliant rubbery stage; the temperature T

_{α}is associated with the glass transition of crystalline hard segments due to the breakdown of hydrogen-bonded interactions of van der Waal’s forces between the rigid and flexible segments of TPU, while at the temperature T

_{β}, the short-range order translation, and reorientation motions within both the soft-phase and hard-phase crystallites, occur [15]. Activation energies of individual relaxation transitions with values of 379.73 kJ·mol

^{−1}for α-transition, 65.53 kJ·mol

^{−1}for β-transition and 45.98 kJ·mol

^{−1}for γ-transition, calculated based on the linearized Arrhenius equation from the frequency-dependent shift of transition temperatures taken as peaks of tanδ(T, f) in our earlier work [17], confirm that at temperature T

_{α}the primary relaxation transition and at temperatures T

_{β}and T

_{γ}the secondary relaxation transitions actually occur [16].

#### 3.2. Analysis of the RBF-ANN Model

^{−7}, which is significantly lower than the target MSE value of 1 × 10

^{6}, indicating that the best-fit function of the training targets was perfectly estimated.

^{−2}is negligibly small compared to the experimental data acquired from the DMA tests. The maximum training data error of 0.87% is shown by tanδ(T), validation data at 0.56 % E′(T, f) and testing data at 4.63 % tanδ(T), which can be considered an excellent result.

_{δ}and above the temperature T

_{f}makes no sense in practical terms, since this temperature range includes the whole temperature range of their lifetime and is usually the standard temperature interval of DMA measurements of TPEs. Frequently, however, it is necessary to know the values of E′, E″ and tanδ at temperatures that are not between the experimental data although they come from the measured temperature interval. ANNs’ excellent interpolation capabilities make it possible to solve this problem.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Drobny, J.G. Handbook of Thermoplastic Elastomers, 2nd ed.; Elsevier: Oxford, UK, 2014. [Google Scholar]
- Olagoke, O.; Kolapo, A. Handbook of Thermoplastics, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Coveney, V.A. Elastomers and Components; Elsevier: Oxforf, UK, 2006. [Google Scholar]
- Prisacariu, C. Polyurethane Elastomers: From Morphology to Mechanical Aspects; Springer: Wien, Austria, 2011. [Google Scholar]
- Spontak, R.J.; Patel, N.P. Thermoplastic elastomers: Fundamentals and applications. Curr. Opin. Colloid Interface Sci.
**2000**, 5, 333–340. [Google Scholar] [CrossRef] - Kopal, I.; Vršková, J.; Labaj, I.; Ondrušová, D.; Hybler, P.; Harničárová, M.; Valíček, J.; Kušnerová, M. The Effect of High-Energy Ionizing Radiation on the Mechanical Properties of a Melamine Resin, Phenol-Formaldehyde Resin, and Nitrile Rubber Blend. Procedia Eng.
**2017**, 11, 2405. [Google Scholar] [CrossRef] [PubMed] - Roylance, D. Engineering Viscoelasticity; MIT: Cambridge, MA, USA, 2001. [Google Scholar]
- Brinson, H.F.; Brinson, L.C. Polymer Engineering Science and Viscoelasticity, 2nd ed.; Springer: Berlin, Germany, 2014. [Google Scholar]
- Randall, D.; Lee, S. The Polyurethanes Book; Wiley: Hoboken, NJ, USA, 2003. [Google Scholar]
- Koštial, P.; Bakošová, D.; Jančíková, Z.; Ružiak, I.; Valíček, J. Thermo-Mechanical Analysis of Rubber Compounds Filled by Carbon Nanotubes. Defect Diffus. Forum
**2013**, 336, 1–10. [Google Scholar] [CrossRef] - Ward, I.M.; Sweeney, J. An Introduction to the Mechanical Properties of Solid Polymers, 2nd ed.; Wiley: Chichester, UK, 2004. [Google Scholar]
- Ashby, M.F.; Jones, H.R.D. Engineering Materials 2. An Introduction to Microstructures, Processing and Design; Elsevier/Butterworth-Heinemann: Amsterdam, The Netherlands, 2005. [Google Scholar]
- Riande, E.; Diaz-Calleja, R.; Prolongo, M.G.; Masegosa, R.M.; Salom, C. Polymer viscoelasticity: stress and strain in practice; Marcel Dekker, Inc.: New York, NY, USA, 2000. [Google Scholar]
- Gabbott, P. Principles and Applications of Thermal Analysis; Blackwell Publishing: Oxford, UK, 2008. [Google Scholar]
- Menard, K. Dynamic Mechanical Analysis: A Practical Introduction, 2nd ed.; CRC Press: Boca Raton, FL, USA, 1999. [Google Scholar]
- Fried, J.R. Polymer Science and Technology, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2014. [Google Scholar]
- Kopal, I.; Harničárová, M.; Valíček, J.; Koštial, P.; Jančíková, Z.K. Determination of activation energy of relaxation events in thermoplastic polyurethane by dynamic mechanical analysis. Materialwiss. Werkstofftech.
**2018**, 49, 627–634. [Google Scholar] [CrossRef] - Mahieux, C.A.; Reifsnider, K.L. Property modeling across transition temperatures in polymers: A robust stiffness-temperature model. Polymer
**2001**, 42, 3281–3291. [Google Scholar] [CrossRef] - Richeton, J.; Schlatter, G.; Vecchio, K.S.; Rémond, Y.; Ahzi, S. Unified model for stiffness modulus of amorphous polymers across transition temperatures and strain rates. Polymer
**2005**, 46, 8194–8201. [Google Scholar] [CrossRef] - Kopal, I.; Bakošová, D.; Koštial, P.; Jančíková, Z.; Valíček, J.; Harničárová, M. Weibull distribution application on temperature dependence of polyurethane storage modulus. Int. J. Mater. Res.
**2016**, 107, 472–476. [Google Scholar] [CrossRef] - Kopal, I.; Labaj, I.; Harničárová, M.; Valíček, J.; Hrubý, D. Prediction of the Tensile Response of Carbon Black Filled Rubber Blends by Artificial Neural Network. Polymers
**2018**, 10, 644. [Google Scholar] [CrossRef] [PubMed] - Aliev, R.; Bonfig, K.; Aliew, F. Soft Computing; Verlag Technic: Berlin, Germany, 2000. [Google Scholar]
- Zhang, Z.; Friedrich, K. Artificial neural networks applied to polymer composites: A review. Compos. Sci. Technol.
**2003**, 63, 2029–2044. [Google Scholar] [CrossRef] - Al-Haik, M.S.; Hussaini, M.Y.; Rogan, C.S. Artificial Intelligence Techniques in Simulation of Viscoplasticity of Polymeric Composites. Polym. Compos.
**2009**, 30, 1701–1708. [Google Scholar] [CrossRef] - Hagan, M.T.; Demuth, H.B.; Beale, M.H.; De Jesús, O. Neural Network Design, 2nd ed.; Martin Hagan: Jersey, NJ, USA, 2014. [Google Scholar]
- Rao, M.A. Neural Networks: Algorithms and Applications; Alpha Science International: Oxford, UK, 2003. [Google Scholar]
- Livingstone, D.J. Artificial Neural Networks Methods and Applications (Methods in Molecular Biology); Humana Press: Totowa, NJ, USA, 2013. [Google Scholar]
- Kopal, I.; Harničárová, M.; Valíček, J.; Kušnerová, M. Modeling the Temperature Dependence of Dynamic Mechanical Properties and Visco-Elastic Behavior of Thermoplastic Polyurethane Using Artificial Neural Network. Polymers
**2017**, 9, 519. [Google Scholar] [CrossRef] [PubMed] - Seidl, D.; Jančíková, Z.; Koštial, P.; Ružiak, I.; Kopal, I.; Kreislova, K. Exploitation of Artificial Intelligence Methods for Prediction of Atmospheric Corrosion. Defect Diffus. Forum
**2012**, 326, 65–68. [Google Scholar] [CrossRef] - Ružiak, I.; Koštial, P.; Jančíková, Z.; Gajtanska, M.; Krišťák, Ľ.; Kopal, I.; Polakovič, P. Artificial Neural Networks Prediction of Rubber Mechanical Properties in Aged and Nonaged State. In Improved Performance of Materials; Öchsner, A., Altenbach, H., Eds.; Springer: Berlin, Germany, 2018; Volume 72, pp. 27–35. [Google Scholar]
- Nguyen, L.T.K.; Keip, M.A. A data-driven approach to nonlinear elasticity. Comput. Struct.
**2018**, 194, 97–115. [Google Scholar] [CrossRef] - Davydov, O.; Oanh, D.T. On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation. Comput. Math. Appl.
**2011**, 62, 2143–2161. [Google Scholar] [CrossRef] [Green Version] - Pislaru, C.; Shebani, A. Identification of Nonlinear Systems Using Radial Basis Function Neural Network. Int. J. Comput. Inf. Syst. Control Eng.
**2014**, 8, 1528–1533. [Google Scholar] - Xu, X.; Gupta, N. Artificial neural network approach to predict the elastic modulus from dynamic mechanical analysis results. Adv. Theory Simul.
**2019**, 2, 1800131. [Google Scholar] [CrossRef] - Xu, X.; Gupta, N. Application of radial basis neural network to transform viscoelastic to elastic properties for materials with multiple thermal transitions. J. Mater. Sci.
**2019**, 54, 8401–8413. [Google Scholar] [CrossRef] - Trebar, M.; Susteric, Z.; Lotric, U. Predicting mechanical properties of elastomers with neural networks. Polymer
**2007**, 48, 5340–5347. [Google Scholar] [CrossRef] - Bhowmick, A.K.; Stephens, H.L. Handbook of Elastomers, 2nd ed.; CRC-Press: Boca Raton, FL, USA, 2000. [Google Scholar]
- Shi, F.; Wang, X.C.; Yu, L.; Li, Y. MATLAB 30 Case Analysis of MATLAB Neural Network; Beijing University Press: Beijing, China, 2009. [Google Scholar]
- Paliwal, M.; Kumar, U.A. Neural networks and statistical techniques: A review of applications. Expert Syst. Appl.
**2009**, 36, 2–17. [Google Scholar] [CrossRef] - Wang, H.; Xu, X. Applying RBF Neural Networks and Genetic Algorithms to Nonlinear System Optimization. Adv. Mater. Res.
**2012**, 605, 2457–2460. [Google Scholar] [CrossRef]

**Figure 2.**Temperature-frequency dependences of average storage modulus, loss modulus and loss tangent of TPU over the temperature range from 146 K to 527 K at constant strain rate of 0.1 Hz and frequencies of 0.5 Hz, 1 Hz, 2 Hz, 5 Hz and 10 Hz.

**Figure 4.**(

**a**) Linear regression plot for training data; (

**b**) Linear regression plot for validation data; (

**c**) Linear regression plot for testing data.

**Figure 5.**(

**a**) Comparison of the training target with simulated training outputs for E′(T, f); (

**b**) Comparison of the training target with simulated training outputs for E″(T, f); (

**c**) Comparison of the training target with simulated training outputs for tanδ(T, f).

**Figure 6.**(

**a**) Comparison of the validation targets with validation outputs for E′(T, f); (

**b**) Comparison of the validation targets with validation outputs for E″(T, f); (

**c**) Comparison of the validation targets with validation outputs for tanδ(T, f).

**Figure 7.**(

**a**) Comparison of the testing targets with testing outputs for E′(T, f); (

**b**) Comparison of the testing targets with testing outputs for E″(T, f); (

**c**) Comparison of the testing targets with testing outputs for tanδ(T, f).

**Figure 8.**(

**a**) Error plot for training data; (

**b**) Error plot for validation data; (

**c**) Error plot for testing data.

**Table 1.**RBF-ANN parameters for an optimized model of temperature-frequency dependence of dynamic thermo-mechanical parameters E′(T, f), E″(T, f) and tanδ(T, f) of TPU.

IL | HL | OL | TF | DDF | PF | MN | Spread | Goal |
---|---|---|---|---|---|---|---|---|

2 | 268 | 3 | Gaussian RBF, linear | dividerand | MSE | 10^{3} | 7 | 10^{−6} |

**Table 2.**Parameters of optimized ANN and results of the training network process, validation and testing of the RBF-ANN model.

Data Division | Samples | MSE | R | Intercept | |
---|---|---|---|---|---|

Training | 0.85 | 1016 | 8.176 × 10^{−7} | 1 | 4.9 × 10^{−7} |

Validation | 0.15 | 180 | - | 0.99999 | 3 × 10^{−4} |

Testing | 1 | 344 | - | 0.99999 | 3.9 × 10^{×5} |

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**MDPI and ACS Style**

Kopal, I.; Harničárová, M.; Valíček, J.; Krmela, J.; Lukáč, O.
Radial Basis Function Neural Network-Based Modeling of the Dynamic Thermo-Mechanical Response and Damping Behavior of Thermoplastic Elastomer Systems. *Polymers* **2019**, *11*, 1074.
https://doi.org/10.3390/polym11061074

**AMA Style**

Kopal I, Harničárová M, Valíček J, Krmela J, Lukáč O.
Radial Basis Function Neural Network-Based Modeling of the Dynamic Thermo-Mechanical Response and Damping Behavior of Thermoplastic Elastomer Systems. *Polymers*. 2019; 11(6):1074.
https://doi.org/10.3390/polym11061074

**Chicago/Turabian Style**

Kopal, Ivan, Marta Harničárová, Jan Valíček, Jan Krmela, and Ondrej Lukáč.
2019. "Radial Basis Function Neural Network-Based Modeling of the Dynamic Thermo-Mechanical Response and Damping Behavior of Thermoplastic Elastomer Systems" *Polymers* 11, no. 6: 1074.
https://doi.org/10.3390/polym11061074