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Fusberta Loparo

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Jul 22, 2024, 6:58:01 AM7/22/24

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Magnetorheological (MR) elastomers are intelligent materials with tunable viscoelastic properties when exposed to magnetic fields1. MR elastomers are composed of matrix elements (e.g., silicone rubber and natural rubber)2 and commonly used micro-sized magnetic particles (e.g., carbonyl iron particles (CIPs))3,4. MR elastomers are suitable for vibration absorber applications due to their stiffness flexibility and damping properties in the presence of a magnetic field5. One example is the use of isolators in civil infrastructure as seismic protection to replace conventional isolators6,7. The MR effect of MR elastomers is an important consideration when designing a device because a higher MR effect means a more comprehensive controllability range8,9. When a sample is subjected to a range of magnetic field densities, the MR effect is calculated based on the measurement of the dynamic modulus. Many studies have proposed various methods to improve MR effect values, particularly the essential parameters, such as the variations of magnetic particle concentration10,11, curing magnetic field12,13, and magnetic particle shapes and sizes14,15,16. As discussed in recent works17,18, different curing conditions can affect material properties by changing the dispersion condition and distance between magnetic particles during the curing process. The curing condition classification can be anisotropic or isotropic with chain like-alignment in the presence of magnetic fields and arbitrary alignment in the absence of magnetic fields19.

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Even though conventional viscoelastic models are easy to handle, the accommodated inputs are limited. For example, Agirre et al.30 developed a new method to consider one more composition related-variable to predict the complex or dynamic modulus of MR elastomer. With the modification, the input variables include frequency, magnetic flux density, and matrix-filler concentration. A similar method is also applied to accommodate five CIP volume fractions31 for anisotropic conditions. However, developing a similar model to accommodate more input variables is time-consuming and requires careful consideration of the additional parameters. Therefore, a model will be beneficial if it can flexibly accommodate more inputs. Rather than conventional methods as part of the parametric models, non-parametric data or data-driven models are preferable solutions to consider more inputs, especially when dealing with a more extensive database.

Therefore, this work proposes a multi-input model using a non-parametric approach that uses machine learning as a potential solution to predict the complex modulus and magnetorheological effect at various curing conditions and compositions that can be considered as multiple outputs cases. The workflow is described as follows. First, the modeling structure of the feedforward neural network is formulated and describes the training algorithm, which is an artificial neural network (ANN) and an extreme learning machine (ELM). Then, the sample preparation and data collection are carried out. Finally, the simulation results are discussed through several analyses before summarizing all sections.

Fabrication conditions also affect the dynamic properties of MR elastomers. The fabrication conditions can be curing magnetic flux density value, magnetic field direction, and temperature. The applied magnetic field direction can influence the particle alignment lock during curing conditions. Consequently, the alignment has a significant effect on the dynamic modulus of MR elastomer21. Curing conditions are classified into two types: isotropic and anisotropic. Isotropic is a condition where the magnetic particle is randomly distributed in the matrix. In contrast, anisotropic is aligned in a chain by applying a certain magnetic field during curing conditions. Both conditions may produce different MR effects and dynamic modulus. In addition, different applications used different curing conditions following its usages. Thus, it is important to consider both isotropic and anisotropic conditions when analyzing the dynamic modulus of MR elastomer.

The three groups of variables represent the factors that influenced the dynamic modulus of MR elastomer to be accommodated in the formulated model. The characterization process is represented by strain amplitude, frequency, and magnetic field in predicting dynamic states on oscillation tests. This work also provides insight on the effect of excitation frequency on-ramp magnetic field specifically to resolve the vibration issue. A strain amplitude of 0.01% is chosen due to the linear viscoelastic (LVE) region in which the microstructure is considered linear due to strong bonding between particle and matrix47. Hence, the variables used to represent the characterization processes are frequency \((f)\) and magnetic flux density \((B)\). Besides, the particle distribution, which is isotropic or anisotropic, is also included as an independent variable in the model, represented by the value of the applied magnetic field during curing conditions \((B_c)\). Then, the concentration of magnetic particle \((W_p)\) representing the composition is also accommodated.

This section discusses the performance of the proposed model based on listed data in Table 5. The effect of excitation frequency on rheological properties of MR elastomer, specifically complex shear modulus, is being predicted by two proposed models based on a machine learning approach, by comparing with experimental data. Table 6 shows the accuracy of two configurations of ANN models and four configurations of ELM models that have potential in terms of accuracy compared to other configurations based on hidden nodes number and activation function. ANN-8 and ANN-10 have the highest accuracy among other ANN configurations, where ANN-8 and ANN-10 denoted 8 and 10 hidden nodes, respectively. On the other hand, ELM-SINE1 and ELM-SINE2 denoted 10,000 and 55,000 hidden nodes with sine activation function whereas ELM-SIG1 and ELM-SIG2 represented 10,000 and 55,000 hidden nodes with a sigmoid activation function, respectively. Table 6 displays the accuracy for all data. According to the RMSE and R2 values, the ELM-SINE2 model has higher accuracy for almost all data sets than other ELM models, whereas, for the ANN model, the ANN-8 model may have better generalization performance than the ANN-10 model. In addition, Ts5 data sets that represent the combination of all unlearned data have the highest accuracy among all models. This proved that the ELM-SINE2 model is the most generous among other models. Figures 3, 4, 5 and 6 depict the results of these models.

The implementation of MR elastomer can be seen in various applications such as vibration absorbers61 and acoustic devices62, which involve frequency variation to utilize the functionality of MR elastomer in dynamic properties. Thus, before it is enforced in these devices, it is crucial to investigate its rheological properties by using a frequency sweep test. This section discusses the modeling performance of complex modulus, specifically on storage and loss modulus based on excitation frequency from 0.1 to 10 Hz. Generally, the complex modulus linearly increases with the applied frequency for both isotropic and anisotropic MR elastomer. Meantime, the stiffness of the MR elastomer increases as the CIP compositions and the magnetic field strength are increased.

The accuracy for Ts2 data sets is almost similar on both models. As shown in Fig. 5, which represents a part of Ts2 data, the ANN model presents excellent consistency compared to the ELM model for both storage and loss moduli. The ELM model produced a wavy pattern on isotropic (Fig. 5a) and anisotropic distribution (Fig. 5c). To summarize the accuracy for each Ts2 data set, Table 8 shows the RMSE value of absolute complex modulus for Ts2 data set on each unlearned magnetic flux density on five CIP weight percentages. In Table 8, the RMSE values of ANN (8hn) and ELM (55000-SINE) models were placed beside each other so that they could be easily compared. From the table, there is a pattern where the error increased by the increment of CIP weight percentages, especially when it reached 70 wt%. where it occurred on both models, at isotropic and anisotropic distributions. Furthermore, the RMSE also increased from interpolation data (0.58 T) to extrapolation data (0.81 T). This is obvious for the isotropic case. Meanwhile, no trend can be described in the anisotropic case. Interpolation data is data placed between the training data ranges, whereas extrapolation data is data located outside of the training data ranges. In general, ANN mostly has higher accuracy except for 50 wt% for isotropic for both 0.58 T and 0.81 T in which ELM can maintain lower accuracy for these data.

On the other hand, the RMSE value for the Ts3 data set of the ELM prediction model, mentioned in Table 7, is lower than the ANN model, generally for both moduli. Figure 6 depicts the results, where ELM shows better performance practically for storage modulus and isotropic distribution (Fig. 6a).

In contrast, the ANN model produces better than the ELM model for anisotropic distribution (Fig. 6c). For the loss modulus, both ELM and ANN models have similar performances. These results indicate that by increasing the magnetic field, the model error becomes more significant. Moreover, the prediction on anisotropic distribution is erratic, starting at low frequency. This had a significant effect on loss modulus. To be more specific, Table 9 presents the RMSE values for each magnetic flux density on unlearned 50 wt% for Ts3 data sets. From the table, the ELM model outperforms the ANN model in which the RMSE values decreased by increasing the magnetic flux density until it stopped at 0.81 T. Moreover, no pattern can be observed for ANN models. On the other hand, the ANN model produced better performances for anisotropic distribution compared to the ELM model practically starting at 0.18 T. In addition, trends were indicating that the RMSE values increased until 0.58 T, then it decreased to 0.81 T. This occurred on both models. To summarize, no models can represent the best one for Ts3 data sets in which the ELM model is great at predicting the isotropic case and the ANN model performed better at the anisotropic case. A detailed analysis is explained in the next section to investigate more about the effect of input variables.