Adaptive neuro-fuzzy estimation of conductive silicone rubber mechanical properties

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Abstract

Conductive silicone rubber has great advantages for tactile sensing applications. The electrical behavior of the elastomeric material is rate-dependent and exhibit hysteresis upon cyclic loading. Several constitutive models were developed for mechanical simulation of this material upon loading and unloading. One of the successful approaches to model the time-dependent behavior of elastomers is Bergstrom–Boyce model. An adaptive neuro-fuzzy inference system (ANFIS) model will be established in this study to predict the stress–strain changing of conductive silicone rubber during compression tests. Various compression tests were performed on the produced specimens. An ANFIS is used to approximate correlation between measured features of the material and to predict its unknown future behavior for stress changing. ANFIS has unlimited approximation power to match any nonlinear functions well and to predict a chaotic time series.

Highlights

► Adaptive neuro-fuzzy estimation of conductive silicone rubber properties. ► Adaptive neuro-fuzzy network to approximate correlation between measured features of the material. ► Adaptive neuro-fuzzy network to predict the conductive silicone rubber future behavior for stress changing. ► A new constitutive model of the conductive silicone rubber. ► A new type of stress prediction model based on artificial neural network.

Introduction

The prediction of the mechanical behavior of elastomeric materials has been an active research area for many years. There have been numerous experimental studies addressing different characteristics of the elastomeric response. Most of the proposed models, however, capture only a subset of the experimentally observed phenomena and are mainly phenomenologically based. One further complication of this general class of models is that they normally contain material dependent functional that can be hard to experimentally determine for a new material.

The mechanical property of elastomers is highly nonlinear and it is essential to implement mathematical constitutive models capable of accurate representation of the stress–strain responses of the materials. One of the successful approaches to model the time-dependent behavior of elastomers is Bergstrom–Boyce (BB) model. Bergstrom and Boyce studied in Bergstrom and Boyce (1998) the large time-dependent behavior of elastomeric materials. A detailed experimental investigation probing the material response of carbon black filled chloroprene rubber subjected to different time-dependent strain histories was presented in that article. Based on the experimental data a constitutive model was developed. The foundation of the model is that the mechanical behavior can be decomposed into two parts: an equilibrium network corresponding to the state that is approached in long time stress relaxation tests; and a second network capturing the nonlinear rate-dependent deviation from the equilibrium state. In Bergstrom and Boyce (1999), Bergstrom and Boyce investigated the influence of filler particles on the equilibrium stress–strain response. A direct comparison with experimental data suggests that the new model generated superior predictions, particularly for large strain deformations. In Bergstrom and Boyce (2001) they discussed a new constitutive model capable of capturing the experimentally observed behavior under different general multiaxial loading conditions. The proposed model was a modification of the previous BB models and is shown to capture the rate dependence and cyclic loading in both elastomers and soft biological tissues.

In the design of engineering structures with elastomeric components, for which repetitive numerical simulation such as finite element analysis (FEA) is often required, it is always desired to implement a sophisticated mathematical constitutive model capable of accurate representation of the load-deformation responses of the elastomeric materials incorporated. However, characterizing a particular elastomer compound for use with FEA can be quite challenging because of the complexity of elastomers behavior and the large deformation that an elastomeric compound generally undertakes.

At present, however, there is an effective way to circumvent this obstacle. Instead of the tantalizing search for analytical solutions, modern techniques of data mining by soft computing methods can be used, enabling extraction of the knowledge implicitly stored in large databases. To find complex relations in the data that even experts may miss, various data mining models have been devised. Among them, neural networks, as general nonparametric and robust models designed to find nonlinear relationships between specified input/output data samples, have progressively gained popularity.

Artificial neural network (ANN) technique has emerged as a powerful tool which can be used for many scientific and/or engineering applications such as process control and system modeling. ANNs are inspired by the nervous biological architecture systems consisting of relatively simple systems working in parallel to facilitate quick decisions.

ANNs are flexible modeling tools with capabilities of learning the mathematical mapping between input and output variables of nonlinear systems. One of the most powerful types of neural network system is adaptive neuro fuzzy inference system (ANFIS). ANFIS shows very good learning and prediction capabilities, which makes it an efficient tool to deal with encountered uncertainties in any system. Fuzzy inference system (FIS) is the main core of ANFIS. FIS is based on expertise expressed in terms of ‘IF–THEN’ rules and can thus be employed to predict the behavior of many uncertain systems. FIS advantage is that it does not require knowledge of the underlying physical process as a precondition for its application. Thus the ANFIS integrates the fuzzy inference system with a back-propagation learning algorithm of neural network.

So far, there are many studies of the application of ANFIS for prediction and real-time identification of many different systems (Ekici and Aksoy, 2011, Inal, 2008, Khajeh et al., 2009). In Lo and Lin (2005) the effectiveness of predicting non-uniformity of the wafer surface with ANFIS was investigated under conditions of the three process parameters. A developed finite element method was used to obtain the training data and testing data about non-uniformity on wafer surface. The combination of FEM and ANFIS methods can be used effectively in the prediction of non-uniformity of the wafer surface. A neuro-fuzzy model was utilized to predict the hardness and porosity of shape memory alloy in Khalifehzadeh, Forouzan, Arami, and Sadrnezhaad (2007) where an ANFIS model predicted the variations of the porosity content and hardness of the synthesized shape memory alloy. Article (Vairappan, Tamura, Gao, & Tang, 2009) presented an improved ANFIS with self-feedback for the applications of time-series prediction. The self-feedback connections are introduced to overcome the limitation of the static nature of ANFIS. An ANFIS model is applied to predict the flow stress in hot deformation process of Ti6000 alloy in Han, Zeng, Zhao, Qi, and Sun (2011). In Karaagac, Inal and Deniz (in press) optimum cure time of the rubber compounds are predicted using ANFIS model. Various principles of the neural network approach for predicting certain properties of polymer composite materials are discussed in Zhang and Friedrich (2003).

An ANFIS model will be established in this study to predict the stress–strain changing of conductive silicone rubber during compression tests. The ANFIS predicted results will be compared to the BB prediction results. The experimental results were obtained from many compression strain tests to obtain accurate representation of the material behavior. These compression tests of conductive silicone rubber has conducted at different strain rates and strains to characterize the voltage changing behavior and understand the deformation mechanisms during deformation process. The constructed ANFIS model exhibits a high performance for predicting stress–strain changing of the conductive silicone rubber during compression tests. The results obtained in this work indicate that ANFIS is more effective method for prediction of stress–strain changing in conductive silicone rubber and have better accuracy and simplicity compared with BB method.

Section snippets

Bergstrom–Boyce model

The Bergstrom–Boyce (BB) model (Bergstrom and Boyce, 1998, Bergstrom and Boyce, 1999, Bergstrom and Boyce, 2001) is an advanced model for predicting the time-dependent, large–strain behaviour of elastomer-like materials. The model has been shown to be accurate for both traditional engineering rubbers, and soft biomaterials.

The BB-model requires the following material parameters:

  • mu: shear modulus of network A

  • λL: locking stretch

  • κ: bulk modulus

  • s: relative stiffness of network B

  • ξ: strain adjustment

Results and discussion

The ANFIS training curves for two types of MFs combinations, 2–2–2–2 and 4–4–4–4, are shown in Fig. 5. This is two extreme cases where for 2–2–2–2 type the training error has maximal value and for 4–4–4–4 type the error was minimal value. It can be noticed that training procedure was lasted for 100 epochs.

Fig. 6 shows the ANFIS prediction errors for all MFs combinations before and after training procedure. Maximal decreasing of prediction error was occurred for the last MFs combination 4–4–4–4.

Conclusion

In this paper a new constitutive model was presented that allows for predictions of the large strain time-dependent behavior of elastomeric materials. Based on the experimental data this constitutive model was developed. The mechanical behavior of elastomeric materials is known to be rate-dependent and to exhibit hysteresis upon cyclic loading. Elastomeric materials can be explained by continuum based mechanical models. These models present a very complicated mechanical behavior that exceed the

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