Mechanics of Time-Dependent Materials

, Volume 21, Issue 3, pp 331–349 | Cite as

Determination of relaxation modulus of time-dependent materials using neural networks

  • Alexandra AulovaEmail author
  • Edvard Govekar
  • Igor Emri


Health monitoring systems for plastic based structures require the capability of real time tracking of changes in response to the time-dependent behavior of polymer based structures. The paper proposes artificial neural networks as a tool of solving inverse problem appearing within time-dependent material characterization, since the conventional methods are computationally demanding and cannot operate in the real time mode. Abilities of a Multilayer Perceptron (MLP) and a Radial Basis Function Neural Network (RBFN) to solve ill-posed inverse problems on an example of determination of a time-dependent relaxation modulus curve segment from constant strain rate tensile test data are investigated. The required modeling data composed of strain rate, tensile and related relaxation modulus were generated using existing closed-form solution. Several neural networks topologies were tested with respect to the structure of input data, and their performance was compared to an exponential fitting technique. Selected optimal topologies of MLP and RBFN were tested for generalization and robustness on noisy data; performance of all the modeling methods with respect to the number of data points in the input vector was analyzed as well. It was shown that MLP and RBFN are capable of solving inverse problems related to the determination of a time dependent relaxation modulus curve segment. Particular topologies demonstrate good generalization and robustness capabilities, where the topology of RBFN with data provided in parallel proved to be superior compared to other methods.


Relaxation modulus Inverse problem Neural network Multilayer perceptron Radial basis function neural network Structural health monitoring 



Authors would like to thank Slovenian Research Agency for financial support in the frame of programs P2-0264 and P2-0241.


  1. Adler, A., Guardo, R.: A Neural Network Image Reconstruction Technique for Electrical Impedance Tomography. IEEE Trans. Med. Imaging 13(4), 594–600 (1994) CrossRefGoogle Scholar
  2. Baddari, K., et al.: Acoustic impedance inversion by feedback artificial neural network. J. Pet. Sci. Eng. 71(3–4), 106–111 (2010) CrossRefGoogle Scholar
  3. Czél, B., Woodbury, K.A., Gróf, G.: Inverse identification of temperature-dependent volumetric heat capacity by neural networks. Int. J. Thermophys. 34(2), 284–305 (2013) CrossRefGoogle Scholar
  4. Demuth, H., Beale, M., Hagan, M.: Neural Network ToolboxTM 6 User’s Guide (2009) Google Scholar
  5. Elshafiey, I., Udpa, L., Udpa, S.S.: Application of neural networks to inverse problems in electromagnetics. IEEE Trans. Magn. 30(5), 0 (1994) Google Scholar
  6. Elshafiey, I., Udpa, L., Udpa, S.S.: Solution of inverse problems in electromagnetics using Hopfield neural networks. IEEE Trans. Magn. 31(1), 852–861 (1995) CrossRefGoogle Scholar
  7. Ferry, J.D.: Viscoelastic properties of polymers, 650 (1980) Google Scholar
  8. Hagan, M.T., Demuth, H.B., Beale, M.H.: Neural network design (1996) Google Scholar
  9. Haj-Ali, R., et al.: Nonlinear constitutive models from nanoindentation tests using artificial neural networks. Int. J. Plast. 24(3), 371–396 (2008) CrossRefzbMATHGoogle Scholar
  10. Haykin, S.: Neural networks: a comprehensive foundation (1999) Google Scholar
  11. ISO 527-1: Plastics—determination of tensile properties—part 1: general principles (2012) Google Scholar
  12. Jung, S., Ghaboussi, J.: Neural network constitutive model for rate-dependent materials. Comput. Struct. 84(15–16), 955–963 (2006) CrossRefGoogle Scholar
  13. Knauss, W.G., Zhao, J.: Improved relaxation time coverage in ramp-strain histories. Mech. Time-Depend. Mater. 11(3–4), 199–216 (2007) CrossRefGoogle Scholar
  14. Lampinen, J., Vehtari, A.: Using Bayesian Neural Network to Solve the Inverse Problem in Electrical Impedance Tomography. In: Proceedings of the Scandinavian Conference on Image Analysis, p. 1 (1999) Google Scholar
  15. Li, M.M., et al.: Intelligent methods for solving inverse problems of backscattering spectra with noise: a comparison between neural networks and simulated annealing. Neural Comput. Appl. 18(5), 423–430 (2009) CrossRefGoogle Scholar
  16. Li, M.M., et al.: RBF neural networks for solving the inverse problem of backscattering spectra. Neural Comput. Appl. 17(4), 391–397 (2008) CrossRefGoogle Scholar
  17. MacKay, D.J.C.: Bayesian Interpolation. Neural Comput. 4(3), 415–447 (1992) CrossRefzbMATHGoogle Scholar
  18. Nguyen, D., Widrow, B.: Improving the Learning Speed of 2-Layer Neural Networks by choosing Initial Values of the Adaptive Weights. In: IJCNN International Joint Conference on Neural Networks, vol. 3, pp. 21–26 (1990) Google Scholar
  19. Samarskii, A.A., Vabishchevich, P.N., De, W.: Numerical Methods for Solving Inverse Problems of Mathematical Physics. Walter de Gruyter, Berlin (2009) Google Scholar
  20. Sammany, M., Pelican, E., Harak, T.A.: Hybrid neuro-genetic based method for solving ill-posed inverse problem occurring in synthesis of electromagnetic fields. Computing 91(4), 353–364 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  21. Sammut, C., Webb, G.: Topology of neural Network. In: Encyclopedia of Machine Learning, pp. 988–989 (2011). Chapter 19 Google Scholar
  22. Saprunov, I., Gergesova, M., Emri, I.: Prediction of viscoelastic material functions from constant stress- or strain-rate experiments. Mech. Time-Depend. Mater. 18(2), 349–372 (2014) CrossRefGoogle Scholar
  23. Sjoberg, J., Ljung, L.: Overtraining, Regularization and Searching for Minimum in Neural Networks, 1–16 (1992) Google Scholar
  24. Tikhonov, A., Arsenin, V.: Solutions of Inverse Problems. John Wiley & Sons, Washington (1977) zbMATHGoogle Scholar
  25. Tscharnuter, D., Jerabek, M., Major, Z., Lang, R.W.: On the determination of the relaxation modulus of PP compounds from arbitrary strain histories. Mech. Time-Depend. Mater. 15(1), 1–14 (2011) CrossRefGoogle Scholar
  26. Werbos, P.: The Roots of Backpropagation: From Ordered Derivatives to Neural Networks and Political Forecasting. Wiley, New York (1994) Google Scholar
  27. Xiao, W., et al.: Real-time identification of optimal operating points in photovoltaic power systems. IEEE Trans. Ind. Electron. 53(4), 1017–1026 (2006) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Center for Experimental MechanicsFaculty of Mechanical Engineering, University of LjubljanaLjubljanaSlovenia
  2. 2.Laboratory of SynergeticsFaculty of Mechanical Engineering, University of LjubljanaLjubljanaSlovenia

Personalised recommendations