Decoupled homogenization of hyperelastic composite with carbon black inclusion

Martyna Poręba-Sebastjan1, Waclaw Kuś1

1Department for Computational Mechanics and Engineering, Silesian University of Technology, Konarskiego 18A, 44-100 Gliwice, Poland


The goal of the paper is to present the application of decoupled homogenization method to the modeling of hyperelastic composite with inclusions. The method presented in the paper is illustrated by numerical analysis of a trunk door seal. The decoupled homogenization method was used to find macroscale properties of hyperelastic material. The method allows for the determination of the equivalent properties of a composite material based on its structure and the results of numerical experiments. Unlike the coupled method, the results are not transferred in every iteration between scales during computations which leads to lower calculation costs. The analyzed micro model consisted of a hyperelastic matrix and stiff inclusions in the form of spheres of carbon black material. The decoupled procedure uses evolutionary algorithm to obtain macro model material properties. The finite element method is used during analyses of micro scale models.

Cite as:

Poreba-Sebastjan, M., & Kus, W. (2020). Decoupled homogenization of hyperelastic composite with carbon black inclusion. Computer Methods in Materials Science, 20, 14-21.

Article (PDF):

Key words:

Decoupled homogenization, Evolutionary algorithm, Composite, Hyperelastic material


Burczyński, T., Kuś, W., Beluch, W., Długosz, A., Poteralski,A., Szczepanik M., 2020, Intelligent computing in optimal design, Springer-Nature.

Fish, J, 2013, Practical multiscaling, Wiley.

Fung, Y.C., 1993, Biomechanics, Mechanical properties of living tissues, Springer, 300-314.

Gorash, Y., Comlekci, T., Hamilton, R., 2015, CAE-based application for identification and verification of hyperelastic parameters, Department of Mechanical & Aerospace Engineering, University of Strathclyde, Glasgow.

Industrial rubber carbon blacks, Continental Carbon: (accessed on 09.10.2019).

Kyungwhan, Y., Kyoungah, C., Sangsig, K., 2018, Effect of carbon black addition on thermal stability and capacitive performances of supercapacitors. Scientific Reports, volume 8, Article number: 11989.

López Jiménez, F., Pellegrino, S., 2012, Constitutive modeling of fiber composites with a soft hyperelastic matrix, International Journal of Solids and Structures, 49(3-4), 635-647.

Madej, Ł., Mrozek, A., Kuś, W., Burczyński, T., Pietrzyk, M., 2008, Concurrent and upscaling methods in multi scale modelling – case studies, Computer Methods in Materials Science, 8(1), 1-15.

Michalewicz, Z., 1996, Genetic algorithms + data structures = evolutionary algorithms, Springer-Verlag, Berlin.

Nonlinear multi-scale modeling of rubber and tires with DIGIMAT, e-Xtream Engineering: (accessed on 09.10.2019).

Prevost, T.P., Balakrishnan, A., Suresh, S., Socrate, S., 2011, Biomechanics of brain tissue, Acta Biomaterialia, 7, 83-95.

Rackl, M., 2015, Curve fitting for ogden, yeoh and polynomial models. ScilabTech Conference, Paris, I(1),1-11.

Rauch, L., Szeliga, D., Bachniak, D., 2015, Identification of multi-inclusion statistically similar representative volume element for advanced high strength steels by using data farming approach. Procedia Computer Science, 51(1), 924-933.

Terada, K., Kato, J., Hirayama, N., Inugai, T., Yamamoto, K., 2013, A method of two-scale analysis with micro-macro decoupling scheme: application to hyperelastic composite materials, Computational Mechanics, 52, 1199–1219.

Thompson, B., 2010, Thermax N990 medium thermal carbon black in nitrile rubber compounds, Cancarb. Ungar, T., Gubicza, J., Ribarik, G., Pantea, C., Zerda, T.W., 2002, Microstructure of carbon blacks determined by Xray diffraction profile analysis, Carbon, 40(6), 929-937.

Xu, L., Yong, X., Ziran, L., Yuanming, X., 2012, Threedimensional numerical simulations on the hyperelastic behavior of carbon-black particle filled rubbers under moderate finite deformation, Computational Materials Science, 55, 157-165.