Modification of exponential based hyperelastic strain energy to consider free stress initial configuration and Constitutive modeling

Document Type: Research Paper


Department of Materials Science and Engineering, K. N. Toosi University of Technology, Tehran, Iran.


In this research, the exponential stretched based hyperelastic strain energy was modified to provide the unstressed initial configuration. To this end, as the first step, the model was calibrated by the experimental data to find the best material parameters. The fitting results indicated material stability in large deformations and basic loading modes. In the second step, the initial pseudo stress value (ISV) was eliminated from the hyperelastic strain energy using a function of the determinant of the deformation gradient. The modified and unmodified models were implemented in ABAQUS/VUMAT user subroutine and the deformation behavior of the natural rubber and the thermoplastic elastomer was predicted. The results obtained from the modified model represented a better agreement with the experimental data, in comparison to those gained by the unmodified model. In order to present the significance of the unstressed initial configuration in engineering applications, the stenting phenomenon in the atherosclerosis human artery was investigated. It was revealed that a uniform stress distribution could be achieved in the artery using the modified model, thereby reducing the possibility of tearing and restenosis.


Main Subjects

[1]           A. F. Bower, 2009, Applied mechanics of solids, CRC press.

[2]           M. Sasso, G. Palmieri, G. Chiappini, D. Amodio, Characterization of hyperelastic rubber-like materials by biaxial and uniaxial stretching tests based on optical methods, Polymer Testing, Vol. 27, No. 8, pp. 995-1004, 2008.

[3]           D. T. Casem, A. K. Dwivedi, R. A. Mrozek, J. L. Lenhart, Compression response of a thermoplastic elastomer gel tissue surrogate over a range of strain-rates, International Journal of Solids and Structures, Vol. 51, No. 11, pp. 2037-2046, 2014.

[4]           L. Li, S. Ruan, L. Zeng, Mechanical properties and constitutive equations of concrete containing a low volume of tire rubber particles, Construction and Building Materials, Vol. 70, pp. 291-308, 2014.

[5]           X. Li, T. Bai, Z. Li, L. Liu, Influence of the temperature on the hyper-elastic mechanical behavior of carbon black filled natural rubbers, Mechanics of Materials, Vol. 95, pp. 136-145, 2016.

[6]           O. A. Shergold, N. A. Fleck, D. Radford, The uniaxial stress versus strain response of pig skin and silicone rubber at low and high strain rates, International Journal of Impact Engineering, Vol. 32, No. 9, pp. 1384-1402, 2006.

[7]           T. Beda, Y. Chevalier, Hybrid continuum model for large elastic deformation of rubber, Journal of applied physics, Vol. 94, No. 4, pp. 2701-2706, 2003.

[8]           T. Beda, An approach for hyperelastic model-building and parameters estimation a review of constitutive models, European Polymer Journal, Vol. 50, pp. 97-108, 2014.

[9]           R. Ogden, G. Saccomandi, I. Sgura, Fitting hyperelastic models to experimental data, Computational Mechanics, Vol. 34, No. 6, pp. 484-502, 2004.

[10]         K. Terada, J. Kato, N. Hirayama, T. Inugai, K. Yamamoto, A method of two-scale analysis with micro-macro decoupling scheme: application to hyperelastic composite materials, Computational Mechanics, Vol. 52, No. 5, pp. 1199-1219, 2013.

[11]         A. Gendy, A. Saleeb, Nonlinear material parameter estimation for characterizing hyper elastic large strain models, Computational mechanics, Vol. 25, No. 1, pp. 66-77, 2000.

[12]         A. Drozdov, Constitutive equations in finite elasticity of rubbers, International Journal of Solids and Structures, Vol. 44, No. 1, pp. 272-297, 2007.

[13]         H. Darijani, R. Naghdabadi, M. Kargarnovin, Hyperelastic materials modelling using a strain measure consistent with the strain energy postulates, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 224, No. 3, pp. 591-602, 2010.

[14]         H. Darijani, R. Naghdabadi, Hyperelastic materials behavior modeling using consistent strain energy density functions, Acta mechanica, Vol. 213, No. 3, pp. 235-254, 2010.

[15]         M. Mansouri, H. Darijani, Constitutive modeling of isotropic hyperelastic materials in an exponential framework using a self-contained approach, International Journal of Solids and Structures, Vol. 51, No. 25, pp. 4316-4326, 2014.

[16]         B. Fereidoonnezhad, R. Naghdabadi, J. Arghavani, A hyperelastic constitutive model for fiber-reinforced rubber-like materials, International Journal of Engineering Science, Vol. 71, pp. 36-44, 2013.

[17]         J. H. Smith, J. J. García, Constitutive modeling of brain tissue using Ogden-type strain energy functions, in Proceeding of, 2140-2147.

[18]         M. Hosseinzadeh, M. Ghoreishi, K. Narooei, Investigation of hyperelastic models for nonlinear elastic behavior of demineralized and deproteinized bovine cortical femur bone, Journal of the mechanical behavior of biomedical materials, Vol. 59, pp. 393-403, 2016.

[19]         N. Elyasi, K. K. Taheri, K. Narooei, A. K. Taheri, A study of hyperelastic models for predicting the mechanical behavior of extensor apparatus, Biomechanics and modeling in mechanobiology, Vol. 16, No. 3, pp. 1077-1093, 2017.

[20]         W.-Q. Wang, D.-K. Liang, D.-Z. Yang, M. Qi, Analysis of the transient expansion behavior and design optimization of coronary stents by finite element method, Journal of Biomechanics, Vol. 39, No. 1, pp. 21-32, 2006.

[21]         H. Zahedmanesh, C. Lally, Determination of the influence of stent strut thickness using the finite element method: implications for vascular injury and in-stent restenosis, Medical & biological engineering & computing, Vol. 47, No. 4, pp. 385, 2009.

[22]         F. AURICCHIO∗, M. Di Loreto, E. Sacco, Finite-element analysis of a stenotic artery revascularization through a stent insertion, Computer Methods in Biomechanics and Biomedical Engineering, Vol. 4, No. 3, pp. 249-263, 2001.

[23]         P. Prendergast, C. Lally, S. Daly, A. Reid, T. Lee, D. Quinn, F. Dolan, Analysis of prolapse in cardiovascular stents: a constitutive equation for vascular tissue and finite-element modelling, Journal of biomechanical engineering, Vol. 125, No. 5, pp. 692-699, 2003.

[24]         A. Karimi, M. Navidbakhsh, M. Alizadeh, A. Shojaei, A comparative study on the mechanical properties of the umbilical vein and umbilical artery under uniaxial loading, Artery Research, Vol. 8, No. 2, pp. 51-56, 2014.

[25]         G. A. Holzapfel, G. Sommer, C. T. Gasser, P. Regitnig, Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling, American Journal of Physiology-Heart and Circulatory Physiology, Vol. 289, No. 5, pp. H2048-H2058, 2005.

[26]         M. Imani, A. M. Goudarzi, D. D. Ganji, A. L. Aghili, The comprehensive finite element model for stenting: the influence of stent design on the outcome after coronary stent placement, Journal of Theoretical and Applied Mechanics, Vol. 51, No. 3, pp. 639-648, 2013.

[27]         N. Eshghi, M. Hojjati, M. Imani, A. Goudarzi, Finite element analysis of mechanical behaviors of coronary stent, Procedia Engineering, Vol. 10, pp. 3056-3061, 2011.

[28]         R. S. Rivlin, D. Saunders, Large elastic deformations of isotropic materials. VII. Experiments on the deformation of rubber, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 243, No. 865, pp. 251-288, 1951.

[29]         R. W. Ogden, 1997, Non-linear elastic deformations, Courier Corporation,

[30]         W. M. Lai, D. H. Rubin, E. Krempl, D. Rubin, 2009, Introduction to continuum mechanics, Butterworth-Heinemann,

[31]         T. Belytschko, W. K. Liu, B. Moran, K. Elkhodary, 2013, Nonlinear finite elements for continua and structures, John wiley & sons,

[32]         R. Ogden, Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids, in Proceeding of, The Royal Society, pp. 565-584.

[33]         A. F. M. Arif, T. Pervez, M. P. Mughal, Performance of a finite element procedure for hyperelastic–viscoplastic large deformation problems, Finite elements in analysis and design, Vol. 34, No. 1, pp. 89-112, 2000.

[34]         F. Etave, G. Finet, M. Boivin, J.-C. Boyer, G. Rioufol, G. Thollet, Mechanical properties of coronary stents determined by using finite element analysis, Journal of Biomechanics, Vol. 34, No. 8, pp. 1065-1075, 2001.

[35]         F. Migliavacca, L. Petrini, V. Montanari, I. Quagliana, F. Auricchio, G. Dubini, A predictive study of the mechanical behaviour of coronary stents by computer modelling, Medical engineering & physics, Vol. 27, No. 1, pp. 13-18, 2005.

[36]         L. Gu, S. Zhao, A. K. Muttyam, J. M. Hammel, The relation between the arterial stress and restenosis rate after coronary stenting, Journal of Medical Devices, Vol. 4, No. 3, pp. 031005, 2010.