Startseite Hyperelastic membrane modelling based on data-driven constitutive relations
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Hyperelastic membrane modelling based on data-driven constitutive relations

  • Victoria Yu. Salamatova EMAIL logo und Alexey A. Liogky
Veröffentlicht/Copyright: 4. Juni 2020

Abstract

We present data-driven modelling of membrane deformation by a hyperelastic nodal force method. We assume that constitutive relations are characterized by tabulated experimental data instead of the conventional phenomenological approach. As experimental data we use synthetic data from the bulge test simulation for neo-Hookean and Gent materials. The numerical study of descriptive and predictive capabilities of our approach demonstrates very good results of the data-driven modelling provided that the input tabulated data are expanded to a wider region of strain characteristics. Two methods for such expansion are suggested and numerically studied. Different loadings of hyperelastic membranes are successfully recovered by our approach.

MSC 2010: 74G15; 74K15; 74B20

Acknowledgment

The authors are grateful to Prof. Yuri V. Vassilevski for fruitful discussions and valuable comments.

  1. Funding: The work was supported by the Russian Science Foundation grant 19-71-10094.

References

[1] S. Avril and S. Evans (Eds.), Material Parameter Identification and Inverse Problems in Soft Tissue Biomechanics. Springer, 2017.10.1007/978-3-319-45071-1Suche in Google Scholar

[2] P. G. Ciarleti (Ed.), Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. Studies in Mathematics and its Applications, Vol. 20. North-Holland, Amsterdam, 1988.P. G. Ciarlet, Mathematical Elasticity, Vol. I. Volume 20 of Studies in Mathematics and its Applications, 1988.Suche in Google Scholar

[3] J. C. Criscione, Rivlin’s representation formula is ill-conceived for the determination of response functions via biaxial testing. In: The Rational Spirit in Modern Continuum Mechanics, 2004. Springer, Dordrecht, pp. 197–215.10.1007/1-4020-2308-1_15Suche in Google Scholar

[4] D. Guan, F. Ahmad, P. Theobald, S. Soe, X. Luo, and H. Gao, On the AIC-based model reduction for the general Holzapfel–Ogden myocardial constitutive law. Biomechanics Modeling in Mechanobiology18 (2019), No. 4, 1213–1232.10.1007/s10237-019-01140-6Suche in Google Scholar

[5] F. P. K. Hsu, C. Schwab, D. Rigamonti, and J. D. Humphrey, Identification of response functions from axisymmetric membrane inflation tests: implications for biomechanics. Int. J. Solids and Structures31 (1994), No. 24, 3375–3386.10.1016/0020-7683(94)90021-3Suche in Google Scholar

[6] J. D. Humphrey, Computer methods in membrane biomechanics. Comput. Methods Biomechanics and Biomedical Engrg. 1 (1998), No, 3, 171–210.10.1080/01495739808936701Suche in Google Scholar PubMed

[7] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, PA, 1995.10.1137/1.9781611970944Suche in Google Scholar

[8] D. A. Knoll and D. E. Keyes, Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Physics193 (2004), No. 2, 357–397.10.1016/j.jcp.2003.08.010Suche in Google Scholar

[9] S. K. Kyriacou, J. D. Humphrey, and C. Schwab, Finite element analysis of nonlinear orthotropic hyperelastic membranes. Comput. Mech. 18 (1996), No. 4, 269–278.10.1007/BF00364142Suche in Google Scholar

[10] M. R. Labrosse (Ed.), Cardiovascular Mechanics. CRC press, Boca Raton, 2018.10.1201/b21917Suche in Google Scholar

[11] K. Lipnikov, Yu. Vassilevski, A. Danilov et al., Advanced Numerical Instruments 3D. URL: http://sourceforge.net/projects/ani3dSuche in Google Scholar

[12] R. Long and C. Y. Hui, Axisymmetric membrane in adhesive contact with rigid substrates: Analytical solutions under large deformation. Int. J. Solids and Structures49 (2012), No. 3-4, 672–683.10.1016/j.ijsolstr.2011.11.008Suche in Google Scholar

[13] L. T. K. Nguyen and M. A. Keip, A data-driven approach to nonlinear elasticity. Computers … Structures194 (2018), 97–115.10.1016/j.compstruc.2017.07.031Suche in Google Scholar

[14] R. W. Ogden, Large deformation isotropic elasticity – on the correlation of theory and experiment for incompressible rubberlike solids. In: Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences326 (1972) No. 1567, 565–584.10.5254/1.3542910Suche in Google Scholar

[15] R. W. Ogden, G. Saccomandi, and I. Sgura, Fitting hyperelastic models to experimental data. Comput. Mechanics34 (2004), No. 6, 484–502.10.1007/s00466-004-0593-ySuche in Google Scholar

[16] R. S. Rivlin and D. W. Saunders, Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences243 (1951) No, 865, 251–288.10.1098/rsta.1951.0004Suche in Google Scholar

[17] V. Y. Salamatova, Finite element method for 3D deformation of hyperelastic materials. Diff. Equations55 (2019), No. 7, 990–999.10.1134/S0012266119070115Suche in Google Scholar

[18] V. Y. Salamatova and A. A. Liogky, Hyperelastic nodal force method for modelling nonlinear membrane deformation. Diff. Equations56 (2020), No. 7 (submitted).10.1134/S0012266120070137Suche in Google Scholar

[19] F. Schroeder, S. Polzer, M. Slažansky̔, V. Man, and P. Skácel, Predictive capabilities of various constitutive models for arterial tissue. J. Mech. Behav. Biomed. Mater. 78 (2018), 369–380.10.1016/j.jmbbm.2017.11.035Suche in Google Scholar PubMed

[20] A. R. Srinivasa, On the use of the upper triangular (or QR) decomposition for developing constitutive equations for Green-elastic materials. Int. J. Engrg. Sci. 60 (2012), 1–12.10.1016/j.ijengsci.2012.05.003Suche in Google Scholar

[21] H. Tangelder and A. Fabri, dD Spatial Searching. In: CGAL User and Reference Manual. CGAL Editorial Board, 5.0.2 edition. https://doc.cgal.org/5.0.2/Manual/packages.html#PkgSpatialSearchingDSuche in Google Scholar

[22] A. B. Tepole, H. Kabaria, K. U. Bletzinger, and E. Kuhl, Isogeometric Kirchhoff–Love shell formulations for biological membranes. Comput. Methods Appl. Mech. Engrg. 293 (2015), 328–347.10.1016/j.cma.2015.05.006Suche in Google Scholar PubMed PubMed Central

[23] Yu. V. Vassilevski, V. Y. Salamatova, and A. V. Lozovskiy, Concise formulas for strain analysis of soft biological tissues. Diff. Equations53 (2017), No. 7, 908–915.10.1134/S0012266117070072Suche in Google Scholar

Received: 2020-04-08
Accepted: 2020-04-15
Published Online: 2020-06-04
Published in Print: 2020-06-25

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 1.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/rnam-2020-0013/pdf
Button zum nach oben scrollen