MECHANICS OF A HYPERELASTIC INFLATED TUBE WITH EMPHASIS ON BIOLOGICAL SOFT TISSUES

Yıl 2022, , 1023 - 1042, 31.12.2022

Ömer Faruk Büyükkaya , Ali Fethi Okyar

https://doi.org/10.17482/uumfd.1066879

Öz

Inflated hollow cylinder is an important problem encountered in a variety of fields in engineering. In industry, tires and fire hoses are pressurized from inside. In biomechanics, veins, arteries and intervertebral discs can also be modeled using the inflated cylinder problem. The soft ground substance of biological tissues in question are incompressible and portray large non-linear deformations under loading. Classical theories of linear elasticity are incapable of modeling such behavior. Instead, continuum mechanics based large displacement formulation and hyperelasticity are necessary to understand the deformation and mechanics of soft materials. In this study, inflation of a cylinder composed of an isotropic neo-Hookean type of material is analyzed in plane strain and generalized plane strain conditions. First, an analytical solution is established using a continuum mechanical framework. Second, the finite element method is employed to model the same problem. The numerical approach is verified by using a mesh sensitivity analysis and validated by using analytical solution. Therefore, the proposed analytical benchmark can quantify the accuracy of any commercial finite element software solution of neo-Hookean tube inflation. As a side result, it was also revealed that the hydrostatic pressure in the tube is more than six times the inflation pressure.

Anahtar Kelimeler

Hyperelasticity, Continuum Mechanics, Finite Element Method, Annular Cylinder, Biological Tissue

Teşekkür

We would like to thank Mr. Cevat Volkan Karadağ (M.Sc., ME) for reading through the entire manuscript and providing valuable comments and corrections.

Biyolojik Dokulara Vurgu ile Şişirilmiş Hiperelastik bir Tüpün Mekaniği

Öz

İçi boş bir silindirin şişirilmesi, mühendisliğin çeşitli alanlarında karşılaşılan önemli bir problemdir. Endüstride, araba lastikleri ve itfaiye hortumları içten basınçlandırılırlar. Biyomekanikte, damarlar ve intervertebral diskler de şişirilmiş silindir problemi kullanılarak modellenebilir. Bahsedilen biyolojik dokuların yumuşak yapısı sıkıştırılamazlar ve yük altında büyük, doğrusal olmayan deformasyon sergilerler. Klasik doğrusal elastik teoriler böyle bir davranışı modellemede yetersiz kalır. Bunun yerine, yumuşak malzemelerin deformasyonu ve mekaniğini anlamak için sürekli ortamlar mekaniği temelli yüksek yer değiştirme formülasyonları gereklidir. Bu çalışmada, izotropik neo-Hookean malzemeden meydana gelen hiperelastik bir tüp düzlemsel gerinim ve genelleştirilmiş düzlemsel gerinim koşulları altında analiz edilmiştir. İlk olarak, her iki koşul için sürekli ortamlar mekaniği çerçevesinde bir analitik çözüm inşa edilmiştir. Sonrasında, aynı problem için sonlu elemanlar yöntemi işleme konulmuştur. Nümerik modelin doğrulaması ağ hassasiyet analizi kullanılarak sağlanmıştır. Analitik çözüm ile doğrulanmıştır. Buradan, öne sürülen analitik çözüm ile herhangi bir ticari sonlu elemanlar yazılımı ile elde edilmiş şişirilen neo-Hookean bir tüpün davranış hassasiyeti belirlenebilecektir. Analitik çözümle gelen ilginç bir yan sonuç, tüpün içerisindeki hidrostatik basıncın şişirme basıncından altı kat fazla çıkması olmuştur.

Anahtar Kelimeler

Hiperelastisite, Sürekli Ortamlar Mekaniği, Sonlu Elemanlar Yöntemi, Anüler Silindir, Biyolojik Doku

Kaynakça

• 1. Arruda, E. M., & Boyce, M. C. (1993). A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 41(2), 389–412. https://doi.org/10.1016/0022-5096(93)90013-6
• 2. Behnke, R., Dal, H., & Kaliske, M. (2011). An extended tube model for thermo-viscoelasticity of rubberlike materials: Parameter identification and examples. PAMM, 11, 353–354. https://doi.org/10.1002/pamm.201110168
• 3. Breslavsky, I. D., Amabili, M., Legrand, M., & Alijani, F. (2016). Axisymmetric deformations of circular rings made of linear and Neo-Hookean materials under internal and external pressure: A benchmark for finite element codes. International Journal of Non-Linear Mechanics, 84, 39–45. https://doi.org/10.1016/j.ijnonlinmec.2016.04.011
• 4. Cook, R. D., & Young, W. C. (1999). Pressurized cylinders and spinning disks. In Advanced Mechanics of Materials. Prentice Hall.
• 5. Dogan, F., & Celebi, M. S. (2016). Quasi-non-linear deformation modeling of a human liver based on artificial and experimental data. The International Journal of Medical Robotics and Computer Assisted Surgery, 12(3), 410–420. https://doi.org/https://doi.org/10.1002/rcs.1704
• 6. Dogan, F., & Serdar Celebi, M. (2010). Real-time deformation simulation of non-linear viscoelastic soft tissues. Simulation, 87(3), 179–187. https://doi.org/10.1177/0037549710364532
• 7. Erdem, A., Usal, M., & Usal, M. (2005). Keyfi fiber takviyeli viskoelastik piezoelektrik bir cismin elektro-termomekanik davranışı için matematiksel bir model. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi, 20(3), 305–319.
• 8. Fung, Y. C. (1981). Bio-viscoelastic solids. In Biomechanics: Mechanical Properties of Living Tissues (2nd ed., pp. 196–260). New York, NY: Springer New York. https://doi.org/10.1007/978-1-4757-1752- 5_7
• 9. Gasser, T. C., Ogden, R. W., & Holzapfel, G. A. (2006). Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. Journal of the Royal Society, Interface, 3(6), 15–35. https://doi.org/10.1098/rsif.2005.0073
• 10. Goodno, B. J., & Gere, J. M. (2016). Applications of plane stress: pressure vessels. In Mechanics of Materials (9th ed.). Cengage Learning.
• 11. Gültekin, O., Dal, H., & Holzapfel, G. A. (2019). On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials. Computational Mechanics, 63(3), 443–453. https://doi.org/10.1007/s00466-018-1602-9
• 12. Mooney, M. (1940). A theory of large elastic deformation. Journal of Applied Physics, 11(9), 582–592. https://doi.org/10.1063/1.1712836
• 13. MSC Software. (2019). MARC/Mentat 2019.0.0. Irvine, CA, USA.
• 14. Ogden, R. W., & Hill, R. (1972). Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 326(1567), 565–584. https://doi.org/10.1098/rspa.1972.0026
• 15. Petekkaya, A., & Tönük, E. (2013). Yumuşak biyolojik dokuların düzlemsel eşyönsüz mekanik davranışının bireye ve noktaya özel belirlenmesi için elipsoid uçlarla yerinde canli (in vivo) indentör deneyleri. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi, 26(1), 63–72.
• 16. Rajagopal, K. R., & Saravanan, U. (2012). Extension, inflation and circumferential shearing of an annular cylinder for a class of compressible elastic bodies. Mathematics and Mechanics of Solids, 17(5), 473–499. https://doi.org/10.1177/1081286511423125
• 17. Rivlin, R. S. (1948). Large elastic deformations of isotropic materials. III. Some simple problems in cyclindrical polar co-ordinates. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 240(823), 509–525. https://doi.org/10.1098/rsta.1948.0004
• 18. Rivlin, R. S., & Rideal, E. K. (1948). Large elastic deformations of isotropic materials IV. further developments of the general theory. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 241(835), 379–397. https://doi.org/10.1098/rsta.1948.0024
• 19. Selvadurai, A. P. S., & Suvorov, A. P. (2017). On the inflation of poro-hyperelastic annuli. Journal of the Mechanics and Physics of Solids, 107, 229–252. https://doi.org/https://doi.org/10.1016/j.jmps.2017.06.007
• 20. Treloar, L. R. G. (1944). Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc., 40(0), 59–70. https://doi.org/10.1039/TF9444000059
• 21. Vogel, A., Rakotomanana, L., & Pioletti, D. P. (2017). Chapter 3 - viscohyperelastic strain energy function. In Y. Payan & J. Ohayon (Eds.), Biomechanics of Living Organs (Vol. 1, pp. 59–78). Oxford: Academic Press. https://doi.org/https://doi.org/10.1016/B978-0-12-804009-6.00003-1
• 22. Wineman, A. (2005). Some results for generalized neo-Hookean elastic materials. International Journal of Non-Linear Mechanics, 40(2–3), 271–279. https://doi.org/10.1016/j.ijnonlinmec.2004.05.007
• 23. Zhu, Y., Luo, X., Ogden, R., Zhu, Y., Luo, X., & Ogden, R. (2010). Nonlinear axisymmetric deformations of an elastic tube under external pressure. European Journal of Mechanics-A/Solids, 29(2), 216–229. https://doi.org/10.1016/j.euromechsol.2009.10.004ï