On Ellipticity of Hyperelastic Models Based on Experimental Data

Cover Page

Cite item

Full Text

Abstract

The condition of ellipticity of the equilibrium equation plays an important role for correct description of mechanical behavior of materials and is a necessary condition for new defining relationships. Earlier, new deformation measures were proposed to vanish correlations between the terms, that dramatically simplifies restoration of defining relationships from experimental data. One of these new deformation measures is based on the QR-expansion of deformation gradient. In this paper, we study the strong ellipticity condition for hyperelastic material using the QR-expansion of deformation gradient.

About the authors

V Yu Salamatova

Moscow Institute of Physics and Technology (State University) ; Sechenov First Moscow State Medical University

Email: salamatova@gmail.com
9 Institutskiy per., 141701 Moscow Region, Russia; 2 build. 4 Bol’shaya Pirogovskaya st., 119991 Moscow, Russia

Yu V Vasilevskii

Institute of Numerical Mathematics of the Russian Academy of Sciences ; Moscow Institute of Physics and Technology (State University) ; Sechenov First Moscow State Medical University

Email: yuri.vassilevski@gmail.com
8 Gubkina st., 119333 Moscow, Russia; 9 Institutskiy per., 141701 Moscow Region, Russia;2 build. 4 Bol’shaya Pirogovskaya st., 119991 Moscow, Russia

References

  1. Лурье А. И. Нелинейная теория упругости. - М.: Наука, 1980.
  2. Сьярле Ф. Математическая теория упругости. - М.: Мир, 1992.
  3. Тыртышников Е. Е. Матричный анализ и линейная алгебра. - М.: Физматлит, 2007.
  4. Трусделл К. Первоначальный курс рациональной механики сплошных сред. - М.: Мир, 1975.
  5. Criscione J. C. Rivlin’s representation formula is ill-conceived for the determination of response functions via biaxial testing// В сб.: «The Rational Spirit in Modern Continuum Mechanics». - Dordrecht: Springer, 2004. - С. 197-215.
  6. Criscione J. C., Humphrey J. D., Douglas A. S., Hunter W. C. An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity// J. Mech. Phys. Solids. - 2000. - 48, № 12. - С. 2445-2465.
  7. Criscione J. C., McCulloch A. D., Hunter W. C. Constitutive framework optimized for myocardium and other high-strain, laminar materials with one fiber family// J. Mech. Phys. Solids. - 2002. - 50, № 8. - С. 1681-1702.
  8. Criscione J. C., Sacks M. S., Hunter W. C. Experimentally tractable, pseudo-elastic constitutive law for biomembranes: I. Theory// J. Biomech. Eng. - 2003. - 125, № 1. - С. 94-99.
  9. Criscione J. C., Sacks M. S., Hunter W. C. Experimentally tractable, pseudo-elastic constitutive law for biomembranes: II. Application// J. Biomech. Eng. - 2003. - 125, № 1. - С. 100-105.
  10. Freed A. D., Srinivasa A. R. Logarithmic strain and its material derivative for a QR decomposition of the deformation gradient// Acta Mech. - 2015. - 226, № 8. - С. 2645-2670.
  11. Hayes M. Static implications of the strong-ellipticity condition// Arch. Ration. Mech. Anal. - 1969. - 33, № 3. - С. 181-191.
  12. Holzapfel G. A. Biomechanics of soft tissue// Handb. Mater. Behav. Models. - 2001. - 3, № 1. - С. 1049- 1063.
  13. Humphrey J. D. Computer methods in membrane biomechanics// Comput. Methods Biomech. Biomed. Eng. - 1998. - 1, № 3. - С. 171-210.
  14. Knowles J. K., Sternberg E. On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics// J. Elasticity. - 1978. - 8, № 4. - С. 329-379.
  15. Kotiya A. A. Mechanical characterisation and structural analysis of normal and remodeled cardiovascular soft tissue// Doctoral diss. - Texas A&M University, 2008.
  16. Martins P., Natal Jorge R. M., Ferreira A. J. M. A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues// Strain. - 2006. - 42, № 3. - С. 135-147.
  17. Merodio J., Ogden R. W. Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation// Int. J. Solids Structures. - 2003. - 40, № 18. - С. 4707-4727.
  18. Rivlin R. S., Saunders D. W. Large elastic deformations of isotropic materials. VII. Experiments on the deformation of rubber// Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. - 1951. - 243, № 865. - С. 251-288.
  19. Sendova T., Walton J. R. On strong ellipticity for isotropic hyperelastic materials based upon logarithmic strain// Int. J. Nonlinear Mech. - 2005. - 40, № 2. - С. 195-212.
  20. Srinivasa A. R. On the use of the upper triangular (or QR) decomposition for developing constitutive equations for Green-elastic materials// Internat. J. Engrg. Sci. - 2012. - 60.- С. 1-12.

Copyright (c) 2019 Contemporary Mathematics. Fundamental Directions

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies