Applications of implicit constitutive theory for describing the elastic response of rocks and concrete

• Autorzy: Bustamante R. , Rajagopal K. R.

Identyfikatory

DOI

10.24423/aom.4154

• Języki publikacji: EN

Abstrakty

EN

An implicit constitutive relation is proposed for elastic bodies, when the gradient of the displacement is assumed to be very small, and as a result the strains are small. The resulting constitutive relation is a non-linear relationship between the linearized strain and the stress. The model is used to fit data for rock and concrete. Some boundary value problems are studied within the context of homogeneous deformations, and also a problem with inhomogeneous deformations is analyzed, namely the inflation of a circular annulus. The predictions of this new implicit constitutive relation are compared with the predictions of the constitutive equations for linearized elastic bodies.

Słowa kluczowe

EN

nonlinear elasticity; small strain; isotropic solids

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2022
• Tom: Vol. 74, nr 6
• Strony: 513--547
• Opis fizyczny: Bibliogr. 22 poz., tab., wykr.

Twórcy

autor

Bustamante R.

• Departamento de Ingeniería Mecánica, Universidad de Chile, Beauchef 851, Santiago Centro, 8370448 Santiago, Chile

autor

Rajagopal K. R.

• Department of Mechanical Engineering, University of Texas A&M, TX 77843-3123, College Station TX, USA

Bibliografia

• 1. J.F. Bell, Mechanics of Solids, Vol. 1, The Experimental Foundations of Solid Mechanics, Springer, Berlin, Heidelberg, 1973.
• 2. K.R. Rajagopal, Rethinking constitutive relations, Lecture Notes, Necas Center for Mathematical Modeling, Charles University, Prague, Czech Republic, 2022.
• 3. T. Saito, T. Furuta, J.H. Hwang, S. Kuramoto, K. Nishino, N. Suzuki, R. Chen, A. Yamada, K. Ito, Y. Seno, T. Nonaka, H. Ikehata, N. Nagasako, C. Iwamoto, Y. Ikuhara, T. Sakuma, Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism, Science, 300, 5618, 464–467, 2003.
• 4. T. Li, J.W. jr. Morris, N. Nagasako, S. Kuramoto, D.C. Chrzan, ‘Ideal’ engineering alloys, Physical Review Letters, 98, 10, 105503, 2007.
• 5. R.J. Talling, R.J. Dashwood, M. Jackson, S. Kuramoto, D. Dye, Determination of (C11-C12) in Ti–36Nb–2Ta–3Zr–0.3O (wt%) (Gum metal), Scripta Materialia, 59, 6, 669–672, 2008.
• 6. N. Sakaguch, M. Niinomi, T. Akahori, Tensile deformation behavior of Ti-Nb-Ta-Zr biomedical alloys, Materials Transactions, 45, 4, 1113–1119, 2004.
• 7. Y.L. Hao, S.J. Li, S.Y. Sun, C.Y. Zheng, Q.M. Hu, R. Yang, Super-elastic titanium alloy with unstable plastic deformation, Applied Physics Letters, 87, 9, 091906, 2005.
• 8. E. Withey, M. Jin, A. Minor, S. Kuramoto, D.C. Chrzan, J.W. Morris, The deformation of gum metal in nanoindentation, Materials Science and Engineering, A 493, 1, 26–32, 2008.
• 9. S.Q. Zhang, S.J. Li, M.T. Jia, Y.L. Hao, R. Yang, Fatigue properties of a multifunctional titanium alloy exhibiting nonlinear elastic deformation behavior, Scripta Materialia, 60, 8, 733–736, 2009.
• 10. Z. Grasley, R. El-helou, M. D’Ambrosia, D. Mokarem, C. Moen, K.R. Rajagopal, Model of infinitesimal nonlinear elastic response of conrete subjeted to uni-axial compression, Journal of Engineering Mechanics, 141, 7, 04015008, 2015.
• 11. N. Cristescu, Rock Rheology, vol. 7, Springer, Cham, 2012.
• 12. C.A. Truesdell, W. Noll, The Non-linear Field Theories of Mechanics, S.S. Antman [ed.], 3rd ed., Springer, Berlin, 2004.
• 13. K.R. Rajagopal, On implicit constitutive theories, Applied Mathematics, 48, 4, 279–319, 2003.
• 14. K.R. Rajagopal, The elasticity of elasticity, Journal of Applied Mathematics and Physics (Zeitschrift für Angewandte Mathematik und Physik), 58, 2, 309–317, 2007.
• 15. K.R. Rajagopal, A.R. Srinivasa, On the response of non-dissipative solids, Proceedings of the Royal Society, A 463, 2078, 357–367, 2007.
• 16. A.J.M. Spencer, Theory of Invariants, [in:] A.C. Eringen [ed.], Continuum Physics I, pp. 239–253, Academic Press, New York, 1984.
• 17. R. Bustamante, C. Ortiz, A bimodular nonlinear constitutive equation for rock, Applications in Engineering Science, 8, 100067, 2021.
• 18. C.A. Truesdell, R. Toupin, The Classical Field Theories, [in:] Handbuch der Physik, Vol. III/1, Springer, Berlin, 1960.
• 19. A.D. Freed, Soft Solids. A Primer to the Theoretical Mechanics of Materials, Birkhäuser, Basel, 2014.
• 20. R. Bustamante, K.R. Rajagopal, A nonlinear model for describing the mechanical behaviour of rock, Acta Mechanica, 229, 1, 251–272, 2018.
• 21. R. Bustamante, K.R. Rajagopal, Solutions of some boundary value problems for a new class of elastic bodies undergoing small strains. Comparison with the predictions of the classical theory of linearized elasticity: Part I. Problems with cylindrical symmetry, Acta Mechanica, 226, 6, 1815–1838, 2015.
• 22. Comsol Multiphysics, Version 3.4, Comsol Inc. Palo Alto, CA, 2007.

Uwagi

The work of R. Bustamante was financed by the grant No. 1210002 provided by Fondecyt (Chile). The work of K.R. Rajagopal has been supported by the National Science Foundation and the Office of Naval Research. Both authors thank Professor Z. Grasley for providing the experimental data used in this work for the modelling of concrete.