Relative Lagrangian Formulation of Finite Thermoelasticity
DOI:
https://doi.org/10.5540/tcam.2021.022.04.00609Keywords:
Relative Lagrangian formulation, thermoelastic solid, small on large deformation, successive linear approximation, boundary value problemAbstract
The motion of a body can be expressed relative to the present configuration of the body, known as the relative motion description, besides the classical Lagrangian and the Eulerian descriptions. When the time increment from the present state is small enough, the nonlinear constitutive equations can be linearized relative to the present state so that the resulting system of boundary value problems becomes linear. This formulation is based on the well-known ``small-on-large'' idea, and can be implemented for solving problems with large deformation in successive incremental manner. In fact, the proposed method is a process of repeated applications of the well-known “small deformation superposed on finite deformation” in the literature. This article presents these ideas applied to thermoelastic materials with a brief comment on the exploitation of entropy principle in general. Some applications of such a formulation in numerical simulations are briefly reviewed and a numerical result is shown.References
B. D. Coleman and W. Noll, “The thermodynamics of elastic materials with heat conduction and viscosity,” Arch. Ration. Mech. Anal., vol. 13, pp. 167–178, 1963.
I.-S. Liu, Continuum Mechanics. Berlin: Springer: Berlin Heidelberg, 2002.
C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics. Berlin: Springer, 2004.
A. E. Green, R. S. Rivlin, and R. T. Shield, “General theory of small elastic defor-mations superposed on finite deformations,” Arch. Ration. Mech. Anal., vol. 211, pp. 128–154, 1952.
I.-S. Liu, “Successive linear approximation for boundary value problems of nonlinear elasticity in relative-descriptional formulation,” Int. J. Eng Sci., vol. 49, pp. 635–645, 2011.
M. E. Gurtin, E. Fried, and L. Anand, The Mechanics and Thermodynamics of Continua. Cambridge: Cambridge University Press, 2010.
I.-S. Liu, “Entropy flux relation for viscoelastic bodies,” Journal of Elasticity, vol. 90, pp. 259–270, 2008.
I. Müller, “On the entropy inequality,” Arch. Ration. Mech. Anal., vol. 26, pp. 118–141, 1967.
I.-S. Liu, “On entropy flux of transversely isotropic elastic bodies,” Journal of Elas-ticity, vol. 96, pp. 97–104, 2009.
I.-S. Liu, “General theory of small elastic deformations superposed on finite defor-mations,” Arch. Ration. Mech. Anal., vol. 46, pp. 131–148, 1972.
I.-S. Liu, “A note on the mooney–rivlin material model,” Continuum Mech. and Thermody, vol. 24, pp. 583–590, 2011.
I.-S. Liu, R. A. Cipolatti, and M. A. Rincon, “Successive linear approximation for finite elasticity,” Computational and Applied Mathematics, vol. 29, pp. 465–478, 2010.
I.-S. Liu, R. A. Cipolatti, M. A. Rincon, and L. A. Palermo, “Successive linear approximation for large deformation–instability of salt migration,” Journal of Elasticity, vol. 114, pp. 19–39, 2014.
R. A. Cipolatti, I.-S. Liu, and M. A. Rincon, “Mathematical analysis of successive linear approximation for mooney-rivlin material model in finite elasticity,” Journal of Applied Analysis and Computation, vol. 2, pp. 363–379, 2012.
R. A. Cipolatti, I.-S. Liu, L. A. Palermo, M. A. Rincon, and R. M. S. Rosa, “A boundary value problem arising from nonlinear viscoelasticity: mathematical analysis and numerical simulations,” Applied Mathematics and Computation, vol. 335, pp. 237–247, 2018.
R. M. S. Rosa, R. A. Cipolatti, I.-S. Liu, L. A. Palermo, and M. A. Rincon, “On the existence, uniqueness and regularity of solutions of a viscoelastic stokes problem modelling salt rocks,” Applied Mathematics and Optimization, vol. 78, p. 403–456, 2018.
P. Massimi, A. Quarteroni, F. E. Saleri, and G. Scrofani, “Modeling of salt tectonics,” Comput. Methods Appl. Mech. Eng., vol. 197, pp. 281–293, 2007.
C. I. Steefel and G. T. Y. et al., “Reactive transport codes for subsurface environmental simulation,” Computational Geosciences, vol. 19, p. 445–478, 2015.
G. T. Yeh, C. H. Tsai, and I.-S. Liu, A Geo-Mechanics Model for Finite Visco-Elastic Materials (GMECH 1.0): Theoretical Basis and Numerical Approximation. PhD thesis, Graduate Institute of Applied Geology, National Central University, Jhongli, Taoyuan 32001, Taiwan, 2013.
G. T. Yeh and C. H. Tsai, A Three-Dimensional Model of Coupled Fluid Flow,Thermal Transport, Hydrogeochemical Transport, and Geomechanics through Multiple Phase Systems (HYDROGEOCHEM 7.1): Theoretical Basis and Numerical Approximation. PhD thesis, Graduate Institute of Applied Geology, National Central University, Jhongli, Taoyuan 32001, Taiwan, 2015.
C. Colon and M. G. T. et al., “Surface salt in kuqa fold-thrust belt, northwestern china: time-lapse surface deformation and mechanical modelling,” in Proceedings of Fringe 2015 Workshop, vol. 1, European Space Agency, 2015.
Downloads
Published
How to Cite
Issue
Section
License
Copyright
Authors of articles published in the journal Trends in Computational and Applied Mathematics retain the copyright of their work. The journal uses Creative Commons Attribution (CC-BY) in published articles. The authors grant the TCAM journal the right to first publish the article.
Intellectual Property and Terms of Use
The content of the articles is the exclusive responsibility of the authors. The journal uses Creative Commons Attribution (CC-BY) in published articles. This license allows published articles to be reused without permission for any purpose as long as the original work is correctly cited.
The journal encourages Authors to self-archive their accepted manuscripts, publishing them on personal blogs, institutional repositories, and social media, as long as the full citation is included in the journal's website version.
