Curvilinear coordinate systems in continuum mechanics
DOI:
https://doi.org/10.23998/rm.83338Keywords:
coordinate transformation, basis, tensor, Christoffel symbols, deformationAbstract
The article examines curvilinear coordinate systems generalized from rectangular linear coordinate systems and transformations theirbetween. The theory is based on the theory of linear coordinate systems. Unambigious definition of the base system of linear coordinate systems is considered a key feature of the generalization. The applications concern important coordinate systems applied in continuum mechanics, transformations between the coordinate systems, change of basis, the derivatives of changed bases, and the so-called Christoffel symbols needed in the derivatives. Also benefits of curvilinear coordinates and their transformations are discussed. The article is based on T. Salmi's unpublished written material.
References
J. Bonnet and R. D. Wood. Nonlinear continuum mechanics for finite element analysis 2nd edition. Cambridge University Press, 2008.
T. Belytschko, W. K. Liu, and B. Moran. Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons, Chichester, 2000.
Y. Basar and D. Weichert. Nonlinear Continuum Mechanics of Solids. Fundamental mathematical and physical concepts. Springer-Verlag, Berlin Heidelberg, 2000.
G. A. Holzapfel. Nonlinear Solid Mechanics. A Continuum Approach for Engineering. John Wiley & Sons, Chichester, 2001.
M. Itskov. Tensor Algebra and Tensor Analysis for Engineers With Applications to Continuum Mechanics. Springer, New York, 2015.
S. Holopainen, R. Kouhia, and T. Saksala. Continuum approach for modeling transversely isotropic high-cycle fatigue. European Journal of Mechanics A/Solids, 60:183–195, 2016. Online: https://doi.org/10.1016/j.euromechsol.2016.06.007
S. Holopainen and T. Barriere. Modeling of mechanical behavior of amorphous solids undergoing fatigue loadings, with application to polymers. Computers and Structures, 199:57–73, 2018. Online: https://doi.org/10.1016/j.compstruc.2018.01.010
V. Huovinen. Havukka-Ahon ajattelija. WSOY, 1952.
I. S. Sokolnikoff. Tensor Analysis. Theory and Applications. John Wiley & Sons, New York, 1951.
C. Truesdell and W. Noll. The Nonlinear Field Theories of Mechanics, Handbuch der Physik, Vol. III/3. Springer-Verlag, Berlin, 1965.
W.H. Greub. Multilinear Algebra. Springer-Verlag, 1967.
A. Borisenko and I. Tarapov. Vector and Tensor Analysis with Applications. Prentice-Hall, New Jersey, 1968.
R. Abraham, J. E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis and Applications. Addison-Wesley, 1983.
J. E. Marsden and T. J. R. Hughes. Mathematical Foundations of Elasticity. Dover publications, Inc., 1993.
T. Apostol. Mathematical Analysis. Addison-Wesley, Lontoo, 1957.
S. Holopainen. Representations of m-linear functions on tensor spaces. Duals and transpositions with applications in continuum mechanics. Mathematics and Mechanics of Solids, 19:168–193, 2014. Online: https://doi.org/10.1177/1081286512456199
J. A. Schouten. Tensor Analysis for Physicists. Dover, New York, 1989.
J. Gerdeen, H. Lord, and R. Rorrer. Engineering Design with Polymers and Composites. Taylor & Francis, New York, 2006.
I. Newton. Philosophiæ Naturalis Principia Mathematica (Principia). London, 1687.
J. Hartikainen, R. Kouhia, M. Mikkola, and J. M¨akinen. Muodonmuutoksen mitat kontinuumimekaniikassa. Rakenteiden Mekaniikka, 49:86–99, 2016. Online: http://rmseura.tkk.fi/rmlehti/2016/nro2/RakMek_49_2_2016_6.pdf
M. Mikkola. Venym¨amitat kontinuumimekaniikassa Hillin-Sethin mukaan. Rakenteiden Mekaniikka, 50:1–16, 2017. Online: https://doi.org/10.23998/rm.63299
M. Mikkola. Venym¨amitat kontinuumimekaniikassa Fingerin ja Piolan mukaan. Rakenteiden Mekaniikka, 50:451–462, 2017. Online: https://doi.org/10.23998/rm.66414
S. Holopainen. Alternative work conjugate pairs of stress and strain in simple shear. In 16th Nordic Seminar on Computational Mechanics (NSCM), 2003. https://www.researchgate.net/conference-event/NSCM_
Nordic-Seminar-on-Computational-Mechanics_2003/40301
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