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Time Derivatives in Material and Spatial Description—What Are the Differences and Why Do They Concern Us?

  • Elena A. Ivanova
  • Elena N. Vilchevskaya
  • Wolfgang H. Müller
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 60)

Abstract

This paper has many, albeit mostly didactic objectives. It is an attempt toward clarification of several concepts of continuum theory which can lead and have led to confusion. In a way the paper also creates a bridge between the lingo of the solid mechanics and the fluid mechanics communities. More specifically, an attempt will be made, first, to explain and to interpret the subtleties and the relations between the so-called material and spatial description of continuum fields. Second, the concept of time derivatives in material and spatial description will be investigated meticulously. In particular, it will be explained why and how the so-called material and total time derivatives differ and under which circumstances they turn out to be the same. To that end, material and total time derivatives will be defined separately and evaluated in context with local fields as well as during their use in integral formulations, i.e., when applied to balance equations. As a special example the mass balance is considered for closed as well as open bodies. In the same context the concept of a “moving observation point” will be introduced leading to a generalization of the usual material derivative. When the total time derivative is introduced the distinction between the purely mathematical notion of a coordinate system and the intrinsically physics-based concept of a frame of reference will gain particular importance.

Keywords

Observation Point Position Vector Material Point Total Derivative Reference Configuration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Adler, P.: Porous Media: Geometry and Transports. Butterworth-Heinemann, USA (1992)Google Scholar
  2. Altenbach, H., Naumenko, K., Zhilin, P.A.: A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions. Contin. Mech. Thermodyn. 15(6), 539–570 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Asaro, R., Lubarda, V.: Mechanics of Solids and Materials. Cambridge University Press, New York (2006)CrossRefzbMATHGoogle Scholar
  4. Batchelor, G.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1970)zbMATHGoogle Scholar
  5. Cornille, P.: Inhomogeneous waves and Maxwell’s equations (Chapter 4). Essays on the Formal Aspects of Electromagnetic Theory. World Scientific, Singapore (1993)Google Scholar
  6. Daily, J., Harleman, D.: Fluid Dynamics. Addison-Wesley, Boston (1966)zbMATHGoogle Scholar
  7. Dang, T.S., Meschke, G.: An ALE-PFEM method for the numerical simulation of two-phase mixture flow. Comput. Methods Appl. Mech. Eng. 278, 599–620 (2014)MathSciNetCrossRefGoogle Scholar
  8. Del Pin, F., Idelsohn, S., Onate, E., R A, : The ALE/Lagrangian particle finite element method: A new approach to computation of free-surface flows and fluid object interactions. Comput. Fluids 36, 27–38 (2007)Google Scholar
  9. Dettmer, W., Peric, D.: A computational framework for free surface fluid flows accounting for surface tension. Comput. Methods Appl. Mech. Eng. 195, 3038–3071 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Dmitrienco, U.: Nonlinear mechanics of continus. Physmatlit, Moscow (2009)Google Scholar
  11. Durst, F.: Fluid Mechanics: An Introduction to the Theory of Fluid Flows. Springer, Berlin (1992)Google Scholar
  12. Eringen, C.: Mechanics of Continua. Robert E Krieger Publishing Company, Huntington, New York, (1980)Google Scholar
  13. Filipovic, N., Akira, Mijailovic A.S., Tsuda, Kojic M.: An implicit algorithm within the arbitrary Lagrangian–Eulerian formulation for solving incompressible fluid flow with large boundary motions. Comput. Methods Appl. Mech. Eng. 195, 6347–6361 (2006)CrossRefzbMATHGoogle Scholar
  14. Fung, Y.: Foundations of Solid Mechanics. Prentice-Hall, Englewood Cliffs (1965)Google Scholar
  15. Gadala, M.: Recent trends in ale formulation and its applications in solid mechanics. Comput. Methods Appl. Mech. Eng. 193, 4247–4275 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Granger, R.A.: Fluid Mechanics. Dover Books on Physics (1995)Google Scholar
  17. Gurtin, M.: An Introduction to Continuum Mechanics. Academic Press Inc, London (1981)zbMATHGoogle Scholar
  18. Hassanizadeh, M., Gray, W.: General conservation equations for multi-phase systems: 3. constitutive theory for porous media flow. Adv. Water Resour. 3, 25 (1980)CrossRefGoogle Scholar
  19. Ilyushin, A.: Continuum mechanics. Moscow University Press, Moscow (1971)Google Scholar
  20. Khoei, A., Anahid, M., Shahim, K.: An extended arbitrary Lagrangian–Eulerian finite element modeling (X-ALE-FEM) in powder forming processes. J. Mater. Process. Technol. 187–188, 397–401 (2007)CrossRefGoogle Scholar
  21. Lamb, H.: Hydrodynamics. Cambridge University Press, New York (1975)zbMATHGoogle Scholar
  22. Landau, L., Lifshitz, E.: Fluid Mechanics, vol. 6, 1st edn. Pergamon Press, Oxford (1959)zbMATHGoogle Scholar
  23. Lojtsanskij, L.: Mechanics of liquid and gas. Moscow (1950)Google Scholar
  24. Malvern, E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall Inc, Englewood Cliffs (1969)zbMATHGoogle Scholar
  25. Mase, G.: Theory and Problems of Continuum Mechanics. McGraw-Hill Book Company, New York (1970)Google Scholar
  26. Milne-Thomson, L.: Theoretical Hydrodynamics. Martin’s Press, Macmillan and Co. LTD, New York (1960)zbMATHGoogle Scholar
  27. Müller, W.H., Muschik, W.: Bilanzgleichungen offener mehrkomponentiger systeme. I. massen- und impulsbilanzen. J. Non-Equilib. Thermodyn. 8(1), 29–46 (1983)CrossRefGoogle Scholar
  28. Nerlich, G.: What Spacetime Explains: Metaphysical Essays on Space and Time. Cambridge University Press, Cambridge (1994)Google Scholar
  29. Ogden, R.: Nonlinear Elasticity with Application to Material Modelling. Polish Academy of Sciences, Warsaw (2003)Google Scholar
  30. Pasipoularides, A.: Heart’s vortex: intracardiac blood flow phenomena. PMPH-USA (2009)Google Scholar
  31. Petrila, T., Trif, A.: Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics (Numerical Methods and Algorithms). Springer, Boston (2005)zbMATHGoogle Scholar
  32. Prandtl, L., Tietjens, O.: Hydro- und Aeromechanik. Springer, Berlin (1929)zbMATHGoogle Scholar
  33. Preisig, M., Zimmermann, T.: Free-surface fluid dynamics on moving domains. Comput. Methods Appl. Mech. Eng. 200, 372–382 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Rouse, H.: Advanced Mechanics of Fluids. Wiley, New York (1959)zbMATHGoogle Scholar
  35. Sarrate, J., Huerta, A., Donea, J.: Arbitrary Lagrangian–Eulerian formulation for fluid-rigid body interaction. Comput. Methods Appl. Mech. Eng. 190, 3171–3188 (2001)CrossRefzbMATHGoogle Scholar
  36. Serrin, J.: Mathematical Principles of Classical Fluid Mechanics. Springer, Berlin (1959)CrossRefGoogle Scholar
  37. Surana, K., Blackwell, B., Powell, M., Reddy, J.: Mathematical models for fluid-solid interaction and their numerical solutions. J. Fluids Struct. 50, 184–216 (2014)CrossRefGoogle Scholar
  38. Truesdell, C.: A First Course in Rational Continuum Mechanics. John’s Hopkins University, Baltimore (1972)Google Scholar
  39. Vuong, A.T., Yoshihara, L., Wall, W.: A general approach for modeling interacting flow through porous media under finite deformations. Comput. Methods Appl. Mech. Eng. 283, 1240–1259 (2015)MathSciNetCrossRefGoogle Scholar
  40. Zhilin, P.A.: Racional’naya mekhanika sploshnykh sred (Rational Continuum Mecanics, in Russian). Politechnic University Publishing House, St. Petersburg (2012)Google Scholar

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  • Elena A. Ivanova
    • 1
    • 2
  • Elena N. Vilchevskaya
    • 1
    • 2
  • Wolfgang H. Müller
    • 3
  1. 1.Peter the Great Saint-Petersburg Plytechnic UniveritySt.-PetersburgRussia
  2. 2.Institute for Problems in Mechanical Engineering, Russian Academy of SciencesSt.-PetersburgRussia
  3. 3.Chair of Continuum Mechanics and Materials TheoryInstitute of Mechanics, Technical University of BerlinBerlinGermany

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