Abstract
In this chapter we introduce the concept of balances in particular the balances of mass momentum angular momentum energy or in other words the conservation laws of classical physics. The balances will be stated in integral form—for a material volume—as well as locally—in regular singular points of the continuum. In particular the effects of changes of non-mechanical energy in time i.e. non-isothermal processes are broadly covered in later sections of this book. This way we go well beyond the scope of traditional continuum mechanics may speak of an introduction to continuum physics instead.
We all must have a balance in our life.
John D. Rockefeller
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- 1.
We distinguish a function of current position and time from its value by a circumflex.
- 2.
In this book functions written in Eulerian notation are identified by a circumflex whereas functions in Lagrangian description are easily spotted by a tilde.
- 3.
This is why we have used capital Greek characters for the indices.
- 4.
A 0 refers to the area of the singular surface in the reference configuration.
- 5.
For a proof one should simply expand the products.
- 6.
From the viewpoint of balance laws it would make more sense to say “mass times acceleration equals force” and to distinguish strictly between cause and effect as Newton did (see Chap. 8).
- 7.
Frequently the stress tensor is symmetric and then we may as well write for the production term \( \sigma_{ij} \partial \upsilon_{i} /\partial x_{j} \equiv {\varvec{\sigma}}\cdot\cdot\; \nabla {\varvec{\upsilon}} \) where the double scalar product for non-neighboring indices has been used, which will be introduced in Eq. (3.10.1).
- 8.
It is easily verified that the specific kinetic energy (energy per unit mass) is given by \( {\boldsymbol{\upsilon}}^{2}/2 \).
- 9.
Note that this convention was not applied to the traction vector, t, probably for historic reasons.
- 10.
The argument was started by two eminent mechanics professors with a strong disposition and admiration for mathematics and clearly geared toward libeling physicists as numbskulls: see the paper (in German) and books by Truesdell [18, 19] as well as the book (also in German) by Szabó [16]. Until today many mechanics professors join the clamor of the Boeotians in a sycophant manner even without being able to give an explanation of what the problem really is.
- 11.
Scattering experiments make some particle physicists believe that the electron is a true point. However, it does have a (quantized) spin of ±½\( \hbar \), \( \hbar = 1.055 \times 10^{- 34} {\text{Js}} \) being the normalized Planck constant (note the units of moment of momentum), which could easily be interpreted in terms of moment of momentum if the electron were only a rotating distributed mass.
- 12.
The typo actually appears twice in §. 28. and §. 29. of Euler’s work so that we may suspect that it was incorrectly written down in his personal notes.
- 13.
We have used index notation in Table 3.2 since it makes it easier to distinguish operations referring to the volume and to the surface, respectively.
- 14.
In German.
References
Euler L (1775) Nova methodus motum corporum rigidorum determinandi. Novi Commentarii Academiae Petropolitanae, pp 208–238
Greve R (2003) Kontinuumsmechanik: Ein Grundkurs für Ingenieure und Physiker. Springer, Berlin
Haupt P (2002) Continuum mechanics and theory of materials, 2nd edn. Springer, Berlin
Irgens F (2008) Continuum mechanics. Springer, Berlin
Liu I-S (2010) Continuum mechanics. Springer, Berlin
McBride AT, Javili A, Steinmann P, Bargmann S (2011) Geometrically nonlinear continuum thermomechanics with surface energies coupled to diffusion. J Mech Phys Solids 59:2116–2133
Moeckel GP (1974) Thermodynamics of an interface. ARMA 57:255–280
Müller I (1973) Thermodynamik Die Grundlagen der Materialtheorie. Bertelsmann Universitätsverlag, Düsseldorf
Müller I (1985) Thermodynamics. Pitman Advanced Publishing Program, Boston
Müller I (1994) Grundzüge der Thermodynamik mit historischen Anmerkungen, 1st edn. Springer, Berlin
Müller WH, Ferber F (2008) Technische Mechanik für Ingenieure, 4. aktualisierte Auflage, Carl Hanser, München
Müller I, Müller WH (2009) Fundamentals of thermodynamics and applications. Springer, Berlin
Müller WH, Muschik W (1983) Bilanzgleichungen offener mehrkomponentiger Systeme I: Massen- und Impulsbilanzen. J Non-Equilib Thermodyn 8:29–46
Muschik W, Müller WH (1983) Bilanzgleichungen offener mehrkomponentiger Systeme II: Energie und Entropiebilanz. J Non-Equilib Thermodyn 8:47–66
Schade H (1970) Kontinuumstheorie strömender Medien. Springer, Berlin
Szabó I (1977) Geschichte der mechanischen Prinzipien. Birkhäuser, Basel
Truesdell C, Toupin R (1960) The classical field theories. In: Flügge S (ed) Encyclopedia of physics, vol III/1, Principles of classical mechanics and field theory. Springer, Berlin, Göttingen, Heidelberg
Truesdell C (1968) Whence the law of moment of momentum. In: Essays in the history of mechanics. Springer, Berlin
Truesdell C (1969) Rational thermodynamics. McGraw-Hill, New York
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Müller, W.H. (2014). Balances (in Particular in Cartesian Systems). In: An Expedition to Continuum Theory. Solid Mechanics and Its Applications, vol 210. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7799-6_3
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