Abstract
This chapter contains the general integral or local balance law in Eulerian and Lagrangian form. Then, this general law is used to derive the mass conservation, the Cauchy stress-tensor, Piola–Kirchhoff tensor, the momentum equation, the angular momentum with the symmetry of Cauchy’s stress-tensor, the balance of energy, and the Clausius–Duhem entropy inequality. All the above laws are written in the presence of a moving singular surface across which the fields exhibit discontinuities. The balance laws considered in this chapter are fundamentals for all the developments of continuum mechanics, also in the presence of electromagnetic fields.
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Notes
- 1.
In [43], W. Noll proves that Cauchy’s hypothesis follows from the balance of linear momentum under very general assumptions concerning the form of the function describing the surface source s.
- 2.
The mean value theorem applies to each component of the vector function, and not to the vector function itself.
- 3.
See the footnote of Sect. 5.1.
- 4.
Cauchy’s stress tensor is not symmetric in the case of a polar continuum.
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Romano, A., Marasco, A. (2014). Balance Equations. In: Continuum Mechanics using Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-1604-7_5
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DOI: https://doi.org/10.1007/978-1-4939-1604-7_5
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