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.
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Notes
- 1.
Eringen’s choice of symbols has been adapted to coincide with the ones used in this article.
- 2.
This is not a verbal quote. For the convenience of the reader it has been adjusted to the symbols used in this paper.
- 3.
For simplicity it is assumed that there is no particle on the surface.
- 4.
We will use symbol \(\delta \) for the material derivative since the notation D introduced in Sect. 2 is often associated with the material description.
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Ivanova, E.A., Vilchevskaya, E.N., Müller, W.H. (2016). Time Derivatives in Material and Spatial Description—What Are the Differences and Why Do They Concern Us?. In: Naumenko, K., Aßmus, M. (eds) Advanced Methods of Continuum Mechanics for Materials and Structures. Advanced Structured Materials, vol 60. Springer, Singapore. https://doi.org/10.1007/978-981-10-0959-4_1
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