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2019 | OriginalPaper | Chapter

5. Moving Reference Frames

Author : Zdeněk Martinec

Published in: Principles of Continuum Mechanics

Publisher: Springer International Publishing

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Abstract

The frame-indifference principle, one of the fundamental concepts of continuum mechanics, means that a physical process remains unchanged when it is observed by different observers.

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previous chapter Fundamental Conservation Principles

next chapter Constitutive Equations

A frame should not be confused with a coordinate system as they are not the same. A frame can only be used to observe motion. But to describe the motion mathematically and perform operations on the vectors, we need a coordinate system. An observer in a frame is free to choose any coordinate system that may be convenient for describing observations made from that frame.

Both are later associated with the present configuration of a body and are, therefore, called the spatial frames.

Note that we distinguish between three different vectors, $\vec x=x_k\vec i_k$, $\vec x^{*}=x_k^*\vec i_k$ and $\vec x^{**}=x_k^*\vec i_k^*$. Vector notation becomes ambiguous if the vectors $\vec x^{*}$ and $\vec x^{**}$ are denoted by the same symbol $\vec x^{*}$; compare (5.1) and (5.9) in this case.

A vector is a geometric object that has, by definition, an invariant property under an arbitrary change of coordinate system. Even though the components of a vector change when a coordinate system changes, they must change in a very specific way. A vector triplet is a triplet of numbers representing a vector in a particular coordinate system.

The most general change of frame $(\vec x,t)\rightarrow (\vec x^{*},t^*)$ is, in addition, characterised by a shift in time

$$\displaystyle \begin{aligned}t^*=t-a, \end{aligned}$$

where a is a particular time. The shift-in-time transformation is trivial and does not affect the derived relationships. For example, the consequence of frame-indifference with respect to the shift of time is that the constitutive functionals do not depend explicitly on the current time. We will consider this as a starting assumption in defining a general form (6.5) of the constitutive equations.

The unstarred and starred observers are also called equivalent observers due to these properties.

This does not affect the generality of what follows; the notion of a frame-indifferent quantity is independent of the chosen reference configuration.

Note again that we distinguish between three different tensors a, a ^∗ and a ^∗∗.

To present this alternative concept, let t ₀ be the instant when the reference configuration is chosen. The observer transformation (5.9) for the reference configuration is

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-05390-1_5/MediaObjects/466538_1_En_5_Equ32_HTML.png

(5.32)

where O(t ₀) and

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-05390-1_5/466538_1_En_5_IEq59_HTML.gif

describe a rigid motion of the observer frame at the instant t ₀, and $\vec X=X_K\vec I_K$, $\vec X^*=X^*_{K^*}\vec I_{K^*}$, where X _K and $x^*_{k^*}$ are the coordinates of a point associated with an event in the reference configuration, which is recorded in the fixed and moving frames, respectively. Then, the deformation gradient in the two starred frames, one related to the present configuration and the other to the reference configuration, can be expressed as

$$\displaystyle \begin{aligned}F_{k^*K^*}^*={\partial\chi_{k^*}^*\over\partial {X^*}_{\hspace{-3pt} K^*}} ={\partial\chi_{k^*}^*\over\partial X_K}{\partial X_K\over\partial {X^*}_{\hspace{-3pt} K^*}} =O_{k^*k}{\partial\chi_k\over\partial X_K} [{\boldsymbol{O}}(t_0)]_{K^*K} =O_{k^*k}F_{kK}[{\boldsymbol{O}}(t_0)]_{K^*K}, \end{aligned}$$

or, multiplying both sides of the last equation by the dyad $\vec i_{k^*}\otimes \vec I_{K^*}$ gives

$$\displaystyle \begin{aligned} {\boldsymbol{F}}^*(\vec X,t)={\boldsymbol{O}}(t)\cdot{\boldsymbol{F}}(\vec X,t)\cdot{\boldsymbol{O}}^T(t_0). \end{aligned} $$

(5.33)

Note that ${\boldsymbol {F}}^*(\vec X,t)$ is the same as in (5.31). The transformation rule (5.33) differs from (5.31), the standard formula obtained under the assumption that the reference configuration is not affected by the change of observer frame, so that O(t ₀) reduces to the identity transformation.

The unit normal vector to the surface is defined by (2.47) as the gradient of an objective scalar function. In view of (5.27), $\vec n$ is then a frame-indifferent vector.

Liu, I.-S., & Sampaio, R. (2014). Remarks on material frame-indifference controversy. Acta Mechanica, 225, 331–348.MathSciNetCrossRef

Truesdell, C., & Noll, W. (1965). The nonlinear field theories of mechanics. In S. Flügge (Ed.), Handbook der Physik (Vol. III/3, pp. 359–366). Berlin: Springer.

Title: Moving Reference Frames
Author: Zdeněk Martinec
Publisher: Springer International Publishing
Book: Principles of Continuum Mechanics
Print ISBN: 978-3-030-05389-5

Electronic ISBN: 978-3-030-05390-1

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-05390-1_5

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