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on 07-Aug-2021 (Sat)

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Flashcard 5525576748300

#continuum-mechanics #reference-configuration
Describe two reasons considering a reference configuration.

In order to identify a particle of a body, we must label the particles. The abstract particle label \(p\), while perfectly acceptable in principle and intuitively clear, is not convenient for carrying out calculations. It is more convenient to pick some arbitrary configuration of the body, say \(\chi_\textrm{ref}\) , and use the (unique) position \(\mathbf{x} = \chi_\textrm{ref} (p)\) of a particle in that configuration to label it instead. Such a configuration \(\chi_\textrm{ref} (p)\) is called a reference configuration of the body. It simply provides a convenient way in which to label the particles of a body. The particles are now labeled by \(\mathbf x\) instead of \(p\).

A second reason for considering a reference configuration is the following: we can study the geometric characteristics of a configuration \(\chi\) by studying the geometric properties of the points occupying the region \(\mathcal R = \chi (\mathcal B)\). This is adequate for modeling certain materials (such as many fluids) where the behavior of the material depends only on the characteristics of the configuration currently occupied by the body. In describing most solids however one often needs to know the changes in geometric characteristics between one configuration and another configuration (e.g. the change in length, the change in angle etc.). In order to describe the change in a geometric quantity one must necessarily consider (at least) two configurations of the body: the configuration that one wishes to analyze, and a reference configuration relative to which the changes are to be measured.

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