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Applications to Continuum Mechanics

  • Mikhail Itskov
Chapter
Part of the Mathematical Engineering book series (MATHENGIN)

Abstract

Let us consider an infinitesimal vector \(\text {d}\varvec{{X}}\) in the reference configuration of a material body and its counterpart \(\text {d}\varvec{{x}}\) in the current configuration. By virtue of the representation for the deformation gradient ( 2.67) we get
$$\begin{aligned} \text {d}\varvec{{x}} = \frac{\partial \varvec{x}}{\partial X ^j} \text {d}X ^j = \left( \frac{\partial \varvec{x}}{\partial X ^j} \otimes \varvec{e}^j \right) \left( \text {d}X ^k \varvec{e}_k\right) =\mathbf {F}\text {d}\varvec{{X}}. \end{aligned}$$
Thus, the deformation gradient \(\mathbf {F}\) maps every infinitesimal vector from the reference configuration to the current one by \(\text {d}\varvec{{x}} = \mathbf {F}\text {d}\varvec{{X}}\).

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Continuum MechanicsRWTH Aachen UniversityAachenGermany

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