# 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 () 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 $$\text {d}\varvec{{x}} = \mathbf {F}\text {d}\varvec{{X}}$$.

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

1. 1.Department of Continuum MechanicsRWTH Aachen UniversityAachenGermany