\[\renewcommand{\u}[1]{\boldsymbol{#1}} \renewcommand{\b}[1]{\mathbb} \newcommand{\set}[2]{\left\{#1\,\big|\,#2\right\}} \newcommand{\pr}[1]{\left(#1\right)} \newcommand{\st}[2][]{#2_{\mathscr{s}#1}} \newcommand{\mt}[2][]{#2_{\mathscr{m}#1}} \newcommand{\lsc}[2][\mathscr{l}]{ { }^{#1}\!#2 } \newcommand{\norm}[1]{\lVert #1 \rVert}\]

Material velocity field

We call \(\mathcal{L}\pr{\mathbb{T},\mathbb{E}}\) the physical velocity vector space and denote it as \(\mathbb{V}\). It can be shown that the set \(\pr{\u{v}_{i}}_{i\in\mathcal{I}}\), where \(\u{v}_{i}\in\mathbb{V}\) and are defined such that \(\u{v}_{i}(\u{\tau})=\tau\u{e}_i\), forms an orthonormal basis for \(\mathbb{V}\). The velocity of a material particle \(\u{X}\) executing its motion in \(\mathbb{E}\) lies in \(\mathbb{V}\). The velocity of \(\u{X}\) at the instant \(\u{\tau}\) is \(\begin{align} \u{x}_{\u{X}}'(\u{\tau})=:\mt[\u{\tau}]{\u{v}}(\u{X}), \end{align}\) where \(\u{x}_{\u{X}}(\u{\tau})=\u{x}_{\tau}(\u{X})\) and \(\u{x}_{\u{X}}'(\u{\tau})\) is the value of the Fréchet derivative of \(\u{x}_{\u{X}}$ at the instant\)\u{\tau}\(. The mapping\)\mt[\u{\tau}]{\u{v}}:\u{\kappa}_{\rm R}(\mathcal{B})\to \mathbb{V}\(is called the Material Velocity field. The material velocity field answers the question: what is the the velocity of the material partial\)\u{X}\(at the time instance\)\u{\tau}$$.

When \(\u{x}_{\u{\tau}}(\u{X})=x_{i}(X_i(\u{X}),\tau)\u{e}_i\), where \(x_i:\mathbb{R}^4\to \mathbb{R}\) and \(X_j:\b{E}_{\rm R}\to \b{R}\), \(X_{j}(\u{X})=\u{X}\cdot\hat{\u{E}}_j\), it follows that

\[\begin{equation} \mt[\u{\tau}]{\u{v}}(\u{X})=x_{i,\tau}(X_i(\u{X}),\tau)\u{v}_{i} \end{equation}\]

\(\textit{Example 1.}\) A vibrating beam fixed with accelerometers will measure the material acceleration field.