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

Length Changes

Total lengths

The length of the material curve \(\mt{\u\gamma}\) is

\[\begin{align} \text{m}(\mt{\u\gamma})&=\int_{0}^{1}\norm{\mt{\u\gamma}'(\xi)}\, d\xi\\ &=\int_{0}^{1}\pr{\mt{\u\gamma}'(\xi)\cdot \mt{\u\gamma}'(\xi)}^{1/2}\, d\xi\\ &=\int_{0}^{1}\pr{\mt{\u n}(\xi)\cdot\mt{\u n}(\xi)}^{1/2}\, d\xi \end{align}\]

The length of the spatial curve \(\st{\u\gamma}\) is

\(\begin{align} \text{m}(\st{\u\gamma})&=\int_{0}^{1}\norm{\gamma_s'}\, d\xi\\ &=\int_{0}^{1}\norm{\u{\varphi}'(\mt{\u\gamma}(\xi))\mt{\u{\gamma}}'(\xi)}\,d\xi\\ &=\int_{0}^{1}\norm{\u{F}\pr{\mt{\gamma}(\xi)}\mt{\u\gamma}'(\xi)}\, d\xi\\ &=\int_{0}^{1}\pr{\u{F}\pr{\mt{\gamma}(\xi)}\mt{\u\gamma}'(\xi)\cdot \u{F}\pr{\mt{\gamma}(\xi)}\mt{\u\gamma}'(\xi)}^{1/2}\, d\xi\\ &=\int_{0}^{1}\pr{\mt{\u\gamma}'(\xi)\cdot \u{F}\pr{\mt{\gamma}(\xi)}^{\rm T}\u{F}\pr{\mt{\gamma}(\xi)}\mt{\u\gamma}'(\xi)}^{1/2}\, d\xi\\ &=\int_{0}^{1}\pr{\mt{\u n}(\xi)\cdot \u{C}\pr{\mt{\gamma}(\xi)}\mt{\u n}(\xi)}^{1/2}\, d\xi \end{align}\) Where \(\mt{\u n}(\xi):= \mt{\u\gamma}'(\xi)\) is called the material tangent vector, and \(\begin{align} \u{C}\pr{\u X}:=\u{F}\pr{\u X}^{\rm T}\u{F}\pr{\u X} \end{align}\) is called the right Cauchy-Green deformation tensor.

Stretch Ratio

We call the smooth mapping \(\u\gamma_{\mathscr{m}}:[0,1]\to \mathbb{E}_{R}\) a material curve. The curve can be arbitrary except for that \(\mt{\u{\gamma}}'(0)\propto \hat{\u{N}}\). Or more specifically, \(\mt{\u{\gamma}}'(0)=N \hat{\u{N}}=\u{N}\), where \(N\in \mathbb{R}_{\ge 0}\).

We can define the parameterized curve \(\mt[\epsilon]{\u{\gamma}}:[0,\epsilon]\to \mathbb{E}_{\rm R}\) as \(\mt[\epsilon]{\u\gamma}(\xi)=\mt{\u\gamma}(\xi)\). It follows that \(\mt[\epsilon]{\u{\gamma}}'(0)\propto \hat{\u{N}}\). We call the smooth mapping \(\u\gamma_{\mathscr{s}}:[0,1]\to \mathbb{E}\),where \(\st{\u\gamma}(\xi)=\u{\varphi}\circ\mt{\u\gamma}(\xi)\) the curve \(\mt{\u{\gamma}}\)’s corresponding spatial curve. The curve \(\st[\epsilon]{\u\gamma}\) is a anologously defiend. It can be shown that the limit \(\begin{align} \lim_{\epsilon \to 0}\frac{\norm{\mt{\u\gamma}(\epsilon)-\mt{\u\gamma}(0)}}{\norm{\st{\u\gamma}(\epsilon)-\st{\u\gamma}(0)}} \end{align}\) only depends on \(\hat{\u{N}}\) and is independent of other particulars of the curve. In fact it can be shown that that the limit equals

\(\begin{align} \lambda(\u{X},\hat{\u{N}}) &=\pr{\hat{\u{N}}\cdot \u{C}\pr{\u X}\hat{\u{N}}}^{1/2} \end{align}\) where \(\lambda(\u{X},\hat{\u{N}})\) is called the stretch ratio at the location \(\u{X}\) in the direction \(\hat{\u{N}}\).

Proof.

\[\begin{align} \mt{\u\gamma}(\epsilon)&=\mt{\u\gamma}(0)+\mt{\u\gamma}'(0)\epsilon+o(\epsilon)\\ \mt{\u\gamma}(\epsilon)-\mt{\u\gamma}(0)&=\mt{\u\gamma}'(0)\epsilon+o(\epsilon)\\ \Delta \u X_{\epsilon}&=\mt{\u\gamma}'(0)\epsilon+o(\epsilon)\\ \Delta \u X_{\epsilon}\cdot \Delta \u X_{\epsilon}&\mt{\u\gamma}'(0)\cdot \mt{\u\gamma}'(0)\epsilon^2+o(\epsilon^2)\\ \norm{\Delta \u X_{\epsilon}}&=\pr{N^2\epsilon^2+o(\epsilon^2)}^{1/2}\\ \st{\u\gamma}(\epsilon)&=\st{\u\gamma}(0)+\st{\u\gamma}'(0)\epsilon+o(\epsilon)\\ \st{\u\gamma}(\epsilon)-\st{\u\gamma}(0)&=\st{\u\gamma}'(0)\epsilon+o(\epsilon)\\ \Delta \u x_{\epsilon}&=\st{\u\gamma}'(0)\epsilon+o(\epsilon)\\ \Delta \u x_{\epsilon}\cdot \Delta \u x_{\epsilon}&=\st{\u\gamma}'(0)\cdot \st{\u\gamma}'(0)\epsilon^2+o(\epsilon^2) \end{align}\]

From the chain rule of differentiation we have that \(\begin{align} \st{\u\gamma}'(\xi)&=\u{\varphi}'(\mt{\u\gamma}(\xi))\circ\mt{\u\gamma}'(\xi)\\ \st{\u\gamma}'(\xi)&=\u{\varphi}'(\mt{\u\gamma}(\xi))\mt{\u\gamma}'(\xi)$\\ \st{\u\gamma}'(0)&=\u{F}(\mt{\u\gamma}(0))\mt{\u\gamma}'(0)$\\ \st{\u\gamma}'(0)&=\u{F}(\u X)\mt{\u\gamma}'(0)$\\ \Delta \u x_{\epsilon}\cdot \Delta \u x_{\epsilon}&=\u{F}(\u{X})\mt{\u\gamma}'(0)\cdot \u{F}(\u{X})\mt{\u\gamma}'(0)\epsilon^2+o(\epsilon^2)\\ \Delta \u x_{\epsilon}\cdot \Delta \u x_{\epsilon}&=\mt{\u\gamma}'(0)\cdot \u{C}(\u{X})\mt{\u\gamma}'(0)\epsilon^2+o(\epsilon^2)\\ \Delta \u x_{\epsilon}\cdot \Delta \u x_{\epsilon}&=N^2\lambda(\u{X},\hat{\u N})^2\epsilon^2+o(\epsilon^2)\\ \norm{\Delta \u x_{\epsilon}}&=\pr{N^2\lambda(\u{X},\hat{\u N})^2\epsilon^2+o(\epsilon^2)}^{1/2} \end{align}\)

Total length of curves