\[\renewcommand{\u}[1]{\boldsymbol{#1}} \renewcommand{\b}[1]{\mathbb} \newcommand{\set}[2]{\left\{#1\,\big|\,#2\right\}} \newcommand{\pr}[1]{\left( #1 \right)}\]

Deformation Gradient

\(\u{\varphi}_{\u{\tau}}(\u{X})=\varphi_{i\u{\tau}}(X_{1}(\u{X}),X_{2}(\u{X}),X_{3}(\u{X}))\hat{\u{e}}_i\), Here \(\varphi_{i\u{\tau}}:\mathbb{R}^3 \to \mathbb{R}\), and \(X_j:\mathbb{E}_{\rm R}\to \mathbb{R}\), where \(X_{j}(\u{X})=\u{X}\cdot\hat{\u{E}}_j\).

\[\u{\varphi}_{\u{\tau}}(\u{X}+\epsilon \hat{\u{N}})=\varphi_{i\u{\tau}}(X_{1}(\u{X}+\epsilon \hat{\u{N}}),X_{2}(\u{X}+\epsilon \hat{\u{N}}),X_{3}(\u{X}+\epsilon \hat{\u{N}}))\hat{\u{e}}_i\] \[\frac{d\u{\varphi}_{\u{\tau}}(\u{X}+\epsilon \hat{\u{N}})}{d\epsilon}=\varphi_{i\u{\tau},j}(X_{1}(\u{X}+\epsilon \hat{\u{N}}),X_{2}(\u{X}+\epsilon \hat{\u{N}}),X_{3}(\u{X}+\epsilon \hat{\u{N}}))\frac{d X_{j}(\u{X}+\epsilon \hat{\u{N}})}{d\epsilon}\hat{\u{e}}_i\] \[\frac{d\u{\varphi}_{\u{\tau}}(\u{X}+\epsilon \hat{\u{N}})}{d\epsilon}=\varphi_{i\u{\tau},j}(X_{1}(\u{X}+\epsilon \hat{\u{N}}),X_{2}(\u{X}+\epsilon \hat{\u{N}}),X_{3}(\u{X}+\epsilon \hat{\u{N}}))N_j\hat{\u{e}}_i\] \[\left.\frac{d\u{\varphi}_{\u{\tau}}(\u{X}+\epsilon \hat{\u{N}})}{d\epsilon}\right|_{\epsilon=0}=\varphi_{i\u{\tau},j}(X_{1}(\u{X}),X_{2}(\u{X}),X_{3}(\u{X}))N_j\hat{\u{e}}_i\] \[\left.\frac{d\u{\varphi}_{\u{\tau}}(\u{X}+\epsilon \hat{\u{N}})}{d\epsilon}\right|_{\epsilon=0}=\varphi_{i\u{\tau},j}(X_{1}(\u{X}),X_{2}(\u{X}),X_{3}(\u{X}))\hat{\u{e}}_i\otimes \hat{\u{E}}_{j} N_k\hat{\u{E}}_{k}\] \[\left.\frac{d\u{\varphi}_{\u{\tau}}(\u{X}+\epsilon \hat{\u{N}})}{d\epsilon}\right|_{\epsilon=0}=\pr{\varphi_{i\u{\tau},j}(X_{1}(\u{X}),X_{2}(\u{X}),X_{3}(\u{X}))\hat{\u{e}}_i\otimes \hat{\u{E}}_{j} }\hat{\u{N}}\] \[\u{F}_{\u{\tau}}(\u{X})=\pr{\varphi_{i\u{\tau},j}(X_{1}(\u{X}),X_{2}(\u{X}),X_{3}(\u{X}))\hat{\u{e}}_i\otimes \hat{\u{E}}_{j} }\]

We can also define the deformation gradient by providing the mapping \(\u{F}_{\u{\tau}}:B_{\rm R} \to L(\mathbb{E}_{\rm R},\mathbb{E})\), such that

\[\begin{equation} \u{F}_{\u{\tau}}(X_1,X_2,X_3)=\pr{\varphi_{i\u{\tau},j}(X_{1},X_{2},X_{3})\hat{\u{e}}_i\otimes \hat{\u{E}}_{j} } \end{equation}\]

The component form of \((\u{F}_{\u{\tau}}(X_1,X_2,X_3))\) w.r.t, \((\hat{\u{e}}_i\otimes \hat{\u{E}}_j)_{i,j\in\mathcal{I}}\) is \(\pr{\varphi_{i\u{\tau},j}(X_{1},X_{2},X_{3})}_{i,j\in\mathcal{I}}\). For short we write the last equation as \((\u{F}_{\u{\tau}}(X_1,X_2,X_3))=\pr{\varphi_{i\u{\tau},j}(X_{1},X_{2},X_{3})}\).When we are not interested in the time dependence then we write the last equation as \((\u{F}(X_1,X_2,X_3))=\pr{\varphi_{i,j}(X_{1},X_{2},X_{3})}\). Even more explicitly the last equation can be written as

\[\begin{equation} (\u{F}(X_1,X_2,X_3))= \begin{pmatrix} \varphi_{1,1}(X_{1},X_{2},X_{3}) & \varphi_{1,2}(X_{1},X_{2},X_{3}) & \varphi_{1,3}(X_{1},X_{2},X_{3}) \\ \varphi_{2,1}(X_{1},X_{2},X_{3}) & \varphi_{2,2}(X_{1},X_{2},X_{3}) & \varphi_{2,3}(X_{1},X_{2},X_{3}) \\ \varphi_{3,1}(X_{1},X_{2},X_{3}) & \varphi_{3,2}(X_{1},X_{2},X_{3}) & \varphi_{3,3}(X_{1},X_{2},X_{3}) \end{pmatrix} \end{equation}\]

We also use \(F_{ij}(X_1,X_2,X_3)\) as an alias for \(\varphi_{i,j}(X_1,X_2,X_3)\). The last two expressions can be abbreviated as \(F_{ij}((X_k))\) and \(\varphi_{i,j}((X_k))\).