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

Deformation

The deformation mapping \(\u{\varphi}_{\tau}:\u{\mathcal{B}}_{\rm R}\to \mathbb{E}\) takes in a material particle \(\u{X}\in \u{\mathcal{B}}_{\rm R}\) and return the spatial position \(\u{\varphi}_{\tau}(\u{X})\) occupied in \(\mathbb{E}\) at the time instance \(\u{\tau}\). We often use \(\u{x}_{\tau}\) as an abbreviation for \(\u{\varphi}_{\tau}(\u{X})\).

When we are considering a specific deformation mapping, i.e., a deformation mapping at a single time instance, we will omit writing the symbol \(\tau\) as a subscript. That is, the deformation mapping will be written simply as \(\u{\varphi}\). In such cases we often use \(\u{x}\) as an alias of \(\u{\varphi}(\u{X})\).

Admissible and inverse deformation mapping

In continuum mechanics we will be interested in a wide class of possible deformation mappings, but we will place some consditions on $\u{\varphi}$. We will say that $\u{\varphi}$ is an admissible deformation mapping if and only if

1. $\u{\varphi}$ is defined for all $\u{X}\in \mathcal{B}_{\rm R}$
2. $\u{\varphi}$ is injective, except perhaps at the boundary of $\mathcal{B}_{\rm R}$ (such as to avoid interpenetration)
3. $\u{\varphi}$ is continuously differentiable, expect perhaps at the boundary of $\mathcal{B}_{\rm R}$ (such as to avoid cracks, dislocations)

If the first two conditions are satisfied then the deformation mapping is called one-to-one. It is then possible to define the inverse mapping \(\u{\varphi}^{-1}~:~\mathcal{B}\times [0,\u{\tau}_{f}]\to \mathcal{B}_{\rm R}\) such that \(\u{\varphi}^{-1}(\u{x},\u{\tau}):=\) gives the material particle \(\u{X}\) that is loacted at \(\u{x}\) at time instance \(\u{\tau}\). Clearly \(\u{\varphi}(\u{\varphi}^{-1}(\u{x},t),t)=\u{x}\).

Deformation Mapping Components

\(\begin{align} \u{\varphi}_{\tau}(\u{X})=\varphi_{i\tau}(X_{1}(\u{X}),X_{2}(\u{X}),X_{3}(\u{X}))\hat{\u{e}}_i, \end{align}\) where \(\varphi_{i\tau}:\mathbb{R}^3\to \mathbb{R}\) and \(X_j:\mathbb{E}_{\rm R}\to \mathbb{R}\), and \(X_{j}(\u{X})=\u{X}\cdot\hat{\u{E}}_j\). The functions \(\varphi_{i\tau}\) are called the components of the deformation mapping. We can also define the deformation mapping by defining the mapping, \(\begin{align} \u{\varphi}_{\tau}:B_{\rm R} \to \mathbb{E}, \end{align}\) where \(\u{\varphi}_{\tau}(X_1,X_2,X_3)=\varphi_{i\tau}(X_{1},X_{2},X_{3})\hat{\u{e}}_i\). This mapping is different from the mapping \(\u{\varphi}_{\tau}:\u{\mathcal{B}}_{\rm R}\to \mathbb{E}\). Despite using the same symbols, it will be clear from context as to which mapping we are talking about.

When we are considering a specific deformation mapping, i.e., a deformation mapping at a single time instance, we will omit writing the \(\tau\) as a subscript. In such a case the above paragraph can be written as, \(\begin{align} \u{\varphi}(\u{X})=\varphi_{i}(X_{1}(\u{X}),X_{2}(\u{X}),X_{3}(\u{X}))\hat{\u{e}}_i, \end{align}\) where \(\varphi_{i}:\mathbb{R}^3\to \mathbb{R}\), and \(X_j:\mathbb{E}_{\rm R}\to \mathbb{R}\), \(X_{j}(\u{X})=\u{X}\cdot\hat{\u{E}}_j\). The functions \(\varphi_{i}\) are called the components of the deformation mapping. We can also define the deformation mapping by defining the mapping \(\u{\varphi}:B_{\rm R}\to \mathbb{E}\), where \(\u{\varphi}(X_1,X_2,X_3)=\varphi_{i}(X_{1},X_{2},X_{3})\hat{\u{e}}_i\). Clearly, when \((X_i)\) are the components/co-ordinates of \(\u{X}\) w.r.t \((\hat{\u{E}}_i)\), \(\u{\varphi}(X_1,X_2,X_3)=\u{\varphi}(\u{X})\). This mapping is different from the mapping \(\u{\varphi}:\u{\mathcal{B}}_{\rm R}\to \mathbb{E}\). Despite using the same symbols, it will be clear from context as to which mapping we are talking about. In many books you will see \(x_i\) being used as an alias for \(\varphi_{i}(X_1,X_2,X_3)\).

It follows from the above introduced notation that the spatial location occupied by the material particle \(\u{X}\) is \(x_i\hat{\u{e}}_i\). Thus, \(\u{x}=x_i\hat{\u{e}}_i\).

examples

Stretching of a cube

Let \(B_{\rm R}=[0,L]^3\), where \(L \in \mathbb{R}_{\ge 0}\). Consider a defromation mapping \(\u{\varphi}:B_{\rm R}\to \mathbb{E}\). The deformation consists in strectching each edge of the cube in the \(a^\text{th}\) direction by a factor of \(\lambda_a\). \(\lambda_a\in (0, \infty)\), which is expressed as \(\begin{align} x_1&=\lambda_1X_1, \\ x_2&=\lambda_2X_2, \\ x_3&=\lambda_3 X_3. \end{align}\) The values \(\lambda_1\),\(\lambda_2\), and \(\lambda_3\) are called the stretch rations.

Pure shear of a cube

For \(\gamma\in \mathbb{R}\), the deformation mapping \(\begin{align} x_1&=X_1+\gamma X_2\\ x_2&=X_2\\ x_3&=X_3, \end{align}\) is called pure shear in the \(\u{X}_1-\u{X}_2\) plane, and \(\gamma\) is the shear deformation.

Anti-plane shear (along the \(\hat{\u{E}}_3\) direction)

Another type of two dimensional deformation has the following form, \(\begin{align} x_1&=X_1\\ x_2&=X_2\\ x_3&=\varphi_3(X_1,X_2). \end{align}\) In this case the deformation is only in the \(\hat{\u{E}}_3\) direction.

Plane strain

(with-in the \(\hat{\u{E}}_1,~\hat{\u{E}}_2\) plane) We often describe deformations of bodies in two-dimensions. One possible type of essentially two-dimensional deformation is, \(\begin{align} x_1&=\varphi_1(X_1,X_2)\\ x_2&=\varphi_2(X_1,X_2)\\ x_3&=X_3. \end{align}\) Because there are no Deformations in the \(\hat{\u{E}}_3\) direction, these types of deformations are called ``plain strain’’.