Section 1.10 exercises
Subsection 1.10.1
Consider the following two-dimensional transformation
- Is the transformation linear?
- Calculate the components of the deformation gradient, its determinant and its inverse.
- Study the transformation over a unit square defined through the following points\begin{equation*} A:\, (0,0)\,,\quad B:\, (1,0)\,,\quad C:\, (1,1)\,,\quad D:\, (0,1)\,. \end{equation*}
Subsection 1.10.2
Consider the following transformation
- Is the transformation linear?
- Calculate the components of the deformation gradient, its determinant and its inverse.
- Study the transformation over a unit cube defined through the following points\begin{equation*} A:\, (0,0,0)\,,\quad B:\, (1,0,0)\,,\quad C:\, (1,1,0)\,,\quad D:\, (0,1,0)\,, \end{equation*}\begin{equation*} E:\, (0,0,1)\,,\quad F:\, (1,0,1)\,,\quad G:\, (1,1,1)\,,\quad H:\, (0,1,1)\,. \end{equation*}
Subsection 1.10.3
Apply to the same unit cube of the previous problem the following transformations
evaluating for each:
- the configuration \(\body\) of the cube;
- the components of the deformation gradient;
- the ammissibility of the transformation.
Subsection 1.10.4
Consider the following transformation
- Calculate the deformation gradient and its determinant.
- Is the transformation admissible for any subset of the Euclidean space?
- Calculate the displacement field \(\vec{u}\text{.}\)
Subsection 1.10.5
Consider the following displacement field
Calculate the deformation gradient and its inverse; verify that the trasnformation is isochoric.
Subsection 1.10.6
Consider the following transformation
being \(\alpha\text{,}\) \(\beta\) and \(\gamma\) generic constants.
- Calculate tensors \(\tens{C}\) and \(\tens{E}\text{.}\)
- Assuming \(\beta=-\cos{\theta}\) and \(\gamma=-\sin{\theta}\) evaluate for which value of \(\alpha\) the strain is zero.
Subsection 1.10.7
Suppose the following value for the deformation gradient evaluated in a point of a body
Calculate tensor \(\tens{C}\) and right stretch tensor \(\tens{U}\text{.}\)
Subsection 1.10.8
The following transformation is assigned
with \(p\text{,}\) \(q\) and \(r\) generic costants.
- Study the deformation of a unit cube defined by the following points\begin{equation*} A:\, (0,0,0)\,,\quad B:\, (1,0,0)\,,\quad C:\, (1,1,0)\,,\quad D:\, (0,1,0)\,, \end{equation*}\begin{equation*} E:\, (0,0,1)\,,\quad F:\, (1,0,1)\,,\quad G:\, (1,1,1)\,,\quad H:\, (0,1,1)\,. \end{equation*}
- Calculate tensors \(\tens{C}\text{,}\) \(\tens{U}\) and \(\tens{E}\text{.}\)
- The stretches \(\lambda\) relative to the reference axes.
- The stretch \(\lambda\) relative to the direction going from point \((0,0,0)\) to point \((1,1,1)\text{.}\) Hint: consider equation (1.8.4).
Subsection 1.10.9
The following transformation is assigned
where \(\alpha\) is a generic costant.
- Study the deformation of a unit cube defined by the following points\begin{equation*} A:\, (0,0,0)\,,\quad B:\, (1,0,0)\,,\quad C:\, (1,1,0)\,,\quad D:\, (0,1,0)\,, \end{equation*}\begin{equation*} E:\, (0,0,1)\,,\quad F:\, (1,0,1)\,,\quad G:\, (1,1,1)\,,\quad H:\, (0,1,1)\,. \end{equation*}
- Calculate the components of tensor \(\tens{C}\text{.}\)
- Calculate the shear angle \(\gamma\) relative to \((\vec{e}_1,\vec{e}_2)\) plane and \((\vec{e}_1,\vec{e}_3)\) plane.
- Calculate the principal stretches \(\lambda_i\) (\(i=1 \dots 3\)) e the principal directions of tensor \(\tens{U}\text{.}\) Hint: \(\tens{C} = \tens{U}^2\text{.}\)
Subsection 1.10.10
Consider the following displacement field
- Calculate the infinitesimal strain tensor \(\tens{\varepsilon}\text{.}\)
- Calculate the principal strains \(\varepsilon_i\) (\(i=1 \dots 3\)) and the principal directions of \(\tens{\varepsilon}\text{.}\)
- Calculate the principal stretches \(\lambda_i\) (\(i=1 \dots 3\)) and principal directions fo tensor \(\tens{U}\text{.}\)
- Compare \(\varepsilon_i\) and \(\lambda_i\) oon the basis of equation (1.9.9).
- Compare the principal directions of \(\tens{\varepsilon}\) and the principal directions of \(\tens{U}\text{.}\)
- Repeat the previous evaluations with respect to the same displacement field but amplified with a factor of \(10^3\text{:}\) what considerations can be drawn?