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Section 1.10 exercises

Subsection 1.10.1

Consider the following two-dimensional transformation

\begin{align*} x_1 \amp= 4 -2X_1-X_2\,,\\ x_2 \amp= 2 +3/2X_1-X_2/2\,. \end{align*}
  1. Is the transformation linear?
  2. Calculate the components of the deformation gradient, its determinant and its inverse.
  3. 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

\begin{align*} x_1 \amp= X_1\,,\\ x_2 \amp= X_3\,,\\ x_3 \amp= -X_2\,. \end{align*}
  1. Is the transformation linear?
  2. Calculate the components of the deformation gradient, its determinant and its inverse.
  3. 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

\begin{align*} \text{(a)}\;\amp x_1 = 1+X_1\,,\; x_2 = X_2/2+5\,,\; x_3=2X_2\,,\\ \text{(b)}\;\amp x_1 = X_1\,,\; x_2 = 3\,,\; x_3=X_3\,,\\ \text{(c)}\;\amp x_1 = X_1\,,\; x_2 = X_2\,,\; x_3=4-X_3\,. \end{align*}

evaluating for each:

  1. the configuration \(\body\) of the cube;
  2. the components of the deformation gradient;
  3. the ammissibility of the transformation.

Subsection 1.10.4

Consider the following transformation

\begin{align*} x_1 \amp= X_1^2\,,\\ x_2 \amp= X_3^2\,,\\ x_3 \amp= X_2 X_3\,. \end{align*}
  1. Calculate the deformation gradient and its determinant.
  2. Is the transformation admissible for any subset of the Euclidean space?
  3. Calculate the displacement field \(\vec{u}\text{.}\)

Subsection 1.10.5

Consider the following displacement field

\begin{align*} u_1 \amp= x_1-x_2/4\,,\\ u_2 \amp= x_1+2x_2\,,\\ u_3 \amp= -3x_3\,. \end{align*}

Calculate the deformation gradient and its inverse; verify that the trasnformation is isochoric.

Subsection 1.10.6

Consider the following transformation

\begin{align*} x_1 \amp= \alpha X_1\,,\\ x_2 \amp= -\left(\beta X_2+\gamma X_3 \right)\,,\\ x_3 \amp= \gamma X_2 - \beta X_3 \,, \end{align*}

being \(\alpha\text{,}\) \(\beta\) and \(\gamma\) generic constants.

  1. Calculate tensors \(\tens{C}\) and \(\tens{E}\text{.}\)
  2. 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

\begin{equation*} \mat{F} = \left[\begin{array}{ccc} 1 \amp 0 \amp 0 \\ 0 \amp 2 \amp 1 \\ 0 \amp 1 \amp 2 \end{array}\right]\,. \end{equation*}

Calculate tensor \(\tens{C}\) and right stretch tensor \(\tens{U}\text{.}\)

Subsection 1.10.8

The following transformation is assigned

\begin{align*} x_1 \amp= p X_1\,,\\ x_2 \amp= q X_2\,,\\ x_3 \amp= r X_3 \,, \end{align*}

with \(p\text{,}\) \(q\) and \(r\) generic costants.

  1. 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*}
  2. Calculate tensors \(\tens{C}\text{,}\) \(\tens{U}\) and \(\tens{E}\text{.}\)
  3. The stretches \(\lambda\) relative to the reference axes.
  4. 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

\begin{align*} x_1 \amp= X_1 + \alpha X_2\,,\\ x_2 \amp= X_2\,,\\ x_3 \amp= X_3 \,, \end{align*}

where \(\alpha\) is a generic costant.

  1. 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*}
  2. Calculate the components of tensor \(\tens{C}\text{.}\)
  3. Calculate the shear angle \(\gamma\) relative to \((\vec{e}_1,\vec{e}_2)\) plane and \((\vec{e}_1,\vec{e}_3)\) plane.
  4. 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

\begin{align*} u_1 \amp= 3.5 \; 10^{-3} X_1 + 2.0 \; 10^{-3} X_2\,,\\ u_2 \amp= 1.0 \; 10^{-3} X_1 - 0.5 \; 10^{-3} X_2\,,\\ u_3 \amp= 0 \,. \end{align*}
  1. Calculate the infinitesimal strain tensor \(\tens{\varepsilon}\text{.}\)
  2. Calculate the principal strains \(\varepsilon_i\) (\(i=1 \dots 3\)) and the principal directions of \(\tens{\varepsilon}\text{.}\)
  3. Calculate the principal stretches \(\lambda_i\) (\(i=1 \dots 3\)) and principal directions fo tensor \(\tens{U}\text{.}\)
  4. Compare \(\varepsilon_i\) and \(\lambda_i\) oon the basis of equation (1.9.9).
  5. Compare the principal directions of \(\tens{\varepsilon}\) and the principal directions of \(\tens{U}\text{.}\)
  6. 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?