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Section 2.6 exercises

Subsection 2.6.1

Let \(\vec{t}_{\vec{n}}\) and \(\vec{t}_{\tilde{\vec{n}}}\) be the tractions relative to two different normals \(\vec{n}\) and \(\tilde{\vec{n}}\) through point \(\vec{x}\text{.}\) Prove that

\begin{equation*} \vec{n} \cdot \vec{t}_{\tilde{\vec{n}}} = \tilde{\vec{n}} \cdot \vec{t}_{\vec{n}} \end{equation*}

only if the stress tensor relative to point \(\vec{x}\) is symmetric.

Subsection 2.6.2

Given the following state of stress

\begin{equation*} \mat{\sigma} = \left[\begin{array}{ccc} 1 \amp 4 \amp -2 \\ 4 \amp 0 \amp 0 \\ -2 \amp 0 \amp 3 \end{array} \right]\,, \end{equation*}

calculate the following:

  1. the components of the traction vector \(\vec{t}_{\vec{n}}\) relative to a plane through the plane passing through the given point and parallel to the plane \(2x_1 + 3x_2+x_3 = 5\text{;}\)
  2. the length of \(\vec{t}_{\vec{n}}\) and its angle with the normal to the plane;
  3. the components of the stress tensor \(\tilde{\vec{\sigma}}\) with respect to a new reference basis \(\{\tilde{\vec{e}}_a\}\text{,}\) \(a=1,2,3\text{,}\) obatined by rotating the initial reference triad by and angle of \(\pi/2\) around axis \(\vec{e}_3\text{.}\)

Subsection 2.6.3

Consider the stress state defined as follows

\begin{equation*} \mat{\sigma} = \left[\begin{array}{ccc} 5x_2x_3 \amp 3x_2^2 \amp 0 \\ 3x_2^2 \amp 0 \amp -x_1 \\ 0 \amp -x_1 \amp 0 \end{array} \right]\,. \end{equation*}

Calculate the components of the traction vector at the coordinate point \((1/2, \sqrt{3}/2, -1)\) belonging to surface with equation \(x_1^2 +x_2^2 + x_3=0\text{.}\) Hint: the direction normal to a surface can be calculated evaluating its gradient.

Subsection 2.6.4

Consider the tensor

\begin{equation*} \mat{\sigma} = \left[\begin{array}{ccc} 7 \amp 0 \amp 14 \\ 0 \amp 8 \amp 0 \\ 14 \amp 0 \amp -4 \end{array} \right] \end{equation*}

and calculate the principal stresses and the principal directions.

Subsection 2.6.5

For a generic point \(\vec{x}\) of a body the following stress components hold

\begin{equation*} \mat{\sigma} = \left[\begin{array}{ccc} 5 \amp 3 \amp -3 \\ 3 \amp 0 \amp 2 \\ -3 \amp 2 \amp 0 \end{array} \right]\,. \end{equation*}

For the same point, consider also the two normals

\begin{equation*} \vec{n} = \left[\begin{array}{c} 0 \\ 1/\sqrt{2}\\1/\sqrt{2} \end{array}\right]\,, \quad \bar{\vec{n}} = \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right]\,. \end{equation*}
  1. Calculate the tractions \(\vec{t}_{\vec{n}}\) and \(\vec{t}_{\bar{\vec{n}}}\) and their normal and tangential components.
  2. Calculate the principal stress and principal directions verifying that \(\vec{n}\) is one of the principal directions.

Subsection 2.6.6

Consider a parallelepiped delimited by the plans \(x_1=\pm a\text{,}\) \(x_2=\pm b\) and \(x_3=\pm c\text{,}\) and the state of stress defined by

\begin{equation*} \mat{\sigma} = \left[\begin{array}{ccc} \alpha\left(x_1-x_2\right) \amp \beta x_1^2 x_2 \amp 0 \\ \beta x_1^2 x_2 \amp -\alpha\left(x_1-x_2\right) \amp 0 \\ 0 \amp 0 \amp 0 \end{array}\right]\,, \end{equation*}

with \(\alpha\) and \(\beta\) generic constants. Calculate:

  1. the principal stress and principal directions in point with coordinates \((a/2, -b/2, 0)\text{;}\)
  2. the traction vectors relative to the three faces intersecting at point \((a, b, c)\text{.}\)

Subsection 2.6.7

Consider the body \(\mathcal{B} = \{\vec{x} \,:\; 0 \leq x_i \leq 1\}\text{,}\) subjected to the bulk load \(\vec{b} = \alpha x_3 \vec{e}_3\) and surface forces applied on \(\partial\mathcal{B}\) and defined by

\begin{equation*} \vec{t} = \left\{\begin{array}{ccl} x_1 x_2 (1-x_1)(1-x_2)\vec{e}_3, \amp\amp \text{sulla faccia}\; x_3 = 0,\\ \vec{0} \amp\amp \text{su tutte le altre facce}. \end{array}\right.\,. \end{equation*}

Calculate the value of \(\alpha\) which allows to obtain a zero value of the resultant of applied bulk and surface forces.

Subsection 2.6.8

Consider the body \(\mathcal{B} = \{\vec{x} \,:\; 0 \leq x_i \leq 1\}\text{,}\) subjected to the bulk force \(\vec{b} = \transp{[0, 0, -\rho g]}\text{,}\) where \(\rho\) is the mass density and \(g\) the gravity acceleratioin. Assuming the following stress tensor field

\begin{equation*} \mat{\sigma} = \left[\begin{array}{ccc} x_2 \amp x_3 \amp 0 \\ x_3 \amp x_1 \amp 0 \\ 0 \amp 0 \amp \rho g x_3 \end{array} \right]\,, \end{equation*}
  1. verify the satisfaction of local equilibrium equations;
  2. calculate the traction relative to the six faces which form the boundary of \(\mathcal{B}\text{;}\)
  3. check the overall balance of the body by calculating the resultant of the forces applied on \(\mathcal{B}\text{.}\)