Section 2.6 exercises
Subsection 2.6.1
Subsection 2.6.2
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 2.6.3
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:
-
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 \(2x_1 + 3x_2+x_3 = 5\text{;}\)
-
the length of \(\vec{t}_{\vec{n}}\) and its angle with the normal to the plane;
-
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 2.6.4
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 2.6.5
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 2.6.6
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*}
-
Calculate the tractions \(\vec{t}_{\vec{n}}\) and \(\vec{t}_{\bar{\vec{n}}}\) and their normal and tangential components.
-
Calculate the principal stress and principal directions verifying that \(\vec{n}\) is one of the principal directions.
Subsection 2.6.7
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:
-
the principal stress and principal directions in point with coordinates \((a/2, -b/2, 0)\text{;}\)
-
the traction vectors relative to the three faces intersecting at point \((a, b, c)\text{.}\)
Subsection 2.6.8
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 2.6.9
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*}
-
verify the satisfaction of local equilibrium equations;
-
calculate the traction relative to the six faces which form the boundary of \(\mathcal{B}\text{;}\)
-
check the overall balance of the body by calculating the resultant of the forces applied on \(\mathcal{B}\text{.}\)
