\frac{\partial \sigma_{xy}}{\partial \theta} = \frac{\sigma_{yy} - \sigma_{xx}}{2} \sin(2\theta) + \sigma_{xy} \cos(2\theta).
Show \text{ that } \sigma_{xy}|_{max/min} = \pm \sqrt{(\frac{\sigma_{xx} - \sigma_{yy}}{2})^2 + \sigma_{xy}^2} \text{ (see notes)
p.45 use}
2. For the motion below, calculate the Strain matrix.
(simple shear)
u_x = Ky, \quad u_y = u_z = 0. \quad K-\text{constant}
For the motion below, calculate the Strain matrix
(uniform uni-axial extension \to tensile test!)
u_x = (\lambda - 1)x, \quad u_y = u_z = 0 \quad \lambda-\text{constant}
3. What the angles that cause max/min shear strain ($d_s$)
and max/min normal strain ($d_p$) ? (in terms of $\epsilon_{xx}, \epsilon_{xy}, \epsilon_{yy}$)
4. \begin{tikzpicture}[scale=0.8]
\draw[thick] (0,0) -- (4,0);
\draw[thick] (0,0) -- (0,-0.5);
\draw[thick] (4,0) -- (4,-0.5);
\draw[thick,->] (4,0) -- (5,0) node[midway,above]{$F$};
\draw[thick] (2,0) circle (0.2);
\draw[thick,->] (2,0.2) -- (2,0.5) node[midway,right]{$A$};
\end{tikzpicture} Assume $\sigma_{xx} = \frac{F}{A}$ with all other
components of $\sigma$ stress identically zero.
What are the principal stresses?
5. For the same stress matrix as problem 4, calculate the
angle where the max normal stress is greatest ($\alpha_p$) and
the angle where the max shear stress is greatest ($\alpha_s$). What is
