mmme1028 leture l2.1

mechanical/mmme1028_statics.md

@@ -445,3 +445,131 @@ b. Use the graphical/trigonometric method o check your answer.
 
 </details>
 
+# Lecture L2.1
+
+## Hooke's Law and Young's Modulus
+
+Hooke's law states that the extension of an object experiencing a force is proportonal to the force.
+
+We can generalize this to be more useful creating:
+
+- Direct stress:
+
+  $$ \sigma = \frac F {A_0} $$
+
+- Direct strain:
+
+  $$ \epsilon = \frac {\Delta L}{L_0} $$
+
+Using these more generalized variables, Young defined Young's Modulus, $E$, which is a universal
+constant of stiffness of a material.
+
+$$ \sigma = E\epsilon $$
+
+<details>
+<summary>
+
+#### Example 1
+
+Calculating Young's Modulus of a Piece of Silicone
+
+</summary>
+
+\begin{align*}
+L_0 &= 4.64 \\
+w_0 &= 0.10 \\
+t_0 &= 150\times10^{-6} \\
+F &= 1.40\times9.81 \\
+L &= 6.33 \\
+w &= 0.086 \\
+t &= 125\times10^{-6} \\
+\\
+\sigma &= \frac F {A_0} = \frac F {w_0t_0} = \frac{1.4\times9.81}{0.1\times150\times10^{-6}} = 915600 \\
+\epsilon &= \frac{\Delta L}{L_0} = \frac{6.33 - 4.64}{4.64} = 0.36422...\\
+E &= \frac \sigma \epsilon = 2513836.686 = 2.5\times10^6 \text{ Pa}
+\end{align*}
+
+</details>
+
+## Stress Strain Curves
+
+![](./images/vimscrot-2021-11-01T09:50:51,184232288+00:00.png)
+
+## Poisson's Ration
+
+For most materiajs, their cross sectionts change when they are stretched or compressed.
+This is to keep their volume constant.
+
+$$ \epsilon_x = \frac {\Delta L}{L_0} $$
+$$ \epsilon_y = \frac {\Delta w}{w_0} $$
+$$ \epsilon_z = \frac {\Delta t}{t_0} $$
+
+Poissons' ratio, $\nu$ (the greek letter _nu_, not v), is the ratio of lateral strain to axial
+strain:
+
+$$ \nu = \frac{\epsilon_y}{\epsilon_x} = \frac{\epsilon_z}{\epsilon_x} $$
+
+<details>
+</summary>
+
+#### Example 1
+
+Measuring Poisson's Ratio
+
+</summary>
+
+\begin{align*}
+L_0 &= 4.64 \\
+w_0 &= 0.10 \\
+t_0 &= 150\times10^{-6} \\
+\\
+L &= 6.33 \\
+w &= 0.086 \\
+t &= 125\times10^{-6} \\
+\\
+\epsilon_x &= \frac {\Delta L}{L_0} = 0.364 \\
+\epsilon_y &= \frac {\Delta w}{w_0} = -0.14 \\
+\epsilon_z &= \frac {\Delta t}{t_0} = -0.167 \\
+\\
+\nu_y &= \frac{\epsilon_y}{\epsilon_x} = \frac{-0.14}{0.364} = -0.38 \\
+\nu_z &= \frac{\epsilon_z}{\epsilon_x} = \frac{-0.167}{0.364} = -0.46 \\
+\end{align*}
+
+It's supposed to be that $\nu_y = \nu_z$ but I guess it's close enough right? lol
+
+</details>
+
+## Typical Values of Young's Modulus and Poisson's Ratio
+
+Material | Young's Modulus / GPa | Poisson's Ratio
+-------- | --------------------- | ---------------
+Steel    | 210                   | 0.29
+Aluminum | 69                    | 0.34
+Concrete | 14                    | 0.1
+Nylon    | 3                     | 0.4
+Rubber   | 0.01                  | 0.495
+
+## Direct Stresses and Shear Stresses
+
+![](./images/vimscrot-2021-11-01T10:35:47,339443980+00:00.png)
+
+- A direct stress acts normal to the surface
+- A shear stress acts tangential to the surface
+
+Shear stress is defined in the same way as direct stress but given the symbol $tau$ (tau):
+
+$$ \tau = \frac F A $$
+
+Shear strain is defined as the shear angle $\gamma$:
+
+$$ \gamma \approx \tan\gamma = {\frac x {L_0}} $$
+
+The shear modulus, $G$, is like Young's Modulus but for shear forces:
+
+$$ \tau = G\gamma $$
+
+## Relationship between Young's Modulus and Shear Modulus
+
+$$ G = \frac E {2(1+\nu)} $$
+
+$G \approx \frac E 3$ is a good approximation in a lot of engineering cases