$ \tau=\text{ shear stress }=\lim_{ \Delta A \to 0 } \left( \frac{\Delta V}{\Delta A} \right)=\frac{dV}{dA}\implies dV=\tau dA $ $ \tau_{ave}=\frac{1}{A}\int_{A} \tau dA\to \boxed{ \tau_{ave}=\frac{V}{A} } $ ![[Pasted image 20260830134700.png|right|370]] $\gamma=\text{ shear strain }=\frac{\pi}{2}-\theta^*$ $ \text{Let }\tan\gamma=\gamma \therefore \boxed{ \gamma\approx \frac{\delta_{s}}{L_{s}} } $ Shear modulus $\boxed{ G=\frac{E}{2(1+v)}=\frac{\tau}{\gamma} }$ - Shear modulus $G$ measures how much a solid material resists changing its shape when a force is pushed or pulled parallel to its surface - Young's Modulus $E$ measures how much a material stretches or bends when you pull or push on it - Poisson's Ratio $\upsilon$ measures how much a material expands/contracts when squeezed/stretched lengthwise ![[Pasted image 20260830135559.png|center|400]] $ A_{c}=\frac{A}{\cos\theta} $ $ \sigma=\frac{F_{n}}{A_{c}}=\frac{P\cos\theta}{A/\cos\theta}=\frac{P}{A}\cos^2\theta=\boxed{ \frac{P}{2A}(1+\cos2\theta) } $ $ \tau=\frac{F_{t}}{A_{c}}=\frac{P\sin\theta}{A\cos\theta}=\frac{P}{A}\cos\theta \sin\theta=\boxed{ \frac{P}{2A}\sin2\theta } $ - Normal stress $\sigma$ acts **perpendicular** to failure surface area - Shear stress $\tau$ acts **parallel** to failure surface area