## Governing Equations for 3D Elastic Bodies ### Equations of Motion Relates acceleration and forces to stress $ \sum_{j=1}^{3} \frac{\partial \sigma_{ij}}{\partial x_j} + \rho g_i = \rho a_i \quad \text{for } i = 1,2,3 $ #### Expanded form $ \begin{align} \frac{\delta\sigma_{11}}{\delta x}+\frac{\delta\sigma_{12}}{\delta y}+\frac{\sigma_{13}}{\delta z}+\rho g_{1}&=\rho a_{1} \\ \frac{\delta\sigma_{21}}{\delta x}+\frac{\delta\sigma_{22}}{\delta y}+\frac{\sigma_{23}}{\delta z}+\rho g_{2}&=\rho a_{2} \\ \frac{\delta\sigma_{31}}{\delta x}+\frac{\delta\sigma_{32}}{\delta y}+\frac{\sigma_{33}}{\delta z}+\rho g_{3}&=\rho a_{3} \end{align} $ ### Strain-Displacement Relation Relates strain and displacement $ [\underline{\epsilon}]=\frac{1}{2}([\nabla \underline{u}]+[\nabla \underline{u}]^T) $ ### Elastic Constitutive Relation Relates stress and strain $ [\underline{\sigma}]=K(\text{tr}\underline{\epsilon})[\underline{1}] + 2G[\underline{\epsilon}'] $ #### Compliance relation Inverting the above equation $ \begin{align} [\underline{\epsilon}]&=\frac{1}{9K}(\text{tr}\underline{\sigma})[\underline{1}]+\frac{1}{2G}[\underline{\sigma}'] \\ \\ &=\frac{1}{E}(-\nu(\text{tr}\underline{\sigma})[\underline{1}] + (1+\nu)[\underline{\sigma}]) \end{align} $ #### Expanded form $ \begin{align} \epsilon_{11} &= \frac{1}{E} (\sigma_{11} - \nu (\sigma_{22} + \sigma_{33})) \\ \epsilon_{22} &= \frac{1}{E} (\sigma_{22} - \nu (\sigma_{33} + \sigma_{11}) )\\ \epsilon_{33} &= \frac{1}{E} (\sigma_{33} - \nu (\sigma_{11} + \sigma_{22}) ) \\ \epsilon_{12} &=\frac{1+\nu}{E}\sigma_{12}\\ \epsilon _{23}&=\frac{1+\nu}{E}\sigma_{23}\\ \epsilon _{31}&=\frac{1+\nu}{E}\sigma_{31}\\ \end{align} $ ## Stress ### Cauchy's Result $ [\underline{t}(\underline{x},\underline{n})]= [\underline{\sigma}(\underline{x})][\underline{n}] $ ### Stress Tensor $\underline{\sigma}$: tensor, physical load state, inputs vector $\underline{n}$ and outputs vector $\underline{t}$ $[\underline{\sigma}]$: matrix, representation of $\underline{\sigma}$ in a given basis $ [\underline{\sigma}] = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{11} & \sigma_{12} & \sigma_{13} \\\end{bmatrix} $ $ [\underline{\sigma}]=[\underline{\sigma}]^T $ ### Stress Decomposition $ [\underline{\sigma}]=\sigma_{M}[\underline{1}]+[\underline{\sigma}'] $ #### Spherical Part Tries to make the material change volume $ \sigma_{M}= \frac{1}{3}tr(\underline{\sigma}) $ #### Deviatoric Part Tries to make the material change shape $ [\underline{\sigma}']= [\underline{\sigma}]-\sigma_{M}[\underline{1}] $ $ tr([\underline{\sigma}'])=0 $ ### Principle Stress $ [\underline{\sigma}]^P= \begin{bmatrix} \sigma_{1}^P & 0 & 0 \\ 0 & \sigma_{2}^P & 0 \\ 0 & 0 & \sigma_{3}^P \end{bmatrix} $ No off-diags, no shear stress Each of $\sigma_{i}^P$ and $\underline{e}_{i}^P$ are eigenvalue eigenvector pairs of $[\underline{\sigma}]$ ### Common Stress States #### Simple Tension $ [\underline{\sigma}] = \begin{bmatrix} 0 & 0 & 0 \\ 0 & \sigma & 0 \\ 0 & 0 & 0 \end{bmatrix} $ #### Simple Shear $ [\underline{\sigma}] = \begin{bmatrix} 0 & 0 & \tau \\ 0 & 0 & 0 \\ \tau & 0 & 0 \end{bmatrix} $ #### Hydrostatic Pressure $ [\underline{\sigma}]=\begin{bmatrix} -p & 0 & 0 \\ 0 & -p & 0 \\ 0 & 0 & -p \end{bmatrix} $ #### Plane Stress One of $\sigma_{i}^P=0$ Can simplify to a 2D problem Does not lead to plane strain in most cases ## Strain ### Displacement Field $ \underline{u}(\underline{X})=\underline{x}-\underline{X} $ $ [\nabla u] = \begin{bmatrix} \frac{\delta u_{1}}{\delta x_{1}} & \frac{\delta u_{1}}{\delta x_{2}} & \frac{\delta u_{1}}{\delta x_{3}} \\ \frac{\delta u_{2}}{\delta x_{1}} & \frac{\delta u_{2}}{\delta x_{2}} & \frac{\delta u_{2}}{\delta x_{3}} \\ \frac{\delta u_{3}}{\delta x_{1}} & \frac{\delta u_{3}}{\delta x_{2}} & \frac{\delta u_{3}}{\delta x_{3}} \end{bmatrix} $ ### Strain Matrix $ \begin{align} [\underline{\epsilon}]&=\frac{1}{2}([\nabla \underline{u}]+[\nabla \underline{u}]^T) \\ \\ &= \begin{bmatrix} \frac{du_{1}}{dX_{2}} & \frac{1}{2}\left( \frac{du_{1}}{dX_{2}}+\frac{du_{2}}{dX_{1}} \right) & \dots\\ \frac{1}{2}\left( \frac{du_{1}}{dX_{2}} +\frac{du_{2}}{dX_{1}}\right) & \frac{du_{2}}{dX_{2}} & \dots \\ \dots & \dots & \dots \end{bmatrix} \end{align} $ - Strain matrix is symmetric - Off-diags reflect angle change, diags reflect length change - Diagonal reflects - Pure rotations and translations give $[\underline{\epsilon}]=0$ $ \epsilon_{ij} \begin{cases} i\neq j : & (\text{angle change between } \underline{e}_{i} \text{ and } \underline{e}_{j})/2 \\ i=j: & \text{relative length change} \end{cases} $ $[\underline{\epsilon}]$ relates the dot product from $\underline{a}$ and $\underline{b}$ to $\underline{\alpha}$ and $\underline{\beta}$ $ [\underline{a}]\cdot([\underline{\epsilon}][\underline{b}])=\frac{1}{2}(\underline{\alpha}\cdot\underline{\beta}-\underline{a}\cdot \underline{b}) $ ### Decomposing Strain $ \begin{align} [\underline{\epsilon}] &= \underbrace{ \frac{1}{3}\text{tr}(\underline{\epsilon})[\underline{1}] }_{ spherical \ part }+ \underbrace{ [\underline{\epsilon}'] }_{ deviatoric \ part } \\ \\ &=\frac{1}{9K}(\text{tr}\underline{\sigma})[\underline{1}]+\frac{1}{2G}[\underline{\sigma}'] \end{align} $ $ [\underline{\epsilon}']=\frac{1}{3}\text{tr}(\underline{\epsilon})[\underline{1}]-[\underline{\epsilon}] $ - Deviatoric part causes shape change - Spherical part causes volume change ### Common Deformations #### Simple Shear $ \underline{\epsilon}=\begin{bmatrix} 0 & \frac{\gamma}{2} & 0 \\ \frac{\gamma}{2} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $ #### Uniform Compaction $ \underline{\epsilon}=\begin{bmatrix} -\frac{\Delta}{3} & 0 & 0 \\ 0 & -\frac{\Delta}{3} & 0 \\ 0 & 0 & -\frac{\Delta}{3} \end{bmatrix} $ $ \text{Volumetric Strain}=tr\underline{\epsilon}= \frac{\text{volume change}}{\text{initial vol}}=-\Delta $ ## Material Properties ### Young's Modulus Resistance to stretching $ E = \frac{\sigma_{11}}{\epsilon_{11}}=\frac{9KG}{3K+G} $ ### Shear Modulus Resistance to shear $ G=\frac{E}{2(1+\nu)} $ $ [\underline{\sigma}']=2G[\underline{\epsilon}'] $ ### Bulk Modulus Resistance to volume change $ K = \frac{E}{3(1-2\nu)} $ $ \sigma_{m}=3K\epsilon_{m}=K(tr\underline{\epsilon})[\underline{1}] $ ### Poisson's Ratio Ratio of lateral contraction to longitudinal strain $ \nu= -\frac{\epsilon_{22}}{\epsilon_{11}}=\frac{3K-2G}{6K+2G} $ As material becomes incompressible, $\frac{K}{G} \to \infty$ and $\nu\to \frac{1}{2}$ ### Lamé Coefficient $ \lambda=K-\frac{2}{3}G $ ## Stress Concentration :sob: ### Thick-Walled Cylinders ![[Pasted image 20240519042131.png|400]] ![[Pasted image 20240519042320.png|295]] Solution after $ \begin{align} \sigma_{rr} &= \frac{ p_{i}-p_{o}\left( \frac{b}{a} \right)^{2} - (p_{i}-p_{o})\left( \frac{b}{r} \right) ^{2}}{\left( \frac{b}{a} \right)^{2}-1} \\ \sigma_{\theta\theta} &= \frac{ p_{i}-p_{o}\left( \frac{b}{a} \right)^{2} +(p_{i}-p_{o})\left( \frac{b}{r} \right) ^{2}}{\left( \frac{b}{a} \right)^{2}-1} \\ \sigma_{zz} &= \frac{ 2\nu\left( p_{i}-p_{o}\left( \frac{b}{a} \right)^{2} +E\epsilon_{zz} \right)}{\left( \frac{b}{a} \right)^{2}-1} \end{align} $ $ u_{r} = \frac{1+\nu}{E} \frac{r}{\left( \frac{b}{a} \right)^{2}-1}((1-2\nu)\left( p_{i}-p_{o}\left( \frac{b}{a} \right)^{2} \right)+(p_{i}-p_{o})\left( \frac{b}{r}^{2} \right)) - \nu r\epsilon_{0} $ ### Stress Concentration Factor ### Failure Criterion ### Von Mises $ \sqrt{ \frac{3}{2} }\sqrt{ (\sigma_{1}')^{2}+(\sigma_{2}')^{2}+(\sigma_{3}')^{2} }=\bar{\sigma}\leq\sigma _{y} $ $ \bar{\sigma}=\sqrt{ \frac{1}{2}((\sigma_{11}-\sigma_{22})^{2}+(\sigma_{22}-\sigma_{33})^{2}+(\sigma_{11}-\sigma_{33})^{2})+3(\sigma_{12}^{2}+\sigma_{23}^{2}+\sigma_{13}^{2}) } $ ### Tresca $ f([\underline{\sigma}])=\frac{\sigma_{1}^P-\sigma_{2}^P}{2}=\tau_{max}\leq \frac{\sigma_{y}}{2}=\tau_{y,\text{tresca}} $ $\tau_{max}$ is maximal shear stress $\tau_{max}=\text{max}|\underline{m}\cdot (\underline{\sigma} \underline{n})|$ $\underline{m}\perp \underline{n}$ ### Fast Fracture Fast fracture occurs when $ \sigma_{1}^P\leq\sigma_{c}=\text{Critical Fracture Stress} $ We use $\sigma_{1}^P$ not von mises because micro cracks are so small that only local tension is relevant