### True Strain and Stress In plasticity we use true strain and stress: $\epsilon = \ln \frac{L}{L_{0}}= \ln(1+e)$ $\sigma = \frac{P}{A} = s(1+e)$ $s = \frac{P}{A_{0}}$ $e = \frac{L}{L_{0}}-1$ ### Metal Plasticity Due to motion of dislocations (as opposed to stretching bonds) - Incompressible - Unaffected by hydrostatic pressure #### Hall-Petch Relation Grain boundaries inhibit dislocation motion. Smaller grains are stronger. $ \sigma_{y} = \sigma_{0} +\frac{k}{\sqrt{ D }} $ $\sigma_{0} =$ "large grain" yield strength $k = \text{fit parameter}$ $d = \text{grain size}$ ### Assumptions Elastic-Perfectly-Plastic: No strain hardening Rigid-Plastic: No elastic deformation Rigid-Perfectly-Plastic: No elastic deformation or strain hardening ### Elastic-Plastic Stress-Strain Response ![[Pasted image 20240518165338.png|300]] ### Elastoplasticity Stress is independent of strain rate in metals when $T<0.35T_{m}$ (melting point in K) | | 1D | 3D | | --------------------------------- | --------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------- | | Kinematic decomposition of strain | $\epsilon = \epsilon^{e} + \epsilon^{p}$ | $\underline{\epsilon}=\underline{\epsilon}^{e} + \underline{\epsilon}^{p}$ | | Elastic constitutive relation | $\sigma = E\epsilon^{e}$ | $\underline{\sigma}=2G(\underline{\epsilon}^{e})'+K(\text{tr}\underline{\epsilon})\underline{I}$ | | Equivalent tensile plastic strain | $\bar{\epsilon}^{p} =\int \lvert \dot{\epsilon}^{p} \rvert\, dt$ | $\bar{\epsilon}^{p}= \int \sqrt{ \frac{2}{3} } \cdot\lvert \dot{{\underline{\epsilon}}}^{p} \rvert\\ \, dt$ | | Yield Condition | $\lvert \sigma \rvert = Y(\bar{\epsilon}^{p})$ | $\bar{\sigma} = Y(\bar{\epsilon}^{p})$ | | Codirectionality | $\text{sign}(\sigma) = \text{sign}(d\epsilon^{p})$ | $\text{sign}(\underline{\sigma}') = \text{sign}(d \underline{e}^{p})$ | | Flow Rule | $\dot{\epsilon}^{p} = \dot{\bar{\epsilon}}^{p} \frac{\sigma}{\bar{\sigma}}$ | $\underline{\dot{\epsilon}}^{p}=\frac{3}{2}\dot{\bar{\epsilon}}^{p}\frac{\underline{\sigma}'}{\bar{\sigma}}$ | $\underline{\dot{\epsilon}}^{p} = \text{strain rate tensor}$ $\dot{\bar{\epsilon}}^{p} = \text{equivalent strain rate}$ $\bar{\sigma}= \text{von mises stress}$ $\underline{\sigma}' = \text{deviatoric stress state}$ $\lvert \underline{A} \rvert = \sqrt{ \sum_{ij} A_{ij}^{2} }$ $\text{sign}(\underline{A}) = \frac{\underline{A}}{\lvert \underline{A} \rvert }$ #### Incremental Form When you know the stress state, you can calculate the strain state as follows: $ d\epsilon_{ij} = d\epsilon_{ij}^{e} + d\epsilon_{ij}^{p} = \underbrace{ \frac{1}{E}((1+\nu)d\sigma_{ij} -\nu d\sigma_{kk}\delta_{ij}) }_{ d\epsilon_{ij}^{e} } +\underbrace{ \frac{3}{2}d\bar{\epsilon}^{P}\frac{\sigma_{ij}'}{\bar{\sigma}} }_{ d\epsilon_{ij}^{p} } $ or $ \dot{\epsilon}= \frac{1}{E}((1+\nu)\dot{\sigma} -\nu ~\text{tr}(\dot{\sigma})\underline{1}) + \frac{3}{2}\dot{\bar{\epsilon}}^{p}\frac{\sigma'}{\bar{\sigma}} $ #### Analog Model ![[Pasted image 20240518224943.png|300]] Slider has static friction $\sigma _y$ and Mr. Plasticity adds weights to it every time it moves (strain hardening). ### Viscoplasticity Metals exhibit viscoplasticity when $T>0.35T_{m}$ due to atomic diffusion affecting plastic flow - No yield strength: $\dot{\bar{\epsilon}}^{p} \neq 0$ whenever $\bar{\sigma} \neq 0$ - hot metals experience creep and stress relaxation - material response becomes rate dependent #### New Flow Rule: $ \dot{\bar{\epsilon}}^{p}=\dot{\epsilon}_{0}\left( \frac{\bar{\sigma}}{S} \right)^{1/m} $ If $T>0.5 T_{m}$, then we have $ \dot{\epsilon}_{0}= A e^{-Q/RT} $ $m =$ strain rate sensitivity parameter, $0<m<1$ $\dot{\epsilon}_{0} =$ reference strain rate $S =$ flow strength (like Y) #### Isothermal tensile loading: $ \dot{\epsilon} = \dot{\epsilon}^{e} + \dot{\epsilon}^{p} = \frac{\dot{\sigma}}{E} + B\sigma^{n} = \frac{\dot{\epsilon}_{0}}{S^{n}} $ #### Analog Model ![[Pasted image 20240506104915.png|400]] $ \dot{\bar{\epsilon}}^{p}=\dot{\epsilon}_{0}\left( \frac{<\bar{\sigma}-Y_{th}(\bar{\epsilon}^{P})>}{S} \right)^{1/m} $ $ <u> = \begin{cases} u & \text{if } u\geq0 \\ 0 & \text{if } u<0 \end{cases} $ $Y_{th}(\bar{\epsilon}^{p}) =$ threshold resistance