## Viscoelasticity ### Math functions #### Heaviside function $ h(t) =\begin{cases} 0 & t\leq 0 \\ 1 & t>0 \end{cases} $ #### Dirac function $ \delta(t) = \dot{h}(t) = \begin{cases} 0 & \text{for } t \neq 0 \\ \infty & \text{for } t=0 \end{cases} $ $ \int_{-\infty}^{\infty} \delta(t) \, dt=1 $ For any function $g(t)$, continuous at $t=0$: $ \int_{-\infty}^{\infty} g(t)\delta(t) \, dt=g(0) $ ### Material Properties #### Relaxation time $ \tau_{R} = \frac{\eta}{E_{2}} $ $\eta=\text{viscosity}$ #### Creep retardation time $ \tau_{C} = \frac{E_{1}+E_{2}}{E_{1}}\tau_{R} $ ### Standard Linear Solid Model for viscoelastic behavior consisting of springs and dampers $ (\eta+E_{1}\tau_{R})\dot{\epsilon}-\tau_{R}\dot{\sigma}=\sigma-E_{1}\epsilon $ ![[Pasted image 20240416173556.png|250]] ### Stress Relaxation Stress spikes and relaxes over time when a sudden strain is applied: $\epsilon=\epsilon_{0}h(t)$ e.g. clamping in a vice $ \begin{align} E_{r}(t) &=\frac{\sigma(t)}{\epsilon_{0}} = E_{2}e^{-t/\tau_{R}} + E_{1} \\ &=E_{\text{re}}+(E_{\text{rg}}+E_\text{re})e^{-t/\tau_{R}} \end{align} $ ### Creep How a material elongates over time after application of sudden stress: $\sigma = \sigma_{0}h(t)$ e.g. hanging a weight $ \epsilon(t)=C e^{-t/\tau_{C}} + \frac{\sigma_{0}}{E_{1}} $ $ J_{c}(t) = \frac{\epsilon(t)}{\sigma_{0}} = \left( \frac{1}{E_{1}+E_{2}}-\frac{1}{E_{1}} \right)e^{-t/\tau_{C}}+\frac{1}{E_{1}} $ ### Boltzmann Superposition Principle Relates stress history and strain history #### Discrete form Input: $ \sigma(t)=\sum_{i=1}^n h(t-t_{i})\Delta\sigma_{i} $ Output: $ \epsilon(t) = \sum_{i=1}^N J_{c}(t-t_{i})\Delta\sigma_{i} $ #### Stress relaxation integral form Input: $ \epsilon(t) = \int _{0^-}^t h(t-\tau)\frac{d\sigma(\tau)}{d\tau} \, d\tau $ Output: $ \sigma(t)=\int _{0^-}^t E_{r}(t-\tau) \frac{d\epsilon(t)}{d\tau} \, d\tau $ #### Creep integral form Input: $ \sigma(t) = \int _{0^-}^t h(t-\tau)\frac{d\epsilon(\tau)}{d\tau} \, d\tau $ Output: $ \epsilon(t)=\int _{0^-}^t J_{r}(t-\tau) \frac{d\sigma(\tau)}{d\tau} \, d\tau $ ### Correspondence Principle Pretend body is elastic and replace $J$ with $J_{c}$ or replace $E$ with $E_{r}$ $ \underline{u}(\underline{x},t) = \underline{u}_{0}(\underline{x})\cdot \frac{J_{c}(t)}{J} $ $ \underline{\sigma}(\underline{x},t) = \underline{\sigma}_{0}(\underline{x})\cdot \frac{E_{r}(t)}{E} $ ### Dynamic Mechanical Analysis (DMA) Viscoelastic response to oscillatory inputs #### Stress Input Stress Input: $ \sigma(t) = \sigma_{0}\cos \omega t $ Strain Output: $ \begin{align} \epsilon(t) &= \epsilon_{0}\cos(\omega t-\delta) \\ &=\sigma_{0} (J' \cos(\omega t)+J''\sin(\omega t)) \end{align} $ ![[Pasted image 20240408112504.png|425]] #### Strain Input Strain Input: $ \epsilon(t) = \epsilon_{0}\cos\omega t $ Stress Output: $ \begin{align} \sigma(t) &= \sigma_{0}\cos(\omega t-\delta) \\ &=\epsilon_{0}(E'\cos(\omega t) -E''\sin(\omega t)) \end{align} $ #### Material Properties ##### Loss Angle Phase lag of DMA, depends on frequency and temp: $\delta=\delta(\omega,T)$ Fully out of phase if $\delta=\frac{\pi}{2}$ ##### Loss Tangent $ \tan\delta = \frac{J''}{J'}=\frac{E''}{E'} $ ##### Storage compliance Measure of how in-phase the strain is with the stress, function of $\omega$ $ J' = \frac{\epsilon_{0}}{\sigma_{0}}\cos\delta $ ##### Loss compliance Measure of how out-of-phase the strain is with the stress, function of $\omega$ $ J'' = \frac{\epsilon_{0}}{\sigma_{0}}\sin\delta $ ##### Storage modulus Represents rate of energy absorbed by material $ E' = \frac{\sigma_{0}}{\epsilon_{0}}\cos\delta $ ##### Loss modulus Represents rate of energy dissipated by material $ E'' = \frac{\sigma_{0}}{\epsilon_{0}}\sin\delta $ #### Loss per cycle Power expended per volume: $ \begin{align} P &= \sigma \dot{\epsilon} \\ &=-\omega\sigma_{0}\epsilon_{0}\sin(\omega t)\cos(\omega t+\delta) \end{align} $ Work done in one cycle with period $T=\frac{2\pi}{\omega}$ (dissipation loss per cycle): $ \begin{align} W&=\int _{0}^T P \, dt \\ &=\pi\sigma_{0}\epsilon_{0}\sin\delta \\ &=\pi\sigma_{0}^{2}J'' \\ \end{align} $ Dissipation depends on $\delta, E'', \text{and } J''$ ![[Pasted image 20240416180522.png|375]]