## Large Deformations ### Stretch Alternative way of expressing strain that is more convenient for large deformations $ \lambda_{i}=\frac{dx_{i}}{dX_{i}} $ ![[Pasted image 20240328140355.png|350]] ### Volume Ratio $ J=\frac{\text{deformed volume}}{\text{original volume}} =\frac{dx_{1}dx_{2}dx_{3}}{dX_{1}dX_{2}dX_{3}}=\lambda_{1}\lambda_{2}\lambda_{3} $ If incompressible: $J=1$ ### Strain-Energy Density $ \psi(\lambda_{1},\lambda_{2},\lambda_{3},{\text{material properties}}) =\frac{\text{energy stored}}{\text{original volume}} =S_{1}\dot{\lambda}_{1}+S_{2}\dot{\lambda}_{2}+S_{3}\dot{\lambda}_{3} $ ### Engineering Stress $ S = \frac{F}{A_{0}}=\frac{ \partial \psi }{ \partial \lambda} $ ### Engineering Strain $ e = \frac{L-L_{0}}{L_{0}}=\lambda-1 $ ### True Stress $ \sigma = \frac{F}{A} = \frac{\lambda}{J} \frac{ \partial \psi }{ \partial \lambda} =s(1+e) $ ### True Strain $ \epsilon =\ln \frac{L}{L_{0}} = \ln(1+e) $ ### Incompressible Rubber Constitutive Relation Due to incompressibility, arbitrary hydrostatic pressure $p$ does no work ($\psi$ is unaffected) and $J=\lambda_{1}\lambda_{2}\lambda_{3}=1$ $ \sigma_{i}=\lambda_{i}\frac{ \partial \psi }{ \partial \lambda i } -p $ ### Neo-Hookean Simplest and most common strain-energy function for rubber $ \psi_{\text{NH}} = C\cdot (\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-3) $ $C$ can be determined from a small deformation stretch or shear test: $C = \frac{E}{6}=\frac{G}{2}$ Experiments show that rubber is stiffer than neo-hookean beyond $\lambda=1.5$ ### Stretch Invariants Constants that are unaffected by rotation $ \begin{align} I_{1} &=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2} \\ I_{2}&=\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{1}^{2} \lambda_{3}^{2} \\ I_{3}&= \lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2} \end{align} $ ### Other Material Models Neo-Hookean: $\psi=C\cdot(I_{1}-3)$ Mooney-Rivlin: $\psi = C_{1} \cdot (I_{1}-3) + C_{2}\cdot(I_{2}-3)$