User:Eas4200c.f08.blue.a/Lecture 22

How to relate strains to $$E,\nu$$

first is to define Hooke's law in three dimensions which is

for normal strains:

$$\epsilon_{xx}=\frac{\sigma_{xx}}{E}-\frac{\nu*\sigma_{yy}}{E}-\frac{\nu*\sigma_{zz}}{E}$$

$$\epsilon_{yy}=\frac{\sigma_{yy}}{E}-\frac{\nu*\sigma_{xx}}{E}-\frac{\nu*\sigma_{zz}}{E}$$

$$\epsilon_{zz}=\frac{\sigma_{zz}}{E}-\frac{\nu*\sigma_{xx}}{E}-\frac{\nu*\sigma_{yy}}{E}$$

for shear strains:

$$\gamma_{xy}=2*\epsilon_{xy}=\frac{\tau_{xy}}{G}$$

$$\gamma_{yz}=2*\epsilon_{yz}=\frac{\tau_{yz}}{G}$$

$$\gamma_{xz}=2*\epsilon_{xz}=\frac{\tau_{xz}}{G}$$

from the symmetry of the strain tensor

$$   \mathbf{\epsilon} = \begin{bmatrix} \epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\ \epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\ \epsilon_{31} & \epsilon_{32} & \epsilon_{33} \end{bmatrix} ~. $$

it is clear that $$\epsilon=[\epsilon_{ij}], \sigma=[\sigma_{ij}]$$ which are both three by three matrices.

next arrange the strain components $$\epsilon_{ij}$$ and the stress components in column matrices. this is also called the voigt notation.

$$ \mathbf{\epsilon_{ij}} = \begin{bmatrix} \epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{23}\\ \epsilon_{31}\\ \epsilon_{12}\\ \end{bmatrix} $$

and

$$ \mathbf{\sigma_{ij}} = \begin{bmatrix} \sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23}\\ \sigma_{31}\\ \sigma_{12}\\ \end{bmatrix}

$$

next hookes law for an isotropic material can be written which is

$$                     \begin{bmatrix} \epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{23}\\ \epsilon_{31}\\ \epsilon_{12}\\ \end{bmatrix}~= $$                            $$                      \begin{bmatrix} \frac{1}{E}&\frac{-\nu}{E}&\frac{-\nu}{E}&0&0&0\\ \frac{-\nu}{E}&\frac{1}{E}&\frac{-\nu}{E}&0&0&0\\ \frac{-\nu}{E}&\frac{-\nu}{E}&\frac{1}{E}&0&0&0\\ 0&0&0&\frac{1}{2*G}&0&0\\ 0&0&0&0&\frac{1}{2*G}&0\\ 0&0&0&0&0&\frac{1}{2*G}\\ \end{bmatrix}~* $$ $$                     \begin{bmatrix} \sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23}\\ \sigma_{31}\\ \sigma_{12}\\ \end{bmatrix}

$$

which can also be written as

$$                     \begin{bmatrix} \epsilon_{xx}\\ \epsilon_{yy}\\ \epsilon_{zz}\\ \gamma_{yz}\\ \gamma_{zx}\\ \gamma_{xy}\\ \end{bmatrix}~= $$                            $$                      \begin{bmatrix} \frac{1}{E}&\frac{-\nu}{E}&\frac{-\nu}{E}&0&0&0\\ \frac{-\nu}{E}&\frac{1}{E}&\frac{-\nu}{E}&0&0&0\\ \frac{-\nu}{E}&\frac{-\nu}{E}&\frac{1}{E}&0&0&0\\ 0&0&0&\frac{1}{G}&0&0\\ 0&0&0&0&\frac{1}{G}&0\\ 0&0&0&0&0&\frac{1}{G}\\ \end{bmatrix}~* $$ $$                     \begin{bmatrix} \sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \tau_{yz}\\ \tau_{zx}\\ \tau_{xy}\\ \end{bmatrix}

$$

The poisson ratio $$\nu$$ for certain materials was also mentioned in class such as cork which is unique for the fact that is has a zero poissons ratio which means that when it is compressed or put under tension there is no change in volume in the directions other than that of the force applied. for example if there was a cylinder of cork of some radius R and it was put under compression the radius would not change. More can be found about the poisson ratio here