User:Eas4200c.f08.blue.a/Lecture 21

As a continuation, the 3rd Ad Hoc point in the engineering derivation is the inconsistency in the assumption on the size of angle $$\alpha$$ : To get line PP', assume $$\alpha$$ is finite and $$\rho = r\cos \alpha$$, then reintroduce the small angle assumption afterwards.

Returning back to the formal derivation:

$$\bar{\epsilon} = \begin{bmatrix} \epsilon_{xx} & \epsilon_ {xy}&\epsilon_{xz} \\ \epsilon_{yx} & \epsilon_ {yy}&\epsilon_{yz} \\ \epsilon_{zx} & \epsilon_{zy} &\epsilon_{zz} \end{bmatrix}$$

This matrix can also be represented by

$$\bar{\epsilon} = \begin{bmatrix} \epsilon_{11} & \epsilon_ {12}&\epsilon_{13} \\ \epsilon_{21} & \epsilon_ {22}&\epsilon_{23} \\ \epsilon_{31} & \epsilon_{32} &\epsilon_{33} \end{bmatrix} = [\epsilon_{ij}]$$ where i,j = 1,2,3

i is the row index and j is the column index

$$\epsilon _{ij} = \frac{1}{2}\left( \frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}\right)$$ where $$x\Leftrightarrow x_{1}$$, $$y\Leftrightarrow x_{2}$$, and ,$$z\Leftrightarrow x_{3}$$

For example $$\epsilon _{11} = \frac{1}{2}\left( \frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{1}}{\partial x_{1}}\right) = \frac{\partial u_{1}}{\partial x_{1}} = \frac{\partial u_{x}}{\partial x_{x}}$$

Hence: only 6 independent components of $$\epsilon$$. Similarly for the stress tensor $$\bar{\sigma }= \left[ \sigma_{ij}\right]$$

Also, there are only 6 independent components of $$\bar{\sigma}$$ in 3-D

A question asked in class was whether or not $$\bar{\epsilon}$$ is related to isotropy of the mater. The answer is NO, isotropic elasticity is related to material behavior.

One last keynote to emphasis is the fact that

$$\epsilon _{xx} = \epsilon _{yy} = \epsilon _{zz} = \gamma _{xy} = 0$$

and from the stress-strain relationship

$$\sigma _{xx}=\sigma _{yy}=\sigma _{zz}=\tau _{yz}=0$$