User:Eas4200c.f08.blue.a/Lecture 23

Reading page 70

$$\left\{\epsilon_{ij} \right\}_{6x1} = \begin{bmatrix} \bar{A}_{3x3} & o\\ 0 & \bar{B}_{3x3} \end{bmatrix}_{6x6}\left\{\sigma_{ij} \right\}_{6x1}$$

The $$\sigma$$ - $$\epsilon$$ relationship: Assuming diagonal coefficient is not zero

$$\left\{\sigma_{ij} \right\}_{6x1} = \begin{bmatrix} \bar{A}^{-1}_{3x3} & 0\\ 0 & \bar{B}^{-1}_{3x3} \end{bmatrix}_{6x6}\left\{\epsilon_{ij} \right\}_{6x1}$$

If $$\begin{bmatrix} \bar{A}^{-1}_{3x3} & 0\\ 0 & \bar{B}^{-1}_{3x3} \end{bmatrix}_{6x6}$$ we can varify that the relationship by $$\bar{c}^{-1}\bar{c} = \bar{I}$$ then $$\bar{c}^{-1}\bar{c} = \begin{bmatrix} \bar{A}^{-1}\bar{A} & 0 \\ 0 & \bar{B}^{-1}\bar{B} \end{bmatrix} = \bar{I}$$ therefore,

$$\left\{\sigma_{ij} \right\} = \begin{bmatrix} \bar{A}^{-1} & 0 \\ 0 & \bar{B}^{-1} \end{bmatrix}\begin{Bmatrix} 0\\ 0\\ 0\\ 0\\ \epsilon_{31}\\ \epsilon_{12} \end{Bmatrix}$$

Road map

Equilibrium Equation for Stress in a nonuniform stress field (section 2.4 in the book)

Consider a one dimensional case as a model.For a nonuniform field, the distribution is not constant, therefore $$f\left(x \right)\neq constant$$



and showing the shear forces acting on this