User:Eas4200c.f08.blue.a/Lecture 27

Continuing from the equation of equilibrium in terms of $$\bar{\sigma}$$

The goal is to find $$\frac{\partial \sigma_{yx}}{\partial y}+\frac{\partial \sigma_{zx}}{\partial z}=0$$ (equation 3.14 in textbook) Lets use indicial notation $$\frac{\partial \sigma_{21}}{\partial x_{2}}+\frac{\partial \sigma_{31}}{\partial x_{3}}=0$$

Recall from section 2.4 that

$$\frac{\partial \sigma_{xx}}{\partial x}+\frac{\partial \tau_{yx}}{\partial y}+\frac{\partial \tau_{zx}}{\partial z}=0$$ (equation 2.21 in textbook)

It follows, using the indicial notation that the three equilibrium equations become

$$\frac{\partial \sigma_{11}}{\partial x_{1}}+\frac{\partial \tau_{21}}{\partial x_{2}}+\frac{\partial \tau_{31}}{\partial x_{3}}=0$$ (equation 2.21 in textbook)

$$\frac{\partial \tau_{12}}{\partial x_{1}}+\frac{\partial \sigma_{22}}{\partial x_{2}}+\frac{\partial \tau_{32}}{\partial x_{3}}=0$$ (equation 2.22 in textbook)

$$\frac{\partial \tau_{13}}{\partial x_{1}}+\frac{\partial \tau_{23}}{\partial x_{2}}+\frac{\partial \sigma_{33}}{\partial x_{3}}=0$$ (equation 2.23 in textbook)

Therefore the summation notation of the equilibrium equations is as follows

$$\sum_{i=1}^{3}{\frac{\partial \sigma_{ij}}{\partial x_{i}}}=0$$ for $$j=1,2,3$$