\left\{ \begin{array}{ll} \frac{\partial F}{\partial z_1} = 0 \\ \frac{\partial F}{\partial z_2} = 0 \\ \frac{\partial F}{\partial z_3} = 0 \end{array} \right.$$ $$C_{11}+C_{22}+C_{33}=B_{11}+B_{22}+B_{33}$$ $$C_{ii}=B_{jj}$$ $$\frac{\partial F_1}{\partial y_1}+\frac{\partial F_1}{\partial y_1}+\frac{\partial F_1}{\partial y_1}=0$$ $$\frac{\partial F_i}{\partial y_i}=0$$ or $$F_{i,i}=0$$ $$C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{ik}\delta{jl}+\delta_{il}\delta_{jk})+\beta(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$$ ekv 1: $$\varepsilon_{ij}=\frac{1+v}{E}\sigma_{ij}-\frac{v}{E}\sigma_{kk}\delta_{ij} $$ $$i=j\Rightarrow\varepsilon_{jj}=\frac{1+v}{E}\sigma_{jj}-\frac{v}{E}\sigma_{kk}\delta_{jj}$$ $$j=k \Rightarrow \varepsilon_{kk}=\frac{1+v}{E}\sigma_{kk}-\frac{3v}{E}\sigma_{kk}=(\frac{1+v-3v}{E})\sigma_{kk}=(\frac{1-2v}{E})\sigma_{kk}$$ $$\Rightarrow \sigma_{kk}=\frac{E}{1-2v}\varepsilon_{kk}$$ sätt i ekv. 1 $$\Rightarrow \varepsilon_{ij}=\frac{1+v}{E}\sigma_{ij}-\frac{v}{1-2v}\varepsilon_{kk}\delta_ij$$ $$\Rightarrow \sigma_{ij}=\frac{E}{1+v}(\varepsilon_{ij}+\frac{v}{1-2v}\varepsilon_{kk}\delta_{ij})$$