Displacement
Obviously, according to the continuous condition, the displacements $\vec{u}$ is continuous at the surface $S$. If it is not, there are holes or overlaps in the body which is contradictory to the continuous condition.
Assume the interface $S$ is $z=0$.
| Solarized dark | Solarized Ocean |
|---|---|
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Derivatives of displacements
The derivatives of the displacements along the tangent direction on the surface is continuous too. The definition of two derivatives on both surface are
\[u_{i,j}^1=\frac{\partial u_i^1}{\partial x_j}=\lim_{\Delta x_i \rightarrow 0} \frac{u^1(x_i + \Delta x_i) - u^1(x_i)}{\Delta x_i} \\ u_{i,j}^2=\frac{\partial u_i^2}{\partial x_j}=\lim_{\Delta x_i \rightarrow 0} \frac{u^2(x_i + \Delta x_i) - u^2(x_i)}{\Delta x_i}, i=1,2,3;j=1,2\]And because of the continuous condition of $\vec u$, we have
\[u^1(x_i + \Delta x_i)=u^2(x_i + \Delta x_i)\\ u^1(x_i) = u^2(x_i).\]Therefore, we have
\[u_{i,j}^1=u_{i,j}^2\]Stress
According to the static condition (without inerial force),
\[\sigma_{ij}^1 n_j - \sigma_{ij}^2 n_j = \sigma_{ij,j}^1 - \sigma_{ij,j}^2 = \rho_1 \ddot{u}_i-\rho_2 \ddot u_i=0\]Therefore, $\sigma_{33}, \sigma_{13}, \sigma_{23}$ are continuous.
Strain
According to the constitutive law
\[\begin{aligned} \varepsilon_{11} &=\frac{1}{E}\left(\sigma_{11}-\nu\left(\sigma_{22}+\sigma_{33}\right)\right) \\ \varepsilon_{22} &=\frac{1}{E}\left(\sigma_{22}-\nu\left(\sigma_{11}+\sigma_{33}\right)\right) \\ \varepsilon_{33} &=\frac{1}{E}\left(\sigma_{33}-\nu\left(\sigma_{11}+\sigma_{22}\right)\right) \\ \varepsilon_{12} &=\frac{1}{2 G} \sigma_{12}\\ \varepsilon_{13} &=\frac{1}{2 G} \sigma_{13}\\ \varepsilon_{23} &=\frac{1}{2 G} \sigma_{23} \end{aligned}\]And definition of strain
\[\begin{aligned} \varepsilon_{1 1} &= \frac{\partial u_{1}}{\partial x_{1}} \\ \varepsilon_{2 2} &= \frac{\partial u_{2}}{\partial x_{2}} \\ \varepsilon_{3 3} &= \frac{\partial u_{3}}{\partial x_{3}} \\ \varepsilon_{1 2} &= \frac{1}{2}\left(\frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial u_{2}}{\partial x_{1}}\right) \\ \varepsilon_{1 3} &= \frac{1}{2}\left(\frac{\partial u_{1}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{1}}\right) \\ \varepsilon_{2 3} &= \frac{1}{2}\left(\frac{\partial u_{2}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{2}}\right) \end{aligned}\]Therefore, $\varepsilon_{11}, \varepsilon_{22}, \varepsilon_{1 2}$ are continuous.
Conclusion
| Contiunuous | Uncontiunous | Undetermined | |
|---|---|---|---|
| Displacements | $u_1,u_2,u_3$ | / | / |
| Stress | $\sigma_{33}, \sigma_{13}, \sigma_{23}$ | $\sigma_{12}$ | $\sigma_{22},\sigma_{33}$ |
| Strain | $\varepsilon_{11}, \varepsilon_{22}, \varepsilon_{1 2}$ | $\varepsilon_{13}, \varepsilon_{23}$ | $ \varepsilon_{33}$ |