Displacement
Obviously, according to the continuous condition, the displacements u ⃗ \vec{u} u S S S
Assume the interface S S S z = 0 z=0 z = 0
Solarized dark Solarized Ocean
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 = ∂ u i 1 ∂ x j = lim Δ x i → 0 u 1 ( x i + Δ x i ) − u 1 ( x i ) Δ x i u i , j 2 = ∂ u i 2 ∂ x j = lim Δ x i → 0 u 2 ( x i + Δ x i ) − u 2 ( x i ) Δ x i , i = 1 , 2 , 3 ; j = 1 , 2 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 u i , j 1 = ∂ x j ∂ u i 1 = Δ x i → 0 lim Δ x i u 1 ( x i + Δ x i ) − u 1 ( x i ) u i , j 2 = ∂ x j ∂ u i 2 = Δ x i → 0 lim Δ x i u 2 ( x i + Δ x i ) − u 2 ( x i ) , i = 1 , 2 , 3 ; j = 1 , 2 And because of the continuous condition of u ⃗ \vec u u
u 1 ( x i + Δ x i ) = u 2 ( x i + Δ x i ) u 1 ( x i ) = u 2 ( x i ) . u^1(x_i + \Delta x_i)=u^2(x_i + \Delta x_i)\\
u^1(x_i) = u^2(x_i). u 1 ( x i + Δ x i ) = u 2 ( x i + Δ x i ) u 1 ( x i ) = u 2 ( x i ) . Therefore, we have
u i , j 1 = u i , j 2 u_{i,j}^1=u_{i,j}^2 u i , j 1 = u i , j 2 Stress
According to the static condition (without inerial force),
σ i j 1 n j − σ i j 2 n j = σ i j , j 1 − σ i j , j 2 = ρ 1 u ¨ i − ρ 2 u ¨ i = 0 \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 σ ij 1 n j − σ ij 2 n j = σ ij , j 1 − σ ij , j 2 = ρ 1 u ¨ i − ρ 2 u ¨ i = 0 Therefore, σ 33 , σ 13 , σ 23 \sigma_{33}, \sigma_{13}, \sigma_{23} σ 33 , σ 13 , σ 23
Strain
According to the constitutive law
ε 11 = 1 E ( σ 11 − ν ( σ 22 + σ 33 ) ) ε 22 = 1 E ( σ 22 − ν ( σ 11 + σ 33 ) ) ε 33 = 1 E ( σ 33 − ν ( σ 11 + σ 22 ) ) ε 12 = 1 2 G σ 12 ε 13 = 1 2 G σ 13 ε 23 = 1 2 G σ 23 \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} ε 11 ε 22 ε 33 ε 12 ε 13 ε 23 = E 1 ( σ 11 − ν ( σ 22 + σ 33 ) ) = E 1 ( σ 22 − ν ( σ 11 + σ 33 ) ) = E 1 ( σ 33 − ν ( σ 11 + σ 22 ) ) = 2 G 1 σ 12 = 2 G 1 σ 13 = 2 G 1 σ 23 And definition of strain
ε 11 = ∂ u 1 ∂ x 1 ε 22 = ∂ u 2 ∂ x 2 ε 33 = ∂ u 3 ∂ x 3 ε 12 = 1 2 ( ∂ u 1 ∂ x 2 + ∂ u 2 ∂ x 1 ) ε 13 = 1 2 ( ∂ u 1 ∂ x 3 + ∂ u 3 ∂ x 1 ) ε 23 = 1 2 ( ∂ u 2 ∂ x 3 + ∂ u 3 ∂ x 2 ) \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} ε 11 ε 22 ε 33 ε 12 ε 13 ε 23 = ∂ x 1 ∂ u 1 = ∂ x 2 ∂ u 2 = ∂ x 3 ∂ u 3 = 2 1 ( ∂ x 2 ∂ u 1 + ∂ x 1 ∂ u 2 ) = 2 1 ( ∂ x 3 ∂ u 1 + ∂ x 1 ∂ u 3 ) = 2 1 ( ∂ x 3 ∂ u 2 + ∂ x 2 ∂ u 3 ) Therefore, ε 11 , ε 22 , ε 12 \varepsilon_{11}, \varepsilon_{22}, \varepsilon_{1 2} ε 11 , ε 22 , ε 12
Conclusion
Contiunuous Uncontiunous Undetermined Displacements u 1 , u 2 , u 3 u_1,u_2,u_3 u 1 , u 2 , u 3 / / Stress σ 33 , σ 13 , σ 23 \sigma_{33}, \sigma_{13}, \sigma_{23} σ 33 , σ 13 , σ 23 σ 12 \sigma_{12} σ 12 σ 22 , σ 33 \sigma_{22},\sigma_{33} σ 22 , σ 33 Strain ε 11 , ε 22 , ε 12 \varepsilon_{11}, \varepsilon_{22}, \varepsilon_{1 2} ε 11 , ε 22 , ε 12 ε 13 , ε 23 \varepsilon_{13}, \varepsilon_{23} ε 13 , ε 23 ε 33 \varepsilon_{33} ε 33
