ONDAS DE RAYLEIGH (Ondas de Superfície)

DEFORMAÇÃO E TENSÃO

O tensor de deformação \varepsilon, pode ser descrito de maneira geral como:

\begin{equation}\color{black}\varepsilon_{i j}=\frac{1}{2}(u_{i,j}+u_{j,i})\end{equation}

Assim:

\begin{equation}\color{black}\varepsilon_{x x}=k^2 \left(- Ae^{-k z \sqrt{1-\frac{c^2}{c_L^2} }}+j \sqrt{1-\frac{c^2}{c_T^2}}+ Be^{-k z \sqrt{1-\frac{c^2}{c_T^2}}} \right) e^{j k(x - c t)}\end{equation}

\begin{equation}\color{black}\varepsilon_{y y}=0\end{equation}

\begin{equation}\color{black}\varepsilon_{z z}=k^2 \left((1-\frac{c^2}{c_L^2}) Ae^{-k z \sqrt{1-\frac{c^2}{c_L^2} }}-j \sqrt{1-\frac{c^2}{c_T^2}} Be^{-k z \sqrt{1-\frac{c^2}{c_T^2} }}\right) e^{j k(x -c t)}\end{equation}

\begin{equation}\color{black}\varepsilon_{x y}=0\end{equation}

\begin{equation}\color{black}\varepsilon_{y z}=0\end{equation}

\begin{equation}\color{black}\varepsilon_{x z}=\frac{1}{2}k^2 \left(2 j \sqrt{1-\frac{c^2}{c_T^2}}Ae^{-k z \sqrt{1-\frac{c^2}{c_L^2} }}+(2-\frac{c^2}{c_T^2}) Be^{-k z \sqrt{1-\frac{c^2}{c_T^2} }}\right) e^{j k(x -c t)}\end{equation}

Pode-se então calcular o tensor de tensões \sigma que, para um meio isotrópico, assume a seguinte expressão: \sigma_{i j}=\lambda \delta_{i j} \varepsilon_{k k}+2 \mu \varepsilon_{i j}:

\begin{equation} \color{black}\sigma_{x x}=\lambda (\varepsilon_{x x}+\varepsilon_{z z})+2 \mu \varepsilon_{x x}= \mu k^2 \left((1-\frac{c^2}{c_T^2}-2(1-\frac{c^2}{c_L^2})-1)Ae^{-k z \sqrt{1-\frac{c^2}{c_L^2} }} + 2 j \sqrt{1-\frac{c^2}{c_T^2}} Be^{-k z \sqrt{1-\frac{c^2}{c_T^2} }}\right) e^{j k(x - c t)}\end{equation}

\begin{equation}\color{black}\sigma_{y y}=0\end{equation}

\begin{equation}\color{black}\sigma_{z z}=\lambda (\varepsilon_{x x}+\varepsilon_{z z})+2 \mu \varepsilon_{z z}=\mu k^2 \left((2-\frac{c}{c_T^2}) Ae^{-k z \sqrt{1-\frac{c^2}{c_L^2} }} + 2 j \sqrt{1-\frac{c^2}{c_T^2}} Be^{-k z \sqrt{1-\frac{c^2}{c_T^2} }}\right) e^{j k(x - c t)}\end{equation}

\begin{equation}\color{black}\sigma_{x y}=0\end{equation}

\begin{equation}\color{black}\sigma_{y z}=0\end{equation}

\begin{equation}\color{black} \sigma_{x z}=2 \mu \varepsilon_{x z}=-\mu k^2 \left(2 j \sqrt{1-\frac{c^2}{c_L^2}} Ae^{-k z \sqrt{1-\frac{c^2}{c_L^2} }} + (2-\frac{c}{c_T^2}) Be^{-k z \sqrt{1-\frac{c^2}{c_T^2} }}\right) e^{j k(x - c t)} \end{equation}