Boundary conditions in terms of potential functions

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Problem 2.11

Using the definitions of stress and strain in problem 2.1 and the potential functions in equation (2.9b), show that the boundary conditions at the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {xy}} -plane separating two semi-infinite solids require that, for a wave traveling in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {xz}} -plane, the following functions must be continuous:


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left({\phi}_{z} -{\chi}_{x} \right),\qquad \left({\phi}_{x} +{\chi}_{z} \right), \end{align} } (2.11a)


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {\mathbf{\lambda \nabla ^{2} \phi}} +{\mathbf{2\mu}}\left({\phi}_{zz} -{\chi}_{xz} \right),\qquad {\mu} \left({\mathbf{2\phi}}_{xz} +{\chi}_{zz} -{\chi}_{xx} \right), \end{align} } (2.11b)

where subscripts denote partial derivatives. These terms are, respectively, the normal and tangential stressses and the normal and tangential displacements.

Background

As stated in problem 2.10, all stresses and displacements must be continuous at an interface between two different media.

Solution

From equations (2.9d) and (2.9e) we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} u=\mathrm{\phi}_{x} +\mathrm{\chi}_{z}, \; w=\mathrm{\phi} -\mathrm{\chi}_{x}. \end{align} }

The normal displacement is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle w} while the tangential displacement is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u} , so these two functions must be continuous.

The normal stress is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\sigma}_{zz}} and equations (2.1b), (2.1h), (2.9e), and (2.9f) show that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{\sigma}_{zz} =\mathrm{\lambda}\Delta +2\mathrm{\mu} \mathrm{\varepsilon}_{zz} =\mathrm{\lambda} \nabla ^{2} \mathrm{\phi} +2\mathrm{\mu} w_{z} =\mathrm{\lambda} \nabla ^{2} \mathrm{\phi} +2\mathrm{\mu} \left(\mathrm{\phi}_{z} -\mathrm{\chi}_{xz} \right), \end{align} }

the last step being obtained by differentiating the above expression for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle w} .

The tangential stress is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\sigma}_{xz}} and equations (2.1c), (2.1i), (2.9d), and (2.9e) give

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{\sigma}_{xz} =\mathrm{\mu \varepsilon}_{xz} =\mathrm{\mu} \left(u_{z} +w_{x} \right)=\mathrm{\mu} \left(2\mathrm{\phi}_{xz} +\mathrm{\chi}_{zz} -\mathrm{\chi}_{xx} \right). \end{align} }

Since the normal and tangential stresses must be continuous, these two functions must also be continuous.

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