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 -plane separating two semi-infinite solids require that, for a wave traveling in the -plane, the following functions must be continuous:



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


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


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

The normal displacement is while the tangential displacement is , so these two functions must be continuous.

The normal stress is and equations (2.1b), (2.1h), (2.9e), and (2.9f) show that

the last step being obtained by differentiating the above expression for .

The tangential stress is and equations (2.1c), (2.1i), (2.9d), and (2.9e) give

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

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