Elastic waves in structural elements are described by \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 u}{\partial x^2} + \mu \left( \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 v}{\partial x \partial y} + \frac{\partial^2 w}{\partial x \partial z} \right) \\ \rho \frac{\partial^2 v}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 v}{\partial y^2} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial z^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 u}{\partial x \partial y} + \frac{\partial^2 w}{\partial y \partial z} \right) \\ \rho \frac{\partial^2 w}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 w}{\partial z^2} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 u}{\partial x \partial z} + \frac{\partial^2 v}{\partial y \partial z} \right) \end{align}

In a thin plate, these equations result in lamb-waves. However, imagine that we make the plate much thinner such that we end up in a very thin sheet. Is it then possible to assume that there is no variation over the thickness ($\partial / \partial z =0$)? If there is no variation over the thickness (z-direction) of the thin sheet, then the equations become \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + (\lambda+\mu) \frac{\partial^2 v}{\partial x \partial y} \\ \rho \frac{\partial^2 v}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 v}{\partial y^2} + \mu \frac{\partial^2 v}{\partial x^2} + (\lambda+\mu)\frac{\partial^2 u}{\partial x \partial y} \\ \rho \frac{\partial^2 w}{\partial t^2} &= \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) \end{align}

Question: Is this a real case? Is this form of the equations used for real systems/structures (if yes, which ones)? Does it have a specific name like plane stress, antishear strain, quasi 2D elastodyanmics ...?

  • $\begingroup$ It is impossible to create a structural membrane with invariant thickness subjected to stress deformation without creating a temporal thinning of the membranous material. The elastic deformation, from my junior understanding, is limited to the modulus of elasticity. $\endgroup$ – Rhodie Jan 23 '20 at 22:35

I've not looked at your specific equations, but in general if you take a plate and make it very thin, that's called a membrane. A membrane is to a plate what a string is to a beam. It has no bending stiffness, and can only carry load through in-plane tension. You can definitely have traveling waves in membranes. E.g. a drum head is a membrane. See https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane for just one example.

  • $\begingroup$ Yes, the membrane is described by having out-of-plane displacements. Hence, the membrane is described only by the third equation in the set. What about the in plane displacements? Do two dimensional plates/membranes/... exist with also in plane wave behavior? $\endgroup$ – Frederic Jan 22 '20 at 9:54
  • $\begingroup$ Yes in theory those motions exist. I've never seen any practical applications, but solid mechanics is a big field with a long history. If you search you'll probably find somebody looked at it at some point. $\endgroup$ – Daniel K Jan 23 '20 at 0:23

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