# Elastic waves in very thin plate

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 ...?

• 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. – Rhodie Jan 23 '20 at 22:35