# Doubt about 3D generalization of the Elastic Field Equations

My only source is $$[1]$$. In $$[1]$$ Finn introduces an example (in a totally undergraduate level discussion) that the problem on waves propagating in a solid cylindrical bar, firstly induces you to explore more about the physical conditions concerning the perturbation of an elastic media.

In this problem he presented the so called Elastic Field Equations given by:

$$\begin{cases} \displaystyle F^{x} = YA \frac{\partial \xi}{\partial x} \\ \\ \displaystyle\frac{\partial F^{x}}{\partial x} = \rho A \frac{\partial^{2}\xi}{\partial t^{2}} \end{cases} \tag{1}$$

Where $$F^{x}$$ is the one dimensional force called tension given by:

$$\vec{\mathfrak{S}}A = \vec{F} \iff \vec{\mathfrak{S}} = \frac{1}{A}\vec{F}$$

where $$\vec{\mathfrak{S}}$$ is the normal tension and $$A$$ is the area of the surface.

$$Y$$ is the Young's Number, $$\rho$$ is the density of the material that constitute the bar*, and $$\xi$$ is the displacement field of each section of the bar (which is a scalar field $$\xi \equiv \xi(x,y,z,t)$$ -that plays the role of a "wave function"-).

Now, I want to rewrite equations $$(1)$$ in 3D. My try is the following equations:

$$\begin{cases} \displaystyle \vec{F} = YA \vec{\nabla}\xi \\ \\ \displaystyle \vec{\nabla} \cdot \vec{F} = \rho A \frac{\partial^{2}\xi}{\partial t^{2}} \end{cases} \tag{2}$$

My motivation for $$(2)$$ is then:

1) Since $$F^{x} = YA \frac{\partial \xi}{\partial x}$$ then is a equation of components of vector fields.

2) Since $$\displaystyle\frac{\partial F^{x}}{\partial x} = \rho A \frac{\partial^{2}\xi}{\partial t^{2}}$$ then the term $$\displaystyle\frac{\partial F^{x}}{\partial x}$$ is one of the terms of a divergent operator.

So, are the $$(2)$$ the correct generalization to $$(1)$$?

$$* * *$$

$$[1]$$ FINN E.J, Fundamental University Physics, v.2, 2ed, 1983.

• The development is much more complicated than than that, involving the kinematics of the 3D strains and the stress tensor. Google something like "Hooke's law derivation in 3D." – Chet Miller Oct 6 '19 at 12:14
• I was awaiting a commentary like that though. But these equations describes nothing at all then? – M.N.Raia Oct 6 '19 at 20:04
• Yes, they are inappropriate to use. You can find the correct equations online, and in any book of Strength or Materials or Theory of Elasticity. – Chet Miller Oct 6 '19 at 22:11