# Relation between stress and resonance

I am working on a project to measure the stress in metals so I am looking for a variable that changes directly with stress. I noticed that when hitting a metal (unstressed) (say iron) by another metal (say copper) a sound is generated with a unique wavelength. When I retry the same experiment, but this time applying some stress to the metal (iron), and hit it again (with copper), will the sound generated have different wavelength?

If what I said is right I aim to use this property to measure the stress in any metal. So if you have any idea on how can I build such a measuring tool please share it with me. Thank you in advance.

• How do you apply the stress to the metal? The resonance depends on boundary conditions so if you attach the metal to some device or restrict it's motion in any way this may change the resonances.
– nasu
Nov 3, 2020 at 20:03
• This method is often used to measure the tension of drive belts after mounting in production, so it is a well known method of measuring stress, whether it will work for metals or how I don't know. EDIT: By "This method" I mean the method of measuring acoustic resonance frequency to determine stress. Nov 3, 2020 at 20:19
• The change in resonant frequency depends on the shape of the complete object, as well as the stress distribution in it. Your concept will work for simple geometry like a rod with axial stress in it (that is how you tune a guitar string!) but for example in a beam with bending or torsional stress, there will be no change in frequency at all. Nov 3, 2020 at 23:43

For an elastic rod under uniaxial stress is valid the relation for the longitudinal vibration:

$$\frac{\partial \sigma_{xx}}{\partial x} = \rho a_x = \rho \frac{\partial^2 u_x}{\partial t^2}$$ where $$\rho$$ is the density and $$u_x(x,y,z)$$ is the displacement at the point to the $$x$$ direction.

As $$\sigma_{xx} = E\epsilon_{xx} = E\frac{\partial u_x}{\partial x}$$, we have a wave equation:

$$\frac{\partial^2 u_x}{\partial x^2} = \frac{\rho}{E} \frac{\partial^2 u_x}{\partial t^2}$$

with a solution of the type: $$u_x = Acos\left(kx - k\sqrt{\frac{E}{\rho}}t\right)$$

$$\sigma_{xx} = -kEAsin\left(kx - k\sqrt{\frac{E}{\rho}}t \right)$$

$$\omega = k\sqrt{\frac{E}{\rho}}$$, so the stress is linearly proportional to the frequency, for the same material and geometry.

But it is only valid for an uniaxial case (and longitudinal vibration). In a generic 3D situation, and where the stress is not constant (a shrinking fit assembly for example) it may be hard to derive a relation frequency x stress state.