# How does Nonlinear Viscous Damping work? In what way are Viscous Dampers typically nonlinear?

Viscoelastic materials like metal cables/wires/strings/webs/beams can be modeled with a variety of combinations of springs and dampers as described here.

For example, one simple common structure to represent these materials is as follows:

The spring, represented by Young's Modulus of Elasticity, has a generally predictable nature to its nonlinearity which is described and shown here:

Is there any general concept by which the nonlinearity of a viscous damping coefficient can be similarly displayed or understood?

In simple terms, say for a piece of metal being stretched/bent:

• Would a viscous damping coefficient be higher or lower at higher velocity of bending/stretching?
• Is there any similar published data or curves that could demonstrate the relationship between the damping coefficient and velocity of bending/stretching?

The only information I can find on nonlinearity of viscous damping tends to pertain to models like the Bouc-Wen Hysteresis Model .

Perhaps that model actually answers my question and I don't realize it? I'm not sure. I think that more pertains just to capturing damping due to displacement (rather than damping due to velocity/viscosity).

Thanks for any ideas or help.

• TLDR: not sure if this will be what you are looking for, but perhaps search for fractional order viscoelasticity. // Fractional order models have been proposed (note that these are still linear). I don't know much about them, including whether or not it is an "widely accepted" technique, although I've encountered fractional order systems elsewhere (they're not that mysterious, all the standard frequency domain techniques work just fine on them, and if you like you can even convert them to integer order with Padé approximants). At one point I ran into some vehicle suspension modeling papers. Commented Jan 30, 2021 at 2:10

The following graph shows you the properties of Polypropylene at high strain rates

As you see here the modulus increases with increasing displacement rate (its not entirely equal to strain rate but its closely related). The effect is due to this viscous damping.

Polymers usually are more strongly affected, in terms of modulus and ultimate tensile strength.

Metallic materials (e.g. steels) are also affected. There were some "older" empirical laws that relate strain rate to stress for the uniaxial loading case. You can find information in articles like this. The one I have come across more often is:

$$\sigma_{\dot{\epsilon}}(\epsilon) = \sigma_{0}(\epsilon)\cdot \left(1 + \left(\frac{\dot{\epsilon}}{a}\right)^m\right)$$

where:

• a, m are coeffiecient experimentally obtained
• $$\sigma_0(\epsilon)$$: is the stress at a given strain $$\epsilon$$ at the standard strain rate (usually set by the standard displacement speed).
• $$\sigma_\dot{\epsilon}(\epsilon)$$: is the stress at strain $$\epsilon$$ and strain rate $$\dot{\epsilon}$$.

This type of formulas are over 100 years old and they were attributed to internal friction i.e. viscous damping.

The following graph is an example on an Mn steel. Notice that the modulus of elasticity does not change significantly, however the ultimate stress exhibits a more noticeable increase.

UPDATE: Nonlinearity and engineers

Non linearity can have many meaning is the context of material science and structural analysis. Normally an engineer's work is limited to a linear region. This is due to necessity and also for practicality. I might seem to go on a tangent but bear with me.

For example, take the stress strain curve of steels. Most engineering work does not take it into account. Engineers usually work just with the yield stress and young's modulus. Notice that the elastic region is much smaller than the rest, however we restrict our selves to that. The reasons are that in most applications, a) we don't want structures that after use get deformed, and b) linear properties can make use of the superposition principle, and calculations are much easier (without even considering the benefits of linear algebra).

Of course there are situations where non-linear considerations are necessary. For example in structures, you can have non-linear behaviour when:

• the material is non linear: this means that the proportional relationship between stress and strain does not hold (i.e. $$\sigma\ne E\epsilon$$
• Non linear due to transient effects: When you take into consideration time (e.g. the load that is applied to the structure changes with time). This effect can be for different reasons, due to inertia masses, damping etc.
• Coulomb Friction effects: Contrary to viscous damping, Coulomb damping (or dry friction) creates a non linear effect (e.g. the direction and magnitude of the friction force change abruptly based on the direction of movement and the net resultant forces)

Although, the above don't necessarily apply to the original post, describe different non linear effects which are quite different. Notice that these non-linear problems seemingly affect a small portion of "real" mechanical and structural engineering applications, compared to their linear counterparts. Additionally, they require high degree of expertise and computational effort. So in essense, it is another application of the Paretto's rule, (i.e. "with only 20% of the effort you can get 80% of the accuracy", or "with only 20% of knowledge in the field you can address 80% of the common problems").

If my understanding is correct you maybe in the other part. I.e. needing to know exert the "extra" 80% to get that 20% increase in accuracy.

• Thanks NMech. That's exactly the kind of info I was looking for. Is it correct then that stress is generally higher at higher strain rate? Does this imply that there is less viscous damping at higher strain rate? I appreciate the formula you provided to indicate this relationship. How would you implement that into viscoelastic model like the Maxwell Burgers model: en.wikipedia.org/wiki/Viscoelasticity#Burgers_model Or is this type of behavior already incorporated in that type of model?
– mike
Commented Jan 30, 2021 at 0:13
• the equation I've shown is an older empirical law. The Standard linear solid model, or the Maxwell Burgers incorporate this in the development of the model. The problem of course is defining the properties. Commented Jan 30, 2021 at 0:32
• Thanks NMech. The question I have though is the modern Standard Linear Model or Generalized Maxwell or Maxwell Burgers etc. all use simple dampers with a set stable damping coefficient for each damper. Perhaps I just have to expand to a higher order Generalized Maxwell to get the behavior I want. I will try that next. But I'm wondering, is it generally presumed that leaving the dampers as individually linear (ie. set coefficients) is a good idea? I wonder because we know Young's Modulus is only linear in a certain range. I presume the same is true of dampers.
– mike
Commented Jan 30, 2021 at 0:46
• The topology of these models is independent to the linearity or non linearity of the individual elements. Commented Jan 30, 2021 at 0:51
• Yes, I understand NMech. And I'm trying to figure out how to best nonlinearize the individual dampers based on their velocities. I'm a bit of a novice at this. Do those articles you provided explain that? Sorry if I'm not making sense or being clear. Thanks again.
– mike
Commented Jan 30, 2021 at 0:52