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.
