## Introduction to Linear and Non-Linear Viscoelasticity

This article covers differences between linear and non-linear viscoelasticity. Most engineers are familiar with linear viscoelasticity, and perhaps also how to use linear viscoelasticity in a FE simulation. But most engineers are not as familiar with non-linear viscoelasticity. In this article I will try to explain some of the practical differences between these two model types. One key take away is that it is as easy to use a non-linear viscoelastic material model (like the Bergstrom-Boyce model) as it is to use a linear viscoelastic model.

### Linear Viscoelasticity

- Based on Bolzmann's superposition principle
- An instantaneous (or long-term) hyperelastic response + a stress relaxation modulus is all that is needed
- The theory is elegant and simple
- Is mathematically equivalent to a linear multi-network rheological model
- The stress-strain response is non-linear, but the viscosity is linear
- Can be combined with time-temperature superpositioning

- Can (most of the time) accurately fit frequency domain DMA data

- Can (most of the time) NOT accurately predict the finite-strain response of polymers

### Non-Linear Viscoelasticity

- Multiple parallel networks
- Each spring is hyperelastic
- Each flow element is non-linear viscoplastic
- Don't need a plastic element to get permanent set
- Examples include: Bergstrom-Boyce model, PolyUMod Three Network Viscoplastic (TNV) model, Abaqus PRF model
- This modular approach is very accurate!
- The "trick" is to pick the right components
- Can predict both frequency and strain amplitude DMA data

- Can accurately predict the finite-strain response of polymers

- With some experience, is as easy to calibrate and use as linear viscoelasticity
- The MCalibration software makes it easy to calibrate non-linear viscoelastic/viscoplastic material models.
- This is where the future is!

## 2 thoughts on “Practical Differences Between Linear and Non-Linear Viscoelasticity”

Yuhai XiangHi Gorgen,

I am confused about the definition of linear viscoelasticity. The linear viscoelasticity should obey: E(t)=sigma(t)/epsilon0; Obviously, hyperelastic response+stress relaxation modulus does not satisfy the definition of linear viscoelasticity.

Is there no consistency for the definition of linear viscoelasticiy?

thanks!

Best,

Yuhai

JorgenYou can define linear viscoelasticity as either: (linear elastic) + (Prony series), or (hyperelastic) + (Prony series). I prefer the second definition, since it is more useful in practice.

/Jorgen