Polymers exhibit a wide range of non-linear behaviours under load making their simulation challenging. In contrast to metals, several non-linear behaviours are introduced by large deformations, rate dependence and time-dependance. The main types of non-linear models used to capture polymer behaviour can be broadly divided into hyperelastic models and viscoelastic models.
1. Hyperelastic Models
Hyperelastic models are used to describe reversible large deformtions such as in rubbers and elastomers.
\[W = W(\mathbf{F}) \quad \text{or} \quad W = W(I_1, I_2, I_3)\]Hyperelastic models describe the large, reversible deformations typical of rubbers and elastomers. These models use strain energy density functions to relate stress and strain. Common examples include:
- Neo-Hookean
- Mooney-Rivlin
- Ogden
- Yeoh
- Arruda-Boyce
These models are widely used for soft polymers and biological tissues.
2. Viscoelastic Models
\[\sigma(t) + \tau_\sigma \,\dot{\sigma}(t) = E_1 \,\varepsilon(t) + E_2 \,\varepsilon(t) + E_2 \tau_\varepsilon \,\dot{\varepsilon}(t)\]Viscoelasticity accounts for time-dependent (creep and relaxation) behavior. Linear viscoelastic models use Prony series, but for nonlinear response, models like:
- Nonlinear Generalised Maxwell
- Bergström-Boyce
- Schapery’s model
are used to capture both rate and history dependence.
3. Viscoplastic and Elastoplastic Models
Some polymers yield and flow irreversibly under load. Nonlinear viscoplastic models (e.g., Perzyna, Anand) and elastoplastic models (e.g., Drucker-Prager, von Mises with hardening) are used for thermoplastics and filled polymers.
4. Damage and Failure Models
To simulate progressive damage, crazing, or fracture, models such as:
- Continuum Damage Mechanics (CDM)
- Cohesive Zone Models (CZM)
- Gurson-Tvergaard-Needleman (GTN)
are applied, especially for predicting failure in toughened polymers and composites.
5. Mullins Effect and Hysteresis
For filled rubbers and soft polymers, the Mullins effect (stress softening) and hysteresis are important. Specialized models extend hyperelasticity to capture these phenomena.
Choosing the right nonlinear model depends on the polymer type, loading conditions, and the simulation’s goals. Accurate material testing and calibration are essential for reliable results.