Viscoplastic Material

The viscoplastic material is available to model solid polymer materials in Nonlinear Mechanical analysis. It is based on the Parallel Rheological Framework (PRF) — an approach that combines nonlinear elasticity, viscous flow, and plasticity to accurately represent the complex mechanical behavior of polymer materials under large strains.

Standard plasticity models such as multilinear isotropic hardening or hyperelastic models alone are generally insufficient for polymers, because polymers exhibit a combination of effects that only emerge under large strains, cyclic loading, or time-dependent conditions. The Viscoplastic model is designed to capture these effects within a single material definition.

When to Use the Viscoplastic Model

Use the Viscoplastic material model when your polymer component exhibits one or more of the following behaviors under the expected loading conditions:

Behavior	Description
Nonlinear elasticity	Stiffness that changes with the level of deformation, typical in rubbers and soft polymers
Mullin’s damage	Stress softening on the first loading cycle compared to subsequent cycles (also called stiffness damage)
Hysteresis	Energy dissipation visible as a difference between loading and unloading stress-strain curves
Permanent set	Residual deformation remaining after load is removed
Creep / viscous flow	Time-dependent deformation under sustained load
Strain-rate dependence	Stiffer response at higher loading rates

Typical polymer families where this model applies include polyethylene (PE), polypropylene (PP), polycarbonate, nylon, PVC, silicone, and thermoplastic elastomers.

Viscoplastic Model Structure

The Viscoplastic model is a three-network PRF model. Each network deforms in parallel and contributes independently to the total stress response. The three networks are:

Network	Role
Primary network	Nonlinear hyperelastic backbone; captures large-strain elastic response and optional stiffness damage (Mullin’s effect)
Viscoelastic network	Time-dependent viscous response; captures creep, stress relaxation, and strain-rate dependence via a Prony series relaxation model
Elastoplastic network	Nonlinear elastic-plastic response; captures permanent set and hysteresis through Arruda-Boyce elasticity combined with bilinear combined hardening plasticity

Primary Network

The primary network defines the elastic equilibrium behavior of the polymer using the Yeoh hyperelastic model, which is a first-invariant-based polynomial well suited to rubbers and soft polymers across a wide strain range.

Hyperelastic Model — Yeoh

Parameter	Symbol	Unit	Description
C10	C₁₀	Pa	First-order stiffness coefficient
C20	C₂₀	Pa	Second-order stiffness coefficient
C30	C₃₀	Pa	Third-order stiffness coefficient
Bulk modulus	\(\kappa\)	Pa	Volumetric stiffness; controls near-incompressibility

Damage Model

An optional damage model can be added to the primary network to capture the Mullin’s effect — a reduction in stiffness observed after loading cycles. Two options are available:

Option	Description
None	No damage; elastic response is unchanged across load cycles
Ogden-Roxburgh	Phenomenological damage model for Mullin’s effect; requires three scalar parameters

When Ogden-Roxburgh is selected, the following parameters must be defined:

Parameter	Unit	Description
r	—	Relative extent of damage; controls how much softening occurs
m	J/m³	Dependence on deformation history
\(\beta\)	—	Factor applied to the maximum deviatoric strain energy in the damage criterion

Viscoelastic Network

The viscoelastic network captures the time-dependent response of the polymer — including creep under sustained load and stress relaxation over time. The elastic part of this network shares the same Yeoh hyperelastic model as the primary network. The viscous part is defined by a Prony series relaxation model.

Option	Description
None	No stress contribution from this network
Prony series	Defines creep response as a sum of exponential terms; each term is defined by a relaxation time and a Prony factor

When Prony series is selected, enter the series as a table with one row per term:

Column	Unit	Description
Relaxation time	s	Time constant for each exponential decay term
Prony factor	—	Amplitude of each term; must be between 0 and 1

Elastoplastic Network

The elastoplastic network captures the irreversible (plastic) deformation of the polymer, including hysteresis in cyclic loading and residual strain after unloading. The elastic part uses the Arruda-Boyce hyperelastic model, which is physically motivated and well-suited to lightly cross-linked polymers and thermoplastic elastomers. Plasticity is modeled with a bilinear flow rule, a von Mises yield criterion and a combined isotropic/kinematic hardening model.

Option	Description
None	No stress contribution from this network
Arruda-Boyce Plastic	Nonlinear elastic-plastic model with combined hardening; requires the parameters below

When Arruda-Boyce Plastic is selected, the following parameters are required:

General Properties

To take into account the weight of the material, the density property is available.

Property	Unit	Description
Density	kg/m³	Mass per unit volume; required for all solid materials

Last updated: May 19th, 2026