## Introduction

All finite element programs support different types of plasticity model. Some of the more common ones include: isotropic hardening, kinematic hardening, combined hardening, Johnson-Cook, and Drucker-Prager. These plasticity models can all be applied to polymers (specifically thermoplastics), but the accuracy of the predictions are often no so good. The goal of this article is to demonstrate how isotropic and kinematic hardening plasticity models work, and how they can be merged into a combined hardening plasticity model. I will also show how easy it is to calibrate the Abaqus combined hardening plasticity model using MCalibration.

## Isotropic Hardening Plasticity

Isotropic hardening plasticity is the most basic form of plasticity, and is sometimes just called “plasticity”. This type of plasticity is typically represented using a piecewise representation of the stress-strain curve. The material model needs the Young’s modulus, Poisson’s ratio, and *N* pairs of stress-strain points (see Figure 1a).

The isotropic hardening plasticity model can accurately fit a monotonic experimental stress-strain curve, but the unloading predictions can be quite bad. Figure 1b shows experimental data for a UHMWPE material, and predictions from the plasticity model. The predicted unloading response is certainly bad. **It is generally true that isotropic hardening plasticity should not be used to predict the unloading or cyclic loading response of polymers.**

The yield stress for isotropic hardening plasticity changes with the plastic strain (see Figure 1c). The unloading is linear elastic until reverse plasticity occurs at a stress that has the same magnitude as is was in the initial loading.

## Kinematic Hardening Plasticity

In a kinematic hardening plasticity model the size of the yield surface does not change, it is only translated as the plastic strain increases. This makes the predicted unloading response more similar to what is seen in thermoplastics (see Figure 2).

## Combined Hardening Plasticity

The yield surface in combined hardening plasticity is given by the following equation. Both the back stress \( \boldsymbol{\alpha}_i\) and the yield stress \(\sigma_y\) are functions of the plastic strain magnitude.

The back stress evolution with plastic strain magnitude is often given by:

For the Abaqus combined hardening plasticity model the isotropic hardening part is given by:

\[ \sigma_y(\varepsilon_p) = \sigma_{y0} + Q_{\infty} \cdot \left[ 1 – e^{-b \varepsilon_p} \right] \]

An exemplar load-unload curve is shown in Figure 3. The figure shows that the linear elastic unloading domain is between isotropic and kinematic hardening. Just as expected.

*Figure 3. Predicted load-unload curve from a combined hardening plasticity model.*

## Application to UHMWPE

Figure 4 shows that the combined hardening plasticity can match the load-unload response of UHMWPE reasonably well. There are other viscoplastic material models that will be more accurate, but this model may be acceptable in some applications.

*Figure 4. Combined hardening plasticity prediction of UHMWPE.*

## Summary

- Combined hardening plasticity is often better than pure isotropic hardening or pure kinematic hardening plasticity.
- Predicts a decreasing tangent modulus (which is not true for many polymers).
- Runs fast (but often not that accurate for polymers).