# How to Predict Stress Reduction After Yielding

In this tutorial I will cover a few different material model options that can be used to model the reduction in the engineering stress that occurs for certain thermoplastics after yielding.

This softening response is often associated with the onset of necking, but it does not have to be. Some polymer soften after yielding, but do not undergo necking.

The key question that I will try to answer is: What material model one should use if softening after yielding occurs in the experimental data.

The focus here is on non-linear viscoplastic material model, which is the class of material models that are most suitable for predicting the large-strain response of thermoplastics. This class of material models are organized in multiple parallel networks, where the network structure is selected to get a specific desirable mechanical response. I most often recommend the Three Network (TN) model and the Three Network Viscoplastic (TNV) models from the PolyUMod library for this purpose.

## Three Network (TN) Model

The TN model consists of three networks. Each network consists of an Arruda-Boyce Eight-Chain hyperelastic element, and two of the networks contain a non-linear viscoplastic flow element. Neither flow element supports yield evolution, but network B has an elastic stiffness that evolves with the plastic strain of network A. This network coupling enables the TN model a limited ability to predict stress softening after yielding. The figure below shows the most accurate prediction of the experimental data that can be obtained using the TN model. The average error i this case is 1.5%, which is not too bad. In other cases, with stronger or faster drop in the stress, the TN model is often less accurate.

## Three Network Viscoplastic (TNV) Model

The Three Network Viscoplastic (TNV) model supports a viscoplastic network with yield evolution. Specifically, the yield stress for networks A and B are multiplied by the factor: $$f_f + (1-f_f) \exp(-\varepsilon_p/\varepsilon_f)$$. The parameter fff specifies the final yield stress multiplication factor at large strains. In this case the average fit to the experimental data has an error 1.7%, which is very good.

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