# Stress Triaxiality and Thermoplastics

## Introduction

The deformation response of all materials depends on the history of the applied stress state. For most polymer materials the deformation is mainly driven by the deviatoric stress state. Recall that the deviatoric stress is given by:

$\text{dev}[\boldsymbol\sigma] = \boldsymbol\sigma – \frac{1}{3} \text{tr}[\boldsymbol\sigma] \mathbf{I},$

or in words, the deviatoric stress is the total stress minus the pressure. For many thermoplastic materials, the stress triaxiality (or pressure) can influence the material response in unexpected ways. There are 3 specific behaviors that I will highlight in this article. First, however, let’s define the term Stress Triaxiality:

$\sigma_{triax} = \frac{ \text{tr}[\boldsymbol\sigma] / 3} {\sigma_{mises}}.$

It is a scalar dimensionless quantity, which is convenient. Note that the arguments presented here also apply to thermoplastic elastomers, thermoplastic vulcanizates, and thermosets, but usually not rubber-like materials.

Uniaxial Tension1/3
Uniaxial Compression-1/3
Simple Shear0
Biaxial Tension2/3
Biaxial Compression-2/3
Triaxial TensionInfinity
Triaxial Compression-Infinity

## Behavior 1: Yield Stress

Many thermoplastics, perhaps even most, exhibit a yield stress that is different between tension and compression. The reason for this is that when the macromolecules are squeezed together by a hydrostatic pressure the shear stress required to cause viscoplastic flow tends to be slightly higher than when there is no (or even negative) hydrostatic pressure. This dependence on the pressure (or equivalently, the stress triaxiality) does not only influence the onset of yielding, but also the rate of viscoplastic flow at larger strains. To capture this behavior I recommend always testing thermoplastic materials in both uniaxial tension and compression, and then using a viscoplastic material model with a flow rate equation that depends on the applied pressure. One example of this is the PolyUMod Three Network Viscoplastic (TNV) model, another example if the TN model. A slightly simplified version of the flow rate equation that is used by the TN model is given by the following equation:

$\dot{\gamma} = \dot{\gamma}_0 \left( \frac{\tau}{\hat{\tau} + a \cdot p} \right)^m$

Here $$\tau$$ is the Mises stress, $$\hat{\tau}$$ is the flow resistance (which can be thought of as the yield stress), a and m are a material parameter, and p is the pressure.

Here are two figures that shows how the p0 parameter in the TNV model influences the viscoplastic flow response in tension and compression. The parameter p0 can be found from experimental data.

## Behavior 2: Volume Change During Plastic Deformation

It is well-known that the large-strain deformation of metals and rubbers is typically volume conserving (isochoric). This is often not the case for thermoplastic materials. For thermoplastics, the Poisson’s ratio at small strains is often between 0.3 and 0.45, but the Poisson’s ratio at larger strains can either increase of decrease depending on the material. When experimentally testing a thermoplastic material in tension it is therefore useful to simultaneously measure the transverse strain (i.e. Poisson’s ratio) as a function of applied strain. This can easily be done using Digital Image Correlation (DIC).

To capture the experimentally observed volume change during plastic deformation it is necessary to use an advanced material model, such as the PolyUMod TNV model. The bb parameter in the TNV model specifically enables accurate predictions of how much of the viscoplastic deformation is volume preserving.

## Behavior 3: Failure Strain

The failure strain of most materials (both metals and polymers) is dependent on the stress triaxiality. A common strategy used in the automotive industry is to measure the failure strain as a function of the stress triaxiality using different loading modes (see table above). This is often the reason parts fail surprisingly quickly close to stress concentrations, such as notches. These locations not only increase the stress, due to the stress concentration factor, but also often have a higher stress triaxiality which often reduce the ductility of the material. Both of these factors can be active at once, which can cause problems.

One common strategy to overcome this problem is to run experimental tests to failure in different loading modes (for example, uniaxial tension, simple shear, biaxial tension), or using specimens with different notch geometries. This way one can determine how the failure strain (or stress) depends on the stress triaxiality. That information can be directly used in some material models, such as the LS-DYNA SAMP-1 model and the PolyUMod TNV model. The TNV model allows you to specify a table of correction factors [f1, f2, f3] of how the failure strain (or stress) depends on the stress triaxiality [T1, T2, T3]. With that information you can then run accurate FE simulations that will tell you the safety factor at each integration point during post processing. You can even activate element deletion if needed.

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