## Introduction

The Drucker Stability condition is a well-known equation that specifies if a material model is stable:

\[ \Delta (J \boldsymbol{\sigma}) : \Delta (\mathbf{E}_{\text{ln}}) \geq 0.\]

That is, the material model is considered stable if the tensorial product of the increment of Kirchhoff stress and the increment of true strain is non-negative.

Daniel Drucker (1918-2001)

## Laws of Physics vs. Drucker Stability

Some simple material models are always Drucker stable. Examples of this include the Neo-Hookean model and the Arruda-Boyce Eight-Chain model. Other material models are only Drucker Stable for certain material parameter sets and strain ranges. It is easy to examine if a material model Drucker stable in uniaxial tension, all that is needed is to plot the Kirchhoff stress as a function of the true strain. As illustrated in the Fig 1, if the stress-strain curve has a negative slope then the material is not Drucker stable.

*Figure 1. Stress-strain predictions from a 3rd-order Yeoh model.*

It is not easy to mathematically determine if a material model is always stable since that will require that the all loading modes to be examines.

Figure 2 shows the stress-strain predictions of two almost identical Yeoh hyperelastic models. The figure to the left is thermodynamically OK, but does not satisfy the Druckers stability condition. The figure to the right does not satisfy either condition. Note that it is not OK to have a negative stress for a positive strain.

*Figure 2. Stress-strain predictions of two different Yeoh models.*

## MCalibration and Abaqus

It is easy to examine the Drucker stability condition using MCalibration, see Figure 3. The following article also shows how you can force a material model to be Drucker stable.

*Figure 3. Drucker stability test in MCalibration.*

If you run an Abaqus FE simulation with a material model that is not Drucker stable then Abaqus writes the following warning to the dat-file:

` ````
``` ***WARNING: THE SPECIFIED HYPERELASTIC MATERIAL IS UNSTABLE. --- MORE
INFORMATION CAN BE OBTAINED BY SETTING *PREPRINT, MODEL=YES

## Summary

- A material model needs to satisfy the laws of physics and thermodynamics, but it does not need to satisfy the Drucker Stability Condition.
- A Drucker Stable material model is well behaved – increasing the strain also increases the stress.
- Most rubber-like materials are Drucker stable.
- Some thermoplastics are not Drucker stable.
- MCalibration can examine the Drucker stability and ensure stability.