# Christensen’s Failure Theory

## Introduction

Richard Christensen wrote the book “The Theory of Materials Failure”  with the aim to “put order and organization into the field of failure characterization and failure criteria”. Prof. Christensen is a Professor Emiritus at Stanford University and has written other very well-received books: “Theory of Viscoelasticity”,  and “Mechanics of Composite Materials”. In this article I will provide some comments on how his failure theory applies to thermoplastics.

## Stress vs Strain

One common question is if stress or strain should be used for predicting failure under monotonic loading. Christensen proposes that stress should be used for the following reasons:

• Mises stress works well for ductile metals.
• Fracture mechanics is based on stress.
• Dislocation motion of metals is based on stress.
• Materials fail under constant stress (creep), but not user constant strain (stress relaxation).
• For fluids, stress is more important than strain.

These are all good reasons, but note that for some polymers the stress strain curve is almost flat after yielding making it more convenient/useful to use a strain-based condition.

## Definition of Failure Stress Christensen differentiates between “Effective Failure” and “Fragmented Failure”, the latter of which is what most people would call failure. This distinction is quite interesting, and I have not heard anyone else discussing that.  All of his theory is based on the “Effective Failure”, which is based on the definition: $$U_{dis} = 2 U_{ref}$$, where $$U_{dis}$$ is the dissipated energy, and $$U_{rec}$$ is the recoverable energy. Note that this definition means that all materials will also fail in compression, even though some thermoplastics (like LDPE) does not really fail in a fragmented way in compression.

## Review of the Mises Failure Condition The Mises failure condition can be written:

$\displaystyle\frac{1}{2} \left[ (\sigma_1 – \sigma_2)^2 + (\sigma_2 – \sigma_3)^2 + (\sigma_3 – \sigma_1)^2 \right] \le T^2.$

In this equation, T is the uniaxial tensile strength. This model predicts the same failure strength in tension and compression.

## Christensen's Failure Condition

The Christensen’s failure condition has 2 parameters: T (uniaxial tensile strength) and C (uniaxial compressive strength), and can be written:

$\displaystyle \left(\frac{1}{T} – \frac{1}{C} \right) \sigma_{ii} + \frac{1}{2TC} \left[ (\sigma_1 – \sigma_2)^2 + (\sigma_2 – \sigma_3)^2 + (\sigma_3 – \sigma_1)^2 \right] \le 1.$

In the ductile limit: $$T \rightarrow C$$, and in the brittle limit: $$T \rightarrow 0$$. Christensen lists that for thermoplastics T/C is often between 0.5 and 0.8.

## PolyUMod Support

The PolyUMod library supports the Christensen’s failure criterial with the following features: (1) strain rate effects; (2) temperature effects; (3) both brittle and ductile failure failure; and (4) no failure in compression.

## Example: UHMWPE

To examine the usefulness of the Christensen failure condition I compared my experimental uniaxial and biaxial failure data for UHMWPE to the predictions from the Christensen failure condition. It is clear that if T=C then Christensen’s failure condition becomes the same as the Mises failure condition. As I showed in my article about thermoplastic failure, the Mises failure condition does not work for UHMWPE.

The Christensen failure condition is more advanced and uses two failure parameters: T and C. If I set T/C = 0.5, then the failure stress in uniaxial tension becomes T, and the failure stress in biaxial tension becomes 0.62 T. This is not in agreement with my experimental data that showed that UHMWPE fails at a higher stress in biaxial loading than in uniaxial loading. In other words, the Christensen’s failure condition is not accurate for UHWMPE (when applied to fragmented failure).

## Summary

• Christensen’s failure theory is different and interesting.
• It is based on very different ideas than traditional failure modeling.
• The proposed failure criteria have not been extensively tested for polymers.
• For UHMWPE, the Christensen’s failure condition does not work as well as a condition based on failure strain with stress triaxiality dependence.