Hello Harry. The "classical" creep constitutive equation is based on the following assumptions (see for example the book of Odqvist):

-The material is isotropic-therefore creep rate tensor is the isotropic function of stress tensor,

-Creep process does not produce an essential change in volume. Second order effects like change in the length by torsion, are negligible. Therefore the creep rate tensor is proportional to the stress deviator. The proportionality factor depends only on the second invariant of the stress deviator: that is the von Mises equivalent stress.

In 70'-90'  many experimental studies (basically tension-torsion tests on tubes) were made to validate this von Mises-Odqvist creep theory. The published results indicate:: the creep rate within the primary and secondary stages can be well described by this theory. The accelerated creep rate may depend additionally on the maximum tensile stress and hydrostatic stress (since damage processes, like growth of creep cavities, are active during tertiary creep .)

Hi Konstantin,

Thanks a lot for your answer. That all makes sense - I understand why, if you make the assumptions that you mentioned, it is logical for the proportionality factor between the creep rate and stress tensors to be the second invariant of the stress deviator. But what about the experimental results from the '70-'90?

I suppose my question is really more of a historical one: was there ever a moment when this creep consitiutive theory was considered "proven" for engineering purposes eg. through a particular large-scale study or statistical meta-analysis of existing results? Or is it simply the case that this theory has been used successfully for a long time, and there has been little evidence which contradicts it despite a large number of experiments?

Hi Harry,

please have a look in Chapter 11 of this book

http://www.sciencedirect.com/science/book/9780408011723

I hope you have access. You find a summary of experiments made on model structures, like a beam or a plate. These all tests were performed to validate the classical creep theory. In addition pleas have a look in

https://www.researchgate.net/publication/277929490_Analysis_of_Inelastic_Behavior_for_High_Temperature_Materials_and_Structures

Starting from page 27, we present an example of a real steam transfer line.

Chapter Analysis of Inelastic Behavior for High Temperature Material...

Hi Konstantin - thanks for those, I will have a read.

Hi Harry

I have nothing really to add to Konstantin's answer as he has said it all.  Except that my first job was to collate multiaxial creep data for steels used in power stations (austenitic and low alloy ferritic).  An internal report:

The Multi-axial Creep of Austenitic Stainless Steels, Nuclear Electric Report TIGM/REP/0014/94

MW Spindler

Nuclear Electric PLC, UK

Shows biaxial creep data (both torsion-tension and internal pressure-tension) for Type 304 and 316 stainless steels, which are isotropic, I will send you a copy privately.  As Konstantin mentioned the von Mises equivalents work perfectly for primary and secondary creep.  However, it is important to take account of the change in true stress to do these comparisons (something Brian Dyson pointed out).  Tertiary creep is a slightly different matter, and appears to accelerate more quickly for higher multiaxiality stress factors (such as max principal/von Mises or hydrostatic over von Mises).    Whether this is due to cavitation or strain localisation in test specimens such as necking is an interesting question.  Each test needs to be carefully analysed and complex tests such as notched bars need quite detailed FEA using elastic-plastic-creep models and large strain analyses.  

Hi Mike, thanks for the detailed answer. To clarify, meant specifically in the case of primary and secondary creep deformation, before any strain localistion or cavitation effects come into play. That report that you mention sounds like exactly what I'm interested in - I'd be very grateful for a copy.

Book on "Mechanics of Constitutive Modeling" by Niels Ottosen and Matti Ristinmaa has a good discussion on the yield criteria. 

Hi Harry,

You are getting some good responses here!

A lot of good work on multiaxiality was done by David Hayhurst - one of his earlier references is:

Creep rupture under multi-axial states of stress. J Mech Phys Solids

Journal of the Mechanics and Physics of Solids 20(6):381-382 · December 1972

The 'Creep in Structures' conferences contain a lot of good material and I would also suggest a look at 'Design for Creep' by Roy Penny and Doug Marriott.

To my recollection, the acceptance of the von Mises stress for plastic flow evolved rather than resulted from a single study.  And in my experience it works well in many situations. 

However, we should not always confine max principal stress effects to a distant tertiary regime.  Type IV (intercritical zone) cracking in weldments reflects a situation where a narrow soft region leads to a locally high hydrostatic stress which inhibits flow and exacerbates damage, causing changes in creep rate as well as rupture life.  It is at this point that we realise that continuum hypotheses are inadequate! 

PS - Mike, might I have a copy of that report - maybe when we meet next?

Hi John, thanks a lot for your answer. It was actually a line in Penny and Marriott's book that got me thinking about this in the first place (p38 in the 2nd ed.):

"[the assumed form of the effective stress] is based on little more than intuition and is accepted in most cases because it works reasonably well. Other criteria have been tried, notably that of Tresca, but with less success."

Before I asked this question I was reading some of the recent papers by David Hayhurst, but I'll go back and have a look at that one now. Thanks also for pointing out the effect on the creep rate in the Type IV region - I'll have a read up on that phenomenon especially.

From all of the (great!) answers given, it certainly sounds like acceptance of the von Mises equivalent stress evolved over time as the majority of experimental results upheld the validity of that theory for most situations. I suppose that if you're starting from the point of view that the von Mises stress works well for low-temperature plasticity in most cases, and the underlying mechanism for that (i.e. dislocation movement) is the same as for primary and secondary creep, then it works in a mechanistic sense as well as a phenomenological one too.

Can the von Mises equivalent stress be applied to the constitutive modelling of  creep of brittle materials such as rocks since generally the rock is anisotropic and there is a change in volume due to microcracking during creep?

Hi Tao Xu,

the von Mises equivalent stress can only be applied for isotropic materials. But you may use all the ideas: flow potential, flow rule, etc. to derive a similar theory for initially anisotropic materials. Richard von Mises proposed such a theory in 1928

http://onlinelibrary.wiley.com/doi/10.1002/zamm.19280080302/abstract

So it is not wrong to say: "the von Mises theory" or "von Mises equivalent stress" for anisotropic material too.

To consider micro-cracks, and phenomena like dilatation, or tension-compression difference, the Continuum Damage Mechanics is a good choice.

Hi Konstantin,

Thanks for your kind reply and sorry for my late response due to long-time no access here. @Konstantin Naumenko