I have come across work of P.J. Yoder (1980) (A strain-space plasticity theory and numerical implementation). He made a strain-space equivalent to pure plastic Coulomb model. His model has the following benefits (according to Yoder P.J.):
Despite dramatic increse in computation efficiency, Yoder's work did not go mainstream. (perhaps you can you spot why?)
I am intrigued by his work as during recent triaxial testing we were able to control sand stiffness through controlling applied strain (deformation) history. Yoder P.J. has explicitly called for a "rationale to the strain space problem", and we appear to have found the missing experimental "rationale" .
We found strain (not stress) dependant soil patterns, which capture and allow full control over loss and recovery of drained sand stiffness. Also, post-liquefaction soil stiffness recovery, and we can impose post-liquefaction soil state during drained loading. We seem to be closing the gap between drained, undrained and partially drained sand, but the new findings explicitly call for a strain-space plasticity formulation.
Any input is greatly appreciated. Any links to strain-space models or strain-dependant observations of sand (such as liquefaction charts by R. Dobry, or plastic spin (strain) tenstor by Y. Dafalias). This topic is not covered by "mainstream" geotechnical curriculum, thus I don't expect many answers. But every answer and question on the rare subject is appreciated.
Dear Tomas,
Some time ago I came across the paper:
Moss W.C.: On the computational significance of the strain space formulation of plasticity theory, Int. J. Numer. Methods Eng. Vol. 20, 1984, 1703-1709.
Why not to try to verify your findings with it.
Regards,
Krzysztof Sternik
Thank you Krzysztof Sternik, The paper of Moss W.C. is quite general, but easy to read and supportive of strain-space plasticity formulation.
I've noticed proposing "strain-space" modelling is usually met with a few raised eyebrows and scepticism, regardless of experimental evidence of it being necessary and plausible.
Perhaps you have a personal opinion on why strain-space formulation is not gaining more popularity?
Hi Tomas
For what it is worth, I am not an expert or even versed in this field but, your strain-space curves look a bit (worse than...) those we find for rubber and similar (though temperature would matter) to those I'd expect for viscoelastic materials.
I may be completely off topic here, but I'd like to show some things that may or may not piece together. (I do not know. It was just a gut feel I got looking at your curves so here flies a few very loose thoughts ...).
The best material model for rubber that I have seen is found here
https://polymerfem.com/content.php?77-bergstrom-boyce-model
http://web.mit.edu/course/3/3.11/www/pset03/Rec9.pdf
http://web.mit.edu/cortiz/www/Arslan_Boyce_Qi_Ortiz_JAMRevisedFinal.pdf
and as energy levels somehow have to come into play, perhaps it is possible to borrow some of the thinking behind modal analysis of shock nonlinearities?
http://web-code-aster.org/V2/spip.php?article750&debut_articles=40
Also, it seems Jim Woodhouse et al have made some progress on friction modeling and, as you mention using a Coulomb model, perhaps it is worth taking a look at this work. https://www.researchgate.net/publication/299591364_The_frequency_response_of_dynamic_friction_Enhanced_rate-and-state_models
Last - a silly question. Have you tried using an InfraRed Camera (or insert temperature probes) to view your experiments? As friction work is involved, I would expect temperature to jump between different states. Perhaps temperature can be used as a way to discriminate between different states? Note that you cannot use ordinary glass as it is opaque for IR. Some plastics are transparent for IR wavelengths though. This reasoning is inspired by Risitanos method that detects the onset of fatigue as a jump in temperature. https://www.researchgate.net/publication/222708445_Thermographic_Methodology_for_Rapid_Determination_of_the_Fatigue_Limit_of_Materials_and_Mechanical_Components?ev=srch_pub&_sg=v9eogpbiNQHNseAFo_DRwM1L7OBGSJJgQZ_rH6Xb2_uXfx7kN0NfhnJ8kOU4Tz3v.JmXAZdk4NWA6UKmRCzl0zU18jz8JYcgUzAKEQpxKmZcG-EvpruN1RUD5gpsMukyZ.wvWp8zbRSrI6Z305mMpXkoPyiMuoUNF0xT3Jn0J947msKwkPeVcfdkfTrYjq0HJu
In the hope of my gossiping not disrupting you too much. :)
Regards
Claes
Article The frequency response of dynamic friction: Enhanced rate-an...
Article Thermographic Methodology for Rapid Determination of the Fat...
Gut feeling is good. The strain space model of Yoder P.J. was inspired by an innovative aluminium model of the time... gut feeling fits perfect here :)
"your strain-space curves look rubber" - I heard that before, when we first noticed capabilities of the triaxial apparatus. The sand sample was squeezed and pulled with amplitudes up to 20% strain. Looking at hysteresis loops of rubber I do notice rubber stabilizes in a similar manner - constant shape of a stiffness loop at a given strain amplitude.
Your model has strain rate dependency. That is nice. We find it extremely relevant for off shore wind turbine foundation design. Sand on the sea floor has enormous additional strength due to dilation component, which is activated only if the loading rate is fast enough (too fast for water to drain).
Energy loss - equivalent to "damping", I assume. Offshore foundation natural frequency drops after a big load, only to gradually recover during smaller load cycles. The new triaxial results provide reliable hysteresis loops, hope they will capture "energy absorption". This should give some insight for modal and frequency domain solutions. They are quite necessary as the offshore wind turbines are slender.
"Have you tried using an InfraRed Camera temperature probes to view your experiments? I would expect temperature to jump between different states." - During undrained failure water reaches -100kPa, which forces it to vaporize (cavitate). Thermodynamics says temperature should drop quire a bit. Yet we ignore it. The sand particles seem to not be sensitive. Once water becomes vapour, it loses stiffness, and sample behave drained, albeit at a much higher confining pressure. At 20m water depth, during undrained yielding, confining pressure will increase by 300kPa, before the sample yields. A significant increase in sand strength.
You mentioned "jumping between states". We did find a way to measure soil states. We quantify a soil state by observing the hysteresis loop which forms during cycles of constant deformation amplitude. We can transition from one soil state to the other by changing deformation amplitude, and waiting for the stiffness loop to re-stabilize. This way we can go back and forth between soil states, getting the exact same stiffness and volumetric response each time. Repeating the same outcome form the same soil state is quite important. This works not only in a triaxial tests, but in small scale foundation testing as well (these things are in reviewing phase).
Wow, that's a long text... got carried away. Not a single reference from my side. Your referencing is superb, though. Thank you for taking the time, Claes.
Hi Tomas
Only happy to be able to help. You know, once in a while even a blind hen finds a corn.
To pester you a bit more. Cavitate - perhaps, perhaps not. It is a comfortable mode of reasoning, but who wants that?
From what you are writing and some of the stuff I have had to pick up on aeroacoustics. You know, there are three kinds of pressure, i) Line Pressure (LP) which is static, does not move mass and is scalar, ii) Kinetic Pressure (KP) which is a vector, moves mass and propagates with the flow velocity and iii) pulsation (sound) pressure which is scalar and does not impose any net mass motion but does come with oscillatory mass flux that is a vector. These three are different animals and one usually get away with ignoring the KP.
Now, if you take 100*Pulsation/LP you get a measure on how nonlinear is wave propagation. For things truly to hold linear one usually assume %LP to be 1% or less. Only, with pumps it never is.
Measuring on plunger pump operation, I experienced strange spikes. At first I thought these to be glitches in cabling but scrutiny did not match the expected signature. After having looked through some 22 GB of measurement data I finally nailed one spike to synchronously arise on three different sensors of two different types. It was real and I was left with a puzzle.
Well, we cannot have that so I scoured some books and finally upon reading a book by Hamblin and Blackstock on nonlinear acoustics, I found the solution to the riddle.
It all boiled down to how much gas was dissolved in the liquid and the effect is not very well known outside the field of those dealing with sonar and ultrasonics. Having found this thing and summarized it with - bubble is trouble - I said to myself, OK, this was pretty nerdy stuff, I will probably never ever see this again - but it keeps popping upp everywhere l look and liquids are involved with large energy.
From what you are writing, again, my gut feel is that you should develop an interest in this as well. Take a look here & welcome to the world of shock waves imposed by wave steepening. Nothing will ever be the same again. You have my apologies. http://qringtech.com/2010/09/15/wave-steepening-increase-peak-pressure-piping-pumps/
Sincerely
Claes
PS - a little bird whispered that your university has a FLIR 7650 state of the art IR camera that is able to do the things I mentioned. (I do too)
Yeah, the water is de-aired very carefully ;) it bubbles like a soda at -95kPa for half an hour... Then we keep it at 25C, -95 kPa for at least 1.5h, often longer - overnight. It's cavitation allright. It's a standard part of triax testing too.
But this duality, of two plastic materials - water and soil plasticity - working independantly is complicating things one step further. No one has completely solved this problem yet. And for a good reason - the data, the experimental measurements are missing. And we seem to have em now. :)
Hi
OK, you have gotten rid of the bubbles in your test. :) The thing is that it takes time form cavitation while a shock wave can accumulate quite fast.
What about bubbles in real life conditions?
Looks like your will have fun with your data. Congrats.
/C
Bubbles in real life conditions? - I use that question to annoy my supervisor. In practice they don't use the undrained strength for anything (yet) - it's just nice to know it is there. It can push the "impact load" resistance up by a factor of 10 (observed in results of small scale foundation testing, where water is not de-aired).
Impact load is not the main concern, though. Waves in the North sea are slow, 0.1Hz is the fastest cycle encountered. A breaking wave is a shock , sure, but the structure is designed to use 50% of it's drained bearing capacity for the most extreme "impact" scenario.
The main problem is deformation accumulated during slow cyclic loading. An offshore wind turbine must stay vertical within 0.5 degree range for over 20 years, during numerous storms, while being affected by wind, waves and moving components of the turbine it self. The whole structure weights up to 500 tones. The deign is done using drained strength and a safety factor of ~2. Then we know there is ~10x additional strength from undrained effects in sand...
Truly excessive safety. A 50 year (design) even hit a full scale foundation a few years ago - and the thing barely noticed it. :)
Hi
There always is some gas dissolved in liquids. The crux of the matter is when %LP is high , say 10% or higher, as a presence of bubbles then can shift the shock formation distance from 100's of meters to centimeters.
Then again, as near static stiffness, you would be dealing with much longer wavelengths than I do which would increase the shock formation distance. Plunger pumps usually run in the ballpark 0.2-2 Hz and you must add order N = 3, ... to this, i.e. an order or two of magnitude faster than typical waves.
For shocks, the rule of thumb is that you get the most energy into your structure when the effective shock duration matches half the period time.
From what I hear, scouring seems to be an issue.... Not much point in getting the foundation stiffness correct only to have it blown away.
/C
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