The best method is numerical modeling of the test in Abacus or flac software and changing parameters step by step

I waited for someone to ask this question for about... 7 year now. Short answer - there is no answer. You are reaching beyond the horizon of "what is known so far". Congratulations. It is a true scientific question.

Based on lab testing results - critical state is imaginary. It is a theoretical concept introduced to help "draw" a curve. As you said yourself - there is no physical observation of it in the literature, the way you would like it to be. Only a few tests where one loading cycle is applied and the test "stops" there. Artificially. Without proving anything. Every such test ever shown - needs a unique modelling calibration. Meaning - you have to know the answer, before the curves can be fitted to that one specific case. New case - new calibration. Randomized testing - is never fitted. Even remotely.

Schofield himself criticized the habit of irresponsible use of "critical state" in his auto biography Disturbed Soil Properties & Geotechnical Design. Where he describes critical state as an "assumption" made to describe one clay specimen, while developing the original Cam-Clay model. Since then, this "idea" has spread far and wide without anyone actually testing it in dynamic tests - as you are looking for. Likely, nobody tested it because of funding or other reasons - same reasons as why you have no access to actually test the "theory" yourself. You see, question is, why don't you have access to the equipment which would allow you to find the answer to your own questions? Free form, curiosity based testing is not financed, now is it?

We are inside a field where the lecturers lecture, and theory is derived based on theory. The contact point with reality is extremely rare. We rely on theories derived from theoretical theory interpretations of theoretical theory concepts... So, the truth could be - you can't find the tests showing the critical state in sand, because it is not a "real" thing. A few rarely cited, forgotten, ignored papers do show it empirically too.

I.vardoulakis had examples of tests where sand is squeezed to 50% strain using the frictionless triaxial apparauts in 1980 (50% is supposed to be theoretically implausible). He observed even loose sand dilates at large deformation levels. His empirical demonstration and revolutionary (yet - unorthodox) test was left not noticed. He was not funded. And he is dead now. And yet, his test opens a whole new scope of empirical testing. You can not only squeeze a sand specimen by 50% strain, you can also pull it back to initial state, and keep testing disturbed soil properties. It's totally plausible, and reveals lots of new soil patterns and properties.

I've checked his tests in my spare time, at nights, in the lab. Back when we still had some unique equipment in the lab from the 90's (a frictionless triaxial apparatus with PID controlled hydraulic piston). And observed the same thing as I.Vardoulakis - critical state in sand does not happen. Even super loose sand dilates during large deformation amplitudes. Hell, take a shovel - try to push loose sand sideways... it locks together, it dilates... Which makes me think most theoreticians had never touched it.

But using a frictionless triaxial apparatus, it is plausible to test peak strength more than once in one specimen. You can even liquefy the specimen and then drain it - multiple times, testing peak strength at different densities, and the stiffness paths in between. In the process observing that the "phase transition line" is not constant. The stress angle at which soil transitions from contractive to dilative response - moves all the time. There is no "unique" state transition line and no "unique" critical state line in sand either. Based on empirical evidence. Based on tests which are not stopped artificially the very moment the first "transition" from drained to undrained is triggered.

If you do 1000 undrained, irregular loading cycles - the state transition angle will move all over the place - in the physical measurements. Perhaps that is also why you will only encounter "constant amplitude" simulations - because the models simply fail to capture the underlying physical causation of the phenomenon governing the problem. And there is no funding for building and operating the type of equipment needed to "decode" the physics of it. It's a question of "fundamental research" not some "profit in 6 months business scheme".

There are some rarely cited, but attention worthy investigations on this topic. Such as the early work of R.Dobry, where he shows liquefying hysteresis loops can be normalized into one line - when strain (deformation) envelope is applied. If hysteresis loops are treated as strain (deformation) functions, the empirical evidence converges to a solution. But in private correspondence with R.Dobry I was told he abandoned that path because industry is selective (only stress based models supported by software already invested into (alternative finding - bed for return of investment)). And researchers only cite things that compliment them (nothing truly new / contradicting will be cited). In a world of "publish or parish" combined with "get your own funding" - there is no place for the type of "fundamental research" that is required, to discover the answer to your question.

My solution would be - code some arduino's, make self a load cell, and a displacement sensor, go to the beach and do your own tests there. There is no place on earth financing "curiosity driven research" or "observation inspired research". You can do science only at nights, if you are allowed to stay in the lab. For during the days you will have to follow orders, do what you're told - for some agenda, some target, some "profit margin".

To some degree I targeted what you ask for in my laboratory tests. Found a way to control soil stiffness with high precision. Demonstrating how to disturb a specimen (test peak strength), then cyclic load it back to initial state (disturb back to initial stiffness path using drained loading cycles). Take a look if you wish. The papers show raw data files, recorded in physical testing. And the test results are not compatible with any of the existing models. The test results show all the constants are moving, and critical state in dynamic loaded sand is merely an illusion. And that's inconvenient. And there have been hints of it in empirical testing since back in the 80's. And the empirical - physical proof has been ignored for more than 40 years now.

Sadly, If you wish to get any social recognition in this field, I can only recommend you never try to do real physical tests, and hide behind books - confirm the opinions, like it's religious. The field is full of anomalies and paradoxes, all the faculties and institutions are at each other's throats globally. Everyone quickly finding "empirical evidence" to contradict the opponents (and ignoring the fact they contradict themselves too) - but no one is converging towards a "common pattern" that would combine drained and undrained loading cycles, durring irregular loads, producing both loss and recovery of soil stiffness in one giant long simulation / or a real physical test run.

As it is, we have Oxford building larger and larger sticks in mud. Repeatedly "discovering" and "demonstrating" the same thing the same way - on increasingly larger scale. And some theory inspired theoreticians who can barely afford a good laptop - thus never touch a real soil specimen in their lifetime.

that's the reality of it. You have a really good question, though. I wish, I dream there was a place on earth where scientific equipment was still being built... rather than removed, and replaced by some "industry standards". Alas, I no longer know of such place. Our equipment from the 90's also has been recently removed and replaced with absurdly bad devices. The old devices looked ugly but boy could they "observe lots of new things". The new ones... can not do any "unique" tests - but do have some touch screens. Now, we can observe "aliens dancing" on the new touch screens... amazing...

That's the reality of this paradigm now. With respect to your question - solution unknown. You have a good question. No one has time or resources to do the required tests, though.

In this reality, your best bet is to find some prestigious, "famous" person, or a financially secure local professor - and follow the leader without raising questions, or getting paid much. As long as they like you, you will be safe. And your answers accepted by public through association - at least in the local space. And if you draw pretty pictures, reviewers will let you pass too. Don't worry about it.

P.S. Just in case someone brave comes by - P.J.Yoder 1980, strain space plasticity. The paper shows how to run elasto-plastic fem using parallel spring systems, which is GPU compatible (super computers). His concepts were repeatedly shown to deliver "equivalent" precision to conventional strain-space models. But now we have physical test that show strain-envelopes govern cyclic loaded soil properties. In addition to GPU (parallel computation cores) that could run these model of improved physical precision - thousands of times faster.

As Mr. Kamal said, the best approach is numerical modeling. I think you need basic geotechnical test results to predict the critical state parameters. You can refer the following articles:

1. Model of Soft Soils under Cyclic Loading by Jing Ni1 ; Buddhima Indraratna, F.ASCE2 ; Xue-Yu Geng3 ; John Phillip Carter4 ; and You-Liang Chen5. . DOI: 10.1061/(ASCE)GM.1943-5622.0000411. © 2014 American Society of Civil Engineers.

2. Critical State Soil Mechanics for Cyclic Liquefaction and Postliquefaction Behavior: DEM study by M.M Rahman et al https://ascelibrary.org/doi/10.1061/%28ASCE%29GT.1943-5606.0002453

3. Santamarina, J. C. and Cho, G. C., “Determination of Critical State Parameters in Sandy Soils—Simple Procedure,” Geotechnical Testing Journal, GTJODJ, Vol. 24, No. 2, June 2001, pp. 185–192.

Dear Mr. Towhid

I highly recommend your to refer to Pr. Md Mizanur Rahman who have done many researches regarding critical state parameters and state parameters from monotonic and cyclic triaxial tests.

https://www.researchgate.net/profile/Md-Mizanur-Rahman-16

Regards.

Abdellah.