L33 Cam-Clay model (Part 2): calculation of elastic and plastic strains

Lecture 33 of PGE 383 (Fall 2020) Advanced Geomechanics at The University of Texas at Austin delivered on 2020/11/4 by DN Espinoza ( / dnegeomechanics .
Topics: Modified Cam-Clay model (MCC), consolidation test, isotropic compressibility coefficients, pre-consolidation stress, elastic ans plastic stiffness matrices, strain localization caused by strain-softening.
Links
MCC in sedimentary basin modeling: doi.org/10.1306/04111110143
MCC for source rocks: library.seg.org/doi/10.1190/g...

[@victoriareis8829]

theres a lot of cement sheath models that uses modified cam clay and only watching your video I could finally understand. Thank you so much.

[@dnicolasespinoza5258]

You are welcome, I'm glad it was useful to you! Many papers put all emphasis on equations and do not talk about the physics and underlying mechanisms behind the math.

[@VahidAzami]

Very beautifully explained. Your lecture was highly helpful for me.

[@mariogalindoq]

Could you explain please why the model doesn't consider the cohesion being that one of the most important characteristic of a clay is the cohesion? Thank you in advance.

[@hugoperalta5362]

great lecture! a pro tip is to watch it in x1.5 speed.

[@dnicolasespinoza5258]

Yeah, I'm a bit slow sometimes! 😂

[@hugoperalta5362]

@@dnicolasespinoza5258 hahahahhaha dont worryyy! Really loved your lectures! Very well explained!

[@PedroAraujop6]

In a FEM 2D model, after calculating the plastic strains, how do I calculate the values for strain in the vertical (axial) and horizontal directions? I believe that with the elastic strains it would be easy as we have a constant poisson ratio.

[@dnicolasespinoza5258]

Itasca offers a fantastic guide on the numerical implementation of the modified Cam-Clay model: docs.itascacg.com/flac3d700/common/models/camclay/doc/modelcamclay.html

[@JosueA_455]

Thank you for your explanation. Could you kindly let me know how you obtained equation number 2? Minute 17.54.

[@dnicolasespinoza5258]

(1) you have to express the differential of void ratio as a function of the variables Vp and Vb, i.e., de = Vb/(Vb-Vp)^2 * dVp - Vp/(Vb-Vp)^2 * dVb, (2) assume dVp=dVb (all deformation comes as change of pore volume), (3) do the math and replace porosity Vp/Vb = e/(1+e), (3) use definition of volumetric strain to replace dVb/Vb for delta eps_p'

[@DivyanshuLal0602]

very beautifully explained ....can you give more references related to the study of compaction and shear band

[@dnicolasespinoza5258]

This is a great paper to get to know more about compaction bands: Pure and shear-enhanced compaction bands in Aztec Sandstone by Eichhbul et al., doi.org/10.1016/j.jsg.2010.02.004

[@JamesDean.]

Thank you so much, Could you explain how's Cam-Clay used to model sands?

[@dnicolasespinoza5258]

Hi Ibrahim, the application is similar. I recommend reading this classic paper for more details: www-civ.eng.cam.ac.uk/geotech_new/people/bolton/mdb_pub/14_Geotechnique_No36_Is1_65_78.pdf

[@pablosorosoto2360]

Wich would be the loading path of an unconfined load test?

[@dnicolasespinoza5258]

Such stress path will start at (p'=0, q=0) and increase with delta_p' = (1/3) * dq. However, that stress path is not possible with the yield surface used in this lecture, valid for an uncemented sediment (unconfined loading test not possible). You would need a yield surface with an intercept q different than 0 at p'=0 (cemented rock), like the one shown here: kzbin.info/www/bejne/oJjGpHuvh7Kbo7M

[@KishanKumar-ih6vo]

Very well explained. What are the assumptions you have considered to obtain explicit relations for plastic strain in MCC model (apart from biot coefficient=1) ? How relevant are these assumptions in actual reservoirs. In the literature I see that for MCC model they solve for plastic variable, stress and hardening parameter simultaneously at each loading step or for example in the attached Ruben Juanes paper for shale they solve for stress and plastic multiplier simultaneously. Thank you,

[@dnicolasespinoza5258]

If you want to learn more about poro-elasto-plasticity I recommend: Coussy, O. (2011). Mechanics and physics of porous solids. John Wiley & Sons.

[@KishanKumar-ih6vo]

@@dnicolasespinoza5258 Thank you for the response Dr Espinoza. I have one more question which is bothering me. What is the equation for increment of preconsolidation pressure dp'0 you substitute in equation 4 (because this term is not present in equation 3) to find plastic multiplier increment (from eqn 3 and 4) ?

[@dnicolasespinoza5258]

@@KishanKumar-ih6vo It comes from the isotropic loading and unloading relationships presented around minute 16:00.

[@zohrea4962]

Do you recommend using Cam-Clay model when modelling the 1-D consolidation test in FEM?

[@dnicolasespinoza5258]

Yes, this is actually problem 2.d of "Weekly Project 8: Soft Sediments Constitutive Models" in dnicolasespinoza.github.io/AdvancedGeomech, for a single point, but can be extended to FEM

[@zohrea4962]

@@dnicolasespinoza5258 What models do you recommend using when simulating loading and unloading on the soil sample? (Mohr - Circle, Drucker Prager or Hyperbolic... )

[@diogocecilio3540]

Thank you for the video, very well explained! Can you give any reference with a simple numerical example?

[@dnicolasespinoza5258]

This is a homework assignment: dnicolasespinoza.github.io/AdvancedGeomech/ [WP8-2]. I will publish the solution in ~October. I remember seeing exercises like this in Braja Das' books (scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=braja+m+das&btnG=&oq=braja+) but cannot recall exactly where : )

[@maria-ds9hq]

Thank you for this nice lecture. i am still confused about how you obtain the hardening parameters from equations 3 , 4 and then how the matrix of plasticity is made? i tried a lot but i couldn't succeed to obtain the equation.

[@dnicolasespinoza5258]

Hi Maria, Eq. 3 is a definition. Eq. 4 is almost the same as Eq. 1 (top) but in the plastic regime past p'_o. Notice that going from changes of void ratios to changes of volumetric strains requires the (1+e) factor (see my reply below to Fedilberto Gonzalez). Last, the plastic stiffness matrix requires taking changes in dp' and dq. The partial derivative equations of the yield surface (in red in my notes) might be a bit misleading because they are hiding q and p' in eta. I recommend a reverse demonstration, assume the matrix is right and try to recover the equations above, particularly for eta=0 and eta=M. I also recommend a numerical solution following exercise 2 here: dnicolasespinoza.github.io/AdvancedGeomech/node10.html . I'm checking my derivation though, for any typo or mistake, I cannot find the original notes now... will update soon. The final matrix should be correct though.

[@maria-ds9hq]

@@dnicolasespinoza5258 thank you so much for your reply.