Animating Soil Models – Visualizations as open education tool for soil constitutive modeling

With this project I aim to facilitate teaching and understanding concepts related to constitutive modeling with the help of visualizations. The following topics are visualized: yield surfaces, stress invariants, Critical State Soil Mechanics and some related models as the Modified Cam Clay model and clay hypoplasticity. The visualizations are shared under the open license CC BY to use them as teaching and learning material.

Acknowledgement: I thank the geotechnical engineers on Twitter, through whom I got the idea for this project. I am grateful for so much feedback and thank you all for your messages regarding the project. I further thank David Mašín for his support here on the platform SoilModels.com and Hans-Peter Schröcker (University of Innsbruck) for suggesting to use asymptote.sourceforge.io for the interactive graphics. I gratefully acknowledge financial support of the University of Innsbruck: ProLehre project, AURORA Challenge Domains. Project duration: 12/2020–11/2021, amount: € 13.808

… include the animations in LaTeX presentations:

Here you can download an example of how to include an animation in your beamer class presentation:
LaTeX_Beamer_Class
For every animation, you can download the related PDF file including all slides below each animation.

The PDF animations can be viewed with Acrobat Reader (except on mobile devices).

For Linux users:
Acroread 9 is availabe as Linux installation.
Okular 21.08.1 enables to watch the animations. However, the control buttons are not available using Okular.
Thanks to Wolfgang Fellin for trying out different PDF viewers for Linux.

… animate soil models: (coming soon)

Animations to visualize the stress invariants in principal stress space:
p’: mean effective stress
q: deviatoric stress
θ: the Lode angle to define the deviatoric direction of a stress state



The interactive graphic below has been created with GeoGebra

You can use the slider for parameter variation of M and choose to display Drucker-Prager. In addition, it is possible to move the principal stress state (red bullet) in the right figure and to rotate the 3D figure. The deviatoric plane corresponds to p = 100 kN/m².

click to enlarge

The interactive WebGL graphics below have been created using asymptote.sourceforge.io
click to enlarge

p-q plane (scaled) for axisymm. tr. comp. is added

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cohesion is added

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vizualise plane stress, σ₃=0

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Rendulic plane is added

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p-q plane (scaled) for axisymm. tr. comp. is added

click to enlarge

cohesion is added

click to enlarge

vizualise plane stress, σ₃=0

click to enlarge

Rendulic plane is added

click to enlarge

p-q plane (scaled) for axisymm. tr. comp. is added

click to enlarge

cohesion is added

click to enlarge

vizualise plane stress, σ₃=0

 

click to enlarge

Rendulic plane is added

click to enlarge

p-q plane (scaled) for axisymm. states is added

click to enlarge

vizualise plane stress, σ₃=0

click to enlarge

p-q plane (scaled) for axisymm. states is added

click to enlarge

vizualise plane stress, σ₃=0

Useful references for 3D failure surfaces are e.g.:

  • Griffiths, D.V. (1990): Failure Criteria Interpretation Based on Mohr Coulomb Friction.
    Journal of Geotechnical Engineering, Vol. 116, Issue 6.
    doi: 10.1061/(ASCE)0733-9410(1990)116:6(986)
  • Griffiths, D.V. and Huang, J. (2009): Observations on the extended Matsuoka–Nakai failure criterion.
    Int. J. Numer. Anal. Meth. Geomech., 33: 1889-1905.
    doi: 10.1002/nag.810
Interactive WebGL graphics, created with asymptote.sourceforge.io

The interactive graphic below has been created with GeoGebra

You can use the yellow sliders for parameter variation. You can also vary the values for the mobilized friction angle φₘ (green) and Hvorslev’s equivalent pressure p’ₑ (black) and display the corresponding asymptotic states. In addition, it is possible to rotate the 3D figure. The ASBS is displayed for 1 < p’ₑ < 100 kN/m².

click to enlarge


The interactive graphic below has been created with GeoGebra

You can use the green sliders for parameter variation. You can also vary the value for the preconsolidation pressure p’₀ (black) and display the corresponding yield surface. In addition, it is possible to rotate the 3D figure. The SBS is displayed for 1 < p’₀ < 100 kN/m².

click to enlarge

This page related to SANISAND has been created in cooperation between Gertraud Medicus (University of Innsbruck, Austria) and Mahdi Taiebat (University of British Columbia, Canada).


The interactive graphics below have been created with GeoGebra

Yield surface: the model uses a Drucker-Prager yield surface with anisotropy. The isotropic size is controlled by model constant m and the anisotropy is controlled by internal variable α. You can adjust the related parameters and visualize the yield surface in the 3D stress Principal Axes space and in the π-plane for the stress-ratio, where co-axiality (i.e., same Principal Axes) of stress and back-stress tensors are implied.

click to enlarge

 


Critical, dilatancy, and bounding surfaces: The model uses a fixed Lode angle dependent critical state surface, and state parameter dependent dilatancy and bounding surfaces. All three surfaces are isotropic. The size of the critical state surface is controlled by model constants M and c, the state parameter ψ adjusts the sizes of the dilatancy and bounding surfaces through model parameters nd and nb, respectively. You can adjust the related parameters and visualize the model surfaces in the 3D principal stress space.

click to enlarge

MATLAB scripts for Drucker-Prager yield surface with anisotropy and all SANISAND surfaces: sanisand.zip

The SANISAND-MSf (Yang, Taiebat & Dafalias, 2022) model includes two novel constitutive ingredients to address primarily the undrained cyclic response:

  • A memory surface (M), resulting in an evolving distance quantity bM, for more precisely controlling stiffness affecting the plastic deviatoric and volumetric strains and ensuing excess pore pressure development in the pre-liquefaction stage.
  • The concept of a semifluidised state (Sf) and the related formulation of an internal degradation variable for plastic modulus and dilatancy, named the ‘strain liquefaction factor’ and symbolled as l, aiming at modelling large shear strain development in the post-liquefaction stage.

The animation of the model simulation for Fig. 13 of the reference paper are illustrated below.


 

The interactive graphic below has been created with asymptote.sourceforge.io

The yield, dilatancy, critical state, bounding and memory surfaces according to SANISAND-MSf model are visualized in the 3D stress space below.

click to enlarge

Key reference:

  • Yang, M., Taiebat, M. & Dafalias Y. F., SANISAND-MSf: a sand plasticity model with memory surface and semifluidised state, Géotechnique, 2022, 72:3, 227-24, doi: 10.1680/jgeot.19.P.363

Other references:

  • Manzari, M. T. & Dafalias, Y. F. (1997). A critical state two-surface plasticity model for sands. Géotechnique 47, No. 2, 255–272.
  • Dafalias, Y. F. & Manzari, M. T. (2004). Simple plasticity sand model accounting for fabric change effects. J. Engng Mech. 130, No. 6, 622–634.
  • Taiebat, M. & Dafalias, Y. F. (2008). SANISAND: simple anisotropic sand plasticity model. Int. J. Numer. Analyt. Methods Geomech. 32, No. 8, 915–948.
  • Dafalias, Y. F. & Taiebat, M. (2016). SANISAND-Z: zero elastic range sand plasticity model. Géotechnique 66, No. 12, 999–1013
  • Barrero, A. R., Taiebat, M. & Dafalias, Y. F. (2020). Modeling cyclic shearing of sands in semifluidized regime. Int. J. Numer. Analyt. Methods Geomech. 44, No. 3, 371–388.

The Modified Cam Clay (MCC) model by Roscoe & Burland (1968) is an elasto-plastic hardening model, assuming associated flow. It includes concepts from Critical State Soil Mechanics as the Normal Compression Line (NCL) and the Critical State Line (CSL).


Click on the images to enlarge them. You can download the GIF files directly. Below each figure, you can also download corresponding PDF files.

State boundary surface of the Modified Cam Clay model

Drained (cd) triaxial tests


Undrained (cu) triaxial tests

normally consolidated:

‘TSP’ indicates the total stress path

Linear-elasticity: How does ν affect the K₀-stress path (the stress path under oedometric compression). Mohr-Coulomb hexagon for φ = 30°, c = 0 is added; inspired by Zheng, Liu & Li (2005): doi: 10.1002/nme.1406 φ –ν inequality, sin φ ⩾1 – 2ν

How does ν affect the stress path of a plane-strain (biaxial) compression test. Linear-elastic, perfectly plastic (Mohr-Coulomb: φ = 30°, c = 0, ψ = 0°) model.

9 Comments
  1. Ahmed Salah Kamel Mohamed

    Hello Prof. Gertraud,

    Thank you so much for the great work. I have a question concerning the elliptical equation used to represent the state of the boundary surface of the cam clay model, may you please guide me to the source of this equation?

    Thank you!

    Ahmed S. Kamel
    PhD Researcher

  2. Alaa Kourdey
    Alaa Kourdey 9 months ago

    Hi Gertraud,

    Could you please provide the programming code for “3D view of the Normal Compression Line (NCL), Critical State Line (CSL) and State Boundary Surface of the MCC model “? Thank you.

    Best wishes,
    Alaa

  3. Ahmed Salah Kamel Mohamed

    Hi Gertraud,

    Thanks a lot for investing time for clarifications, this is a very appreciated efforts, actually I meant the cam clay model in p’, q, v space. I think I will figure it out from the mentioned sources.

    Thank you again for your prompt answer!

    Kind regards,
    Ahmed.

  4. Gertraud Medicus Author
    Gertraud Medicus 8 months ago

    Hi Alaa, hi Ahmed,

    sorry for the late reply.

    Here, I uploaded the script for the state boundary surface of the MCC model in p’-q-e space:
    https://soilmodels.com/wp-content/uploads/2020/12/sbs_MCC_pqe_soilmodels.txt

    The surface has been created using: https://asymptote.sourceforge.io
    Just use the Asymptote Web Application and paste the lines in here:
    http://asymptote.ualberta.ca/

    Kind regards,
    Gertraud

  5. A S M RIYAD
    A S M RIYAD 8 months ago

    Hello Prof. Gertraud,

    Thank you so much for the great work. I have a question concerning the elliptical equation used to represent the state of the boundary surface of the bounding surface plasticity model, may you please guide me to the source of this equation?

    Thank you!

    Riyad
    Ph.D. Researcher

  6. Gertraud Medicus Author
    Gertraud Medicus 8 months ago

    Hello Riyad,

    Thanks for your feedback.

    If you are looking for the state boundary surface of the Modified Cam Clay model, the comments above should help. For a description of the MCC model and related equations, I suggest: Muir Wood (1990), Soil Behaviour and Critical State Soil Mechanics, Cambridge University Press

    Regards, Gertraud

    • A S M RIYAD
      A S M RIYAD 8 months ago

      Thanks a lot for your reply, Prof. Gertraud. Actually, I am looking for the details of the bounding surface plasticity model (initially developed by Prof. Yannis Dafalias). Have you developed any script for the state boundary surface of the bounding surface plasticity model in the p’-q-v space during drained loading?

      • Gertraud Medicus Author
        Gertraud Medicus 8 months ago

        Thanks for clarification. Sorry, I didn’t read your question properly.
        So far, I have not been dealing with the boundary surface plasticity model.
        If I will, I will come back to your questions.
        All the best, Gertraud

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