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A Modified Two-Surface Plasticity Model for Saturated and Unsaturated Soils

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Abstract

A modified two-surface critical state plasticity model for saturated and unsaturated soil is presented in this study. The key modification in new model is inclusion of an alternative yield surface used to simulate the behavior of unsaturated soils in addition to corresponding saturated conditions. Moreover, a numerical technique is used to obtain an incremental stress–strain response from loading curves. Modification is applied continuously in each incremental step to return the final stress states and hardening parameters to the yield surface. Results revealed that the adopted modeling approach can predict two independent sets of laboratory unsaturated experiments under various conditions satisfactorily. The model can well predict the augmentation of shear strength due to a rise in suction under different net stress levels. In addition, the influence of net stress on volumetric tendency was reliably simulated under drained conditions. The distinct feature of this model in capturing the suction-induced hardening and volumetric transition from dilation to contraction should be highlighted. It is also important to note that according to the results of hypothetical simulations, the stress–strain behavior of saturated coarse-grained soils can be rationally captured for a wide range of confining stresses and densities under both drained and undrained conditions.

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Abbreviations

BR:

v

\(A_{0}\) :

State constant in the simplest case

\(A_{d}\) :

Function of the state

C :

Apparent cohesion

\(c^{c, b, d}\) :

Critical state variables

\(C_{z}\) :

Fabric dilatancy constant

\(C_{{\text{h}}}\) :

Hardening constant

\(C_{m}\) :

Positive quantity and a function of state variables

D :

Dilation coefficient

e :

Porosity

\(e_{{\text{c}}}\) :

Critical porosity ratio

\(e_{{\text{c - ref}}}\) :

Model constant

\(\dot{e}^{{\text{p}}}\) :

Deviatoric plastic strain rate tensor

\(f\) :

Yield function

\(G\) :

Shear modulus

\(G_{0}\) :

Initial shear modulus

\(h\) :

Positive scalar-valued

\(h_{0}\) :

Positive constant

\(I\) :

Unit quadratic tensor

\(K\) :

Bulk volumetric module

\(K_{{\text{p}}}\) :

Plastic modulus

\(m\) :

Angle’s tangent attributed to yield surface opening

\(\dot{m}^{p}\) :

Isotropic hardening

\(M^{{\text{b}}}\) :

Bounding stress ratio

\(M^{{\text{c}}}\) :

Critical stress ratio

\(M^{{\text{d}}}\) :

Dilatancy stress ratio

\(M_{{\text{c}}}^{{\text{b}}}\) :

Slope of bounding surface on the q-p space

\(M_{{\text{c}}}^{{\text{c}}}\) :

Slope of critical state surface on the q-p space

\(M_{{\text{c}}}^{{\text{d}}}\) :

Slope of dilatancy surface on the q-p space

\(n\) :

Active loading

\(N\) :

Volumetric component size

\(n^{b}\) :

Factor of drained peak stress state

\(n^{d}\) :

Factor of phase transformation

\(p = (\sigma_{1} + 2\sigma_{3} ){/}3\) :

Confining stress

\(P\) :

Confining pressure

\(P_{{{\text{at}}}}\) :

Atmospheric pressure

\(P_{{\text{c}}}\) :

Critical confining stress

\(P\) :

Confining pressure

\(q = \sigma_{1} - \sigma_{3}\) :

Deviatoric stress

\(s\) :

Deviatoric stress tensor

S :

Matric suction

\(\left| u \right|\) :

Tensor size

\(u:v\) :

Sum of the product of two adjacent tensors

\(z\) :

Fabric-dilatancy tensor-valued variable

\(z_{\max }\) :

Highest value of z

\(\alpha\) :

Angle’s tangent attributed to bisector surface

\(\alpha_{\theta }^{c, b, d}\) :

Deviatoric stress ratios that are functions of Lode angle

\({\varvec{\alpha}}\) :

Deviatoric back-stress ratio tensor/axis direction of the yield surface

\(\dot{\user2{\alpha }}^{p}\) :

Kinematic hardening

\({\varvec{\alpha}}_{{{\text{in}}}}\) :

The value of \({\varvec{\alpha}}\) at the initiation of a loading process

\({\varvec{\alpha}}^{c, b, d}\) :

Deviatoric back-stress ratio tensors related to critical, boundary, and dilative surfaces

\({\varvec{\alpha}}_{{\varvec{\theta}}}^{{\varvec{b}}}\) :

Current state of stress image on the boundary surface

\({\varvec{\alpha}}^{c. b. d}\) :

Deviatoric back-stress ratio tensors related to critical, boundary, and dilative surfaces

\(\dot{\varepsilon }_{{{\text{pl}}}}\) :

Plastic strain rate

\(\varepsilon^{{\text{p}}}\) :

Plastic strain

\(\eta = q{/}p\) :

Deviatoric stress ratio

\(\theta\) :

The Lode angle of n

\(\lambda\) :

Model constant

\(\nu\) :

Poisson’s ratio

\(\xi\) :

Model constant

\(\sigma_{1} . \sigma_{2} . \sigma_{3}\) :

Principal effective stress

\(\partial f{/}\partial \sigma\) :

The gradient of yield surface

\(\partial g{/}\partial \sigma\) :

The gradient of potential function

\(\psi = e - e_{c}\) :

State parameter

\(\omega\) :

Direction of triaxial shearing

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Acknowledgements

The inevitable role of late Prof. Sadrnejad in supervising this research cannot be denied. Although we were unfortunate to have his consent for the final manuscript, we believe that he had been one of the key contributors to this work. May his soul rest in peace and be rewarded according to a life of scientific contribution.

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Authors and Affiliations

  1. K. N. Toosi University of Technology, Tehran, Iran

    Mohammad Vahdani

  2. Memorial University of Newfoundland, St. John’s, Canada

    Mohammad Mahdi Hajitaheriha

  3. University of Calgary, Alberta, Canada

    Amin Hasani Motlagh

  4. Sharif University of Technology, Tehran, Iran

    Hamed Sadeghi

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  1. Mohammad Vahdani
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  2. Mohammad Mahdi Hajitaheriha
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  3. Amin Hasani Motlagh
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  4. Hamed Sadeghi
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Correspondence to Hamed Sadeghi.

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Vahdani, M., Hajitaheriha, M.M., Hasani Motlagh, A. et al. A Modified Two-Surface Plasticity Model for Saturated and Unsaturated Soils. Indian Geotech J 52, 865–876 (2022). https://doi.org/10.1007/s40098-022-00601-7

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