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A macro-meso nonlinear strength criterion for frozen soil

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Abstract

Different from other geotechnical materials, the strength characteristics of saturated frozen soil have nonlinear characteristics that first increase and then reduce with the increasing confining pressure, and are sensitive to varying temperature due to the existence of ice inclusions. Based on analysis on the breakage of frozen soil, it is believed that the proportion of each material component and ice-soil cementation breakage are the internal mechanisms to control the macroscopic strength. On the basis of this analysis, a multiscale strength criterion is proposed to account for this coupling mechanism. Firstly, the linear strength criterion of soil skeleton and ice inclusions in broken/unbroken states are defined, based on which the strength criterion of the bonded elements (unbroken material aggregate) and frictional elements (broken material aggregate) are deduced through the yield design theory and linear comparison composite. Secondly, the failure of ice-soil cementation is reflected by the transformation of the bonded elements to frictional elements, thus the multi-scale nonlinear strength criterion of representative volume element is derived by combining the binary medium model and strength homogenization method. These results have been verified by the test results of different types of saturated frozen soils.

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Abbreviations

\({F}^{bs}\) and \({F}^{fs}\) :

The strength criterion of unbroken soil aggregates and broken soil aggregates, respectively

\({F}^{bi}\) and \({F}^{fi}\) :

The strength criterion of unbroken ice aggregates and broken ice aggregates, respectively

\({c}_{i}\) :

The volume fraction of the ice inclusion for saturated frozen soil

\({p}^{b}\)/\({p}^{bs}\)/\({p}^{bi}\) :

Mean stress of bonded elements/unbroken soil skeleton/unbroken ice inclusion

\({q}^{b}\)/\({q}^{bs}\)/\({q}^{bi}\) :

Deviatoric stress of bonded elements/unbroken soil skeleton/unbroken ice inclusion

\({k}^{bi}\)/\({k}^{bs}\) :

Strength parameters for unbroken ice inclusion/unbroken soil skeleton

\({G}^{bs}\) (\({K}^{bs}\)) and \({G}^{bi}\) (\({K}^{bi}\)):

The shear modulus (bulk modulus) of unbroken soil skeleton and unbroken ice inclusion, respectively

\({\varepsilon }_{br}^{s}\) and \({\varepsilon }_{br}^{i}\) :

The failure strains of unbroken soil skeleton and unbroken ice inclusion, respectively

\({\varepsilon }_{s}^{bs}\) (\({\varepsilon }_{s}^{bi}\)) and \({\varepsilon }_{v}^{bs}\) (\({\varepsilon }_{v}^{bi}\)):

The shear strain and volumetric strain for unbroken soil skeleton (unbroken ice inclusion) at failure point, respectively

\({\sigma }_{1}\) and \({\sigma }_{3}\) :

The axial stress and confining pressure, respectively

\(A\)/\(B\) :

Stress concentration coefficient for bonded elements

\({M}_{C}\)(\({M}_{S}\)) and \({S}_{C}\)(\({S}_{S}\)):

Parameter of D-P criterion of broken ice inclusion (broken soil skeleton)

\({r}_{g}\) :

The ratio of the shear moduli of the broken ice inclusion and broken soil skeleton

\(a\) :

The linear parameter for D-P criterion

\({\varphi }^{b}\) :

The strain energy of bonded elements

\({\pi }^{b}\) :

The dissipated energy of bonded elements

\({K}^{b}\) and \({G}^{b}\) :

The bulk modulus and shear modulus of the bonded elements, respectively

\({G}^{fi}\) and \({G}^{fs}\) :

The ratio of the shear moduli of the broken ice inclusion and broken soil skeleton

\({Y}_{b}\) :

The nonlinear function of bonded elements

\({\pi }^{f}\) :

The dissipated energy of frictional elements

\({\psi }^{f}\) :

The strain energy of frictional elements

\({p}^{f}\) and \({q}^{f}\) :

The mean stress and deviatoric stress of the frictional elements at failure point, respectively

\({\varepsilon }_{v}^{f}\) and \({\varepsilon }_{s}^{f}\) :

The volumetric and shear strains of the frictional elements, respectively

\({K}^{f}\) and \({G}^{f}\) :

The bulk modulus and shear modulus of the frictional elements, respectively

\(\tau \) :

The prestress of frictional elements

\({Y}_{f}\) :

The nonlinear function of the frictional elements

\({\varphi }^{br}\) :

The strain energy of the broken process

\({\pi }^{br}\) :

The dissipated energy of the broken process

\({Y}_{br}\) :

The nonlinear function of the broken process

\({\psi }_{hom}\) :

The strain energy of saturated frozen soil

\({K}^{hom}\) and \({G}^{hom}\) :

The bulk modulus and shear modulus of saturated frozen soil, respectively

\({\widetilde{\Pi }}_{hom}\) :

The dissipated energy of saturated frozen soil

\(P\) and \(Q\) :

The mean stress and deviatoric stress of saturated frozen soil

\(M\) and \(S\) :

The strength parameter of frictional elements

\(\mathcal{g}\) and \(\mathcal{B}\) :

The breakage parameters of the material

\({P}_{0}\) :

The reference value of the mean stress

\({\lambda }_{v}\) :

The breakage ratio for saturated frozen soil

\({\sigma }_{ij}^{b}\)/\({\varepsilon }_{ij}^{b}\) :

The stress/strain tensor of bonded elements for saturated frozen soil

\({\sigma }_{ij}^{br}\) :

The broken stress

\({\varepsilon }_{ij}^{br}\) :

The broken strain

\({\sigma }_{ij}^{f}\) and \({\varepsilon }_{ij}^{f}\) :

The stress and strain of frictional elements

\({\tau }_{ij}\) :

The prestress tensor of frictional elements

\({\delta }_{ij}\) :

The Kronecker delta

\({\tau }_{ij}^{hom}\) :

The prestress tensor of saturated frozen soil

\({C}_{ijkl}^{f}\) :

The elastic stiffness tensor of frictional elements

\({C}_{ijkl}^{b}\) :

The elastic stiffness tensor of bonded elements

\({C}_{ijkl}^{hom}\) :

The elastic stiffness tensor of saturated frozen soil

\({A}_{ijkl}^{c}\) :

The strain concentration tensor of frictional elements

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Acknowledgements

The authors appreciate the funding of the National Natural Science Foundation of China (Grant No. U22A20596), Independent research project of state key laboratory of frozen soil engineering (NO. SKLFSE-ZQ-54), the financial support from China Scholarship Council (NO.202004910812), and the Key Laboratory of Geotechnical and Underground Engineering in Xi'an, Shaanxi Province (2060521002).

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Authors and Affiliations

  1. School of Architecture and Civil Engineering, Xi’an University of Science and Technology, Xi’an, 710054, China

    Pan Wang & Bin Zhi

  2. State Key Laboratory of Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou, 730000, China

    Pan Wang, Bingtang Song & Qihao Yu

  3. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu, 610065, China

    Enlong Liu

  4. China Railway Qinghai-Tibet Group Company, Xining, 810000, Qinghai, China

    Jinchang Wang & Rongning Sun

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  1. Pan Wang
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Correspondence to Enlong Liu.

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Wang, P., Liu, E., Zhi, B. et al. A macro-meso nonlinear strength criterion for frozen soil. Acta Geotech. (2024). https://doi.org/10.1007/s11440-023-02197-5

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