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Improvement to the intergranular strain model for larger numbers of repetitive cycles

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Abstract

The analysis of geotechnical problems involving saturated soils under cyclic loading requires the use of advanced constitutive models. These models need to describe the main characteristics of the material under cyclic loading and undrained conditions, such as the rate of the pore water pressure accumulation and the stress attractors. When properly doing so, the models are expected to be reliable for their use in boundary value problems. In this work, an extension of the widely implemented intergranular strain model by Niemunis and Herle (Mech Cohes Frict Mater 2(4):279–299, 1997) is proposed. The modification is aimed to improve the capabilities of the model when simulating a number of repetitive cycles, where a proper reduction of the strain accumulation is expected. For validation purposes, the reference model and proposed improvement are compared against some monotonic and cyclic triaxial tests. The results indicate that the intergranular strain improvement model provides a more realistic prediction of the accumulation rates under cyclic loading, without spoiling the advantages of the original formulation.

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Acknowledgements

J. Duque and D. Mašín authors appreciate the financial support given by the INTER-EXCELLENCE Project LTACH19028 by the Czech Ministry of Education, Youth and Sports. J. Duque appreciates the financial support given by the Charles University Grant Agency (GAUK) with Project No. 200120. D. Mašín acknowledges institutional support by Center for Geosphere Dynamics (UNCE/SCI/006).

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Appendices

Appendix 1: Notation

Table 6 summarizes some symbols, notation and tensorial operations. Indicial notation is used (e.g., \(A_{ij}\)). \(\delta _{ij}\) is the Kronecker delta, also represented with (\(1_{ij}=\delta _{ij}\)).

Table 6 Summary of some symbols and operations

Components of the effective stress tensor \({{\varvec{\sigma }}}\) or strain tensor \({{\varvec{\varepsilon }}}\) in compression are negative. Roscoe variables are defined as \(p=-\sigma _{ii}/3\), \(q=\sqrt{\frac{3}{2}}\parallel {{\varvec{\sigma }}}^{{\rm dev}}\parallel\), \(\varepsilon _{\rm v}=-\varepsilon _{ii}\) and \(\varepsilon _{\rm s}=\sqrt{\frac{2}{3}}\parallel {{\varvec{\varepsilon }}}^{{\rm dev}}\parallel\). The stress ratio \(\eta\) is defined as \(\eta =q/p\). The deviator stress tensor is defined as \({{\varvec{\sigma }}}^{{\rm dev}}={{\varvec{\sigma }}}+p\,{\mathbf {1}}\).

Appendix 2: Summary of the hypoplastic model for sands by Von Wolffersdorff

In this Appendix, a summary of the constitutive relations of the hypoplastic model for sands by Von Wolffersdorff [40] is provided, see Table 7:

Table 7 Constitutive relations of the hypoplastic model by Von Wolffersdorff [40]

Appendix 3: Simulations of monotonic loading

In this Appendix, some simulations of the HP + ISI model under monotonic loading are shown. The tests correspond to three oedometric compression tests with an unloading–reloading cycle, 8 undrained and 5 drained monotonic triaxial tests, with variation of the initial mean effective pressure. Simulation results are presented in Figs. 7, 8 and 9, respectively, and showed a satisfactory agreement with the experimental data.

Fig. 7
figure 7

Simulations of three oedometric tests OED8-10 with an unloading–reloading cycle, e-Log \(\sigma _1\) space. Initial conditions are given in Table 1

Fig. 8
figure 8

Simulations of undrained monotonic triaxial tests with isotropic consolidation (\(q_0 = 0\) kPa) under extension and compression with variation of the initial mean effective pressure \(p_0=\{100,200,300,400\}\) kPa. Initial conditions are given in Table 1

Fig. 9
figure 9

Simulations of drained monotonic triaxial tests with isotropic consolidation (\(q_0 = 0\) kPa) and variation of the initial mean effective pressure \(p_0=\{50,100,200,300,400\}\) kPa. Initial conditions are given in Table 1

Appendix 4: Short guide for the determination of parameters \(\chi _0\), \(\chi _{{\rm max}}\) and \(C_\Omega\)

In this section, a short guide for the determination of the new parameters is given:

  • Parameter \(\chi _0\) controls the accumulation rate on the first cycles. Its calibration can be performed simulating the first cycles (e.g., \(N<5\)) on an undrained cyclic triaxial test. An example of the influence of this parameter is given in Fig. 10a.

  • Parameter \(\chi _{{\rm max}}\) controls the rate of the accumulation after a larger number of repetitive cycles with small strain amplitudes (e.g., \(N>10\)). It should be adjusted to reproduce the reduction of the strain accumulation rate observed in the experiments. Figure 10b provides an example of its effect on the normalized pore water pressure accumulation.

  • Parameter \(C_\Omega\) controls how fast the strain accumulation rate reduces and the model changes the IS exponent \(\chi\) from \(\chi =\chi _0\) to \(\chi =\chi _{{\rm max}}\). It should be adjusted to reproduce the observed behavior of the pore water pressure accumulation on undrained cyclic triaxial tests, see Fig. 10c.

Fig. 10
figure 10

Accumulation of the normalized pore water pressure \(p^{{\rm acc}}_{\rm w}/p_0\) in the undrained cyclic triaxial test TCUI12. Influence of parameters: a \(\chi _0\), b \(\chi _{{\rm max}}\) and c \(C_\Omega\)

A detailed guide for the calibration of the hypoplastic and intergranular strain parameters can be found in Herle and Gudehus [11] and Niemunis [24], respectively. The calibration procedure of parameter \(\gamma _\chi\) is explained in detail in Wegener and Herle [32], considering that \(\gamma =\gamma _\chi \chi\).

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Duque, J., Mašín, D. & Fuentes, W. Improvement to the intergranular strain model for larger numbers of repetitive cycles. Acta Geotech. 15, 3593–3604 (2020). https://doi.org/10.1007/s11440-020-01073-w

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