On the simulation of multidimensional cyclic loading with intergranular strain

Abstract

A sample of soil is subjected to multidimensional cyclic loading when two or three principal components of the stress or strain tensor are simultaneously controlled to perform a repetitive path. These paths are very useful to evaluate the performance of models simulating cyclic loading. In this article, an extension of an existing constitutive model is proposed to capture the behavior of the soil under this type of loading. The reference model is based on the intergranular strain anisotropy concept and therefore incorporates an elastic locus in terms of a strain amplitude. In order to evaluate the model performance, a modified triaxial apparatus able to perform multidimensional cyclic loading has been used to conduct some experiments with a fine sand. Simulations of the extended model with multidimensional loading paths are carefully analyzed. Considering that many cycles are simulated ($N>30$), some additional simulations have been performed to quantify and analyze the artificial accumulation generated by the (hypo-)elastic component of the model. At the end, a simple boundary value problem with a cyclic loading as boundary condition is simulated to analyze the model response.

Appendices

Appendix 1: Hypoplastic model from Wolffersdorff

The general equation of the hypoplastic model by Wolffersdorff [59] can be written as:

$$\begin{aligned} \dot{{\varvec{\sigma }}}=\bar{\mathsf{E}}:(\dot{{{\varvec{\varepsilon }} }}- \dot{\bar{\varvec{\varepsilon }}}^p) \end{aligned}$$

(23)

whereby $\bar{\mathsf{E}}$ is the “linear” stiffness and $\bar{{\varvec{\varepsilon }}}^p$ is the hypoplastic strain rate defined in the sequel. According to the ISA + HP nomenclature, the tensor $\mathsf{E}$ is referred as the mobilized stiffness and $\bar{{\varvec{\varepsilon }}}^p$ as the mobilized plastic strain rate. The definition of $\bar{\mathsf{E}}$ reads [59]:

$$\begin{aligned} \bar{\mathsf{E}}=f_bf_e\dfrac{1}{\hat{{\varvec{\sigma }}} :\hat{{\varvec{\sigma }}}} (F^2\mathsf{I}+a^2\hat{{\varvec{\sigma }}}\hat{{\varvec{\sigma }}}) \end{aligned}$$

(24)

whereby $\hat{{\varvec{\sigma }}}= {\varvec{\sigma }}/\mathrm{tr}{\varvec{\sigma }}$ is the relative stress, $f_b$, $f_e$, F and a are scalar factors and $\mathsf{I}$ is the fourth-order tensor for symmetric second-order tensors. The scalar factor F is responsible for the Matsuoka–Nakai shape of the critical state surface and is defined as:

$$\begin{aligned} F=\sqrt{\dfrac{1}{8}\tan ^2(\psi )+\dfrac{2-\tan ^2(\psi )}{2+2\sqrt{2}\tan (\psi )\cos (3\theta )}}-\dfrac{1}{2\sqrt{2}\tan (\psi )} \end{aligned}$$

(25)

whereby the factors a, $\theta$ and $\psi$ are defined as:

$$\begin{aligned} a&=\dfrac{\sqrt{3}(3-\sin (\varphi _c))}{2\sqrt{2}\sin (\varphi _c))}\nonumber \\ \tan \psi&=\sqrt{3} \Vert \hat{{\varvec{\sigma }}}^* \Vert \nonumber \\ \cos (3\theta ) &= \sqrt 6 \frac{{{\text{tr}}(\widehat{\sigma }^{*} \widehat{\sigma }^{*} \widehat{\sigma }^{*} )}}{{(\widehat{\sigma }^{*} :\widehat{\sigma }^{*} )^{{3/2}} }} \end{aligned}$$

(26)

The tensor ${\varvec{\sigma }}^*$ is the deviator stress tensor and $\varphi _c$ is the critical state friction angle. The model incorporates the characteristic void ratios corresponding to the maximum $e_i$, minimum $e_d$ and critical $e_c$, respectively. They follow the function proposed by Bauer [5] depending on the mean pressure p:

$$\begin{aligned} e_i&=e_{i0}\exp \left( -\left( 3p/h_s\right) ^{n_B}\right) \nonumber \\ e_d&=e_{d0}\exp \left( -\left( 3p/h_s\right) ^{n_B}\right) \nonumber \\ e_c&=e_{c0}\exp \left( -\left( 3p/h_s\right) ^{n_B}\right) \end{aligned}$$

(27)

where $e_{i0}$, $e_{d0}$ and $e_{c0}$ are parameters representing the characteristic void ratios at $p=0$ and $h_s$ and $n_B$ are additional parameters to fit these curves. The scalar functions $f_e$ and $f_b$ read:

$$\begin{aligned} f_e&=\left( \dfrac{e_c}{e}\right) ^\beta \nonumber \\ f_b&=\dfrac{h_s}{n}\left( \dfrac{1+e_i}{e_i}\right) \left( \dfrac{e_{i0}}{e_{c0}}\right) ^\beta \left( -\dfrac{\mathrm{tr{\varvec{\sigma }}}}{h_s}\right) ^{1-n}\left[ 3+a^2-\sqrt{3}a\left( \dfrac{e_{i0}-e_{d0}}{e_{c0}-e_{d0}}\right) ^\beta \right] ^{-1} \end{aligned}$$

(28)

whereby $\beta$ and $\alpha$ are material parameters The mobilized plastic strain rate $\dot{\bar{\varvec{\varepsilon }}}^p$ is defined as:

$$\begin{aligned} \dot{\bar{\varvec{\varepsilon }}}^p=-\bar{\mathsf{E}}^{-1} : \bar{\mathbf {N}} \parallel \dot{{{\varvec{\varepsilon }} }}\parallel \end{aligned}$$

(29)

whereby tensor $\bar{\mathbf {N}}$ reads:

$$\begin{aligned} \bar{\mathbf {N}}=f_df_bf_e\dfrac{Fa}{\hat{{\varvec{\sigma }}}:\hat{{\varvec{\sigma }}}}(\hat{{\varvec{\sigma }}}+\hat{{\varvec{\sigma }}}^*) \end{aligned}$$

(30)

and the factor $f_d$ follows the relation:

$$\begin{aligned} f_d=\left( \dfrac{e-e_d}{e_c-e_d}\right) ^\alpha \end{aligned}$$

(31)

Details of these functions are explained in [37]. The required parameters are briefly described in the following appendix.

Appendix 2: Short guide to determine the ISA + HP parameters

The ISA + HP model (without the proposed extension) requires the calibration of 12 parameters. In this appendix, a short guide for their determination is provided.

• The critical state friction angle $\varphi _c$ can be adjusted with points of a triaxial compression test after a vertical strain of $\varepsilon _1>25\,\%$. The critical state slope within the $p-q$ space can be calibrated with the relation $q/p=6\sin \varphi _c/(3-\sin \varphi _c)$ for these points.

• The maximum void ratio at $p=0$ denoted with $e_{i0}$ can be obtained through the standardized minimum density test (ASTM D4254-14).

• The exponent $n_B$ can be adjusted to match the elastic stiffness dependence with the mean pressure $G\sim p^{1-n_B}$ through the results of resonant column test for different confining pressures. If the experiments are scarce, some values from the literature can be adopted, e.g., $n_B=0.5$ [47].

• The granular hardness $h_s$ can be adjusted to simulate the oedometric compression stiffness under very loose states $e\approx e_i$ where $e_i=e_i(p)$ is the maximum void ratio curve. A method to determine $h_s$ given some oedometric results is described by Herle and Gudehus [21].

• The critical state void ratio at $p=0$ denoted with $e_{c0}$ can be adjusted from points lying at the critical state ($\varepsilon _1>25\,\%$ with triaxial compression) in the $e-p$ space with very low pressure $p<20\hbox { kPa}$. When data are scarce, one may adopt the approximation $e_{c0}\approx 0.9 e_{i0}$.

• The dilatancy exponent $\alpha$ is calibrated with the behavior of medium-dense and dense samples sheared through drained triaxial compression. This parameter controls the dilatancy rate of the volumetric strains after reaching the phase of transformation line. A relation to determine $\alpha$ with drained triaxial test is described by Herle and Gudehus [21].

• The barotropy exponent $\beta$ is adjusted to dense samples compressed under oedometric conditions. Herle and Gudehus [21] provided an equation to determine this parameter.

• The parameter R defines the size of the elastic locus in terms of strain increments. For the secant shear stiffness $G^\mathrm{sec}$, this can be interpreted as the strain range at which no degradation occurs. Many experiments point a value of approximately $\parallel \Delta {\varvec{\varepsilon }}\parallel \approx 10^{-5}$ for sands, but as mentioned in [17], a small value of this parameter may lead to numerical difficulties when dealing with finite element simulations. Hence, a value of $\parallel \Delta {\varvec{\varepsilon }}\parallel > 5\times 10^{-5}$ is recommended.

• The parameter $\beta _h$ controls the needed strain increment to eliminate the influence of the intergranular strain effect in the model. In other words, it controls the size of the strain amplitude at which no “small strain effects” is simulated by the model. The equation relating this strain amplitude with the parameter $\beta _h$ was provided in [17] and reads:

  $$\begin{aligned} \beta _h=\dfrac{\sqrt{6}R(\log (4)-2\log (1-r_h))}{6\Delta \varepsilon _s-\sqrt{6} R(3+r_h)} \end{aligned}$$

  (32)

  where $\Delta \varepsilon _s$ is the deviatoric strain amplitude and $r_h\approx 0.99$ is a factor which defines how close is tensor $\mathbf {c}$ to its bounding condition $r_h=\parallel \mathbf {c}\parallel /\parallel \mathbf {c}_b\parallel$.

• The parameter $\chi$ controls the degradation curve shape of the secant shear modulus $G^\mathrm{sec}$. Its calibration can be performed simulating some cycles of triaxial test as explained in [17].

• The parameter $C_a$ controls how fast the plastic accumulation rate reduces upon the cycles. It can be adjusted with a cyclic undrained triaxial test with the behavior of the accumulated pore pressure $p^\mathrm{acc}_w$ versus the number of cycles N. The first portion of this curve, with approximately $N<10$, can be adjusted through parameter $C_a$ by trial and error. An example of its calibration is given in Fig.14c.

• The parameter $\chi _{\max }$ controls the accumulation rate when the number of consecutive cycles is large, of about $N>10$. It can be adjusted with a cyclic undrained triaxial test with the behavior of the accumulated pore pressure $p^\mathrm{acc}_w$ versus the number of cycles N. An increasing number of $\chi _{\max }$ would return a lower value of N to reach failure at the critical state line. It can be adjusted by trial and error after fixing $C_a$. An example of its calibration is given in Fig. 14d.