ISA-Hypoplasticity accounting for cyclic mobility effects for liquefaction analysis

Abstract

The hypoplastic model for sands proposed by Wolffersdorff (Mech Cohes Frict Mater 1: 251–271, 1996) combined with the intergranular strain anisotropy by Fuentes and Triantafyllidis (Int J Numer Anal Meth Geomech 39: 1235–1254, 2015) is herein extended to account for cyclic mobility effects to allow for the simulation of liquefaction phenomena. The extension is based on the introduction of an additional state variable that permits the detection of cyclic mobility paths. The simulation capabilities of the model is compared with undrained triaxial tests of Karlsruhe fine sand. At the end, a finite element simulation of an offshore monopile embedded in sand, exposed to environmental forces from the Caribbean Sea, is constructed and analyzed.

Appendices

Notation and conventions

The notation and convention of the present work is as follows: Italic fonts denote scalar magnitudes (e.g., a, b), bold lowercase letters denote vectors (e.g., $\mathbf {a}, \mathbf {b}$), bold capital letters denote second-rank tensors (e.g., $\mathbf {A}$, ${\varvec{\sigma }}$), and special fonts are used for fourth-rank tensors (e.g., $\mathsf{E}, \mathsf{L}$). Indicial notation can be used to represent components of tensors (e.g., $A_{ij}$), and their operations follow the Einstein’s summation convention. The Kronecker delta symbol is represented by $\delta _{ij}$, i.e., $\delta _{ij}=1$ when $i=j$ and $\delta _{ij}=0$ otherwise. The symbol $\mathbf {1}$ denotes the Kronecker delta tensor ($1_{ij}=\delta _{ij}$). The unit fourth-rank tensor for symmetric tensors is denoted by $\mathsf{I}$, where $\mathsf{I}_{ijkl}=\frac{1}{2}\left( \delta _{ik}\delta _{jl}\right.$$\left. +\delta _{il}\delta _{jk}\right)$. Multiplication with two dummy indices (double contraction) is denoted with a colon “ : ” (e.g., $\mathbf {A}:\mathbf {B}=A_{ij}B_{ij}$). The symbol “$\otimes$” represents the dyadic product (e.g., $\mathbf {A}\otimes \mathbf {B}=A_{ij}B_{kl}$). The brackets $\parallel \bigsqcup \parallel$ extract the Euclidean norm (e.g., $\parallel \mathbf {A}\parallel =\sqrt{A_{ij}A_{ij}}$). Normalized tensors are denoted by $\overrightarrow{\bigsqcup }=\frac{\bigsqcup }{\parallel \bigsqcup \parallel }$, or in general as $\sqcup ^{\rightarrow }$. The superscript $\bigsqcup ^\mathrm{dev}$ extracts the deviatoric part of a tensor (e.g., $\mathbf {A}^\mathrm{dev}=\mathbf {A}-\frac{1}{3}(\mathrm {tr}\mathbf {A})\mathbf {1}$). Components of the effective stress tensor ${\varvec{\sigma }}$ or strain tensor ${\varvec{\varepsilon }}$ in compression are negative. Roscoe variables are defined as $p=-\sigma _{ii}/3$, $q=\sqrt{\frac{3}{2}}\parallel {\varvec{\sigma }}^\mathrm{dev}\parallel$, $\varepsilon _v=-\varepsilon _{ii}$ and $\varepsilon _s=\sqrt{\frac{2}{3}}\parallel {\varvec{\varepsilon }}^\mathrm{dev}\parallel$. The stress ratio $\eta$ is defined as $\eta =q/p$. The deviator stress tensor is defined as ${\varvec{\sigma }}^\mathrm{dev}={\varvec{\sigma }}+p\,\mathbf {1}$ and the stress-ratio tensor with $\mathbf {r}={\varvec{\sigma }}^\mathrm{dev}/p=\sqrt{\frac{2}{3}}\,\eta \,\overrightarrow{{{\varvec{\sigma }}^\mathrm{dev}}}$.

Empirical relation for shear degradation curve by Wichtmann and Triantafyllidis [47]

The empirical relation provided by Wichtmann and Triantafyllidis [47] is:

$$\begin{aligned} G_\mathrm{max}=74000\dfrac{1+D_r}{(11.6-D_r)^2}\left( \dfrac{p}{p_\mathrm{atm}}\right) ^{0.48}p_\mathrm{atm} \end{aligned}$$

(23)

with the relative density $D_r=(e_\mathrm{max}-e)/(e_\mathrm{max}-e_\mathrm{min})$ and the reference stress $p_\mathrm{atm}=100$ kPa. For Karlsruhe fine sand $e_\mathrm{max}=1.054$ and $e_\mathrm{min}=0.677$. The secant shear modulus $G_\mathrm{sec}$ is computed with the empirical relation provided by Wichtmann and Triantafyllidis [47] :

$$\begin{aligned} \dfrac{G_\mathrm{sec}}{G_\mathrm{max}}=\dfrac{1}{1+\Delta \gamma /\gamma _r(1+a\exp \left( -\Delta \gamma /\gamma _r\right) )} \end{aligned}$$

(24)

where $\Delta \gamma$ is the shear strain amplitude, $a=1.070\ln (c_u)$ is a constant, $\gamma _r=\tau _\mathrm{max}/G_\mathrm{max}$ is the reference strain, $\tau _\mathrm{max}=p\sin (\varphi _p)$ is the maximum shear stress and $\varphi _p=34^\circ \exp (0.27D_r^{1.8})$ is the peak friction angle. $c_u$ is the uniformity coefficient ($c_u=D_{60}/D_{10}$). For Karlsruhe fine sand, $c_u=1.5$ and therefore $a=0.433$.

For drained triaxial conditions, the strain amplitude $\parallel \Delta {\varvec{\varepsilon }}\parallel$ is computed with the following approximation $\parallel \Delta {\varvec{\varepsilon }}\parallel =\sqrt{(\Delta \varepsilon _1)^2(1+2\nu ^2)}$ where $\nu$ is the Poisson ratio. For same conditions, it can be shown that the relation $\Delta \gamma =\Delta \varepsilon _1(1+\nu )$ holds. For the computations with Karlsruhe fine sand, a value of $\nu =0.3$ was used.

The resulting parameters for $e_0=0.85$ and $p=200$ kPa are $\gamma _r=9.29\times 10^{-4}$, $G_\mathrm{max}=130058$ kPa. For $e_0=0.85$ and $p=300$ kPa are $\gamma _r=1.14\times 10^{-3}$ and $G_\mathrm{max}=158002$ kPa.

ISA-hypoplastic model for sands

“Appendix C” presents a summary of the constitutive equations of the ISA-hypoplastic model. Details of the equations below are found in [11, 35, 49].

$$\begin{aligned} \dot{\varvec{\sigma }}= & {} \mathsf{M}:\dot{{{\varvec{\varepsilon }} }}\end{aligned}$$

(25)

$$\begin{aligned} \mathsf{M}= & {} \left\{ \begin{array}{lllll} m (\mathsf{L}^\mathrm{hyp} + \rho ^{\chi } \mathbf {N}^\mathrm{hyp}\mathbf {N}) &{} \text {for } F_H= 0\qquad \text {(plastic)}\\ m_{R}\mathsf{L}^\mathrm{hyp} &{} \text {for } F_H < 0\qquad \text {(elastic)} \end{array} \right. \end{aligned}$$

(26)

where $\mathsf{L}^\mathrm{hyp}$ and $\mathbf {N}^\mathrm{hyp}$ are the (fourth rank) linear and (second rank) nonlinear stiffness, respectively, $\mathbf {N}=(\mathbf {h}-\mathbf {c})^{\rightarrow }$ is the IS flow rule, $m_R$ is a parameter, and $\rho$, m, $\chi$ and $F_H$ are scalar functions defined in the sequel. The IS yield surface function $F_H$ is defined as:

$$\begin{aligned} F_H=\parallel \mathbf {h}-\mathbf {c}\parallel -R/2 \end{aligned}$$

(27)

where $\mathbf {h}$ is the IS tensor, $\mathbf {c}$ is the back-IS tensor, and R is a parameter. Factors m, $y_h$ and $\rho$ are defined as:

$$\begin{aligned} m&=m_R+(1-m_R)y_h \end{aligned}$$

(28)

$$\begin{aligned} y_h&=\rho ^{\chi }\langle \mathbf {N}:\dot{{{\varvec{\varepsilon }} }}\rangle \end{aligned}$$

(29)

$$\begin{aligned} \rho&=1-\dfrac{\Vert \mathbf {d}_b \Vert }{2R} ,\quad \text {with}\quad \mathbf {d}_b= \mathbf {h}_b-\mathbf {h},\quad \text {and}\quad \mathbf {h}_b= R\mathbf {N}\end{aligned}$$

(30)

The evolution equation for the IS tensor $\mathbf {h}$ is:

$$\begin{aligned} \dot{\mathbf {h}}=\dot{\varvec{\epsilon }}-\dot{\lambda }_H {\mathbf {N}} ,\quad \text {with}\quad \mathbf {N}=(\mathbf {h}-\mathbf {c})^{\rightarrow } ,\quad \text {and}\quad \dot{\lambda }_H=\dfrac{\langle \mathbf {N}:\dot{{{\varvec{\varepsilon }} }}\rangle }{1+\mathbf {N}:\bar{\mathbf {c}}} \end{aligned}$$

(31)

where $\dot{\lambda }_H$ is the plastic multiplier of the IS model. The evolution equation for tensor $\mathbf {c}$ is:

$$\begin{aligned} \dot{\mathbf {c}}=\dot{\lambda }_H\bar{\mathbf {c}} ,\quad \text {with}\quad \bar{\mathbf {c}}=\beta _h(\mathbf {c}_b-\mathbf {c})/R ,\quad \text {and}\quad \mathbf {c}_b=(R/2)\overrightarrow{\dot{{{\varvec{\varepsilon }} }}} \end{aligned}$$

(32)

where $\beta _h$ is a factor, which takes the value of $\beta _h=\beta _\mathrm{hmax}$ for the condition $\mid \overrightarrow{\mathbf {h}_b}:\overrightarrow{\mathbf {d}_b}\mid =0$, and $\beta _h=\beta _{h0}$ for $\mid \overrightarrow{\mathbf {h}_b}:\overrightarrow{\mathbf {d}_b}\mid =1$.

$$\begin{aligned} \beta _h=\beta _{\mathrm{hmax}}+(\beta _{h0}-\beta _\mathrm{hmax})\mid \overrightarrow{\mathbf {h}_b}:\overrightarrow{\mathbf {d}_b}\mid \end{aligned}$$

(33)

The internal variable $\dot{\varepsilon }_\mathrm{acc}$ evolves according to:

$$\begin{aligned} \dot{\varepsilon }_\mathrm{acc}=\dfrac{c_a}{R}(1-y_h-\varepsilon _\mathrm{acc})\parallel \dot{{{\varvec{\varepsilon }} }}\parallel \end{aligned}$$

(34)

Function $\chi$ is defined as:

$$\begin{aligned} \chi = \chi _0+\varepsilon _\mathrm{acc}(\chi _\mathrm{max}-\chi _0) \end{aligned}$$

(35)

The equations of the reference hypoplastic model by Wolffersdorff [49] are given below:

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

(36)

$$\begin{aligned} {\mathbf {N}}^\mathrm{hyp}= & {} f_df_bf_e\dfrac{Fa}{\hat{{\varvec{\sigma }}}:\hat{{\varvec{\sigma }}}}(\hat{{\varvec{\sigma }}}+\hat{{\varvec{\sigma }}}^\mathrm{dev}) \end{aligned}$$

(37)

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

(38)

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

(39)

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

(40)

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

(41)

The set of parameters are $\varphi _c$, $h_s$, $n_B$, $e_{i0}$, $e_{c0}$, $e_{d0}$, $\alpha$, $\beta$, R, $\chi _0$, $\chi _\mathrm{max}$, $\beta _{h0}$, $\beta _\mathrm{hmax}$, $c_a$. The state variables are e, $\mathbf {c}$, $\mathbf {h}$ and $\varepsilon _\mathrm{acc}$.

Inspection of hypoplastic flow rule tensor $\mathbf {m}$

Substitution of Eqs. 36 and 37 in tensor $\mathbf {m}=-(\mathsf{L}^\mathrm{hyp})^{-1}:\mathbf {N}^\mathrm{hyp}$ gives (see procedure in [29]):

$$\begin{aligned} \begin{aligned} \mathbf {m}&=-(\mathsf{L}^\mathrm{hyp})^{-1}:\mathbf {N}^\mathrm{hyp}\\&=-\dfrac{1}{F^2}\left[ \mathsf{I}-\dfrac{\hat{{\varvec{\sigma }}}\hat{{\varvec{\sigma }}}}{(F/a)^2+\hat{{\varvec{\sigma }}}:\hat{{\varvec{\sigma }}}}\right] :f_d a^2\left( \dfrac{F}{a}\right) (\hat{{\varvec{\sigma }}}+\hat{{\varvec{\sigma }}}^\mathrm{dev}) \end{aligned} \end{aligned}$$

(42)

Splitting tensor $\mathbf {m}$ into volumetric and deviatoric components $\mathbf {m}^\mathrm{vol}=\mathsf{I}^\mathrm{vol}:\mathbf {m}$ and $\mathbf {m}^\mathrm{dev}=\mathsf{I}^\mathrm{dev}:\mathbf {m}$, where $I^\mathrm{vol}_{ijkl}=1/3\delta _{ij}\delta _{kl}$ and $I^\mathrm{dev}_{ijkl}=I_{ijkl}-I^\mathrm{vol}_{ijkl}$ gives:

$$\begin{aligned} \mathbf {m}^\mathrm{vol}&=\mathsf{I}^\mathrm{vol}:\mathbf {m}=\dfrac{1}{3 \sqrt{3}}\dfrac{a}{F}f_\sigma \mathbf {1}\end{aligned}$$

(43)

$$\begin{aligned} \mathbf {m}^\mathrm{dev}&=\mathsf{I}^\mathrm{dev}:\mathbf {m}=\dfrac{a}{F} (f_\sigma +1)\hat{{\varvec{\sigma }}}^\mathrm{dev} \end{aligned}$$

(44)

Where fuction $f_\sigma$ is defined as:

$$\begin{aligned} f_\sigma =\dfrac{(F/a)^2-\parallel \hat{{\varvec{\sigma }}}^\mathrm{dev}\parallel }{(F/a)^2+\parallel \hat{{\varvec{\sigma }}}\parallel } \end{aligned}$$

(45)

Simulations of monotonic loading

In this “Appendix”, some simulations of the Karlsruhe fine sand under monotonic loading are presented. Simulations versus experiments are shown in Figs. 18 and 19. Parameters of Table 1 were used for the simulations.

Fig. 18

Simulation of oedometer compression with one cycle unloading–reloading with different initial void ratios

Fig. 19

Simulation of undrained triaxial tests with different initial void ratios