Flow rule in a high-cycle accumulation model backed by cyclic test data of 22 sands

Abstract

The flow rule used in the high-cycle accumulation (HCA) model proposed by Niemunis et al. (Comput Geotech 32: 245, 2005) is examined on the basis of the data from approximately 350 drained long-term cyclic triaxial tests (N = 10⁵ cycles) performed on 22 different grain-size distribution curves of a clean quartz sand. In accordance with (Wichtmann et al. in Acta Geotechnica 1: 59, 2006), for all tested materials, the “high-cyclic flow rule (HCFR)”, i.e., the ratio of the volumetric and deviatoric strain accumulation rates $\dot{\varepsilon}_{\rm{v}}^{{\rm acc}}/\dot{\varepsilon}_{\rm{q}}^{{\rm acc}}$, was found dependent primarily on the average stress ratio η ^av = q ^av/p ^av and independent of amplitude, soil density and average mean pressure. The experimental HCFR can be fairly well approximated by the flow rule of the modified Cam-clay (MCC) model. Instead of the critical friction angle $\varphi_{\rm{c}}$ which enters the flow rule for monotonic loading, the HCA model uses the MCC flow rule expression with a slightly different parameter $\varphi_{\rm{cc}}$. It should be determined from cyclic tests. $\varphi_{\rm{cc}}$ and $\varphi_{\rm{c}}$ are of similar magnitude but not always identical, because they are calibrated from different types of tests. For a simplified calibration in the absence of cyclic test data, $\varphi_{\rm{cc}}$ may be estimated from the angle of repose $\varphi_{\rm{r}}$ determined from a pluviated cone of sand (Wichtmann et al. in Acta Geotechnica 1: 59, 2006). However, the paper demonstrates that the MCC flow rule with $\varphi_{\rm{r}}$ does not fit well the experimentally observed HCFR in the case of coarse or well-graded sands. For an improved simplified calibration procedure, correlations between $\varphi_{\rm{cc}}$ and parameters of the grain-size distribution curve (d ₅₀,   C _u) have been developed on the basis of the present data set. The approximation of the experimental HCFR by the generalized flow rule equations proposed in (Wichtmann et al. in J Geotech Geoenviron Eng ASCE 136: 728, 2010), considering anisotropy, is also discussed in the paper.

Appendices

Appendix 1: Generalized flow rule

In [10], the direction of accumulation m _ij of the HCA model has been generalized in order to describe (inherent or induced) anisotropy. A second-order structure tensor

$$ a_{ij} = \breve{\sigma}_{ij}^{*}/\breve{p} $$

(14)

has been introduced with $\breve{\sigma}_{ij}$ lying on a stress line for which the flow rule is assumed purely volumetric, i.e., $m_{ij}(a_{ij}) = (\delta_{ij})^{\rightarrow} \cdot \breve{\sigma}_{ij}^{*}$ is the deviatoric part of $\breve{\sigma}_{ij}$ and $\breve{p}$ is the corresponding mean pressure. The isotropic flow rule can be recovered by setting a _ij  = 0. For the critical state, the flow rule is purely deviatoric. For an intermediate stress σ _ij , an interpolation is used. Given a _ij , the stress σ _ij is projected radially onto the deviatoric plane expressed by p = 1. Next, the projected stress σ _ij /p is decomposed as follows (Fig. 10a):

$$ \sigma_{ij}/p = \delta_{ij} + r_{ij} = \delta_{ij} + \sigma_{ij}^{*}/p $$

(15)

The so-called conjugated stress t _ij is found from

$$ t_{ij} = \delta_{ij} + a_{ij} + \lambda(r_{ij} - a_{ij}) $$

(16)

It should lie on the critical surface at p = 1 (Fig. 10b). For that purpose, the scalar multiplier λ must be determined from the condition that the conjugated stress t _ij satisfies

$$ t_{ii} (t^{-1})_{ii} = Y_{\rm{c}} \quad \hbox{or} \quad (t^{-1})_{ii} = -Y_{\rm{c}}/3 $$

(17)

Having found λ, the generalized flow rule is calculated from

$$ m_{ij} = \frac{\delta_{ij} (1 - \lambda^{-n_g}) + \lambda^{-n_g} (t_{ij}^{*})^{\rightarrow}} {\sqrt{(1 - \lambda^{-n_g})^{2} + (\lambda^{-n_{g}})^{2}}} $$

(18)

wherein n _g is an interpolation parameter (a material constant). Linear interpolation is obtained with n _g  = 1.

Fig. 10

Generalized flow rule. a Projection of σ _ij on the deviatoric plane at p = 1. b Definition of conjugated stress t _ij

As an example, the flow rule m _ij for an axisymmetric stress with diagonal components

$$ \sigma_{ij} = \hbox{diag}(\sigma_{1},\sigma_{3},\sigma_{3}) $$

(19)

and for a transversal isotropy

$$ a_{ij} = \hbox{diag}(a, -a/2, -a/2). $$

(20)

is derived. The parameter a and the stress ratio η _iso for which the accumulation is purely volumetric are interrelated via

$$ a = -2/3 \eta_{{\rm iso}}. $$

(21)

Furthermore,

$$ r_{ij} = \hbox{diag}(r, -r/2, -r/2) \quad \hbox{with}\quad r = -2/3 \eta $$

(22)

holds. From two solutions of Eq. (17) ,

$$ \begin{aligned} \lambda_{1|2} &= \frac{1}{2 (a-r)^{2} Y_{\rm{c}}} \left[-9 a + 3 \left(3 r\right.\right. \\ & \pm \left.\left.\sqrt{(a-r)^{2} (Y_{\rm{c}} -9) (Y_{\rm{c}} -1)}\right)\right.\\ & \left.+ (2 a+1) (a-r) Y_{\rm{c}}\right] \end{aligned} $$

(23)

the positive one is chosen as λ. Finally, the strain rate ratio ω is calculated from

$$ \omega = \frac{\dot{\varepsilon}_{\rm{v}}^{{\rm acc}}}{\dot{\varepsilon}_{\rm{q}}^{{\rm acc}}} = \frac{1 - \lambda^{-n_{g}}}{\lambda^{-n_{g}}} \quad \hbox{or} \quad \omega = - \frac{1 - \lambda^{-n_{g}}}{\lambda^{-n_{g}}} $$

(24)

for triaxial compression and extension, respectively. For an isotropic material (η _iso = a = 0) Eq. (23) simplifies to

$$ \lambda_{1|2} = \frac{1}{2 r Y_{\rm{c}}} \left[3 \left(3 \pm \sqrt{(Y_{\rm{c}} -9) (Y_{\rm{c}} -1)}\right)- Y_{\rm{c}}\right] $$

(25)

Appendix 2: Notation

δ _ij :

Kronecker delta (1 for i = j, 0 for i ≠ j)

e :

Void ratio

$\varepsilon_1$ :

Axial strain

$\varepsilon_{3}$ :

Lateral strain

$\varepsilon_v$ :

Volumetric strain ($= \varepsilon_{1} + 2 \varepsilon_{3}$ for triaxial tests)

$\varepsilon_{\rm{q}}$ :

Deviatoric strain ($= 2/3 (\varepsilon_{1} - \varepsilon_{3}$) for triaxial tests)

$\varepsilon^{{\rm ampl}}$ :

Strain amplitude

$\varepsilon^{{\rm acc}}$ :

Residual (accumulated) strain

$\varepsilon_{\rm{v}}^{{\rm acc}}$ :

Accumulated volumetric strain

$\varepsilon_{\rm{q}}^{{\rm acc}}$ :

Accumulated deviatoric strain

$\dot{\varepsilon}^{{\rm acc}}$ :

Intensity of strain accumulation

$\dot{\varepsilon}_{\rm{v}}^{{\rm acc}}$ :

Rate of volumetric strain accumulation

$\dot{\varepsilon}_{\rm{q}}^{{\rm acc}}$ :

Rate of deviatoric strain accumulation

$\varepsilon_{ij}^{{\rm av}}$ :

Average strain tensor

$\dot{\varepsilon}_{ij}^{{\rm av}}$ :

Trend of average strain

$\dot{\varepsilon}_{ij}^{{\rm acc}}$ :

Rate of strain accumulation

$\dot{\varepsilon}_{ij}^{{\rm pl}}$ :

Plastic strain rate

E _ijkl :

“Elastic stiffness” of HCA model

$\varphi_{\rm{c}}$ :

Critical friction angle

$\varphi_{\rm{cc}}$ :

MCC flow rule parameter calibrated from cyclic tests

$\varphi_{\rm{ccg}}$ :

Parameter of generalized flow rule

$\varphi_{\rm{r}}$ :

Angle of repose

F :

Correction factor for M

η :

Stress ratio =q/p

η ^av :

Average stress ratio

I _D :

Relative density

I _D0 :

Initial value of relative density

m _v :

Volumetric part of flow rule (=m _ii )

m _q :

Deviatoric part of flow rule ($=\sqrt{2/3}\|m_{ij}^*\|$)

M :

Critical stress ratio / Stress ratio with $\dot{\varepsilon}_{\rm{v}}^{{\rm acc}} = 0$

M _c :

Critical stress ratio for triaxial compression

M _cc :

Stress ratio with $\dot{\varepsilon}_{\rm{v}}^{{\rm acc}} = 0$ for triaxial compression

M _ec :

Stress ratio with $\dot{\varepsilon}_{\rm{v}}^{{\rm acc}} = 0$ for triaxial extension

m _ij :

Direction of strain accumulation (high-cyclic flow rule, HCFR)

n _g :

Parameter of generalized flow rule

N :

Number of cycles

p :

Effective mean pressure (=(σ ₁ + 2 σ ₃)/3 in triaxial tests)

p ^av :

Average effective mean pressure

q :

Deviatoric stress (=σ ₁ − σ ₃ in triaxial tests)

q ^ampl :

Deviatoric stress amplitude

σ ₁ :

Effective axial stress

σ ₃ :

Effective lateral stress

σ _ij :

Effective Cauchy stress tensor

σ ^av _ij :

Average effective stress tensor

$\dot{\sigma}_{ij}^{{\rm av}}$ :

Trend of average effective stress

$\breve{\sigma}_{ij}$ :

Stress for which flow rule is purely volumetric

ω :

Strain rate ratio (= $\dot{\varepsilon}_v^{{\rm acc}}/\dot{\varepsilon}_q^{{\rm acc}}$)

$\bar{\omega}$ :

Average strain rate ratio of a test

$\zeta$ :

Amplitude-pressure ratio

$\|\sqcup\|$ :

Norm of $\sqcup$

$\sqcup^{*}$ :

Deviatoric part of $\sqcup$

$\sqcup^{\rightarrow}$ :

Normalization, i.e., $\sqcup/\|\sqcup\|$