Monotonic and cyclic tests on kaolin: a database for the development, calibration and verification of constitutive models for cohesive soils with focus to cyclic loading

Abstract

A database with about 60 undrained monotonic and cyclic triaxial tests on kaolin is presented. In the monotonic tests, the influences of consolidation pressure, overconsolidation ratio, displacement rate and sample cutting direction have been studied. In the cyclic tests, the stress amplitude, the initial stress ratio and the control (stress vs. strain cycles) have been additionally varied. Isotropic consolidation leads to a failure due to large strain amplitudes with eight-shaped effective stress paths in the final phase of the cyclic tests, while a failure due to an excessive accumulation of axial strain and lens-shaped effective stress paths was observed in the case of anisotropic consolidation with $q^{\text{ ampl }}< |q^{\text{ av }}|$. The rate of pore pressure accumulation grew with increasing amplitude and void ratio (i.e. decreasing consolidation pressure and overconsolidation ratio). The “cyclic flow rule” well known for sand has been confirmed also for kaolin: With increasing value of the average stress ratio $|\eta ^{\text{ av }}| = |q^{\text{ av }}|/p^{\text{ av }},$ the accumulation of deviatoric strain becomes predominant over the accumulation of pore water pressure. The tests on the samples cut out either horizontally or vertically revealed a significant effect of anisotropy. In the cyclic tests, the two kinds of samples exhibited an opposite inclination of the effective stress path. Furthermore, the horizontal samples showed a higher stiffness and could sustain a much larger number of cycles to failure. All data of the present study are available from the homepage of the first author. They may serve for the examination, calibration or improvement in constitutive models dedicated to cohesive soils under cyclic loading, or for the development of new models.

Appendix: Generalized flow rule considering anisotropy

The equations for the generalized flow rule applied in Sect. 4.4 are briefly outlined in the following. The generalized flow rule has been proposed in [129] as a component of the high-cycle accumulation (HCA) model of Niemunis et al. [87]. It allows for considering anisotropy. The equations do not distinguish between an inherent and an induced anisotropy. A second-order anisotropy tensor

$$\begin{aligned} \mathbf{a}= & {} {\varvec{\sigma }}^*/p \end{aligned}$$

(4)

is introduced with ${\varvec{\sigma }}$ being a stress for which the flow rule is assumed purely volumetric, $\mathbf{m}(\mathbf{a})\,=\,\vec{\mathbf{1}}$. The deviatoric part of ${\varvec{\sigma }}$ is denoted as ${\varvec{\sigma }}^*$. The isotropic flow rule can be recovered by setting $\mathbf{a}= {\mathbf{0}}$. For the critical state, the flow rule is purely deviatoric. For an intermediate stress ${\varvec{\sigma }},$ an interpolation is used. Given $\mathbf{a}$, the stress ${\varvec{\sigma }}$ is projected radially onto the deviatoric plane expressed by $p = 1$. Next, the projected stress ${\varvec{\sigma }}/p$ is decomposed as follows (Fig. 30a):

$$\begin{aligned} {\varvec{\sigma }}/p= & {} {\mathbf{1}}+ \mathbf{r}\,=\, {\mathbf{1}}+ {\varvec{\sigma }}^*/p \end{aligned}$$

(5)

The conjugated stress $\mathbf{t}$ is found from

$$\begin{aligned} \mathbf{t}= & {} {\mathbf{1}}+ \mathbf{a}+ \lambda (\mathbf{r}- \mathbf{a}) \end{aligned}$$

(6)

It should lie on the critical surface at $p = 1$ (Fig. 30b). For that purpose, the scalar multiplier $\lambda$ must be determined from the condition that the conjugated stress $\mathbf{t}$ satisfies

$$\begin{aligned} \text{ tr }\,\mathbf{t}\, \text{ tr }\,(\mathbf{t}^{-1}) = Y_c \,\,\,\,\,\hbox {or} \,\,\,\,\, \text{ tr }\,( \mathbf{t}^{-1}) = -Y_c/3 \end{aligned}$$

(7)

with $Y_c$ calculated from:

$$\begin{aligned} Y_{c}= & {} \frac{9 - \sin ^2{\varphi _{cc}}}{1 - \sin ^2{\varphi _{cc}}} \end{aligned}$$

(8)

Therein $\varphi _{cc}$ is a material constant being similar (but not necessarily identical) to the critical friction angle $\varphi _c$ derived from monotonic shear tests. Having found $\lambda$ the generalized flow rule is calculated from

$$\begin{aligned} \mathbf{m}= \frac{1}{\sqrt{(1 - \lambda ^{-n})^2 + (\lambda ^{-n})^2}} \left[ \vec{\mathbf{1}}{(1 - \lambda ^{-n})} + {\lambda ^{-n}} (\mathbf{t}^*)^{\rightarrow } \right] \end{aligned}$$

(9)

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

Fig. 30

a Projection of stress ${\varvec{\sigma }}$ on the surface $p = 1$, b determination of the conjugated stress $\mathbf{t}$

As an example, the flow rule $\mathbf{m}$ for the axisymmetric stress state in a triaxial test with the diagonal components

$$\begin{aligned} {\varvec{\sigma }}= & {} {\text{ diag }}(\sigma _1,\sigma _3,\sigma _3) \end{aligned}$$

(10)

and for a transversal isotropy

$$\begin{aligned} \mathbf{a}= & {} {\text{ diag }}(a, -a/2, -a/2). \end{aligned}$$

(11)

is derived. The parameter a and the stress ratio $\eta _{\text{ iso }}$ for which the accumulation is purely volumetric are interrelated via

$$\begin{aligned} a= & {} -2/3 ~ \eta _{\text{ iso }}. \end{aligned}$$

(12)

Furthermore,

$$\begin{aligned} \mathbf{r}= & {} {\text{ diag }}(r, -r/2, -r/2) \,\,\,\, \hbox {with} \,\,\,\, r = -2/3 \, \eta \end{aligned}$$

(13)

holds. From two solutions of Eq. (7)

$$\begin{aligned} \lambda _{1|2}= & {} \frac{1}{2 (a-r)^2 Y_c} \left[ -9 a \right. \nonumber \\&\left. +\,3 \left( 3 r \pm \sqrt{(a-r)^2 (Y_c -9) (Y_c -1)}\right) \right. \nonumber \\&\left. +\,(2 a+1) (a-r) Y_c\right] \end{aligned}$$

(14)

the positive one is chosen as $\lambda$. Finally, the strain rate ratio $\omega = \dot{\varepsilon }_v^{\text{ acc }}/\dot{\varepsilon }_q^{\text{ acc }}$ shown in Fig. 19 is calculated from

$$\begin{aligned} \omega= & {} \frac{1 - \lambda ^{-n}}{\lambda ^{-n}} \,\,\, \hbox {or} \,\,\, \omega \,=\, - \frac{1 - \lambda ^{-n}}{\lambda ^{-n}} \end{aligned}$$

(15)

for triaxial compression or extension, respectively.