An experimental database for the development, calibration and verification of constitutive models for sand with focus to cyclic loading: part I—tests with monotonic loading and stress cycles

Abstract

For numerical studies of geotechnical structures under earthquake loading, aiming to examine a possible failure due to liquefaction, using a sophisticated constitutive model for the soil is indispensable. Such a model must adequately describe the material response to a cyclic loading under constant volume (undrained) conditions, amongst others the relaxation of effective stress (pore pressure accumulation) or the effective stress loops repeatedly passed through after a sufficiently large number of cycles (cyclic mobility, stress attractors). The soil behaviour under undrained cyclic loading is manifold, depending on the initial conditions (e.g. density, fabric, effective mean pressure, stress ratio) and the load characteristics (e.g. amplitude of the cycles, application of stress or strain cycles). In order to develop, calibrate and verify a constitutive model with focus to undrained cyclic loading, the data from high-quality laboratory tests comprising a variety of initial conditions and load characteristics are necessary. The purpose of these two companion papers was to provide such database collected for a fine sand. The database consists of numerous undrained cyclic triaxial tests with stress or strain cycles applied to samples consolidated isotropically or anisotropically. Monotonic triaxial tests with drained or undrained conditions have also been performed. Furthermore, drained triaxial, oedometric or isotropic compression tests with several un- and reloading cycles are presented. Part I concentrates on the triaxial tests with monotonic loading or stress cycles. All test data presented herein will be available from the homepage of the first author. As an example of the examination of an existing constitutive model, the experimental data are compared to element test simulations using hypoplasticity with intergranular strain.

Appendix: Equations of the constitutive model used for the element test simulations

1.1 Notation

Scalar variables are denoted by characters with normal letters (e.g. e), second-order tensors are formatted fat (e.g. ${\varvec{\sigma }}$, ${\mathbf{h}}$), and fourth-order tensors are given in sans-serif font (e.g. ${\mathsf{L}}, {\text{I}}$). A dyadic product is denoted by ${\mathbf{a}}\otimes {\mathbf{b}}$ (i.e. $a_{ij} \, b_{kl}$ in index notation), a single contraction by ${\mathbf{a}}\cdot {\mathbf{b}}\,\hat{=}\, a_{ik} \, b_{kj}$ and a double contraction by ${\mathbf{a}}: {\mathbf{b}}\,\hat{=}\, a_{kl} \, b_{kl}$. The Euclidean norm is defined as $\Vert {\mathbf{a}}\Vert = \sqrt{\mathbf{a}: \mathbf{a}}$, the trace of a tensor as $\text{tr}\,{({\mathbf{a}})} \,\hat{=}\, a_{kk}$ and the deviator as ${\mathbf{a}}^* = {\mathbf{a}}- \text{tr}\,{({\mathbf{a}})}/3 \, {\mathbf {1}}$ with the second-order identity tensor ${\mathbf {1}} \,\hat{=} \delta _{ij}$. The Kronecker symbol $\delta _{ij}$ is equal to 1 for $i = j$ and 0 for $i \ne j$. A normalization is denoted by an arrow above the respective symbol $\overrightarrow{\mathbf{a}} = {\mathbf{a}}/\Vert {\mathbf{a}}\Vert$ , and a division by the trace of the tensor is identified by a roof $\hat{\mathbf{a}} = {\mathbf{a}}/\text{tr}\,({\mathbf{a}})$.

1.2 Basic hypoplastic model after von Wolffersdorff [51]

The basic equation of the hypoplastic model proposed by von Wolffersdorff [51] reads:

$$\dot{{\varvec{\sigma }}}={\mathsf{L}}: \dot{{\varvec{\varepsilon }}} + {\mathbf{N}}\Vert \dot{{\varvec{\varepsilon }}}\Vert = \underbrace{\left( {\mathsf{L}}+ {\mathbf{N}}\frac{\dot{{\varvec{\varepsilon }}}}{\Vert \dot{{\varvec{\varepsilon }}}\Vert }\right) }_{{\mathsf{M}}} : \dot{{\varvec{\varepsilon}}}$$

(7)

with Jaumann stress rate $\dot{{\varvec{\sigma }}}$, strain rate $\dot{{\varvec{\varepsilon }}}$ and the linear and nonlinear stiffness tensors ${\mathsf{L}}$ and ${\mathbf{N}}$:

$${\mathsf{L}}=f_b \, f_e \, \frac{1}{\hat{{\varvec{\sigma }}} : \hat{{\varvec{\sigma }}}} \, \left( F^2 \, {\text{I}}\,+\, a^2 \, \hat{{\varvec{\sigma }}} \otimes \hat{{\varvec{\sigma }}}\right)$$

(8)

$${\mathbf{N}}=f_b \, f_e \, f_d \, \frac{F \, a}{\hat{{\varvec{\sigma }}} : \hat{{\varvec{\sigma }}}} \, \left( \hat{{\varvec{\sigma }}} + \hat{{\varvec{\sigma }}}^*\right)$$

(9)

Therein, $I_{ijkl} = 0.5 (\delta _{ik} \delta _{jl} + \delta _{il} \delta _{jk})$ is a fourth-order identity tensor. The parameters a and F in Eqs. (8) and (9) describe the failure criterion of Matusoka and Nakai [32]:

$$a=\frac{\sqrt{3} \, (3 - \sin {\varphi _c})}{2 \, \sqrt{2} \, \sin {\varphi _c}}$$

(10)

$$F=\sqrt{\frac{1}{8} \, \tan ^2{\psi } + \frac{2 - \tan ^2{\psi }}{2 + \sqrt{2} \tan {\psi } \cos {(3 \theta )}} } \,-\, \frac{\tan {\psi }}{2 \, \sqrt{2}}$$

(11)

$$\tan {\psi }=\sqrt{3} \, \Vert \hat{{\varvec{\sigma }}}^*\Vert$$

(12)

$$\cos {(3 \theta )}=-\sqrt{6} \, \frac{\text{tr}\,{\left( \hat{{\varvec{\sigma }}}^* \cdot \hat{{\varvec{\sigma }}}^* \cdot \hat{{\varvec{\sigma }}}^*\right) }}{\left[ \hat{{\varvec{\sigma }}}^* : \hat{{\varvec{\sigma }}}^*\right] ^\frac{3}{2}}$$

(13)

$\varphi _c$ is the critical friction angle (material constant). The barotropy and pyknotropy factors read:

$$f_d=\left( \frac{e - e_d}{e_c - e_d}\right) ^\alpha$$

(14)

$$f_e=\left( \frac{e_c}{e}\right) ^\beta$$

(15)

$$f_b=\frac{\left( \frac{e_{i0}}{e_{c0}}\right) ^\beta \frac{h_s}{n} \frac{1 + e_i}{e_i} \left( \frac{3 p}{h_s}\right) ^{1-n}}{3 + a^2 - a \, \sqrt{3} \, \left( \frac{e_{i0} - e_{d0}}{e_{c0} - e_{d0}}\right) ^\alpha }$$

(16)

with material constants $\alpha ,\, \beta ,\, h_s$ and n. The pressure dependence of the void ratios $e_d,\, e_c$ and $e_i$, corresponding to the densest, the critical and the loosest possible state, is described by (Bauer [4]):

$$\frac{e_i}{e_{i0}}=\frac{e_c}{e_{c0}} \,=\, \frac{e_d}{e_{d0}} \,=\, \exp {\left[ -\left( \frac{3 p}{h_s}\right) ^n\right] }$$

(17)

with the void ratios $e_{i0},\, e_{c0}$ and $e_{d0}$ (material constants) at pressure $p = 0$. The material parameters $e_{i0}, \,e_{c0},\, e_{d0},\, \varphi _c,\, h_s,\, n,\, \alpha$ and $\beta$ used for the simulations are summarized in the first eight columns of Table 2.

1.3 Extension of hypoplastic model by intergranular strain according to Niemunis and Herle [36]

In order to eliminate an excessive accumulation of strain (ratcheting) of the original hypoplastic model proposed by von Wolffersdorff [51] in the case of a cyclic loading, Niemunis and Herle [36] introduced the additional state variable “intergranular strain” $\mathbf{h}$, which memorizes the last part of the previous strain path. A measure of the mobilization of the intergranular strain is $\rho = \Vert {\mathbf{h}}\Vert /R$ with a material constant R describing the range of an elastic locus. Depending on the actual strain rate $\dot{{\varvec{\varepsilon }}}$ in relation to the direction of the intergranular strain $\overrightarrow{\mathbf{h}}$, the stiffness ${\mathsf{M}}$ in Eq. (7) is increased according to:

$$\begin{aligned} {\mathsf{M}}=\,& {} \left[ \rho ^\chi \,m_T + (1-\rho ^\chi )\,m_R\right] \,{\mathsf{L}} \\&+\left\{ \begin{array}{ll} \rho ^\chi (1-m_T){\mathsf{L}}:\overrightarrow{\mathbf{h}} \otimes \overrightarrow{\mathbf{h}} + \rho ^\chi \,\mathbf{N}\otimes \overrightarrow{\mathbf{h}} &{}\text{for}\,\overrightarrow{\mathbf{h}}:\dot{{\varvec{\varepsilon }}} > 0\\ \rho ^\chi (m_R-m_T){\mathsf{L}}:\overrightarrow{\mathbf{h}} \otimes \overrightarrow{\mathbf{h}} &{} \text{ for }\,\overrightarrow{\mathbf{h}}:\dot{{\varvec{\varepsilon }}} \,\le\, 0 \end{array} \right. \end{aligned}$$

(18)

with material constants $m_T,\, m_R$ and $\chi$. The evolution of the rate $\dot{\mathbf{h}}$ of intergranular strain obeys:

$$\begin{aligned} \dot{\mathbf{h}}= & {} \left\{ \begin{array}{ll} \left( {\text{I}}- \overrightarrow{\mathbf{h}} \otimes \overrightarrow{\mathbf{h}} \varrho ^{\beta _r}\right) : \dot{{\varvec{\varepsilon }}} &{} \text{for} \, \overrightarrow{\mathbf{h}}:\dot{{\varvec{\varepsilon }}} > 0\\ \dot{{\varvec{\varepsilon }}} &{} \text{for} \,\overrightarrow{\mathbf{h}}:\dot{{\varvec{\varepsilon }}} \,\le\, 0\\ \end{array} \right. \end{aligned}$$

(19)

with another material constant $\beta _r$. If a sufficiently large monotonic strain is applied after a change in the strain path direction, the comparatively low stiffness of the original hypoplastic model according to Eq. (7) is regained. The material parameters $R,\, m_T,\, m_R,\, \chi$ and $\beta _r$ used for the simulations are summarized in the last five columns of Table 2.