Undrained mechanical behavior of saturated completely decomposed granite: experimental investigation and constitutive modeling

Abstract

In this study, the undrained mechanical behavior of saturated completely decomposed granite (CDG) with different weathering degrees is investigated. To this end, a series of consolidation undrained (CU) triaxial compression tests are conducted on saturated CDG, and the effects of weathering degree on the main undrained mechanical properties are analyzed. Based on the experimental results, a poromechanical model is then established with the concept of effective plastic stress in a poroplasticity framework. Plastic distortion is described using a particular yield surface and a nonassociated plastic potential, which are both functions of the effective plastic stress and a subtly unified smooth hardening/softening variable. As an original contribution, an enhanced semi-implicit return mapping (ESRM) algorithm is developed to integrate the proposed model. This algorithm is based on a semi-implicit return mapping procedure and is combined with a new adaptive substepping technique. The model is subsequently implemented and validated by comparing the numerical simulation results with the experimental data. The main undrained mechanical characteristics of saturated CDG with different weathering degrees are well reproduced. A discussion follows regarding the parameter sensitivity analysis and robustness of the ESRM algorithm. Interestingly, the high accuracy of the ESRM algorithm is almost step-size independent, and the computational efficiency is also greatly improved.

Appendices

Appendix A: Derivation of hardening modulus $H_{\text {har}}$

The loading/unloading conditions of plasticity are expressed in the so-called Kuhn-Tucker form, i.e.,

$$\begin{aligned} \lambda ^p \ge 0,\quad \mathcal {F}<0,\quad \lambda ^p \mathcal {F}=0 \end{aligned}$$

(A.1)

It follows the condition in incremental form

$$\begin{aligned} \lambda ^p\text{d} \mathcal {F}=0 \end{aligned}$$

(A.2)

Under plastic loading, one has $\lambda ^p \ge 0$ and $\text {d}\mathcal {F}=0$, so that

$$\begin{aligned} \text{d} \mathcal {F}=\frac{\partial \mathcal {F}}{\partial \varvec{\sigma }^{\prime }}: \text{d} \varvec{\sigma }^{\prime }+\frac{\partial \mathcal {F}}{\partial \alpha } \text{d} \alpha =0 \end{aligned}$$

(A.3)

On the other hand, the effective stress increment can be obtained by differentiating Eq. (13), i.e.,

$$\begin{aligned} \text{d} \varvec{\sigma }^{\prime }=\mathbb {C}_{d}:\left( \text{d} \varvec{\varepsilon }-\text{d} \varvec{\varepsilon }^p\right) \end{aligned}$$

(A.4)

Substituting Eqs.(A.4) and (22) into (A.3), one finally has

$$\begin{aligned} \lambda ^p=\frac{\partial \mathcal {F}}{\partial \varvec{\sigma }^{\prime }}: \mathbb {C}_{d}: \text{d} \varvec{\varepsilon } / H_{\text {har}} \end{aligned}$$

(A.5)

with the hardening modulus

$$\begin{aligned} H_{\text {har}}=-\frac{\partial \mathcal {F}}{\partial \alpha } \frac{\partial \alpha }{\partial \gamma ^p} \frac{\partial \mathcal {G}}{\partial q}+\frac{\partial \mathcal {F}}{\partial \varvec{\sigma }^{\prime }}: \mathbb {C}_{d}: \frac{\partial \mathcal {G}}{\partial \varvec{\sigma }^{\prime }} \end{aligned}$$

(A.6)

Appendix B: SRM with IPCP in substep

It is noted that the yield criterion in effective stress space $\mathcal {F}(\varvec{\sigma }^{\prime }, \alpha )$ could also be rewritten with the function of $\varvec{\varepsilon }$ and $\varvec{\varepsilon }^{p}$ with the aid of Eqs. (13) and (19), i.e., $\mathcal {F}(\varvec{\varepsilon }, \varvec{\varepsilon }^{p})$. During the plastic correction procedure, the yield function can be linearized using the first-order Taylor expansion as follows:

$$\begin{aligned} \mathcal {F}^{j+1}=\mathcal {F}^j+\frac{\partial \mathcal {F}^j}{\partial \varvec{\varepsilon }^{p, j}}: \delta \varvec{\varepsilon }^{p, j} \approx 0 \end{aligned}$$

(B.1)

where superscript j is used to represent the inter iterations related to the plastic correction. Besides, for the sake of clarity, the subscript substep loading m is omitted. Combined with Eq. (22), one further obtains

$$\begin{aligned} \mathcal {F}^{j+1}=\mathcal {F}^j+\delta \lambda ^{p, j} \frac{\partial \mathcal {F}^j}{\partial \varvec{\varepsilon }^{p, j}}: \frac{\partial \mathcal {G}^j}{\partial \varvec{\sigma }^{\prime , j}} \approx 0 \end{aligned}$$

(B.2)

Then one has

$$\begin{aligned} \delta \lambda ^{p, j}=-\frac{\mathcal {F}^j}{\frac{\partial \mathcal {F}^j}{\partial \varvec{\varepsilon }^{p, j}}: \frac{\partial \mathcal {G}^j}{\partial \varvec{\sigma }^{\prime , j}}} \end{aligned}$$

(B.3)

The plastic strain and effective stress can be updated as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta \varvec{\varepsilon }^{p,j}=\sum _{i=1}^j{\mathrm {\delta }\lambda ^{p,i}\frac{\partial \mathcal {G} ^i}{\partial \varvec{\sigma }^i}}\\ \varvec{\varepsilon }^{p,j}=\varvec{\varepsilon }_{k}^{p}+\Delta \varvec{\varepsilon }^{p,j}\\ \varvec{\sigma }^{\prime , j}=\varvec{\sigma }'_k+\mathbb {C}_{d}:\left( \Delta \varvec{\varepsilon }-\Delta \varvec{\varepsilon }^{p,j+1} \right) \\ \end{array}\right. } \end{aligned}$$

(B.4)

The IPCP stops when $|\mathcal {F}^{j+1}|\le \epsilon _{\text {local}}$, with $\epsilon _{\text {local}}$ being the local convergence tolerance.