Large post-liquefaction deformation of sand, part I: physical mechanism, constitutive description and numerical algorithm

Abstract

This paper presents a theoretical framework for predicting the post-liquefaction deformation of saturated sand under undrained cyclic loading with emphasis on the mechanical laws, physical mechanism, constitutive model and numerical algorithm as well as practical applicability. The revealing mechanism behind the complex behavior in the post-liquefaction regime can be appreciated by decomposing the volumetric strain into three components with distinctive physical background. The interplay among these three components governs the post-liquefaction shear deformation and characterizes three physical states alternating in the liquefaction process. This assumption sheds some light on the intricate transition from small pre-liquefaction deformation to large post-liquefaction deformation and provides a rational explanation to the triggering of unstable flow slide and the post-liquefaction reconsolidation. Based on this assumption, a constitutive model is developed within the framework of bounding surface plasticity. This model is capable of reproducing small to large deformation in the pre- to post-liquefaction regime. The model performance is confirmed by simulating laboratory tests. The constitutive model is implemented in a finite element code together with a robust numerical algorithm to circumvent numerical instability in the vicinity of vanishing effective stress. This numerical model is validated by fully coupled numerical analyses of two well-instrumented dynamic centrifuge model tests. Finally, numerical simulation of liquefaction-related site response is performed for the Daikai subway station damaged during the 1995 Hyogoken-Nambu earthquake in Japan.

Appendix: Local stress integration algorithm

• 1. Obtain the strains and stress quantities as well as volumetric strain due to change in mean effective stress at the beginning of the step $\left( {\varepsilon_{\text{vc}} } \right)_{\text{n}}$ and other history variables. Moreover, get strain increment by geometric update from a global converged state.

  $$ \Updelta {\varvec{\upvarepsilon }}_{n + 1} = {\varvec{\upvarepsilon }}_{n + 1} - {\varvec{\upvarepsilon }}_{n} $$

  (64)

• 2. Elastic predictor: Assume that all the strain increments are elastic, the trial stress state can be predicted as

  $$ \left( {\varepsilon_{\text{vc}} } \right)_{n + 1}^{\text{tr}} = \left( {\varepsilon_{\text{vc}} } \right)_{n} + \left( {\Updelta \varepsilon_{\text{v}} } \right)_{n + 1} $$
  $$ p_{n + 1}^{\text{tr}} = f\left( {\left( {\varepsilon_{\text{vc}} } \right)_{n + 1}^{\text{tr}} } \right) $$
  $$ {\mathbf{s}}_{n + 1}^{\text{tr}} = {\mathbf{s}}_{n} + 2G_{n + 1} \Updelta {\mathbf{e}}_{n + 1} $$

• 3. Check for plastic yielding: is $\Updelta f^{\text{tr}} = \Updelta {\mathbf{s}}_{n + 1}^{\text{tr}} :{\mathbf{n}}_{L} - \Updelta p_{n + 1}^{\text{tr}} {\mathbf{r}}_{n + 1}^{\text{tr}} :{\mathbf{n}}_{L} > 0?$

  NO: Update strains and stress quantities and EXIT

  YES: Perform plastic correction (Step 4).

• 4. Plastic corrector for yielding states: by Newton–Raphson iterative procedure. Initialization of iterative variables:

  $$ {\text{k}} = 0,\Updelta {\varvec{\upvarepsilon }}_{n + 1}^{p(0)} = 0,\Updelta {\varvec{\upvarepsilon }}_{n + 1}^{e(0)} = \Updelta {\varvec{\upvarepsilon }}_{n + 1} ,\Updelta L^{(0)} = 0,\Updelta \lambda^{(0)} = 0,\Updelta f^{(0)} = \Updelta f^{\text{tr}} $$

  wherein the variable in the superscript embraced by brackets indicates iterative number.

  Iterative procedure on k

  $$ \delta \lambda^{(k)} = \frac{{\Updelta f^{(k)} }}{{H + 3G - KD\left( {{\mathbf{r}}:{\mathbf{n}}_{L} } \right)}} $$
  $$ \Updelta \lambda^{(k + 1)} = \Updelta \lambda^{(k)} + \delta \lambda^{(k)} ,\Updelta L^{(k + 1)} = \Updelta L^{(k)} + H\delta \lambda^{(k)} $$
  $$ \left( {\Updelta \varepsilon_{\text{v}}^{\text{p}} } \right)_{n + 1}^{(k + 1)} = \Updelta \lambda^{(k + 1)} D,\left( {\Updelta {\mathbf{e}}^{\text{p}} } \right)_{n + 1}^{(k + 1)} = \Updelta \lambda^{(k + 1)} {\mathbf{n}}_{L} , $$
  $$ \left( {\Updelta \varepsilon_{\text{v}}^{\text{e}} } \right)_{n + 1}^{(k + 1)} = \left( {\Updelta \varepsilon_{\text{v}} } \right)_{n + 1} - \left( {\Updelta \varepsilon_{\text{v}}^{\text{p}} } \right)_{n + 1}^{(k + 1)} , $$
  $$ \left( {\Updelta {\mathbf{e}}^{\text{e}} } \right)_{n + 1}^{(k + 1)} = \Updelta {\mathbf{e}}_{n + 1} - \left( {\Updelta {\mathbf{e}}^{\text{p}} } \right)_{n + 1}^{(k + 1)} , $$
  $$ \left( {\varepsilon_{\text{vc}} } \right)_{n + 1}^{(k + 1)} = \left( {\varepsilon_{\text{vc}} } \right)_{n} + \left( {\Updelta \varepsilon_{\text{v}}^{\text{e}} } \right)_{n + 1}^{(k + 1)} $$
  $$ p_{n + 1}^{(k + 1)} = f\left( {\left( {\varepsilon_{\text{vc}} } \right)_{n + 1}^{(k + 1)} } \right), $$
  $$ {\mathbf{s}}_{n + 1} = {\mathbf{s}}_{n} + 2G_{n + 1}^{(k + 1)} \left( {\Updelta {\mathbf{e}}^{\text{e}} } \right)_{n + 1}^{(k + 1)} $$

  where H is plastic modulus, G and K are elastic moduli, D is the total shear-dilatancy rate, ∆L is the increment of loading index.

• 5. Convergence check: Calculate the residual value of yielding function

  $$ \Updelta f^{(k + 1)} = \Updelta {\mathbf{s}}_{n + 1}^{(k + 1)} :{\mathbf{n}}_{L} - \Updelta p_{n + 1}^{(k + 1)} {\mathbf{r}}_{n + 1}^{(k + 1)} :{\mathbf{n}}_{L} - \Updelta L^{(k + 1)} $$

  Is $\left| {\Updelta f^{(k + 1)} } \right| \le TOL1$ and $\left| {p_{n + 1}^{(k + 1)} - p_{n + 1}^{(k)} } \right| \le TOL2?$

  NO: $k \leftarrow k + 1,$ continue iteration

  YES: Update strains, stress and internal variables, then EXIT

  $$ {\mathbf{s}}_{n + 1} = {\mathbf{s}}_{n + 1}^{(k + 1)} ,\quad p_{n + 1} = p_{n + 1}^{(k + 1)} ,\left( {\varepsilon_{vc} } \right)_{n + 1} = \left( {\varepsilon_{vc} } \right)_{n + 1}^{(k + 1)} . $$