Characteristic limitations of advanced plasticity and hypoplasticity models for cyclic loading of sands

Abstract

Numerous studies in the literature are concerned with proposing new constitutive models for sands to simulate cyclic loading. Despite considerable progress in this area, there are various limitations on their simulation capabilities that are either overlooked or not communicated clearly by the developers. A number of these limitations are rather crucial for the end users, and therefore, providing discussion and analysis of them would be of great value for both applications and future developments. The present work is devoted to discussing seven characteristic limitations, which are frequently observed in cyclic loading simulations. Four advanced constitutive models are considered in this study: two bounding surface elastoplasticity and two hypoplasticity models—with the models in each category following a hierarchical order of complexity. Relevant cyclic loading experimental test data on Toyoura and Karlsruhe fine sands support the analysis. The key issues discussed include stress overshooting, one-way ratcheting in cyclic strain accumulation, liquefaction strength curves, stress attractor in strain-controlled shearing, hypoelasticity, cyclic oedometer stiffness, and effect of drained preloading. The presented results elaborate on the specific limitations and capabilities of these rather advanced models in simulating several essential aspects of cyclic loading of sands.

Appendices

Appendix

Notation and variables

The notation and convention is as follows: scalar magnitudes (e.g., a, b) are denoted by italic fonts, vectors (e.g., ${\mathbf {a}}, {\mathbf {b}}$) with bold lowercase fonts, second-rank tensors (e.g., ${\mathbf {A}}$, ${\mathbf {B}}$) with bold capital letter or bold symbols, higher ranked tensors with special fonts (e.g., $\mathsf{E}, \mathsf{L}$). Components of these tensors are denoted through indicial notation (e.g., $A_{ij}$). $\delta _{ij}$ is the Kronecker delta, also represented with ($1_{ij}=\delta _{ij}$). The unit fourth-rank tensor for symmetric tensors is denoted by $\mathsf{I}$, where $\mathsf{I}_{ijkl}=\frac{1}{2}\left( \delta _{ik}\delta _{jl}\right.$ $\left. +\delta _{il}\delta _{jk}\right)$. The following operations hold: ${\mathbf {A}}:{\mathbf {B}}=A_{ij}B_{ij}$, ${\mathbf {A}}\otimes {\mathbf {B}}=A_{ij}B_{kl}$, $\parallel {\mathbf {A}}\parallel =\sqrt{A_{ij}A_{ij}}$, $\overrightarrow{\bigsqcup }=\frac{\bigsqcup }{\parallel \bigsqcup \parallel }$, ${\mathbf {A}}^\mathrm{dev}={\mathbf {A}}-\frac{1}{3}(\mathrm {tr}{\mathbf {A}}){\mathbf {1}}$, $\hat{{\mathbf {A}}}=\frac{{\mathbf {A}}}{\mathrm {tr}({\mathbf {A}})}$. Components of the effective stress tensor ${{\varvec{\sigma }}}$ or strain tensor ${\varvec{\varepsilon }}$ in compression are negative. Roscoe variables are defined as $p^\prime =-\sigma _{ii}/3$, $q=\sqrt{\frac{3}{2}}\parallel {{\varvec{\sigma }}}^{\mathrm{dev}}\parallel$, $\varepsilon _v=-\varepsilon _{ii}$ and $\varepsilon _s=\sqrt{\frac{2}{3}}\parallel {\varvec{\varepsilon }}^{\mathrm{dev}}\parallel$. The stress ratio $\eta$ is defined as $\eta =q/p^\prime$.

Summary of constitutive equations for models for cyclic loading

Table 5 provides a summary of the constitutive equations of the DM04 model by Dafalias and Manzari [7] and the SANISAND-MSf model by Yang et al. [78]. A detailed guide for the calibration of the DM04 and SANISAND-MSf parameters can be found in Dafalias and Manzari [7], Taiebat and Dafalias [62] and Yang et al. [78].

Table 5 Constitutive relations of the elastoplastic models

The constitutive equations of the hypoplastic model for sands by Von Wolffersdorff [75] and the two different intergranular strain approaches by Niemunis and Herle [44] and Fuentes et al. [22] are presented in Table 6. A detailed guide for the calibration of the hypoplastic and conventional intergranular strain parameters can be found in Herle and Gudehus [26] and Niemunis and Herle [44], respectively. On the other hand, a guide for the calibration of the ISA-hypoplasticity parameters can be found in Fuentes [17] and Fuentes et al. [22].

Table 6 Constitutive relations of the hypoplastic models