Research highlights : : Fabric-based Constitutive Modeling of Sand Anisotropy


Constituitve modeling donnut

Overview

Naturally occuring materials such as soils and sands are commonly anisotropic. Modeling the anisotropic behavior of sand is important to a wide range of key geostructures. The centerpiece of our continuum-based constitutive characterization of sand anisotropy is the use of a physically based a second-order fabric tensor, an anisotropic parameter defined by the joint invariant with the stress tensor and/or the consideration of fabric evolution law. The fabric tensor employed characterize the physical arrangements of particles, interparticles contacts, the geometric property of the void space. The fabric evolution law is to account for the rearrangements of the fabric structure in sand during the loading course.

Our fabric-based constitutive approach has been specifically applied to describing the following ascepts of anisotropic behavior in sand: constituitve responeses during different loading paths, strength anisotropy, anisotropic behavior in fiber-reinforced sand and cemented sand.


1. Constitutive modeling of sand anisotropy - highlighting the role of fabric and its evolution

We proposed a three-dimensional elasto-plastic model constructed within the framework of the Anisotropic Critical State Theory to predict the anisotropic behavior of sand. The novelty of the model is four-fold:

While reproducing the typical anisotropic sand behavior observed in laboratory tests well, the model can also capture and explain the non-coaxial behavior in sand in a reasonable and natural manner (see Fig.1).

Non coaxial sand behaivor
Fig.1 Illustration of non-coaxial sand response by non-coincidence of principal directions of the stress and fabric tensors.
As shown in Fig.1(a) the existing angle between relative orientations of stress and fabric tensor indicates non-coaxiality. The explicit yield surface in Gao et al. (2013) is plotted in the deviatoric π-plane of the stress ratio space, for three cases of Toyoura sand (Case 1: α=0°, F0=0.9; Case 2: α=45°, F0=0.9; Case 3: F0=0.0, A=0). Due to explicit dependence of yield function on the anisotropic paremeter A, Case 1 & 2 present different yield surfaces because of different relative orientations of α. Comparing the outer normals at the same stress state for Case 2 & the isotropic case 3, the difference of the two normal directions indeed provides a graphical measure of non-coaxiality because the normal for the isotropic case 3 will be coaxial with the stress, while the normal for the anisotropic case 2 will not. Shown also in (b) is the fabric anisotropy induced non-coaxiality observed at the intersection point of the yield surface with axis r1. The normal to the yield surface for both cases 1 and 3 is parallel to the axis, indicating coaxiality (Case 1 due to α=0° and case 3 for isotropy), while for case 2 it has a clear deviation from axis r1 measured as non-coaxiality.

The model has also been applied to simulating the shear localization problem in sand wherein the unique role of fabric evolution was highlighted. More recently, we have further extended the model to cover both monotonic and cyclic loading case [see Gao and Zhao (2015)] and to account for both anisotropic elasticity and plasticity [see Zhao and Gao (2015)].

References:

2. Characterization of strength anisotropy in Geomaterials - A systematic approach

anisotropic strength of glass beads
Fig.2 Prediction of strength anisotropy for glass beads by our generalized anisotropic Lade failure criterion [see Gao & Zhao (2012)]
anisotropic strength of santa monica beach sand
Fig.3 Prediction of strength anisotropy for Santa Monica Beach sand by our generalized anisotropic failure criterion presented in Gao, Zhao & Yao (2010)

We proposed a systematic approach to generalize an isotropic failure criterion to describe strength anisotropy in geomaterials. A salient ingradient of the method involves the inclusion of the degree of cross anisotropy and the anisotropic parameter, A, defined by the joint invariant of the derivatoric stress tensor and the deviatoric fabric tensor, into the frictional characteristic of the isotropic criterion:

Generalized anisotropic failure criterion

where s and d denote respectively the deviatoric stress tensor and the deviatoric fabric tensor. Taking the Lade's criterion as an example, the generalization can be done as:

Generalized anisotropic Lade failure criterion

The generality and rigorousness of the approach and the generalized criterion have been demonstrated by accurate predictions of the strenghth for a wide range of geomaterials, including completely decomposed granite, glass beads (see Fig. 2), natural clays (natural Pietrafitta clay, natural Pisa clay, San Francisco Bay mud), sands (Toyoura sand, Cambria sand, dense/dry-pluviated Santa Monica Beach sand: see predictions in Fig. 3), silty sand (Nevada II/ATC) and rocks (Touremire shale, Angers schist).

References:

3. Evaluating strength of fiber-reinforced sand

Fiber-reinforced Muskegon Dune sand
Fig.5 Prediction of strength for fiber-reinforced Muskegon Dune sand (test data, Maher and Gay, 1990)





Stablizing inclined slope with fibers
Fig.6 Schematic of optimal fiber orientation for stablization of an inclined slope with fibers

Fiber addition may help to enhance the strenght of soil, stablize near-surface soil layers and mitigate the risk of soil liquefaction. A proper failure criterion to evaluate the strength of fiber-reinforced soil (FRS) is needed. We have proposed a general anisotropic failure criterion to highlight the effect of isotropically/anisotropically distributed fibers on the strength of FRS. The following fabric tensor is defined to quantify the fiber distribution in a FRS:

fabric tensor in fiber-reinforced sand

where the different quantities are defined according to the spherical representive volume shown below:
spherical representative volume element for frs

Fig.4 Spherical representative volume element for fiber-reinforced sand

Similar to the A defined in the research above, an anisotropic variable defined by the joint invariant of the deviatoric stress tensor and a deviatoric fiber distribution tensor was introduced to the criterion to quantify the fiber orientation with respect to the strain rate/stress direction at failure. The failure criterion further considers the fiber concentration and other factors such as aspect ratio of the fifer, presenting in the following form:

failure criterion for fiber-reinforced sand

An instructive agreement was found between our predictions with test data, as is shown in Fig. 5. We have also discussed its applicablity to practical engineering problems such as inclined slope stabilization (Fig. 6).

Reference:

4. Constitutive modeling of cemented sand

stress strain for cemented Ottawa sand
Fig.7 Predicted stress strain relation for cemented Ottawa sand in drained triaxial compression (data from Wang and Leung, 2008)
void change of cemented Ottawa sand
Fig.8 Predicted void change for cemented Ottawa sand in drained triaxial compression (data from Wang and Leung, 2008)

Artifically cemented sand (ACS) has been widely used in practice for soil improvement and liquefaction mitigation. The behavior of ACS is a combined response affected by both its bonding and fabric. We proposed a novel constiutitve model to describe the combined effect of bonding and fabric anisotropy on the mechanical response of ACS. The yield surface was based on an extension of our recently proposed anisotropic failure criterion by including a hardening parameter as follows:

expanding yield surface with hardening for ACS

In conjunction with a fabric tensor to characterize the sand fabric, the triaxial tensile strength was adopted as a macroscopic representation of the inter-particle bonding. A debonding law was proposed by assuming the de-bounding process is driven by the development of plastic deformation, while the soil fabric is kept constant in the study to account for inherent anisotropy.

As demonstrative examples, the model reproduces well the typical responses of cemented Ottawa sand (see Figs. 7 and 8) and multiple-sieving-pluviated (MSP) Toyoura sand (see Figs. 9 and 10). Further investigation can be carried out by incoporating the fabric evolution law into the model to consider the fabric change along with the debonding process.

References:
stress strain for MSP Toyoura sand volume change for MSP Toyoura sand
Fig.9 Predicted stress strain relation for MSP Toyoura sand in drained triaxial compression (data from Miura and Toki, 1984) Fig.10 Predicted volumetric change for MSP Toyoura sand in drained triaxial compression (data from Miura and Toki, 1984).