Strain Gradient Plasticity
I have been interested in developing strain gradient plasticity theories relevant to geomaterials and geomechanics. The various theories I developed and used have been based on the early Toupin-Mindlin form of strain gradient elasticity. In particular, I have combined the strain gradient theory with damage mechanics to simulate the localized failure in geomaterials (see, Zhou et al., 2002) wherein the strain gradient theory introduces a constitutive regularization to maintain the characteristics of governing equation at bifurcation point, as such to avoid such issues as spurious mesh dependency problem in numerical simulation of strain localization phenomenon.
Cavity expansion in micromorphic media - To be updated.
Thermomechanics approaches for gradient plasticity - To be updated.
Constitutive modeling in geomechanics
Stregnth anisotropy and failure criterion - To be updated.
Constitutive modeling of anisotropic granular behavior - To be updated.
Nonlinear finite element algorithms and computations
Developing robust, efficient and accurate stress integration approaches constitutes important parts of finite element implementation of constitutive models for soils. This is especially true when complex soil behavior is to be modeled and the constitutive relation becomes complex. Typical complex models involve multiple yield surface, plastic yielding within yield surface, kinematic and isotropic hardening laws and anisotropy. We have extended an early explicit stress integration scheme proposed by Sloan et al. (2001) such that it presents a general-purpose form and can handle many complex soil models in a generic way (see, Zhao et al., 2005). This stress integration scheme is based on the explicit Euler method with automatic substepping and error control, and employs the classical elastoplastic stiffness matrix and requires only the first derivatives of the yield function and plastic potential. The robustness, efficiency and accuracy of the scheme have been testified by examining its performance on two complex soil models on structured clay. The complex models using the proposed integration scheme has also been applied to the prediction of settlement in the consolidation process of a trial embankment.
Limit and shakedown analysis
Shakedown of pavement materials - To be updated.
Generalization of limit theory to strain gradient plasticity - To be updated.
Micromechanics of geomaterials
Discrete Element Method (DEM) has now been widely used to simulate soil behavior from the microscopic scale. Experiments on soils that traditionally have to be done in soil labs can now be done by DEM - the so called "virtual lab" and "virtual test". It is still, however, a long way to go for DEM to be used for direction simulation of practical engineering problems, The animation in Figure 1 shows how DEM is used to prepare a packing of 1200 mono-dispersed spherical particles into a cylinder for a triaxial compression test. The particle system was generated by using gravational deposition method. Layer by layer, soil particles are generated and deposited into the container under the force of gravitation. Balance state is found by molecular dynamics. The figure has been rendered by POV-Ray.
Fig.1 Fill a jar with colorful chocolate (jk...).
The real soil, however, rarely consists of mono-disperse particles, but of polydisperse in nature. Grain size distribution (GSD) has long been used in soil mechanics and soil science as an important indication of this non-homogeneity in particle size. Meanwhile, soil is a pressure-sensitive material in which the pores can be changed by the translational/rotational movements of particles under external loads. As such, void ratio or porosity plays a crucial rule in characterizing the mechanical response of a soil. In order to simulate a granular soil, GSD and void ratio have to be taken into account in order for the simulation yields any meaningful results. Shown below in Figure 2 are figures for a packing of total 3200 spherical particles with diameter ranging from 0.2 to 0.6 unit packed in a cubic box (6 by 6 by 6 units). The figure on the right is the packing of particles within their Voronoi tessellation cells. The particles were generated by DEM and these cells by using the VORO++ code developed by Dr Chris Rycroft of UC Berkeley/Lawrence Berkeley Laboratory.
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| Fig.2a A packing of 3200 spheres with diameter ranging from 0.2 to 0.6 units in a cubic box of 6 by 6 by 6 units. | Fig.2b The particles with their Voronoi tessellation cells |
Fig.3 Examples of sand particles imaged using a variable-pressure scanning electron microscope in the backscattered electron model (photo source: www.kpal.co.uk).
The introduction of grain size distribution and void ratio definitely makes a DEM modeling more realistic than otherwise. However, it is far from enough. A convincing simulation should have to take into account the particle shape as well. In above figures spherical particles have been used. Using spherical particle shape does lend to numerous advantages for numerical manipulation and computation of the particle system. Real particles in a soil, except in rare occasions such as riverbed sand, are always non-rounded, or rounded but in non-spherical shape. Figure 3 shows examples of sand particles imaged using a variable-pressure scanning electron microscope in the backscattered electron mode (photo source: www.kpal.co.uk). Important shape characteristics of a particle include form (reflected by the degree of particle elongation or flatness), roundness (reflected by the degree of sharpness of corners and edges), sphericity (reflected by the degree to which the external envelope of the particle approximates that of a true sphere); and irregularity (reflected by the number and size of projections and indentations). Visual comparison can be made by using the textural diagrams shown in Figure 4 developed by Sneed and Folk (1958) (see also, Blott and Pye, Sedimentology, 55:31-63, 2008). In the figure S/I characterizes the degree of flatness and I/L the degree of elongation of the particle.
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| Fig.4a Textural diagram for the characterization of elongation | Fig.4b Textural diagram for flatness |
A more realistic way to generate soil particles should therefore be able to reflect the above geometric properties of them, at least in an approximate way. Instead of using purely spherical particles, ellipsoidal or super-ellipsoidal shape serves good candidate for this purpose, especially for rounded soil particles. Using the alterative ways of particle simulation, meanwhile, makes it possible to better capture and interpret many other important physical or mechanical properties of soils, such as anisotropy and plastic deformation. Example candidates of ellipsoids or superellipsoids are show in Figure 5. With particle shape being taken into account, however, questions remains as to how to efficiently generate a large number of particles with non-spherical shapes, to pack them together to the desirable state, and then to use the final packing for DEM simulations. These will be my research focus in the near future.
Fig.5 Examples of ellipsoids and superellipsoids for the simulation of rounded soil particles by DEM.
Any DEM modeling on soils/granular materials without considering the above mentioned aspects of real soils will essentially yield qualitative results at best, and attempts using DEM for practical engineering application without respecting the scaling law of particles will produce results of practically little use. Difficult however, continuous development of robust of DEM tools for soils/granular materials, based on the points outlined above, could lead to very useful results if successful. The outcome of this research could be useful not only for geotechnical and structural engineering, but also for many other relevant areas such as material science and industries on food/drug/powder production, packing and transportation.