Editorial for Special Issue “Fractal and Fractional in Geomaterials”

Geomaterials, such as clay, sand, rockfill and ballast, etc., in the field are usually exposed to complex physical or mechanical conditions, where anomalous behaviors, e.g., state-dependent non-associated flow, non-Fickian diffusion, or non-Darcy seepage, usually take place. In recent years, fractal laws and fractional mechanics have been developed as robust tools for solving such complex or anomalous behavior of different materials.

This Special Issue contains 16 published papers. In [1], a set of nuclear magnetic resonance tests are carried out on different granular materials, including calcareous and quartz sands, where it is found that the grain size significantly affected their pore size distributions (PSDs). As the grain size increases, the heterogeneity and fractal dimension of PSD increase remarkably.

A simplified unified model of the relative permeability coefficient of unsaturated soil with fractal dimension is developed by Tao et al. in [2]. A strong correlation of the model parameters with the fractal dimension of soil is suggested. Through model comparison, it is found that the predicted results are consistent with the measured values.

In [3], a simplified model for predicting the relationship between the saturated permeability coefficient and air-entry value is established, by combining the Tao-Kong model and the fractal model of the soil–water characteristic curve.

To consider the solute transport in highly heterogeneous media, a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion and advection–dispersion is proposed in [4]. A finite difference approximation is then proposed to solve the problem in 1D domains, where two scenarios were examined.

An elastoplastic interface model using the fractional plastic flow rule is developed by Xu et al. in [5], where the dependence of the cyclic mobilisation of the soil–structure interface on its material state and fabric is captured.

In [6], the fractal theory and discrete element method are introduced to quantify the fractal dimension of a particle size distribution and understand the scale effect in soil–rock mixtures.

The effect of the fractal distribution of particle size on the critical state characteristics of calcareous sand is investigated through laboratory tests and theoretical analysis by Shen et al. in [7]. The critical state lines of calcareous sand in the q–p′ plane are unique, regardless of the fractal dimension, whereas those in the e–(p′)^α plane rotate anticlockwise as the fractal dimension increases. Modified constitutive relations modified with the fractal dimension are also provided.

In [8], a series of nuclear magnetic resonance tests on frozen soils are carried out to study the effect of freezing on soil’s micro-pore structure and fractal characteristics. A larger pressure during freezing–thawing or a higher freezing temperature can result in a lower fractal dimension of the soil’s structure.

A series of physical model tests are conducted by Yuan et al. [9] to study the effect of groundwater depth on pile–soil mechanical properties cyclic loads. Combined with fractal theory, the relationships between the pile top displacement, cyclic stiffness, and cyclic steps are evaluated.

In [10], the effect of non-plastic fines and stress anisotropy on the dynamic shear modulus of sand with different contents of non-plastic fines are investigated by using macro- and micro-laboratory tests. A unified expression for the shear modulus of binary mixtures is proposed.

In [11], a fractal relation between the hydraulic conductivity and fractal dimension is derived analytically, based on the capillary model of porous soil. It is found that an increased fractal dimension will increase the connectivity, increase the hydraulic gradient, and reduce the hydraulic conductivity.

The permeability and pore structure of the concrete–rock interfacial transition zone are studied using multi-scale experiments, by Yue et al. [12], including NMR, SEM-EDS, and XRD. A fractal permeability model is proposed.

In [13], to study the effects of the relative density and grading on the particle breakage of granular materials, a series of monotonic drained triaxial tests are performed on granular materials with different initial gradings and relative densities. Due to particle breakage, the grading curves of granular materials after triaxial tests can be simulated by a power-law function with a fractal dimension.

A series of discrete element simulation of triaxial tests on breakable particles within a flexible membrane are carried out by Chen et al. [14]. A strong correlation between the particle breakage ratio and fractal dimension is found.

In [15], a series of SEM tests are carried out on overconsolidated soil to study the effects of consolidation pressure and overconsolidation ratio, etc., on the micro-pore structure and fractal characteristics of soil. The fractal dimension of soil can well represent the complex characteristics of its microstructure.

The authors develop a fractal-entropy model to predict the hydraulic conductivity of granular soils [16], where the pore size distribution (PSD) is discretized based on fractal entropy, while the effective diameter of soil is computed using the grading entropy theory.