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A simple method is proposed to determine the surface fractal dimension of swelling clay minerals. When swelling clay mineral is immersed into the water solution, swelling deformation and swelling pressure will occur. The swelling deformation and swelling pressure are proportional to the amount of water volume absorbed by swelling clay minerals. The water volume absorbed by swelling clay minerals is related to its surface fractality. The correlation of the normalized water volume by the volume of swelling clay minerals (V_w/V_m) to vertical overburden pressure (p) is obtained as V_w/V_m = Kp^D_s-3, from which the surface fractal dimension of swelling clay minerals can be estimated. The surface fractal dimension (D_s) of Wyoming bentonite is 2.64, obtained from the swelling tests. The surface fractal dimension of Wyoming bentonite determined from the swelling tests nearly equals to that obtained from the nitrogen adsorption isotherms, and is larger than that measured using the mercury intrusion porosimetry.

The correlation between the void ratio of swelled montmorillonite and the vertical overburden pressure can be expressed as . The surface fractal dimension D_s of five bentonites were estimated from the swelling deformation tests according to this fractal correlation. The reliability of surface fractal dimension obtained from the swelling deformation test was confirmed by nitrogen adsorption test, with identical values of surface fractal dimension obtained from both tests. The surface fractal dimension can also be used to estimate the swelling deformation of bentonite, after calculating the swelling coefficient K from the parameters of diffuse double layer (DDL) model in the osmotic swelling phase. Comparison of the model predictions with a number of experimental results on swelling deformation of both Na dominant and Ca dominant bentonites suggests that the surface fractal model works excellent in the cases tested.

This study employs swelling deformation test for estimating the fractal dimension of salt-modified bentonite soil by measuring the void ratio of montmorillonite ($e_m$) and the effective stress ($p_e$). For the compacted Tsukinuno bentonite, a typical Na-montmorillonite, the fractal dimension is tested by the swelling deformation test conducted in sodium chloride solutions with different concentrations. According to the results, the fractal dimension of the specimens is essentially constant irrespective of the concentration of inundating NaCl solution. The strong correlation of the results from the swelling deformation test with those from the nitrogen adsorption test indicates that the swelling deformation test is an effective laboratory method for testing the fractal dimension of compacted bentonite in salt solution.

In this paper, we suggest a design strategy for multilayer hydrogel beams with controllable swelling deformation and bending, which can be used in soft devices. We adopt a constitutive model that considers the entanglement chains of the hydrogel and an analytical method for solving multilayer hydrogel beams is developed. First, the reliability of the design strategy is provided by comparing analytical solutions and numerical results. Then, guided by the above design strategy, we quantitatively investigate hydrogel beams with gradient distribution and study the effects of material and distribution on the deformation and bending behavior of the structures. The results show that under non-gradient distribution, all structures with an odd number of total layers only undergo in-plane expansion, while all those with an even number of total layers undergo expansion and bending. However, this law will be broken at the gradient distribution. The introduction of entanglement chains and gradient distribution strategy increase the range of adjustment for deformation and bending. This work is expected to provide new insights into the design of multilayer hydrogel beams for material–structure–function integration.