This paper proposes a simple method for studying the evolution of particle breakage. A two-parameter statistical model is first proposed to describe the distribution of new particles generated from breakage of an initially uniformly graded sample. A Markov chain model is then proposed to describe the particle size evolution of non-uniformly graded samples. The two-parameter statistical model is shown to be able to capture different breakage patterns such as asperity breakage and particle splitting observed during breakage of uniformly graded samples. In the Markov chain model, it is shown that the concept of ‘effective breakage probability’, proposed to define the relative breakage extent of the whole sample, is comparable to a well-established breakage index in the literature. Both models are successfully validated against experimental data, where the characteristics of the model parameters and the engineering implications are discussed.

Purpose: Prediction of excess pore water pressure and estimation of soil parameters are the two key interests for consolidation problems, which can be mathematically quantified by a set of partial differential equations (PDEs). Generally, there are challenges in solving these two issues using traditional numerical algorithms, while the conventional data-driven methods require massive data sets for training and exhibit negative generalization potential. This paper aims to employ the physics-informed neural networks (PINNs) for solving both the forward and inverse problems.Design/methodology/approach: A typical consolidation problem with continuous drainage boundary conditions is firstly considered. The PINNs, analytical, and finite difference method (FDM) solutions are compared for the forward problem, and the estimation of the interface parameters involved in the problem is discussed for the inverse problem. Furthermore, the authors also explore the effects of hyperparameters and noisy data on the performance of forward and inverse problems, respectively. Finally, the PINNs method is applied to the more complex consolidation problems.Findings: The overall results indicate the excellent performance of the PINNs method in solving consolidation problems with various drainage conditions. The PINNs can provide new ideas with a broad application prospect to solve PDEs in the field of geotechnical engineering, and also exhibit a certain degree of noise resistance for estimating the soil parameters.Originality/value: This study presents the potential application of PINNs for the consolidation of soils. Such a machine learning algorithm helps to obtain remarkably accurate solutions and reliable parameter estimations with fewer and average-quality data, which is beneficial in engineering practice.