Though porous materials are naturally available, artificially synthesized porous media have recently attracted significant research attention to meet specific engineering applications. Synthesis of porous media requires a thorough understanding of their dynamic responses. Numerous models have been proposed to study the dynamic characteristics of porous media. In particular, the Biot theory has been widely studied and improved to address different challenges, thus allowing the coverage of wider frequency responses and observation of the second type of longitudinal waves that could not previously be observed in saturated porous media. Subsequent research studies have focused on further developing these models and extending their applications. The Biot theory is widely used in poroelasticity, forming the basis for solving various engineering problems related to porous media. This can be attributed to its experimentally measurable parameters and the ease in refining its mathematical expressions.
Several strategies, including the mixture theory for infiltrated porous media, have been proposed to explore the theoretical applicability of Biot theory. Remarkably, models based on the mixture, volume averaging, and homogenization theories are consistent with Biot theory. Nevertheless, despite the successful practical applications of Biot theory, classical Biot theory is based on idealized assumptions that are yet to be fully clarified. For instance, it does not consider the size effects and the effects of the internal heterogeneous structure that are crucial in determining the overall accuracy of the theoretical models. Incorporating nonlocal elasticity theory into the Biot model framework is a promising approach for overcoming the limitations of classical Biot theory. It accounts for the size effects of the porous media, enabling the prediction and interpretation of the negative and positive dispersion relations that could not be predicted by classical Biot theory. However, a comprehensive modification to overcome the mentioned defects associated with the assumptions is lacking.
Motivated by the previous results, a group of researchers at East China Jiaotong University: Dr. L.H. Tong, Dr. Haibin Ding, Dr. J.W. Yan, Dr. Changjie Xu and Dr. Z. Lei proposed a generalized Biot theory model to study the physical characteristics of materials and structure of porous media based on a higher-order strain gradient nonlocal poroelasticity. Their main aim was to increase its applications in practical engineering by addressing the limitations of classical Biot theory. The work is currently published in the research journal, International Journal of Engineering Science.
In their approach, the theoretical framework of Bio theory was preserved. However, the classical theory was improved by introducing three new parameters: scale factor, nonlocal and two length parameters, to capture the strain gradient and size effects. The proposed model also took into account the heterogeneity effects and the size effects to describe the physic characteristics of the porous media. The governing equations and the differential-form constitutive relations were obtained via a variational method. Finally, the feasibility of the proposed methods was validated by using it to analyze the characteristics of propagating waves in porous media.
Results showed that, unlike classical Biot theory, the proposed theory predicted both positive and negative dispersion relations in saturated porous media because the nonlocality accounted for the interactions amongst solid grains. The wave propagation analysis revealed the softening and hardening effects exhibited by the nonlocal parameter and scale factor, respectively. The softening effects showed a linear relationship with the nonlocal parameter. Moreover, both scale factor and nonlocal parameters exhibited a remarkable influence on the quality factors, especially at a high-frequency range.
In summary, the study reported a higher-order strain gradient nonlocal poroelasticity theory to investigate specific physical characteristics of porous media, taking into account the strain gradient and size effects. This model solved most of the issues associated with the previous models and allowed the analysis of both fast and slow waves. The dispersion relations predicted by this theory were in good agreement with the experimental observation based on Biot theory. In a statement to Advances in Engineering, the authors explained their study would advance the wide frequency range interpretation of dynamical behaviors of porous media.
Tong, L., Ding, H., Yan, J., Xu, C., & Lei, Z. (2020). Strain gradient nonlocal Biot poromechanics. International Journal of Engineering Science, 156, 1-17.