此觀念已漸為此領域國際知名學者所採納(Goddard, J.D., From granular matter to generalized continua. P. Mariano et al. (eds), Mathematical models for granular matter, Lecture Notes in Mathematics, Springer, 2006)，然此分析工作尚於初始階段。本研究將採取不同於其他研究學者之方法，由基礎之微連續體的定義著手，首先提出將顆粒材料模擬為混合體的模型。此混合體非指大小尺寸顆粒之混合，而是以顆粒排列之不同所架構之混合體。其次，顆粒材料可藉由混合體之觀念而模擬為一微連續體。我們將依此微連續體的架構推導顆粒材料之運動與組成方程式。
Abstract: A granular material represents the gathering of macroscopic particles. Under the framework of discrete description, a vast amount of numerical calculations and an appropriate averaging mean to find the macroscopic physical quantities are required. On the contrary, the continuum method is an alternative, which provides an effective tool to analyze the motion of granular media. However, the traditional continuum mechanics needs some modification such that the complicated phenomena of granular systems can be well captured. The project discusses the possibility, viability, and application of modeling a dry granular material as a microcontinuum.
Recently, this concept has gradually adopted by notable researchers (such as: Goddard, J.D., From granular matter to generalized continua. P. Mariano et al. (eds), Mathematical models for granular matter, Lecture Notes in Mathematics, Springer, 2006), and the study of which is still on its infant. With a different point of view in this project, we propose a mixture model of a granular material in view of the basic definition of microcontinnum. Instead of the different components with different size of particles, a new index to account for the arrangement of particles will be introduced to distinguish the different components of a granular mixture. Furthermore, the mixture notion helps a granular material to be treated as a microcontinuum. Then the balance and constitutive equations for the granular microcontinuum can be fully derived.
Since there is a link between the mixture and mesoscopic notations, the mesoscopic theory for granular materials could also be constructed. It is believed that the derived equations can be reduced to the Goodman-Cowin equation, describing the bulk motion for granular materials, and another part related to the shear deformation. In addition, we will compare our theory with different existing macroscopic theories of granular materials. Accompanying with the theoretical analysis, the numerical approach of molecular dynamics and a series of devised experiments will be studied to confirm the validity of our theory. We also look for other practical applications of our theory, and wish to provide a connection between the microscopic and macroscopic mechanics.