Unsaturated soil is a three-phase media and is composed of soil grain, water and gas. In this paper, the consolidation problem of unsaturated soil is investigated based on the theory of mixture. A theoretical formula ...Unsaturated soil is a three-phase media and is composed of soil grain, water and gas. In this paper, the consolidation problem of unsaturated soil is investigated based on the theory of mixture. A theoretical formula of effective stress on anisotropic porous media and unsaturated soil is derived. The principle of effective stress and the principle of Curie symmetry are taken as two fundamental constitutive principles of unsaturated soil. A mathematical model of consolidation of unsaturated soil is proposed, which consists of 25 partial differenfial equations with 25 unknowns. With the help of increament linearizing method, the model is reduced to 5 governing equations with 5 unknowns, i.e., the three displacement components of solid phase, the pore water pressure and the pore gas pressure. 7 material parameters are involved in the model and all of them can he measured using soil tests. It is convenient to use the model to engineering practice. The well known Biot 's theory is a special case of the model.展开更多
Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius tran...Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations f<sub>m</sub> mapping onto . Even for the simplest entity, the pre-image by f<sub>m</sub> of a unique point, there is no way of visualization. Pre-images by f<sub>m</sub> of figures from C are like ghost figures in C<sup>m</sup>. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in C<sup>m </sup>are well known and vector calculus in C<sup>m</sup> is familiar, yet the pre-image by f<sub>m</sub> of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in C<sup>m</sup>. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.展开更多
文摘Unsaturated soil is a three-phase media and is composed of soil grain, water and gas. In this paper, the consolidation problem of unsaturated soil is investigated based on the theory of mixture. A theoretical formula of effective stress on anisotropic porous media and unsaturated soil is derived. The principle of effective stress and the principle of Curie symmetry are taken as two fundamental constitutive principles of unsaturated soil. A mathematical model of consolidation of unsaturated soil is proposed, which consists of 25 partial differenfial equations with 25 unknowns. With the help of increament linearizing method, the model is reduced to 5 governing equations with 5 unknowns, i.e., the three displacement components of solid phase, the pore water pressure and the pore gas pressure. 7 material parameters are involved in the model and all of them can he measured using soil tests. It is convenient to use the model to engineering practice. The well known Biot 's theory is a special case of the model.
文摘Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations f<sub>m</sub> mapping onto . Even for the simplest entity, the pre-image by f<sub>m</sub> of a unique point, there is no way of visualization. Pre-images by f<sub>m</sub> of figures from C are like ghost figures in C<sup>m</sup>. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in C<sup>m </sup>are well known and vector calculus in C<sup>m</sup> is familiar, yet the pre-image by f<sub>m</sub> of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in C<sup>m</sup>. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.
