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Meaning of TIJ concept from the viewpoint of Satake’s fabric tensor

The concept of tij has been proposed to describe simply and uniquely the stress-strain behaviors of frictional materials such as soils in general three-dimensional stress condition. Using this concept, any kinds of one-dimensional models and/or three-dimensional models formulated by using (p and q) can be easily extended to general three-dimensional models. Since the method to apply this concept to elastoplastic constitutive modelling have been presented in the previous papers, its physical meaning is discussed here referring to Oda’s microscopic investigation and Satake’s fabric tensor theorem.
Recently, discrete elements method (DEM) becomes popular as a tool for the investigation of granular materials, and it is shown that as the stress ratio increases, the average of the normal directions to the inter-particle contacts gradually concentrate in the direction of the major principal stress (1). However, Oda already found it more than 40 years ago by microscopic observation of real sand, and Satake pointed out based on this experimental evidence that the principal values of the so-called fabric tensor ij, which represents the relative distribution of the number of vectors normal to the inter-particle contacts, is approximately proportional to the square root of the corresponding principal stresses. Further, employing the fabric tensor, Satake proposed the modified stress tensor for analyzing the induced anisotropy of granular materials.
On the other hand, Nakai and Mihara proposed the modified stress tensor tij which is defined as from continuum approach on three-dimensional soil behavior. Here, aik is the symmetric tensor which principal values are given by the direction cosines of the spatially mobilized plane (SMP) and are inversely proportional to the square root of the respective principal stresses. So, it can be seen that aik corresponds to the inverse of Satake’s fabric tensor ik.
In this symposium, the meaning of the tij concept which is derived from macroscopic study is discussed referring to Satake’s achievements, and the usefulness of this concept is presented.

Author(s):

Teruo Nakai    
Geo-Research Institute
Japan

Hossain Md. Shahin    
Nagoya Institute of technoligy
Japan