Discussion: Microstructure in shear band observed by microfocus X-ray computed tomography Free

The authors wish to thank Dr Smart for paying attention to the crucial and interesting topic that we emphasised in our technical note. Oda & Kazama (1998) have studied in detail the development of microstructures in a shear band, by means of the microscope-thin section method. The present technical note was intended to supplement the previous study by further investigating the three-dimensional micro-structure of a shear band by X-ray computed tomography. In order to facilitate a reply to the questions raised, we need to repeat briefly the main aspect of the microstructure developed in a shear band, discussed in detail by Oda & Kazama (1998).

In the strain-hardening of sand, a columnar structure is gradually developed such that the elongation direction is, on average, parallel to the axial direction (Fig. 7). That is, strain-hardening is achieved by the development of the columnar structure, which eventually leads to so-called induced anisotropy. This is because increasing axial loads are transmitted mainly through the columns, as will be seen later. Similar columnar structures were observed in physical tests by Wakabayashi (1957), Drescher (1976), Oda et al. (1982) and Allersma (1987), and also in numerical simulation tests by Iwashita & Oda (1998). As the axial load reaches a peak, a shear band (or bands) is (are) formed, and the columns buckle or bend, inevitably leading to strain-softening after the peak. In the schematic illustration of Fig. 7, the centres of two particles c and d move to new positions along trajectories such as c → c′ → c″ and d → d′ → d.″ Two important points to be made are that: (a) large voids gradually grow between two buckling columns ac and bd so that extensive dilatation takes place in association with accumulation of shear strain in the shear band; and (b) non-spherical particles rotate concurrently with the buckling in response to the macroscopic rotational strain in the shear zone.

Dr Smart has pointed out that ‘thin columns separated by voids could carry only axial loads; and those illustrated seem to be too slender to be stable.’ The authors readily agree that the buckling columns are mechanically unstable, as they are surrounded by large voids. It can naturally be argued that such an unstable structure cannot persist for long in the shear zone where particles are actively moving, but will rather easily collapse after a short time. This process accounts for why strain-softening takes place after the peak load. If this were to occur, the large voids would also collapse, resulting in a locally compacted volume (negative dilatancy). Importantly, however, more columns would buckle elsewhere in the band. As sand is sheared until a steady state (called the residual state) is attained, collapse is balanced with generation of new buckling columns such that the average void ratio is kept constant in the band.

It is also of particular importance to note that the buckling columns, which are barely stable by means of arching action, can carry axial loads only if rotational resistance works at contacts such that contact moments can be transmitted from particle to particle. Otherwise, the buckling columns, as well as the large associated voids, would collapse quite easily. In fact, the buckling columns must be stabilised towards axial load by rotational resistance. Sand particles are irregularly shaped, their surfaces appear very rough under a scanning electron microscope, and they are covered with a thin film of weathered (soft) matter. These facts may suggest that each particle is in contact with its neighbours through surfaces, and that rotational resistance can work to some extent at least.

Iwashita & Oda (1998) and Oda & Iwashita (2000) have shown that not only buckling columns but also the large voids associated with them can be numerically simulated using the distinct element method (DEM), in a manner quite similar to those of Fig. 7. Interestingly enough, this can be done successfully only when the rotational resistance at contacts is taken into account in DEM. We briefly summarise their results below, because our hypothesis that the dilatancy of sand stems from the buckling movement of columns (see Fig. 7) was at least partially founded on the results of numerical simulations.

The simulations were carried out using conventional DEM, except that rotational resistance at contacts was taken into account. A total of 15 840 circular rods consisting of equal numbers of rods with 4 mm, 5 mm and 6 mm radii were generated at random inside a two-dimensional loading frame. After the assembly had been subjected to uniform ambient pressure, the axial stress was increased step by step while keeping the lateral stress constant, for a biaxial compression test (for more details, see Iwashita & Oda, 1998). Fig. 8 shows three results taken from an assembly sheared beyond peak stress (failure). The following findings are worth noting here.

• Fig. 8(a) shows the distribution of local shear strain γ. It has been observed that shear strain begins to concentrate in multiple bands a little before peak stress. After the peak, as can be seen in this figure, the shear strain is concentrated mostly in one of the bands to grow a major (persistent) shear band.

• Fig. 8(b) is to help visualise how contact forces are transmitted from particle to particle at the peak. The black circles denote particles stressed more than average (highly stressed particles), whereas the white ones denote particles stressed more than half the average, but less than average. The remaining particles are omitted. The major shear band in Fig. 8(a) can be easily detected along the diagonal extending from top left to bottom right. Outside the shear band, the highly stressed particles (black) are linked in chains, and tend to align in columns whose elongation directions are more or less parallel to the major stress (vertical) direction. These columns look like principal stress trajectories, as the major (axial) stress is mainly transmitted through them. On the other hand, in Fig. 8(a), inside the shear band where shear strain is highly concentrated, chains formed by the highly stressed particles decrease in number, and the elongation direction deviates consistently from the vertical by 20–30°.

• Fig. 8(c) shows the distribution of local volumetric strain, which is accumulated from the beginning. Outside the shear band, the volumetric strain is weakly positive (contractive), and is more or less uniformly distributed. It should be noted, however, that the volumetric strain changes widely from −25% to +25% in the shear band. The large dilatational strain (−25%) probably stems from the formation of extremely large voids between the buckling columns (see also Fig. 8(b)), whereas the large compressive strain (+25%) may stem from collapse of the buckling columns. (Note, however, that all the large open spaces in Fig. 8(b) between the buckling columns do not necessarily mean ‘real voids’, but are rather filled with very lightly loaded particles, which were omitted from the plot.)

• Particle rotation does not take place much outside the shear band. Although its magnitude is small, rotation occurs equally in both clockwise and counterclockwise directions so that the average remains almost zero (Oda & Iwashita, 2000). Particle rotation is pronounced in the shear band. More importantly, it is directed either clockwise or counterclockwise, such that the average rotation is in keeping with the macroscopic rotational strain in the shear zone.

A few micro-deformation mechanisms of dilatancy have been proposed in recent decades. It should be realised that all these proposed models basically rely on the common assumption that deformation of sand occurs as a result of frictional sliding at contacts, analogous to frictional sliding between two rigid blocks. It should be noted, however, that microscopic sliding directions at contacts may be different from the macroscopic direction. Newland & Allely (1957), for example, have asserted that this difference causes the dilatancy associated with shear strain. One problem with their approach is that they paid little attention to the important role of particle rotation in the micro-deformation mechanism. Our observations strongly suggest that the dilatation, which takes place rather locally in a shear band, is inevitably tied to the extensive rotation of particles. The dilatation model of Fig. 7 is proposed on the basis of the observations summarised in (a) to (d) above. In our technical note, we have argued that the images obtained from micro-focus X-ray computed tomography (e.g. Figs 4 to 6) are in good accordance with the proposed dilatation model of Fig. 7.

Dr Smart has also pointed out a possibility that the microstructures, which are called buckling columns by the authors, are a series of complementary failure ‘planes’. If so, particle orientation in the complementary failure ‘planes’ should be oblique to that in the main failure zone. Particle orientation in the complementary failure ‘planes’ is, however, in line with that in the main failure zone. In this respect, therefore, the authors cannot agree with him. We frankly admit, however, that more study by means of micro-focus X-ray computed tomography is needed to reach a final conclusion.