Volume changes during shear

During a small time interval d£, the lid of the shearbox moves horizontally by a distance dx. If, during this time, it also moves vertically upward by distance dy, it is travelling at an angle ψ=tan⁻¹(dy/dx) to the horizontal (Figure 2.17). ψ is known as the angle of dilation, and is an indication of the rate at which the sample is changing in volume as it is sheared.

If ψ is positive, the lid of the shearbox is moving upward and the sample is increasing in

Soil strength 87 volume, or dilating. If ψ is negative, the lid of the shearbox is moving downward and the sample is reducing in volume, or compressing.

More formally, ψ is defined as the negative of the rate of increase of volumetric strain with shear strain.

ψ=tan⁻¹(−dε_vol/dγ) (2.9)

The negative sign in equation (2.9) is required so that a negative (i.e. expansive) change in volume corresponds to a positive rate of dilation.

For the shearbox test, dε_vol (compression positive) =−dy/h₀ and dy=dx/h₀ (writing equa-tions (2.6) and (2.7) in incremental form), so that ψ=tan⁻¹(dy/dx), as in the text above.

So far, we have avoided the question of why the soil should change in volume (dilate or compress) as it is sheared. This behaviour arises because the soil is essentially a particulate material. The particles must take up a suitable arrangement of packing—corresponding to what is known as a critical void ratio—before continued shearing can take place. If the particles are initially more densely packed than the critical void ratio, some loosening will have to occur before steady shear can take place.

A loosening of the packing requires an increase in void ratio. This corresponds to an increase in the overall volume—that is, dilation—and an upward movement of the shear-box lid. If the particles are initially more loosely packed than the critical void ratio, densifi-cation of the sample will take place before the appropriate arrangement for steady shear is reached. Densification requires a reduction in void ratio, which corresponds to a compres-sion of the sample and a downward movement of the shearbox lid.

The concept of a degree of packing at which steady shear can take place can be illustrated initially with reference to an imaginary assemblage of ball bearings. In Figure 2.18(a),

Figure 2.17 Dilation.

Figure 2.18 Conceptual model for (a) compression and (b) dilation during shear.

88 Soil

the ball bearings are initially loosely packed, with the ball bearings in the upper layer rest-ing on the peaks of those in the lower layer. On shearrest-ing, the ball bearrest-ings in the upper layer will have to move downward into the troughs, resulting in a reduction in overall volume (compression). In Figure 2.18(b), the ball bearings are initially densely packed, with the upper layer nestling in the troughs between the ball bearings in the lower layer. On shearing, the ball bearings in the upper layer will have to climb out of the troughs, resulting in an increase in overall volume (dilation).

The diagrams shown in Figure 2.18 have only two layers of ball bearings in regular packing arrangements, and are therefore much too simple for anything except a basic illus-tration of the concept. A real assemblage of ball bearings might have only a single particle size, but will be randomly packed. A soil will be randomly packed with a variety of particle sizes. The result of this is that, after dilation or contraction to a critical void ratio, continued shearing of a soil (or an assemblage of ball bearings) can take place at constant volume, without the lumpiness (as the top layer of ball bearings continues to fall into and climb out of the troughs in the lower layer) implied by the simple model of Figure 2.18.

Figure 2.19 gives a more realistic visualization of the rearrangement of real soil parti-cles during shear. Figure 2.19(a) shows an initially dense sample during the early stages of shear: the average particle velocity is inclined upwards, implying dilation. Figure 2.19(b) shows an initially loose sample: the average particle velocity is downward, implying com-pression. Figure 2.19(c) shows a sample deforming at the critical void ratio: some of the soil particles are moving upward and others downward, but the average movement is hori-zontal.

The dilation of a dense sand when sheared can be demonstrated quite graphically by means of a rubber bulb, filled with sand, on the end of a tube (Figure 2.20). The sand is sat-urated, and some excess water is added so that the tube is initially about three-quarters full of clear water (Figure 2.20(a)). When the bulb is squeezed, the sand is sheared and dilates.

This draws water into the bulb to fill the additional void space, resulting in a fall in the water

Figure 2.19 Visualization of rearrangement of soil particles during shear. (Redrawn with from Bolton, 1991.) permission from Bolton, 1991.)

Soil strength 89

Figure 2.20 Demonstration of dilation.

level in the tube (Figure 2.20(b)). It follows from the behaviour shown in Figure 2.20 that, in a saturated soil, a tendency to dilate is accompanied by the development of nega-tive pore water pressures. It is these neganega-tive pore water pressures which are responsible for the drawing in of water from the surroundings. Conversely, a tendency to contract on shearing will be associated with the generation of positive pore water pressures, which lead to the expulsion of water from the soil in order to attain the required reduction in specific volume.