4. Energy Dissipation mechanisms and Densification modeling
4.2. Energy dissipation mechanisms in granular media
Energy loss in saturated soils is attributed mainly to three main mechanisms:
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i) The frictional sliding at grains contact surfaces, ii) The viscous drag of the pore fluid moving relative to the soil skeleton, and iii) Particle breakage.
4.2.1. Frictional dissipation mechanism
The mechanical behavior of materials composed of discrete elastic grains in direct contact was the subject of many researches aiming to develop a mathematical theory that can predict stress-strain relations, stress distributions, vibrations, and wave propagation phenomena for such materials. Because of the extraordinary complex nature of the problem the grains have been idealized as like spheres with radius Rg in regular arrays, and the local forces and deformations at the contact surfaces between adjacent grains were considered (Mindlin 1949), (Mindlin et al. 1951), (Mindlin 1954), and (Deresiewicz 1974).
σ τ
Y T’
Y
Y Y
T’
T’
T’
σ
σ
σ τ
τ
τ
Figure 4-1 Simplified cubic model of equal spheres
The physics of energy dissipation by friction are explained through a simple cubic array of identical quartz spheres (Fig. 4-1). The model is subjected to an isotropic
confining pressure σ, and shear stress τ. Fig. 4-2 shows the forces at any one of the four contact points around the representative central sphere of Fig. 4-1. A normal force Y and a tangential force T’ must be transmitted through the contact. The relations between these forces and the overall stresses σ and τ acting on the array are:
4 g2
Y
σ = R (4-1)
'
4 g2
T
τ = R (4-2)
Y
Y Rg
Rg
T’
T’
Figure 4-2 Elastic spheres under normal and shear loads
The tangential force T’ produces a tangential displacement δ between the centers of the spheres. This tangential displacement is the direct cause of the shear strain of the whole array. If the horizontal displacement between the centers of the spheres δ is plotted
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against the tangential force T’ as shown in Fig. 4-3, we get a force-displacement curve of the yielding type (curve 0P). If the tangential load T’ is cycled between two fixed values T’*, -T’* (T’* < Cf * Y), where Cf is the coefficient of friction of the contact surface (constant) while maintaining Y constant, a hysteresis loop is formed as shown in Fig. 4-3.
This hysteresis loop is similar to the experimental loops measured in sands subjected to cyclic shear loading. The area enclosed by the loop measures the energy spent by friction in the annulus of slip, and for the case considered here (T’* < Cf * Y), the loop is stable (i.e. it repeats itself cycle after cycle).
0 δ
T ’ T
-T
P
’*
’*
Figure 4-3 Theoretical hysteresis loop due to oscillating tangential force for two spheres in contact (Deresiewicz 1974)
The solution for this problem can be found in Mindlin (1949), (Mindlin et al.
1951), (Mindlin 1954), (Deresiewicz 1974), and (Dobry et al. 1982). The previous discussion reveals that energy is dissipated through friction before gross sliding across
the entire contact surface occurs, as liquefaction requires complete breakdown of the soil structure, which inherently involves slippage of contact surfaces as the particles
rearrange.
4.2.2. Viscous dissipation mechanism
Viscous drag is the force resisting the relative movement of fluid and solid and is similar to the frictional force between two solids. Hall and Richart (1963) studied the effect of different parameters, such as amplitude; confining pressure; and pore fluid, on the energy dissipated in granular material through a series of resonant column test where the specimens were excited at their first mode of vibration and then set free, while the rotations amplitude being recorded. The decay in the rotational amplitude is due to energy dissipation in the soil. Considering the frictional and viscous mechanisms of energy dissipation, the relative contribution of both mechanisms can be compared through tests on similar saturated and dry specimens in terms of the logarithmic
decrement. It was concluded that the energy dissipated by viscous mechanism increases as the amplitude of the rotations decreases. While tests done by Hardin (1965) on dry sand indicated that the energy dissipated (frictional mechanism only) is independent of the loading frequency.
However, based on results extrapolation it was concluded that the portion of energy dissipated by frictional mechanism increases with increasing strain and becomes the dominant mechanism in both saturated and dry specimens subjected to large strains.
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Therefore, in summary, for small strains the viscous mechanism of energy dissipation is significant, and the portion of energy dissipated due to that mechanism is proportional to the frequency of the applied load. While for large strains, the amount of energy dissipated in soil can be attributed mostly to the frictional mechanism (dominant mechanism), which is independent of loading frequency (Hall and Richart 1963).
4.2.3. Particle breakage dissipation mechanism
High stresses in some problems may result in particle breakage. The contribution of this mechanism is relatively insignificant in DC and will not be discussed further.
The energy dissipated per unit volume of soil can be evaluated throughout cyclic high-strain laboratory tests. Appendix B presents research results of the energy required to cause liquefaction for sands and non-plastic silty sands.