# THEORETICAL AND EXPERIMENTAL INVESTIGATION OF EFFECTIVE DENSITY AND PORE FLUID INDUCED DAMPING IN SATURATED GRANULAR MATERIALS

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# Introduction

## 1.1 Background

Shear modulus and damping are two important soil dynamic properties.  These two parameters play crucial roles in ground motion analyses in geotechnical earthquake engineering.  For example, Fig. 1-1 shows a deposit of homogeneous soil layer on top of bedrock.  The fundamental frequency, fo , of the soil layer can be estimated as Vs                                                                                                               (1-1) fo  4H where H is the soil layer thickness and Vs is shear wave velocity which can be calculated as GVs   (1-2) where is the soil density and G is shear modulus.  Eq. (1-1) indicates that fo depends on the shear wave velocity and layer thickness.  For a typical earthquake ground motion, the dominant frequency, f g , is generally in the range of 1 – 5 Hz (Kramer 1996).  If f g is close to fo , dynamic amplification will occur and large ground motions (e.g., acceleration, velocity, and displacement) will be expected.  On the other hand, if f g is significantly different than fo and other modes of natural frequencies, deamplification may occur and the resulted ground motions will be small.  As the seismic wave travels between the bedrock and ground surface as shown in Fig. 1-1, wave energy is dissipated through soil damping.  Higher soil damping will result in smaller ground motions as more energy is dissipated during the wave propagation.  Therefore, shear modulus and damping are considered as input parameters for ground motion analyses.     Fig. 1-1.  Seismic wave propagation in homogeneous soil layer   In the current geotechnical engineering research and practice, shear modulus is generally calculated based on shear wave velocities measured from various field and laboratory tests involving shear waves, such as the seismic cone penetration tests (SCPT), bender element (BE) tests, and resonant column (RC) tests, using the following equation  G  Vs2                                                                          (1-3) This equation is based on the theory of elasticity in a continuum, where  is the density of the continuum (single-phase) under all conditions.  Soil is a multi-phase system, consisting of a solid phase, liquid phase (e.g., water), and gas phase (e.g., air).  For dry soil, dry density d is used in Equation (1-3); for saturated soil, saturated density sat is generally used.  However, the use of sat for saturated soil assumes no relative motion between pore fluid and solid skeleton.  The validity of this assumption depends on the magnitude of fluid motion relative to solids during shear wave excitations.  For soils with low permeability (e.g., silts and clays) and under low-frequency excitations (e.g., seismic waves), this assumption is generally valid.  However, relative motion may be important for soils with high permeability (e.g., sands and gravels) and under high-frequency excitations based on Biot theory (Biot 1956), rendering this assumption invalid.  Qiu and Fox (2008) proposed the concept of “effective soil density”, eff , that is related to the fraction of pore fluid that moves with solid skeleton during shear wave propagation.  This effective density is always between d and sat and is the theoretically correct value to use in Equation (1-3) to calculate shear modulus based on measured shear wave velocity.  Qiu and Fox (2008) provided analytical solutions of eff based on Biot theory (Biot 1956).  However, this analytical solution has not been rigorously validated against laboratory test data for different soils. Damping is a consequence of energy dissipation due to sliding and rolling at particle contacts, and the loss and creation of particle contacts when there is particle rearrangement.  This form of energy dissipation is generally considered as “skeleton damping” (Ellis et al. 2000) and is the only source of material damping in dry soil.  For saturated soil, in addition to skeleton damping, energy is also dissipated due to the relative motion and viscous drag between pore fluid and solid skeleton (i.e., viscous coupling).  Therefore, saturated soils exhibit higher damping than the same soils in their dry condition.  This has been experimentally observed by various researchers (e.g., Hall and Richart 1963; Bolton and Wilson 1990; Ellis et al. 1998 and 2000).  In geotechnical engineering research and practice, however, pore fluid induced damping is generally neglected due to the lack of quantitative assessment of its values in various soils.  Qiu and Fox (2006) and Qiu (2010) provided analytical solutions of pore fluid induced damping in saturated soils during shear wave excitation.  These studies suggest that pore fluid induced damping depends on soil types and may have significant contribution to the total damping for coarse sands and gravels, in particular at small strain levels.  However, these findings have not been validated by any experimental test data.

## 1.2 Objectives of Research

The objectives of this research are to quantify effective density and pore fluid induced damping in granular materials for small strain shear waves using BE and RC tests, and to conduct additional analytical study on pore fluid induced damping in saturated soils under quasi-static compressional excitations.  This study can potentially improve the accuracy of how small strain shear modulus and damping are evaluated, especially in highly permeable granular materials (e.g., coarse sands and gravels) under highfrequency excitations (e.g., BE tests), which may improve the accuracy of current ground motion analyses in geotechnical earthquake engineering.  The findings of this study will be of significant value to geotechnical earthquake engineering and soil dynamics. Ultimately, the benefit will be the reduction of losses to society as a result of earthquakes.

## 1.3 Organization of Dissertation

Chapter 2 presents effective density for small strain shear waves in saturated granular materials.  It presents an experimental investigation consisting of RC and BE tests on various granular materials in dry and saturated conditions for the concept of effective density.  This chapter is based on a manuscript submitted to the Journal of Geotechnical and Geoenvironmental Engineering, ASCE. Chapter 3 presents analytical solutions and their comparison with RC test results for pore fluid induced damping in saturated granular materials.  It presents two analytical solutions and compares them with a closed-form analytical solution readily available for RC test in literature.  Furthermore, solution of pore fluid induced damping based on the free vibration decay method is compared with RC test results of various granular materials in dry and saturated conditions.  This chapter is based on a manuscript submitted to the Soil Dynamics and Earthquake Engineering. Chapter 4 presents energy dissipation in nearly saturated soil columns during quasistatic compressional excitations, which is of particular relevance to slow phenomena (e.g., consolidation).  Different components of energy dissipation in a saturated soil column are derived and compared.  This chapter is based on a paper published in the Journal of Engineering Mechanics, ASCE. Chapter 5 draws final conclusions of this study and presents suggestions for future work.

## References

Biot, M.A. (1956). “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range. ІІ. Higher Frequency Range.” J. Acoust. Soc. Am., 28(2), 168-191. Bolton, M. D., and Wilson, J. N. (1990). “Soil stiffness and damping.” Structural dynamics, W. B. Kratzig, D. E. Beskos, and I. G. Vardoulakis, eds., Balkema, Rotterdam, The Netherlands, 209–216. Ellis, E. A., Soga, K., Bransby, M. F., and Sato, M. (1998). “Effect of pore fluid viscosity on the cyclic behavior of sands.” Proc., Centrifuge 98, T. Kimura, O. Kusakabe, and J. Takemura, eds., Balkema, Rotterdam, The Netherlands, 217–222.. Ellis,E.A., Soga, K., Bransby,M.F. and Sato, M. (2000). Resonant Column Testing of Sands with Different Viscosity Pore Fluids, J. Geotech. Geoenviron. Eng., 126(1), 10-17. Hall, J.R. and F.E. Richart (1963). “Dissipation of Elastic Wave Energy in Granular Soils.” J. Soil Mech. and Found. Div., 89(6), 27-56. Kramer, S.L. (1996). Geotechnical Earthquake Engineering, Prentice Hall, Upper Saddle River, NJ. Qiu, T. and Fox, P.J. (2006). “Hydraulic damping of saturated poroelastic soils during steady-state vibration.” J. Eng. Mech., 132(8), 859-870. Qiu, T. and Fox, P.J. (2008). “Effective Soil Density for Propagation of Small Strain Shear Waves in Saturated Soil.” J. Geotech. Geoenviron. Eng., 134(12), 1815-1819. Qiu, T. (2010). “Analytical Solution for Biot Flow-Induced Damping in Saturated Soils during Shear Wave Excitations.” J. Geotech. Geoenviron. Eng., 136(11), 1501-1508.

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