**5. Evaluation of S-wave velocity**

170 Wave Processes in Classical and New Solids

(Sawangsuriya et al. 2006).

**Identifying wave travel time arrivals** 

distance and the distance to the wall varied (Sawangsuriya et al. 2006). Results matched well with a simple refraction model used to determine the shortest travel time from the two possible wave paths. A sketch of the model and a summary of experimental results are shown in Fig. 9. Results suggest that for a given medium and PVC wall having finite wave velocities, the bender elements need to be placed at a minimum distance (d) greater than 0.4 L to avoid receiving refracted signals. As a consequence, it is recommended that with the presence of rigid boundaries, the distance of bender elements to the boundary (d) shall be

**Figure 8.** Limiting distances to avoid reflected P-wave arrivals when testing with boundaries. The lines in the plot represent the upper limits, the recommended test dimensions should fall under these lines

**Figure 9.** (a) Sketch of the test setup and the model for data evaluation and (b) experimental results from Kaolinite specimens (water content = 22%, unit weight = 9.7 kN/m3) (Sawangsuriya et al. 2006).

A selection of the travel time for a wave travelling from a transmitter bender to a receiver bender has been controversial. Generally, the travel time can be determined either in the

greater than approximately 0.4 L to avoid refracting wave from arriving first.

The S-wave velocity in particulate geomaterials depends on the state and history of the effective stresses, void ratio, degree of saturation, and type of particles. Based on experimental evidence and analytical studies, the velocity-effective stress relationship for granular geomaterials is expressed as a power function having two physically-meaningful parameters: a coefficient α and an exponent β. These parameters represent the S-wave velocity at a given state of stress and its variation with stress changes. The velocity-stress power relationship for granular media under isotropic loading is expressed as (Roesler 1979, White 1983):

$$V\_s = \alpha \left(\frac{\sigma\_o}{p\_r}\right)^{\beta} \tag{2}$$

Wave Propagation Methods for Determining Stiffness of Geomaterials 173

13.28 8 *C e <sup>n</sup>* (5)

(6)

Experimentally, the velocity-stress power relation is re-written as:

*s*

*V AFe*

 ' *o*

where A is a fitting parameter which accounts for the properties of the grains and the function F(e) depends on the initial void ratio e of the geomaterial (Mitchell and Soga 2005). For round-grained particles, Hardin and Black (1968) proposed the following function:

**Figure 10.**Velocity-stress response of specimen prepared with different methods (Sawangsuriya et al. 2007a).

*r*

 

*p*

where Vs is the S-wave velocity, σ'o is the isotropic effective stress, pr=1 kPa is a reference stress, α and β are experimentally determined parameters. Fig. 10 illustrates an example of the velocity-stress relationship as determined by S-wave velocity measurements at different stress states for dry medium sand prepared by different methods exhibits a power function (see Fig. 10) along with the experimentally determined coefficient α and exponent β (Sawangsuriya et al. 2007a). As shown in Fig. 10, for a dry medium sand subjected to isotropic loading, the velocity-stress power relation in Eq. (2) closely fits the data and as expected, the S-wave velocity increases as the confining pressure increases.

Several researchers have attempted to quantify the physical meaning of these experimentally determined parameters. Duffy and Mindlin (1957), Hardin and Black (1968), Santamarina et al. (2001), Fernandez and Santamarina (2001), and Fratta and Santamarina (2002) described the physical meaning of the parameters α and β: the coefficient α relates to the type of packing (i.e., void ratio or coordination number), the properties of the material, and fabric changes, while the exponent β relates to the effects of contact behavior. Both parameters indicate the effects of stress history, cementation and rock weathering in the formation. For example, dense sands, overconsolidated clays, and soft rocks have higher coefficient α and lower exponent β. In case of loose sands, normally consolidated clays, and clays with high plasticity, the coefficient α becomes lower while the exponent β becomes higher. Santamarina et al. (2001) suggested an inverse relationship between α and β values for various granular media, ranging from sands and clays to lead shot and steel spheres:

$$
\beta \approx 0.36 - \frac{\alpha}{700 \frac{m}{s}} \tag{3}
$$

In addition, Hardin and Black (1968) suggested that the coefficient α can be separated into two coefficients: one accounts for the grain characteristic (or nature of grains), and the other accounts for the properties of the packing (i.e., void ratio or coordination number). The separation into these different components is justified on particulate material models. For example for random particles of uniform spheres with Hertzian contacts, the relationship between isotropic stresses and S-wave velocity is (Chang et al. 1991, Santamarina et al. 2001):

$$V\_s = \underbrace{\left(\frac{\sqrt{\pi}}{4 \cdot \rho^\prime} \cdot \frac{5 - 4\nu^\prime}{2 - \nu^\prime}\right)^{\not{p}\_1^\prime} \left(\frac{G^\prime}{1 - \nu^\prime}\right)^{\not{p}\_3^\prime}}\_{\textit{particle\ properties}} \cdot \underbrace{\left[\frac{1.7 - e}{\left(1 + e\right)^{\not{p}\_2^\prime}}\right]^{\not{p}\_3^\prime}}\_{\textit{active}} \cdot \underbrace{\left(\sigma^\prime\right)^{\not{p}\_6^\prime}}\_{\textit{stress}}\tag{4}$$

where ρ', G', and ν' are the density, shear modulus and Poisson's ratio of the particles. This equation is derived assuming that the number of contacts per particle or coordination number Cn is related to the void ratio e by the following equation (Chang et al. 1991):

$$C\_n = 13.28 - 8e$$

Experimentally, the velocity-stress power relation is re-written as:

172 Wave Processes in Classical and New Solids

'

(2)

*o*

*r*

*p*

 

where Vs is the S-wave velocity, σ'o is the isotropic effective stress, pr=1 kPa is a reference stress, α and β are experimentally determined parameters. Fig. 10 illustrates an example of the velocity-stress relationship as determined by S-wave velocity measurements at different stress states for dry medium sand prepared by different methods exhibits a power function (see Fig. 10) along with the experimentally determined coefficient α and exponent β (Sawangsuriya et al. 2007a). As shown in Fig. 10, for a dry medium sand subjected to isotropic loading, the velocity-stress power relation in Eq. (2) closely fits the data and as

Several researchers have attempted to quantify the physical meaning of these experimentally determined parameters. Duffy and Mindlin (1957), Hardin and Black (1968), Santamarina et al. (2001), Fernandez and Santamarina (2001), and Fratta and Santamarina (2002) described the physical meaning of the parameters α and β: the coefficient α relates to the type of packing (i.e., void ratio or coordination number), the properties of the material, and fabric changes, while the exponent β relates to the effects of contact behavior. Both parameters indicate the effects of stress history, cementation and rock weathering in the formation. For example, dense sands, overconsolidated clays, and soft rocks have higher coefficient α and lower exponent β. In case of loose sands, normally consolidated clays, and clays with high plasticity, the coefficient α becomes lower while the exponent β becomes higher. Santamarina et al. (2001) suggested an inverse relationship between α and β values for various granular media, ranging from sands and clays to lead shot and steel spheres:

0.36

In addition, Hardin and Black (1968) suggested that the coefficient α can be separated into two coefficients: one accounts for the grain characteristic (or nature of grains), and the other accounts for the properties of the packing (i.e., void ratio or coordination number). The separation into these different components is justified on particulate material models. For example for random particles of uniform spheres with Hertzian contacts, the relationship between isotropic stresses and S-wave velocity is (Chang et al. 1991, Santamarina et al. 2001):

700 *<sup>m</sup> s*

(3)

5 2

*e*

6

(4)

*effective*

<sup>1</sup> <sup>1</sup> <sup>3</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>1</sup>

5 4 ' ' 1.7 '

*particle properties stress packing properties*

4 '2 ' 1 ' <sup>1</sup> *s o*

 

where ρ', G', and ν' are the density, shear modulus and Poisson's ratio of the particles. This equation is derived assuming that the number of contacts per particle or coordination

number Cn is related to the void ratio e by the following equation (Chang et al. 1991):

*G e <sup>V</sup>*

 

*s*

*V*

expected, the S-wave velocity increases as the confining pressure increases.

$$V\_s = A \cdot F(e) \cdot \left(\frac{\sigma\_o^\cdot}{p\_r}\right)^\beta \tag{6}$$

where A is a fitting parameter which accounts for the properties of the grains and the function F(e) depends on the initial void ratio e of the geomaterial (Mitchell and Soga 2005). For round-grained particles, Hardin and Black (1968) proposed the following function:

**Figure 10.**Velocity-stress response of specimen prepared with different methods (Sawangsuriya et al. 2007a).

$$F\left(e\right) = \frac{\left(2.17 - e\right)^2}{1 + e} \tag{7}$$

Wave Propagation Methods for Determining Stiffness of Geomaterials 175

σ'o=35 kPa

σ'o=69 kPa

σ'o=103 kPa

σ'o=138 kPa

σ'o=207 kPa

σ'o=172 kPa

**Figure 12.** S-wave traces from bender element tests. The arrow corresponds to the estimated S-wave

0 250 500 750 1000 1250 150

Time [μs]

0 500 1000 1500

0 250 500 750 1000 1250 150

0 250 500 750 1000 1250 150

Normalized amplitudes

Input signal

arrival. The first deflection corresponds to the near-field effect (Sawangsuriya et al. 2007a).

The evaluation of S-wave velocity using bender elements has been commonly performed on oedometer and standard triaxial samples. Normally, the top and bottom caps of an oedometer cell and a conventional triaxial cell are modified to incorporate bender elements and their electrical connections (see Fig. 11 – Sawangsuriya et al. 2007a). As shown in Fig. 11, the evaluation of S-wave velocity using bender elements is performed on standard triaxial specimens and subjected to isotropic loading. The top and bottom caps of a conventional triaxial cell are modified to incorporate bender elements and their electrical connections (see Fig. 11a). Examples of S-wave traces from bender element tests are shown in Fig. 12 along with the S-wave arrival as indicated by the arrows. As shown in Fig. 12, the S-wave arrival increases with confining pressure.

Laterally mounted benders were also used in triaxial apparatus (Pennington 1999). The horizontal pairs of bender elements are installed to study the anisotropy of the sample (see Fig. 13- Sukolrat 2007). In horizontal bender system, two similar bender elements were mounted orthogonally; one pair is used for S-wave transmitters and the other pair is for Swave receivers. The bender elements are placed diagonally opposite one another and oriented so that the shear stiffness in both HV and HH directions are synchronized (depending on different polarization direction).

(a) (b)

**Figure 11.** Modified triaxial cell with bender element measurement system: (a) details of bender elements and (b) assembled modified triaxial cell (Sawangsuriya et al 2007a).

confining pressure.

(depending on different polarization direction).

<sup>2</sup> (2.17 ) 1 *<sup>e</sup> F e*

The evaluation of S-wave velocity using bender elements has been commonly performed on oedometer and standard triaxial samples. Normally, the top and bottom caps of an oedometer cell and a conventional triaxial cell are modified to incorporate bender elements and their electrical connections (see Fig. 11 – Sawangsuriya et al. 2007a). As shown in Fig. 11, the evaluation of S-wave velocity using bender elements is performed on standard triaxial specimens and subjected to isotropic loading. The top and bottom caps of a conventional triaxial cell are modified to incorporate bender elements and their electrical connections (see Fig. 11a). Examples of S-wave traces from bender element tests are shown in Fig. 12 along with the S-wave arrival as indicated by the arrows. As shown in Fig. 12, the S-wave arrival increases with

Laterally mounted benders were also used in triaxial apparatus (Pennington 1999). The horizontal pairs of bender elements are installed to study the anisotropy of the sample (see Fig. 13- Sukolrat 2007). In horizontal bender system, two similar bender elements were mounted orthogonally; one pair is used for S-wave transmitters and the other pair is for Swave receivers. The bender elements are placed diagonally opposite one another and oriented so that the shear stiffness in both HV and HH directions are synchronized

**Figure 11.** Modified triaxial cell with bender element measurement system: (a) details of bender

(a) (b)

elements and (b) assembled modified triaxial cell (Sawangsuriya et al 2007a).

*e* 

(7)

**Figure 12.** S-wave traces from bender element tests. The arrow corresponds to the estimated S-wave arrival. The first deflection corresponds to the near-field effect (Sawangsuriya et al. 2007a).

Wave Propagation Methods for Determining Stiffness of Geomaterials 177

The current state of a sample relative to Go is defined by: (i) existing normal stresses in the ground which is also known as the mean effective principle stress or confining pressure (σo'), (ii) the overconsolidation ratio (OCR), and (iii) the void ratio (e) or the density of the geomaterial (ρ). By taking all parameters into account, a general expression as proposed by

(1 ) ' ( ) () ( ) *k nn G A OCR f e p <sup>o</sup> a o*

where A is a dimensionless material constant coefficient, k is a overconsolidation ratio exponent, f(e) is a void ratio function, pa is the reference stress or atmospheric pressure (~100 kPa) expressed in the same units as Go and σo', and n is a stress exponent. A number of studies have been conducted to estimate these parameters by relating with other physical

Parameter Dependency Typical value References

Determined by regression analysis for individual test

Vary from 0 to 0.5 (for PI<40, k=0; PI>40, k = 0.5)

density See Table 3 See Table 3

Approximately 0.5 at small strains

Go = A·f(e)·(σo')n

(8)

See Table 2

Hardin and Black (1968), Hardin and Drnevich (1972)

Hardin and Richart (1963), Hardin and Black (1966, 1968), Drnevich et al. (1967), Seed and Idriss (1970), Silver and Seed (1971), Hardin and Drnevich (1972), Kuribayashi et al. (1975), Kokusho (1980)

different investigators for Go of geomaterials is of the following form:

geomaterial properties as summarized in Table 2 and Table 3.

Grain characteristics or nature of grains, fabric or microstructure of geomaterial

k Plasticity index (PI)

f(e) Properties of packing and

<sup>N</sup>Contact between particles

and strain amplitude

**Table 2.** Parameters describing a current state of sample for Go

**Current state** 

A

**Figure 13.** Modified stress path cell to incorporate vertical and horizontal bender system and local strain measuring system: (a) Installation of measuring devices, (b) triaxial stress path cell during testing (Sukolrat, 2007).
