**Anisotropy**

Anisotropic Go of geomaterials is generally described in terms of stress-induced anisotropy and inherent anisotropy (Stokoe et al. 1985).The stress-induced anisotropy results from the anisotropy of the current stress condition and is independent of the stress and strain history of the geomaterial. The inherent or fabric anisotropy results from structure or fabric of the geomaterial that reflects the deposition or forming process (e.g. aging, cementation). Both stress-induced anisotropy and fabric anisotropy of Go depend on the direction of loading (Mitchell and Soga 2005). Such fabric anisotropy of Go can be evaluated based on S-wave measurements using bender elements (Sawangsuriya et al. 2007b). For example, Go is a function of the principal effective stresses in the directions of wave propagation and particle motion and is independent of the out-of-plane principal stress (Stokoe et al. 1995). The inherent anisotropy can be evaluated by measuring body wave velocities propagating through the specimen subjected to isotropic states of stress (i.e., mean effective stress). For the stress-induced anisotropy, the measurements are taken from specimen subjected to anisotropy states of stress (i.e., changes in vertical stress while maintaining average principal stresses).

Under anisotropic states of stress, the representative stiffness values can be different, depending on the measurement conditions and the sample preparation procedures. The anisotropy of the stress state induces anisotropy of small-strain stiffness. An empirical equation for Go under anisotropic stress condition is expressed as (Roesler 1979, Stokoe et al. 1985):

$$\mathbf{G}\_o = \mathbf{A} \text{(OCR)}^k F(e) p\_a^{1-n} \sigma\_i^{\prime n\_i} \sigma\_j^{\prime n\_j} \tag{9}$$

where σi' is the effective normal stress in the direction of wave propagation, σj' is the effective normal stress in the direction of particle motion, and n = ni + nj.

#### **Degree of saturation**

178 Wave Processes in Classical and New Solids

Round-grained Ottawa sand Angular-grained crushed quartz

Reconstituted NC Kaolinite (PI = 20) and undisturbed NC clays

Reconstituted NC Kaolinite (PI = 35) Reconstituted NC Bentonite (PI = 60)

Undisturbed NC clay

**Anisotropy** 

Type of geomaterials A f(e) n References

(2.17-e)2/(1+e)

0.5

3,270 (2.97-e)2/(1+e) 0.5 Hardin and Black (1968)

0.5

0.5

0.5 Hardin and Black (1968)

(1975)

(1977)

Marcuson and Wahls (1972)

(1996)†

(2.97-e)2/(1+e)

Clean sand 41,600 0.67-e/(1+e) 0.5 Shibata and Soelarno

Clean sand 9,000 (2.17-e)2/(1+e) 0.4 Iwasaki et al. (1978) Toyoura sand 8,400 (2.17-e)2/(1+e) 0.5 Kokusho (1980) Clean sand 7,000 (2.17-e)2/(1+e) 0.5 Yu and Richart (1984) Ticino sand 7,100 (2.27-e)2/(1+e) 0.4 Lo Presti et al. (1993) Clean sand 9,300 1/e1.3 0.45 Lo Presti et al. (1997)

(2.97-e)2/(1+e)

(4.4-e)2/(1+e)

(PI = 40~85) 90 (7.32-e)2/(1+e) 0.6 Kokusho et al. (1982)

Remolded clay (PI = 0~50) 2,000~4,000 (2.97-e)2/(1+e) 0.5 Zen et al. (1978)

Clay deposits (PI = 20~150) 5,000 1/e1.5 0.5 Shibuya and Tanaka

Remolded clay (PI = 20~60) 24,000 1/(1+e)2.4 0.5 Shibuya et al. (1997)† Sand and clay 6,250 1/(0.3+0.7e2) 0.5 Hardin (1978) Several soils 5,700 1/e 0.5 Biarez and Hicher (1994)

Anisotropic Go of geomaterials is generally described in terms of stress-induced anisotropy and inherent anisotropy (Stokoe et al. 1985).The stress-induced anisotropy results from the anisotropy of the current stress condition and is independent of the stress and strain history of the geomaterial. The inherent or fabric anisotropy results from structure or fabric of the geomaterial that reflects the deposition or forming process (e.g. aging, cementation). Both stress-induced anisotropy and fabric anisotropy of Go depend on the direction of loading (Mitchell and Soga 2005). Such fabric anisotropy of Go can be evaluated based on S-wave measurements using bender elements (Sawangsuriya et al. 2007b). For example, Go is a

Note: Go and σo' are in kPa, † using effective vertical stress (σv') instead of σo' , Cu = Coefficient of Uniformity

**Table 3.** Function and constants in proposed empirical equations on Go

Clean sand (Cu < 1.8) 14,100 (2.17-e)2/(1+e) 0.4 Iwasaki and Tatsuoka

6,900

3,270

4,500

445

Early studies on the influence of the degree of saturation on Go described a coupled motion of the solid particles and the fluid (Biot 1956, Hardin and Richart 1963, Richart et al. 1970). According to Biot's theory, no structural coupling exists between the solid particle and the fluid (the fluid has no shearing stiffness), the coupling in the shearing mode is only developed by the relative motions of the solid and fluid as indicated by the term involving the apparent additional mass density and thus Go can be expressed as:

$$\mathbf{G}\_o = \upsilon\_s^2 \left(\rho + \frac{\rho\_f \rho\_a}{\rho\_f + \rho\_a}\right) \tag{10}$$

where ρ is the mass density of the solid particles, ρf is the mass density of fluid, and ρa is the mass density of an additional apparent mass. In a real geomaterial, ρa varies with the grain size and permeability; however, the total mass density of the saturated geomaterial could be substituted into the mass density term of Eq. (10) to take into account the coupling effect of the mass of the fluid. The shear wave velocity of saturated geomaterial is therefore less than that of dry geomaterial because the added apparent mass of water moving along with the geomaterial skeleton (i.e., the drag of the water in the pores). Recent studies by Santamarina et al. (2001) and Inci et al. (2003) indicated that the response of Go by varying the degree of saturation demonstrates three phases of behavior and is attributed to contact-level capillary forces or suction. A sharp increase in Go is observed at the beginning of the drying process, followed by a period of gradual increase in measured Go, and a final sharp increase in Go at the end of the drying period.

#### **Aging**

A time-dependent nature of Go of geomaterials has been reported by several investigators (Afifi and Woods 1971, Marcuson and Wahls 1972, Afifi and Richart 1973, Stokoe and Richart 1973, Trudeau et al. 1974, Anderson and Woods 1975, Anderson and Woods 1976, Anderson and Stokoe 1978, Isenhower and Stokoe 1981, Athanasopoulos 1981, Kokusho 1987). Results of these investigations indicate that Go tends to increase with the duration of time under a constant confining pressure after the primary consolidation is complete due to a time effect results from strengthening of particle bonding. The time dependency of Go increase can be characterized by two phases: (i) an initial phase due to primary consolidation and (ii) a second phase in which Go increases about linearly with the logarithm of time and occurs after completion of primary consolidation, also referred as the long-term time effect (Fig. 14). The second phase of secondary consolidation occurs after primary phase when Go increases continuously with time. The rate of secondary increase in Go is related to thixotropic changes in the clay structure and is determined to be linear when plotted versus the logarithm of time. To incorporate this long-term time effect, the change in Go with time can be expressed by:

$$
\Delta \mathbf{G} = \mathbf{N}\_G \mathbf{G}\_{1000} \tag{11}
$$

Wave Propagation Methods for Determining Stiffness of Geomaterials 181

stiffness behavior of cemented materials is controlled by the cementation and the materials become brittle, whereas at high confinement the behavior is controlled by the state of stress

and resembles an uncemented material, which becomes more ductile.

Go

Go

**Figure 14.** Phases of Go versus confinement time (Anderson and Stokoe 1978).

**Figure 15.** Effect of aging on Go (Anderson and Stokoe 1978).

and

$$N\_G = \left(\frac{1}{G\_{1000}}\right) \left(\frac{\Lambda G}{\log\_{10}(t\_2/t\_1)}\right) \tag{12}$$

where ΔG is the increase in Go over one logarithm cycle of time, G1000 is the value of Go measured after 1,000 minutes of application of constant confining pressure following the primary consolidation, and NG is the aging increment coefficient, which indicates an increase of Go within one logarithmic cycle of time.

The duration of primary consolidation and the magnitude of the secondary increase, as defined by change in Go per logarithmic cycle of time, vary with geomaterial types and stress conditions (i.e., confining pressure). For sands, the rate of increase in Go is relatively small (1 to 3% per log cycle of time) but for clays the effect is quite remarkable as illustrated in Fig. 15.

#### **Cementation**

Cementation occurs either naturally due to the precipitation or formation of salts, calcite, alumina, iron oxides, silicates, and aluminates or artificial stabilization processes produced by adding lime, cement, asphalt, fly ash, or other bonding agents to geomaterials. The effect of cementation on Go have been evaluated by Clough et al. (1981), Acar and El-Tahir (1986), Saxena et al. (1988), Lade and Overton (1989), Baig et al. (1997), Fernandez and Santamarina (2001), Yun and Santamarina (2005). Go of cemented materials increases with increasing cement content and confining pressure (Fig. 16). Additionally at low confinement, the stiffness behavior of cemented materials is controlled by the cementation and the materials become brittle, whereas at high confinement the behavior is controlled by the state of stress and resembles an uncemented material, which becomes more ductile.

180 Wave Processes in Classical and New Solids

Go with time can be expressed by:

increase of Go within one logarithmic cycle of time.

A time-dependent nature of Go of geomaterials has been reported by several investigators (Afifi and Woods 1971, Marcuson and Wahls 1972, Afifi and Richart 1973, Stokoe and Richart 1973, Trudeau et al. 1974, Anderson and Woods 1975, Anderson and Woods 1976, Anderson and Stokoe 1978, Isenhower and Stokoe 1981, Athanasopoulos 1981, Kokusho 1987). Results of these investigations indicate that Go tends to increase with the duration of time under a constant confining pressure after the primary consolidation is complete due to a time effect results from strengthening of particle bonding. The time dependency of Go increase can be characterized by two phases: (i) an initial phase due to primary consolidation and (ii) a second phase in which Go increases about linearly with the logarithm of time and occurs after completion of primary consolidation, also referred as the long-term time effect (Fig. 14). The second phase of secondary consolidation occurs after primary phase when Go increases continuously with time. The rate of secondary increase in Go is related to thixotropic changes in the clay structure and is determined to be linear when plotted versus the logarithm of time. To incorporate this long-term time effect, the change in

1000 10 2 1

*G tt* 

log ( / ) *<sup>G</sup> <sup>G</sup> <sup>N</sup>*

where ΔG is the increase in Go over one logarithm cycle of time, G1000 is the value of Go measured after 1,000 minutes of application of constant confining pressure following the primary consolidation, and NG is the aging increment coefficient, which indicates an

The duration of primary consolidation and the magnitude of the secondary increase, as defined by change in Go per logarithmic cycle of time, vary with geomaterial types and stress conditions (i.e., confining pressure). For sands, the rate of increase in Go is relatively small (1 to 3% per log cycle of time) but for clays the effect is quite remarkable as illustrated

Cementation occurs either naturally due to the precipitation or formation of salts, calcite, alumina, iron oxides, silicates, and aluminates or artificial stabilization processes produced by adding lime, cement, asphalt, fly ash, or other bonding agents to geomaterials. The effect of cementation on Go have been evaluated by Clough et al. (1981), Acar and El-Tahir (1986), Saxena et al. (1988), Lade and Overton (1989), Baig et al. (1997), Fernandez and Santamarina (2001), Yun and Santamarina (2005). Go of cemented materials increases with increasing cement content and confining pressure (Fig. 16). Additionally at low confinement, the

1

*G NGG* <sup>1000</sup> (11)

(12)

**Aging** 

and

in Fig. 15.

**Cementation** 

**Figure 14.** Phases of Go versus confinement time (Anderson and Stokoe 1978).

**Figure 15.** Effect of aging on Go (Anderson and Stokoe 1978).

#### **Temperature**

Effect of temperature on time-dependent changes in Go was reported in Bosscher and Nelson (1987), Fam et al. (1998). The dependency of Go on temperature suggests that higher temperatures cause the stiffness increase with time. Fam et al. (1998) presented the evolution in velocity with time for coarse-grained granular salt specimen under a constant effective stress and subjected to a temperature step (heating-cooling cycle) as illustrated in Fig. 17. The rate of increase in velocity with time increases at higher temperatures (Fam et al. 1998). Bosscher and Nelson (1987) studied Go of frozen Ottawa 20-30 sand as a function of the confining pressure, the degree of ice saturation, the relative density, and the temperature. They found that Go of frozen sand is higher than that of non-frozen state. At temperatures near the melting point of ice, Go can be significantly influenced by the confining pressure, the degree of ice saturation, and the relative density (Bosscher and Nelson 1987).

Wave Propagation Methods for Determining Stiffness of Geomaterials 183

**7. Application of wave propagation methods in geoengineering quality** 

The quality of the engineered earth fills depends on the suitability and compaction of the materials used. Earthwork compaction acceptance criteria are typically based on adequate dry density of the placed earthen materials achieved through proper moisture content and compaction energy. For instance, compaction specifications often require achievement of an in-situ dry density of 90-95% of the maximum value obtained from laboratory standard or modified Proctor test. According to this approach, by achieving a certain dry density using an acceptable level of compaction energy assures attainment of an optimum available level of structural properties and also minimises the available pore space and thus future

The question of the achieved structural property, which is the ultimate objective quality control, however remains unfulfilled. Dry density is only a quality index used to judge compaction acceptability and is not the design parameter or relevant property for engineering purposes. For compacted highway, railroad, airfield, parking lot, and mat foundation subgrades and support fills, the ultimate engineering parameter of interest is often the shear modulus of geomaterials, which is a direct structural property for determining load support capacity and deformation characteristic in engineering design.

Shear modulus of compacted geomaterials depends on density and moisture but also on fabric and texture of geomaterials, which varies along the roadway route. The conventional approach of moisture-density control, however, does not reflect the variability of the texture and microstructure of geomaterials and thus their shear modulus. Even if the structural layers satisfy a compaction quality control requirement based on density testing, a large variability in shear modulus can still be observed (Sargand et al. 2000; Nazarian and Yuan, 2000). Additionally, the comparison between density and modulus tests suggests that conventional density testing cannot be used to define subtle changes in the stiffness of the compacted earth fills (Fiedler et al. 1998). Shear modulus is a more sensitive measure of the texture and fabric uniformity than density. The stiffness non-uniformity is directly related to progressive failures and life-cycle cost, direct stiffness testing which can be conducted independently and in conjunction with conventional moisture-density testing is anticipated to reduce variability and substantially enhance quality control of the earthwork

Ismail and Rammah (2006) proposed a test setup and procedure by which the small-strain shear modulus can be measured accurately by propagating elastic shear wave through the compacted geomaterial during laboratory compaction test. They designed a test setup in such a way that it can be readily incorporated into the conventional compaction mould as shown in Fig. 18. In addition, their procedures can be adopted in compaction works e.g. road construction, embankments, and earth structures. Consequently, incorporating the shear modulus into laboratory compaction tests, will guarantee fulfilment of the design

criteria in term of dry density-moisture-modulus relationship.

**control process** 

moisture changes.

construction.

**Figure 16.** Effect of cementation on Go (Acar and El-Tahir 1986).

**Figure 17.** Effect of temperature on time-dependent changes in velocity for a coarse-grained granular salt specimen under a constant vertical load (Fam et al. 1998).
