**Current state**

176 Wave Processes in Classical and New Solids

(Sukolrat, 2007).

relationship Go=ρ·Vs

behavior.

**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

**6. Small-strain shear modulus based on bender element measurement** 

According to elastic theory using the measured S-wave velocity (Vs) and total mass density of the specimen (ρ = γ/g), the small-strain shear modulus (Go) can be calculated with the

variety of geotechnical design applications and can be applied to all kinds of static (monotonic) and dynamic geotechnical problems at small strains (Richart et al. 1970, Jardine et al. 1986, Burland 1989). Note that the term "small-strain" is typically associated with the shear strain range below the elastic threshold strain (10=3-10-2%). Within the small strain range where the deformations or strains are purely elastic and fully recoverable, the shear modulus is independent of strain amplitude and reaches a nearly constant limiting value of the maximum shear modulus. In this strain region, most geomaterials exhibit linear-elastic

A number of factors affecting Go have been extensively investigated and reported. These include the current state of the sample (e.g. stress state, overconsolidation ratio, density, void ratio, and microstructure), anisotropy, degree of saturation, aging, cementation, and

temperature. Such factors can be briefly explained as the followings:

2. Go is an important and fundamental geomaterial property for a

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

$$\mathbf{G}\_o = \mathbf{A(OCR)}^k f(e) p\_a^{(1-n)}(\sigma\_o^{\cdot})^n \tag{8}$$

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 geomaterial properties as summarized in Table 2 and Table 3.


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

Go = A·f(e)·(σo')n


Wave Propagation Methods for Determining Stiffness of Geomaterials 179

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

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.

<sup>1</sup> ( ) () ' ' *<sup>j</sup> <sup>i</sup> k n <sup>n</sup> <sup>n</sup> G A OCR F e p <sup>o</sup> a ij*

where σi' is the effective normal stress in the direction of wave propagation, σj' is the

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

2 *f a*

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

 

 

*f a*

effective normal stress in the direction of particle motion, and n = ni + nj.

the apparent additional mass density and thus Go can be expressed as:

*o s*

*G v*

 

(9)

(10)

principal stresses).

**Degree of saturation** 

the end of the drying period.

1985):

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
