# Dynamic Characterization of Cohesive Material Based on Wave Velocity Measurements
## 1 Introduction  
The determination of seismic velocities, as well as the elasticity modulus and structural properties of porous materials plays an extremely important role in the development of various engineering projects [1]. The small strain and elastic moduli, such as the shear modulus (\(G_{\mathrm{max}}\)) and Young's modulus (\(E_{\mathrm{max}}\)), are key parameters during, for example, the site response analysis of an earthquake, the design of machine foundations and soil dynamics problems [2]. [PERSON] _et al._[3] even emphasized the significance of \(G_{\mathrm{max}}\) in static deformation analysis of geotechnical problems. Usually, the above-mentioned elastic modulus of soil deposits is measured _in situ_ by means of so-called seismic exploration [4]. In order to evaluate the shear modulus (G) or Young's modulus (\(E\)) values in the laboratory, triaxial or torsional shear testing machines, which employ what is generally known as the static loading methods [5], are applied. Other kinds of evaluation methods, commonly referred to as vibration test methods, are those relying on applying wave motions to the test specimens and thenobserving their behavior during the resonance, including free oscillation time. A great example is a resonant column apparatus. Apart from this, there are also pulse transmission techniques, which include the ultrasonic pulse test, bender element (BE), _etc._ They allow calculating \(G_{\text{max}}\) at small strains on the basis of the wave velocity [6]. Some of the laboratory tests and field studies, such as cross-hole seismic [7; 8], down-hole seismic [9], suspension logging [10], seismic cone [11], seismic flat dilatometer [12] and spectral analysis of seismic waves [13], are indirect tests [14]. Compared to direct tests, the indirect tests enable measuring quantities different from the more desired ones and relate them to each other through mathematical relationships. Seismic techniques are classified as indirect testing methods for small strain stiffness [7]. They yield profiles of wave propagation velocity. When we assume that the behavior of the material is linear-elastic, this means that elastic stiffness relates to wave propagation velocity following the equations below:  
\[\upsilon_{P}=\sqrt{\frac{\lambda+2G}{\rho}} \tag{1}\]  
\[\upsilon_{S}=\sqrt{\frac{G}{\rho}} \tag{2}\]  
where \(\upsilon_{P}\) is the propagation velocity of pressure, which might as well be called a primary (P-) wave velocity, \(\upsilon_{S}\) is the propagation velocity of shear or the secondary (S-) wave and \(\lambda\) and \(G\) are [PERSON]'s constants (another term for \(G\) is the shear modulus). For engineers, it is more convenient to express [PERSON]'s constant as a function of [PERSON]'s modulus and [PERSON]'s ratio in accordance with these equations:  
\[G=\frac{E}{2(1+\
u)} \tag{3}\]  
\[\lambda=\frac{\
u E}{(1+\
u)(1-2\
u)} \tag{4}\]  
Figure 1 illustrates the particle motion in P- and S-waves. Over the years, bender elements have become increasingly popular in order to determine the S-wave velocity [9; 10] through experiments. Nowadays, they are commonly available [11; 12]. Bender elements consist of two sheets produced from piezoceramic plates, which are rigidly bonded, to a center brass shim or stainless steel plate. Because of their piezoelectric properties, they are capable of converting mechanical excitation into electrical output and _vice versa_. When excited by an input voltage, the bender element changes its shape, which is accompanied by mechanical excitation. Because of this fact, it acts as a signal transmitter. Being subjected to mechanical excitation causes the emission of an electrical output by another bender element, which is, in this way, acting as a signal receiver [9]. Figure 1: P-wave (top) and S-wave (bottom) particle motion. The particle motion in P-waves is longitudinal, while in S-waves, it is transverse. The particle motion vector in the plane perpendicular to the direction of propagation is referred to as polarization (only in the case of S-waves) [8].