Strain shear waves

Current information on dynamic soil properties indicates that the soil properties used. in> the soil structure interaction (SSI) analyses for the K-Reactor building were estimated from the SPT blow count data from boring K-5. These data, in turn, were correlated with low-strain, shear wave velocity data for sand, which was obtained from the open literature. A comparison of, the data w cross-hole measurements obtained from other areas, of SRS indicates that, on the average, the low strain data estimates are reasonable. As described in Section 8.4.1 of this SER, the latest SASSI calculations, which include soil layering effects, indicate that the response. spectra calculated using the uniform property data are conservative. 

The parameters for the model were originally evaluated for oil shale, a material for which substantial fracture stress and fragment size data depending on strain rate were available (see Fig. 8.11). In the case of a less well-characterized brittle material, the parameters may be inferred from the shear-wave velocity and a dynamic fracture or spall stress at a known strain rate. In particular, is approximately one-third the shear-wave velocity, m has been shown to be about 6 for various brittle materials (Grady and Lipkin, 1980), and k can then be determined from a known dynamic fracture stress using an analytic solution of (8.65), (8.66) and (8.68) in one dimension for constant strain rate. 

Example 2.8 Calculate the kinetic and strain energy densities for the shear wave examined in the Example 2.7. 

Solution Since the shear wave is z-polarized and jr-propagating, the only nonzero derivative in Equation 2.39 is du dx = —jkuj. Thus, the strain energy density is given... 

This indicates a linear dependence of the shear wave on the distance X2 and consequently a time dependence of the strain in phase with the displacement and the stress. In other words, a thin slab is nearly consistent with a linear response. 

The most popular dynamic test procedure for viscoelastic behavior is the application of an oscillatory stress of small amplitude. This shear stress applied produces a corresponding strain in the material. If the material were an ideal Hookean body, the shear stress and shear strain rate waves would be in phase (Fig. 14A), whereas for an ideal Newtonian sample, there would be a phase shift of 90° (Fig. 14B), because for Newtonian bodies the shear strain is at a maximum, when a maximum of stress is present. The shear strain, when assuming an oscillating sine fimction, is at a maximum in the middle of the slope, because there is the steepest increase in shear strain due to the change in direction. For a typical viscoelastic material, the phase shift will have a value between >0° and <90° (Fig. 14C). 

Elastic response This occurs when the maximum of the stress amplitude is at the same position as the maximum of the strain amplitude (no energy dissipation). In this case, there is no time shift between the stress and strain sine waves. Viscous response This occurs when the maximum of the stress is at the point of maximum shear rate (i.e., the inflection point), where there is maximum energy dissipation. In this case, the strain and stress sine waves are shifted by (referred to as the phase angle shift, 5, which in this case is 90°). 

Plastic deformation in crystalline solids occurs mostly as a result of dislocation motion and multiplication. Dislocations move on specific planes in certain directions when the applied stress exceeds the critical value of shear stress. In general, dislocations move with velocities that increase as the rate of deformation increases. However, due to the relativistic effect, dislocations are limited to move at velocities less than the shear wave velocity. Under high strain rate loading (10Vs-10 /s), dislocations accommodate this velocity restriction by high... 

In materials with high dislocation mobility such as copper, dislocation patterns proceed through the rapid motion of dislocations in a very small volume of the specimen [36]. Under high strain rate deformation conditions, it is expected that the dislocations move at subsonic speed or even as fast as the shear wave velocity. The random motion of dislocations on their slip planes causes random changes not only in the local dislocation densities, but also in the dislocation velocities. 

Figure 6a shows the transmission hne representing a viscoelastic layer [64]. Every layer is represented by a T . The apphcation of the Kirchhoff laws to the Ts reproduces the wave equation and the continuity of stress and strain. The detailed proof is provided in [4]. To the left and to the right of the circuit are open interfaces (ports). These can be exposed to external shear waves. They can also be connected to the ports of neighboring layers (Fig. 6b). Alternatively, they may just be short-circuited, in case there is no stress acting on this surface (left-hand side in Fig. 6c). Finally, if the stress-speed ratio Zl (the load impedance, see below) of the sample is known, the port can be short-circuited across an element of the form AZl, where A is the active area (right-hand side in Fig. 6c). Figure 6c shows a viscoelastic layer which is also piezoelectric. This equivalent circuit was first derived by Mason [4,55]. We term it the Mason circuit. The capacitance, Co, is the electric capacitance between the electrodes. The port to the right-hand side of the transformer is the electrical port. The series resonance frequency is given by the condition that the impedance of the acoustic part (the stress-speed ratio, aju) be zero, where the acoustic part comprises all elements connected to the left-hand side of the transformer.

The four commonly used techniques to extract information on the viscoelastic behavior of suspensions are creep-compliance measurements, stress-relaxation measurement, shear-wave velocity measurements, and sinusoidal oscillatory testing (25-27). In general, transient measurements are aimed at two types of measurements, namely, stress relaxation, which is to measure the time dependence of the shear stress for a constant small strain, and creep measurement, which is to measure the time dependence of the strain for a constant stress. 

In recent years, experimental investigation of the depolarized Rayleigh scattering of several liquids composed of optically anisotropic molecules has confirmed the existence of a doublet-symmetric about zero frequency change and with a splitting of approximately 0.5 GHz (see Fig. 12.1.1). The existence of this doublet had been predicted on the basis of a hydrodynamic theory several years previously by Leontovich (1941). This theory assumes that local strains set up by a transverse shear wave are relieved by collective reorientation of individual molecules. Later, Rytov (1957) formulated a more general hydrodynamic theory for viscoelastic fluids that reduces to the Leontovich theory in the appropriate limit. The theories of Rytov and Leontovitch are different from the present two-variable theory, in that the primary variable is the stress tensor and not the polarizability. 

Shear Wave Propagation. A pulse shearometer (Rank Bros.) was used to measure the propagation velocity of a shear wave through the weak gels formed by the solutions of HMHEC in dilute NaCl. The polymer concentration range studied was 0.5-2.0%. With this apparatus, the frequency of the shear wave is approximately 1200 rad s" and the strain is <10 . At this strain, n pst systems behave in a linear viscoelastic fashion, and the wave-rigidity modulus, G is... 

Due to their high sensitivity to strain, temperature variation, vibration, and acoustic waves, embedded extrinsic Fabry-Perot interferometric optical fiber sensors have been developed to detect delamination, based on changes in the acoustic properties of the materials before and after delamination [41]. Impact events and corrosion cracking generate ultrasonic waves, which can be characterized using elliptical core fiber sensors [40]. The in-line Fabry-Perot interferometer seems well suited for the local detection of shear waves and the characterization of impact-induced damage. 

Field tests which are most commonly employed in geotechnical earthquake engineering can be grouped into those that measure low-strain properties and those that measure properties at intermediate to high strains. Low-strain field tests t5q)ically induce seismic waves in the soil and seek to measure the velocities at which these waves propagate. Due to low strain amplitudes the measured shear wave velocity (V ) along with soil density (p) is used to compute low-strain shear modulus. 

Geophysical crosshole tests (CHT) may be conducted in parallel cased boreholes to evaluate the profiles of compression wave (Vp) and shear wave (Vs) velocities (Wightman et al. 2003). The shear wave data allow the direct assessment of the small-strain shear modulus (Go = Pt V where pt = total mass density). The fundamental stiffness Go serves as the initial stiffness of soils, thus the beginning of all shear stress vs. shear strain curves, applicable to both monotonic and dynamic problems (Atkinson, 2000 Clayton 2011). In fact, this well-known fact is also missing from many textbooks, even though Go has been shown relevant to practical foundation problems for over 2 decades (e.g., Burland 1989). 

Mechanical spectroscopy is an ideal technique for investigation of the viscoelastic properties of materials. It involves the application of a sinusoidal oscillation of strain (7) and frequency (oj) to the material. A perfectly elastic material will have a stress wave exactly in phase with the applied strain wave. A purely viscous material will have a stress wave exactly 90° out of phase with the applied strain since at the maximum deflection the rate of strain (shear rate) is zero. This is illustrated in Figure 2.2 [5]. 

Shear Wave Velocity Greater than or equal to 304.8m/sec (l,000ft/sec) based on low-strain best-estimate soil properties over the footprint of the nuclear island at its excavation depth... 

FIG. 5-14. Stroboscope photograph of a wave of shear strain double refraction in a 1% solution of sodium deoxyribonucleate at 2S°C, frequency 125 Hz. The driving plate is oscillated vertically shear waves propagated horizontally to the right produce patterns of strain double refraction. Each boundary between black and white provides the same information the inclination of the base lines is specified by the angle between the axes of the Babinet compensator and the analyzing Polaroid (from reference 117). ... 

It is common practice to measure elastic constants and their temperature dependence experimentally, either via static (e.g. three- or four-point bending, XRD or neutron scattering in connection with strain gauges) or via dynamic tests (e.g. sound velocity methods or resonant frequency techniques). It is well known from solid state physics, cf e.g. [Ashcroft Mermin 1976, Kittel 1988], that the elastic constants determine the velocity of sound waves in solids. For example, the velocity of transversal waves (shear waves) Vj, is given by... 

When adding a seismic sensor (usually a geophone) to the CPT cone, it is possible to measure the shear wave velocity in the soil. This shear wave is generate at the surface. The shear wave velocity can be correlated to the low strain stiffness of a soil. It is a useful tool to evaluate the liquefaction potential of the soil. For further information on this technique reference is made to the ISS-MGE publication seismic cone downhole procedure to measure shear wave velocity . 

---