Complex Elastic Constants Research Articles

Apparent complex Young's modulus of a longitudinally vibrating viscoelastic rod

Longitudinal vibration of a viscoelastic rod with a finite lateral dimension is theoretically analyzed on the basis of the approximate Love theory. The frequency range where the Love theory gives good approximation and its accuracy in that range are determined. The theory predicts that the wave propagation in a viscoelastic rod is not governed solely by the complex Young's modulus of the material at higher frequencies, due to the lateral motion, but by its apparent value. It is shown that the apparent dynamic Young's modulus is smaller and the apparent loss factor is larger than the corresponding actual values for the material. The differences between the apparent and actual values depend on the lateral dimension to wavelength ratio and on the complex elastic constants as well.

Complex elastic, dielectric and piezoelectric constants by domain wall damping in ferroelectric ceramics

A model, describing losses in ferroelectric ceramics by the damped motion of 90° domain walls has been developed. The dependence of the complex dielectric, elastic and piezoelectric constants on the material properties P0 and S0 (spontaneous polarization and deformation), on the wall properties and on the directional distribution of the domains is derived. Measurements confirm this phenomenological, two dimensional theory, thus indicating the domain wall dampaing being the dominant loss mechanism in these ceramics.

Electromechanical Properties in the Composites of Epoxy Resin and PZT Ceramics

The complex dielectric, elastic and piezoelectric constants were measured for the composites of epoxy resin and ferroelectric PZT ceramics at 10 Hz as a function of temperature, and the magnitudes and loss tangents of these constants were compared with the theoretical predictions. The piezoelectric constants for a two phase system with piezoelectric spherical inclusions were expressed in terms of the properties of the constituents by use of the known expressions of the dielectric and elastic constants. The observed values of the d-constant (electric displacement/stress) were about 3/4 of the predicted. In the local mode dispersion region of epoxy resin near -30°C, the g-constant (electric field/stress) showed a larger temperature variation than the theoretical prediction.

Measurement of elastic constants and internal frictions on single-crystal MgO by rectangular parallelepiped resonance.

A measurement of complex elastic constants is made on two single crystal MgO specimens using a resonance method. The results on the real part of the elastic constants are consistent with those reported by CHANG and BARSCH (1969) and SPETZLER (1970) except for a small difference in C12. The elastic constants at a theoretical density of 3.584g/cm3 are C11= 2.978±0.004, C12=0.970±0.005, C44=1.563±0.002, Cs=(C11-C12)/2= 1.004±0.001, and K=(C11+2·C12)/3=1.639±0.006 in Mbar at 293K. The small deviations of the elastic constants in different specimens from the above valucs are well related to the deviation of the actual density from the theoretical one. The internal frictions relating to respective elastic constants are Q-111=0.23±0.61, Q-112=-0.35±1.79, Q-144=1.87±0.16, and Q-1s=0.51±0.14 in units of 10-4 at room temperature and 1MHz. The dominant dis-sipation is not caused by the shear on a plane of easy glide {110} , but the shear on {100}.

Piezoelectric Properties in Polarized Poly(vinyl Fluoride)

Poly(vinyl fluoride) films were elongated twice the original length by rolling, subjected to a d.c. field of 350 kV/cm at 80°C for 90 minutes, and cooled down to room temperature for 2 hours. The piezoelectric strain-constant d of the electret films was about 1×10-8 cgs esu at room temperature. The complex piezoelectric, elastic, and dielectric constants showed two dispersions at the low and high temperatures, respectively. The maximum value of piezoelectric loss factor tan δd is roughly the same as the sum of the maximum values of elastic and dielectric loss factor tan δE and tan δε. This fact may be explained by assuming that the piezoelectric constant h is a real quantity.

Stress waves in a linear viscoelastic cylinder

Steady-state solutions of stress waves travelling in a linear viscoelastic cylinder of infinite length are obtained by a method almost similar to Pochhammer-Chree's. Torsional, longitudinal, and flexural waves are considered. The results acquired are in the form in which complex elastic moduli and propagation constant appear in place of corresponding quantities in elastic solutions. Corrections of the Rayleigh type for phase velocity and attenuation constant are also derived.

On Correlating the Loss Factors of Cylindrical and Spherical Resonators

Theoretical results are presented which relate the Q factors (internal friction) of solid resonators to two loss parameters λi and μi. The latter are assumed to be characteristic of the energy loss in the homogeneous, isotropic material constituting the resonators. Derivations start from the appropriate frequency—or velocity dispersion—equation in which complex elastic constants, λr+jλi and μr+jμi, are substituted. The following cases are treated: torsional, fundamental longitudinal and fundamental bending modes of solid cylinders, and radial modes of solid and hollow spheres. Deviations from constancy of Q are found in regions where velocity dispersion occurs. Computed results are tabulated which show the dependence of Qμ, upon λ1/μ1 (range 0.1–10) and the propagation factor τ (range 0–5 for cylinders); these apply to duralumin. Similar computations are used to interpret experimental results on polystyrene reported by Biesterfeldt, Lange, and Skudrzyk (see reference 22 in text), and an inconsistency is found in the values of λi calculated from the fundamental longitudinal and the fundamental bending modes of rods.

Dispersion and Attenuation of Elastic Waves in a Cylindrical Bar

In the interests of determining the complex elastic constants of a given material (aluminum alloy) over a wide frequency range, the frequency and Q of a large multitude of overtones of a long (3-m), thin (12-in. diameter) bar were measured. These include the longitudinal mode and torsional mode. The frequency range covered was from about 0.5 kc to 100 kc. The elastic constants λ and μ are found to be relatively independent of frequency. The dispersion curve is in reasonable agreement with the Pochhammer-Chree equations for an infinite bar. The Q of the longitudinal mode decreases from about 260 000 at the lowest frequency to 140 000 at the highest.

Vibrations of a System with a Finite or an Infinite Number of Resonances

The study is based on the general differential equation of a mechanical system. Driving force and solution are expressed as a series of natural functions of the dissipationless system. Dissipation is taken into account by introducing complex elastic constants. Each Fourier coefficient of the driving force is written as the product of an excitation constant and the total driving force. The excitation constants can then be included in the mode parameters of the system and the solution, written as the product of the total driving force and the “mechanical impedance” of the system. The solution is exact and useful at the lower frequencies. A simple solution for higher frequencies can be derived by replacing the series by an integral and evaluating this integral. Three important parameters can thus be obtained: the driving-point impedance that describes the velocity of the driving point and the reaction of the system to the driving force, the effective impedance that represents the average velocity amplitude over the system, and the effective dissipation resistance with respect to this average-velocity amplitude. The expression derived for the high-frequency driving-point impedance then turns out to represent, also, the general background level at lower frequencies. The solution for the driving-point impedance in the intermediate frequency range is obtained by adding to this background the contribution of the natural mode that has its resonant frequency closest to the frequency of the force; the effective impedance, on the other hand, is given with good approximation as the impedance of the mode whose resonance frequency is closest to the frequency of the force; and the effective dissipation resistance is given by the resistive component of this mode impedance. The mode parameters are computed for rods, membranes, plates, and shells for three situations—a point force, a linear force distribution, and a force distribution given by a Fourier integral. The results make it possible to handle complicated mechanical systems with almost the same ease as simple mass points, and to compute their sound radiation.