Dynamic Modulus and Damping

From a dynamic test, for example, the vibrating reed test, or the DMA test, a complex modulus E is defined as 

E is a measure of the phase angle lag in the strain cycle. If the energy loss is calculated for one cycle of stress and strain (i.e., for one hysteresis loop) for t varying from 0 to 2ji, and for the stress and strain given by Equations 2.34 and 2.35, respectively, it can be shown that the energy loss per unit volume dissipated per cycle in the form of heat energy is 

Similarly, a complex creep compliance can be found, namely that 

It can also be shown that the following relationships exist  

A damping measure that is commonly used is the logarithmic decrement A. It is defined as 

---

Dynamic Mechanical and Thermomechanical Analysis. A DuPont Model 981 DMA was used to determine the dynamic modulus and damping characteristics of baseline and irradiated specimens. Transverse composite samples 1.27 cm x 2.5 cm were used so that the modulus and damping data were primarily sensitive to matrix effects. Data were generally determined from -120°C through the glass transition temperature (Tg) of each material using a heating rate of 5°C/min. 

The dynamic modulus and damping of a Maxwell unit as a function of frequency are schematically shown in Figure 3.23. The nature of the curves in Figure 3.23 can be explained qualitatively as follows. 

A technique for performing dynamic mechanical measurements in which the sample is oscillated mechanically at a fixed frequency. Storage modulus and damping are calculated from the applied strain and the resultant stress and shift in phase angle. 

Dynamic Mechanical Tests. Plasticizer efficiency, can be measured, not only be the lowering of T , but also by temperature dependence of typical dynamic mechanical properties, such as modulus and damping. 

Figure 11. Schematic diagram of a torsion pendulum for measuring dynamic shear modulus and damping. A typical damped oscillation curve is illustrated at the bottom of the figure (25)...

For T measurements, calorimetry (DSC-990 Du-Pont), linear and volume cubic thermal expansion (TMS-2 Perkin-Elmer), thermo-stimulated discharge (TSD) technique, dynamic mechanical Young modulus and damping (DMA-981 DuPont) and dielectric (102-105 Hz) measurements have been used as experimental techniques. All methods gave very well correlated values of T 19). 

From an experimental point of view, knowledge of the modes is important because it makes it possible to optimize the location of the detector and supports in the experimental devices in order to minimize effects due to spurious stiffness, inertia, or damping (5, p. 90) (see also Chap. 7). By making = —A. -f m, as corresponds to a free vibration experiment, expressions for the real and imaginary parts of the dynamic modulus and for the loss tangent at the resonance frequency = can be obtained. The resulting equations are... 

Physical Property Evaluation. A Rheovibron (direct reading dynamic viscoelastometer, model RHEO-200, manufactured by Toyo Measuring Instruments Co.) was used to measure the temperature dependence of the complex modulus and damping, tan 6, at 11 Hz. 

Half way between conventional thermomechanical analysis and dynamic mechanical analysis is the technique of dynamic force (or load) TMA. This method uses a standard TMA instrument but the force is changed between two values in a stepwise (or sometimes sinusoidal) fashion. The dimensional changes of the specimen are monitored as a function of time (and temperature) but no attempt is made to determine the modulus and damping properties of the material. 

A modified DuPont 980 dynamic mechanical analyzer, which measures modulus and damping, was used to study composite systems and to evaluate glass transition temperatures. 

Dynamic thermomechanometry (or dynamic mechanical analysis, DMA) A technique in which the dynamic modulus and/or damping of a substance under oscillatory load is measured as a function of temperature whilst the substance is subjected to a controlled temperature program. 

Dynamic mechanical analysis (DMA) provides putative information on the viscoelastic properties - modulus and damping - of materials. Viscoelasticity is the characteristic behaviour of most materials in which a combination of elastic properties (stress proportional to strain rate) are observed. A DMA simultaneously measures both elastic properties (modulus) and viscous properties (damping) of a material. 

Figure 6.5 Dynamic mechanical behavior of polyurethane nanocomposite in the presence of various types of nanoparticles, (a and b) Storage modulus and damping factor in the presence of alumina [84], (c and d) storage modulus and damping factor in the presence of CNTs [86],...

Dynamic mechanical analysis measures changes in mechanical behavior, such as modulus and damping as a function of temperature, time, frequency, stress, or combinations of these parameters. The technique also measures the modulus (stiffness) and damping (energy dissipation) properties of materials as they are deformed under periodic stress. Such measurements provide quantitative and qualitative information about the performance of materials. The technique can be used to evaluate reinforced and unreinforced polymers, elastomers, viscous thermoset liquids, composite coating and adhesives, and materials that exhibit time, frequency, and temperature effects or mechanical properties because of their viscoelastic behavior. 

A number of tests are possible to evaluate the behavior of soils under dynamic loads such as wave or earthquake loads. Dynamic tests generally are strength tests with the sample subjected to some sort of cyclic loading. Tests can be performed to evaluate variations of strength, modulus, and damping, with variations in rate and magnitude of cyclic stresses or strains. Small strain parameters for earthquake loading cases can be evaluated from resonant column tests. 

Dynamic, isothermal, and stepwise temperature control makes it possible to design flexible temperature programs. Digital filtering via Fourier analysis enhances the signal-to-noise ratio, which helps to resolve small tan 8 values. Comprehensive, multidimensional calibration of the DMA system leads to reproducible test results for the modulus and damping (viscoelastic) behavior of a sample. 

Figure 6.20 (a) Dynamic resonance method (DRM) head (a parallelepiped beam is observed at the centre of the head). (Courtesy P. Gadaud, ENSMA Poitiers.) (b) Young s modulus and damping coefficient Q" =2 rtan S as... 

One of the advantages of these dynamic soil-structure interaction analyses is that the soil layers are modeled to reflect the idealized site stratigraphy each soil layer can be either modeled as a linear elastic material with strain-compatible shear modulus and damping values or characterized via soil constitutive models that represent soil nonlinearity and hysteretic response at small strains. However, the use of the nonlinear constitutive models requires careful selection of input parameters and thus more advanced testing to define those input parameters. The nonlinear behavior and the frequency content of the free-field environment contribute to the stmctural racking behavior. 

Dynamic mechanical analysis revealed single peaks in both loss modulus and damping factor curves versus temperature, in the majority of samples. PEG-rich semi-IPN specimens showed broad transitions near the PEG melting temperature. The PAA rich specimen showed two peaks, one near the glass transition for PAA and the other near that of the equimolar specimen. Similar peaks were seen with full IPN, except that PAA rich samples showed one broad peak instead of two peaks. Non-stoichiometric samples, therefore, consisted of a 50/50 complex phase with excess amorphous PEG or PAA constituting a separate phase mixed with the complex. This microphase separation appeared less severe in full IPN. 

The principal effects of carbon black on the dynamic properties of elastomers were established as early as 1942. These can be summarized as an increase in both the elastic modulus and damping, compared with the unfilled material, and a pronounced strain-amplitude dependency, which is not found to a significant level in unfilled rubbers. It is true that any filler normally used in elastomers at least increases the elastic modulus, but the reason for the importance of carbon black is that no filler does it better. Not only does carbon black increase the modulus, it also improves the general strength and fatigue properties of elastomeric materials to a level which changes them from a curious hyperelastic novelty into a useful engineering material. 

In dynamic thermomechanometry the dynamic modulus and/or damping of a substance under an oscillatory load is measured as a function of temperature while the substance is subjected to a controlled temperature. The frequency response is then studied at various temperatures. Torsional braid analysis is a particular case of dynamic thermomechanometry in which the substance is supported on a braid. These are all sophisticated versions of thermomechanical methods. The word dynamic here, as noted above, means oscillatory and this term can be used as an alternative to modulation. In DMA the sample is oscillated at its resonant frequency, and an amount of energy equal to that lost by the sample is added in each cycle to keep the sample oscillation at a constant amplitude. 

---