Loss modulus defined

This shows that (co) is composed of two frequency-dependent components (co) is the real part in phase with the strain called the storage modulus, and E"(Csi) is the loss modulus defined as the ratio of the component 90° out of phase with the stress to the stress itself. Hence, E (cxi) measures the amoimt of stored energy and "(co), sometimes called the imaginary part, is actually a real quantity measuring the amoimt of energy dissipated by the material. 

Experimentally DMTA is carried out on a small specimen of polymer held in a temperature-controlled chamber. The specimen is subjected to a sinusoidal mechanical loading (stress), which induces a corresponding extension (strain) in the material. The technique of DMTA essentially uses these measurements to evaluate a property known as the complex dynamic modulus, , which is resolved into two component parts, the storage modulus, E and the loss modulus, E . Mathematically these moduli are out of phase by an angle 5, the ratio of these moduli being defined as tan 5, Le. 

However, it is interesting to perform a more direct comparison of the experimental results to check whether some differences between the mechanical and dielectric behaviours could exist as a function of temperature. The appropriate quantity is E" for the mechanics and, for the dielectric response, it is the dielectric loss modulus, m" (defined as e"/ sa + s"2)). Figure 112 shows the temperature dependence of E" and m" at 1 Hz, obtained by superposing the low-temperature part of the j3 transition. 

The storage modulus is proportional to the amount of energy which is stored in the material elastically, whereas the loss modulus corresponds to the energy that is dissipated during one load cycle. Both quantities are combined in the damping factor tan 8 which is defined as... 

The real part is called the storage modulus and the imaginary part, which defines the energy dissipation, is called the loss modulus. A comparison between the... 

It is necessary to state more precisely and to clarify the use of the term nonlinear dynamical behavior of filled rubbers. This property should not be confused with the fact that rubbers are highly non-linear elastic materials under static conditions as seen in the typical stress-strain curves. The use of linear viscoelastic parameters, G and G", to describe the behavior of dynamic amplitude dependent rubbers maybe considered paradoxical in itself, because storage and loss modulus are defined only in terms of linear behavior. 

The data obtained for the PMMA sample were reduced and are displayed in the nomograph form of Figure 3. The form of the WLF equation used is the "universal" WLF equation with T replaced by Tg and defined as the temperature at the peak value of loss modulus for the 0.01 Hz curve. The constants and C2 were assigned the values of 17.4 and 51.6 respectively. 

Equation (7.144) defines G uj) as the storage modulus and G uj) as the loss modulus. Equation (7.144) can be related to the previous equation for the stress in oscillatory shear using the trigonometric identity for the sine of a sum ... 

Through use of classical network theories of macromolecules, G has been shown to be proportional to crosslink density by G = nKT -i- Gen, where n is the nnmber density of crosslinkers, K is the Boltzmann s constant, T is the absolnte temperature, and Gen is the contribution to the modulus because of polymer chain entanglement (Knoll and Prud Homme, 1987). The loss modulus (G") gives information abont the viscous properties of the fluid. The stress response for a viscous Newtonian fluid would be 90 degrees out-of-phase with the displacement but in-phase with the shear rate. So, for an elastic material, all the information is in the storage modulus, G, and for a viscous material, aU the information is in the loss modulus, G". Refer to Eigure 6.2, the dynamic viscosities p and iT are defined as... 

In dynamic mechanical analysis for thermosetting systems the principal use is to monitor the glass-transition temperature (Tg). In a DMTA Tg is defined as the maximum in loss modulus, loss compliance or loss tangent. For example. Figure 3.57 (Prime, 1997a) shows the various transitions for a cured magnetic ink coating that had previously been cured under air and N2. 

These differences are shown in the following examples where measurements of the dynamic moduli, G and G" are used to monitor the structure of gel networks. Measurements are performed by imposing an oscillatory shear field on the material and measuring the oscillatory stress response. The stress is decomposed into a component in phase with the displacement (which defines the storage modulus G ) and a component 90 out of phase (which defines the loss modulus G"). The value of G indicates the elastic and network structure in the system (15, 17, 18) and can be interpreted by using polymer kinetic theories. 

These are most easily represented by the equation E = E + iE". where E is the ratio between (the amplitude of the in-phase stress component strain, a/e) and E" is the loss modulus (the amplitude of the out-of-phase component. strain amplitude). Similarly for (7 and K and the ratio between the Young s modulus E and the shear modulus C includes Poisson s ratio u, for an isotropic linear elastic solid with a uniaxial stress. (Poisson s ratio is more correctly defined as minus the ratio of the perpendicular. strain to the plane strain, or for one orthogonal direction 22 which equals the. 3.3 strain if the sample is... 

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