Complex dynamic modulus

The strength of the filler skeleton may be characterized by the complex dynamic modulus measured at low frequencies [24]. The authors note that when c > ccr the yield point should be viewed as a sum of two components ... 

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. 

Rheological properties of filled polymers can be characterised by the same parameters as any fluid medium, including shear viscosity and its interdependence with applied shear stress and shear rate elongational viscosity under conditions of uniaxial extension and real and imaginary components of a complex dynamic modulus which depend on applied frequency [1]. The presence of fillers in viscoelastic polymers is generally considered to reduce melt elasticity and hence influence dependent phenomena such as die swell [2]. 

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

We note that, in principle, the main physical discussions related to filler networking in this paper do not change if a sinusoidal tensile or uniaxial compres-sional stress (amplitude 0) is imposed on the rubber material. In some examples the complex dynamic modulus is then denoted with E = E + iE" and the compliance with C = C - iC". All theoretical considerations use the shearing modulus G. ... 

Fig. 1 a,b. Strain amplitude dependence of the complex dynamic modulus E E l i E" in the uniaxial compression mode for natural rubber samples filled with 50 phr carbon black of different grades a storage modulus E b loss modulus E". The N numbers denote various commercial blacks, EB denotes non-commercial experimental blacks. The different blacks vary in specific surface and structure. The strain sweeps were performed with a dynamical testing device EPLEXOR at temperature T = 25 °C, frequency f = 1 Hz, and static pre-deformation of -10 %. The x-axis is the double strain amplitude 2eo... 

Comparison of the Complex Dynamic Modulus as Measured by Three Apparatus... 

Each device considered in this chapter determines the complex dynamic modulus from a theoretical description of the measurement. These descriptions or solutions are derived by making assumptions about the experiment. The degree to which these assumptions are realized determines the accuracy of the measurement. The most important and restrictive assumptions are those concerning the boundary conditions, sample geometry and stress state. 

DLUBAC ET AL. Complex Dynamic Modulus Measured hy Three Apparatus 51... 

The elastic moduli of a lossy material can also be represented as complex quantities [3,4]. The complex dynamic modulus M is related to the corresponding complex sound speed c as follows [3],... 

Generation of Master Curves. Modulus and loss factor data were processed into a reduced frequency plot in the following manner modulus curves at different temperatures were shifted along the frequency axis until they partially overlapped to obtain a best fit minimizing the sum of the squares of a second order equation (in log modulus) between two sets of modulus data at different temperatures. This procedure was completely automated by a computer program. The modulus was chosen to be shifted rather than the loss factor because the modulus is measured more accurately and has less scatter than the loss factor. The final result is a constant temperature plot or master curve over a wider range of frequency than actually measured. Master curves showing the overlap of the shifted data points will not be presented here, but a typical one is found in another chapter of this book (Dlubac, J. J. et al., "Comparison of the Complex Dynamic Modulus as Measured by Three Apparatus"). 

These equations are often used in terms of complex variables such as the complex dynamic modulus, E = E + E", where E is called the storage modulus and is related to the amount of energy stored by the viscoelastic sample. E" is termed the loss modulus, which is a measure of the energy dissipated because of the internal friction of the polymer chains, commonly as heat due to the sinusoidal stress or strain applied to the material. The ratio between E lE" is called tan 5 and is a measure of the damping of the material. The Maxwell mechanical model provides a useful representation of the expected behavior of a polymer however, because of the large distribution of molecular weights in the polymer chains, it is necessary to combine several Maxwell elements in parallel to obtain a representation that better approximates the true polymer viscoelastic behavior. Thus, the combination of Maxwell elements in parallel at a fixed strain will produce a time-dependent stress that is the sum of all the elements ... 

Similar to the dynamic mechanical measurements described in Section 13.7, the fluid here is subjected to an oscillatory stress, and the oscillatory response is analyzed. The complex dynamic modulus, G ((fl), is then resolved into two components ... 

The Autovibron system is designed to measure the temperature dependence of the complex modulus (E ), dynamic storage modulus (E ), dynamic loss modulus (E") and dynamic loss tangent (tan 6) of viscoelastic materials at specific selected frequencies (0.01 to 1 Hz, 3.5, 11, 35, 110 Hz) of strain input. During measurement, a sinusoidal tensile strain is imposed on one end of the sample, and a sinusoidal tensile stress is measured at the other end. The phase angle 6 between strain and stress in the sample is measured. The instrument uses two transducers for detection of the complex dynamic modulus (ratio of maximum stress amplitude to maximum strain amplitude) and the phase angle 6 between stress and strain. From these two quantities, the real part (E ) and the imaginary part (E ) of the complex dynamic modulus (E ) can be calculated. 

In rheology of polymers complex dynamic modulus, G, is of special importance. It is introduced to describe periodic deformations with frequency, cq and defined according to ... 

Another way of describing this type of response is to use the similarity between Eq. (24.42) and the decomposition of a number in the complex plane into its real and imaginary components. We can thus define a complex dynamic modulus E in the following way... 

DMA Deformation (cyclical change In length) Complex dynamic modulus... 

In order to investigate the viscoelastic behavior of crosslinked EVA, rheological measurements were made to determine at what temperatures the phase transitions occur and their effect on the dynamic mechanical modulus. The complex dynamic modulus E expression is given by Eq. (6). 

Complex modulus (complex dynamic modulus) n. A property of viscoelastic materials subjected to periodic variation... 

The complex stress is r = r + ix", which is the sum of a real part of the stress and an imaginary part the complex strain is Y — y + iy", where i is the operator -1 that signifies the rotation of 90° between x and x and y and y". The shear modulus can also be represented by a complex variable, ie, the complex dynamic modulus G, which is the ratio of the complex stress and complex strain G = x ly. The dynamic modulus can also be resolved into two components or vectors G and G ) G = G + iG", where equation 15 holds, and where G = G cos5 and G = G sinS. 

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