Imaginary modulus

This presentation format leads to the terminology EI = real modulus or storage modulus 2 = imaginary modulus or loss modulus. 

The most obvious problem of non-linearity is the definition of a modulus. For a linear viscoelastic material we need to define not only a real and an imaginary modulus but also a spectrum of relaxation times if we are fully to describe the material - although it is more usual to quote either an isochronous modulus or a modulus at a fixed frequency. We must, for a full description of a non-linear material give the moduli (and relaxation times) as a function of strain as well this will not usually be practicable so we satisfy ourselves by quoting the modulus at a given strain. The question then arises as to whether this... 

The <0 is the Brillouin shift in radians per second. The reduced real and imaginary modulus data are shown in Figures 4a and 4b. The plateau in (M — K)Joo extends to wt/sJoo 10 . In the region of solid like behavior reduction fails below 220 K. However, the juxtaposition of the low- and high-angle reduced data and the extent of the plateau region over some three decades in (ot sJqo lends support to the reduction procedure used. The solid curve is a single-relaxation model with a lumped relaxation time, T, of T = 0.06 tm- The dashed curve is based on a two-relaxation-time model with ti = 0.09 tm and to = 0.002 tm. 

Loss modulus G - the component of applied shear stress, which is 90° out of phase with the shear strain, divided by the strain. Syn viscous modulus, hysteretic modulus, imaginary modulus, and out-of-phase modulus. 

Users of DMA will be familiar with the typical outputs of such instruments, namely M (storage or real modulus, E or G ), M" (loss or imaginary modulus, E" or G") and tan 8, which is the ratio M"/M. One of the reasons that DMA is such a powerful technique for exploring the properties of polymeric materials and others that show time-dependent behaviour is that aU ofthe above parametersare influenced greatly by the materials relaxation... 

Loss modulus (Yoimg s ( "). shear (G"), bulk K")) It represents the viscous part of the deformation experienced in a viscoelastic material. It is commonly referred to as the out-of-phase and imaginary modulus (see Argand diagram representation ). It is proportional to the amount of energy lost and irrecoverable in a sample. Unit Pascals (Pa). 

Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide... 

This behavior is usually analy2ed by setting up what are known as complex variables to represent stress and strain. These variables, complex stress and complex strain, ie, T and y, respectively, are vectors in complex planes. They can be resolved into real (in phase) and imaginary (90° out of phase) components similar to those for complex modulus shown in Figure 18. 

The question of the actual form of the relation between toughness and crack speed is still rather unclear. It is tempting to relate 4> V) to the imaginary part of the elastic modulus, but the size parameter required to relate V to a time or... 

The imaginary component, "(f), is the dilational viscosity modulus. This arises when the demulsifier in the monolayer is sufficiently soluble in the bulk liquid, so that the tension gradient created by an area compression/expansion can be short circuited by a transfer of demulsifiers to and from the surface. It is 90° out of phase with the area change. 

Equivalent Circuit Analysis. IS measurements yield values of V and Z the real and imaginary components of the impedance, as a function of f, the AC frequency. The data are usually displayed as Nvauist plots (Z, vs. Z ) or Bode plots (impedance modulus,... 

This expression describes the variation of the complex modulus with frequency for a Maxwell model. It is normal to separate the real and imaginary components of this expression. This is achieved by multiplying through by (1 — icut) to give... 

IKL Write a digital computer program that gives the real and imaginary parts, log modulus, and phase angle for the transfer functions ... 

FORMAT FREQUENCY REAL IMAGINARY LOG MODULUS ANGLE ) WR1TE(6,93)... 

Figure 2. Fourier transformed EXAFS (modulus and imaginary part) of the oxygen exposed samples full lines observed dotted lines regressed with equation (3). Note that the dotted line is mostly covered by the full line.

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