Real modulus

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

An example of the utility of these expressions is given in Figure 1 where the log (real modulus) is plotted against temperature for two test specimens of EPDM. The first test specimen is represented by o s, while the second specimen is represented by the dashed lines, which are 95% confidence limits calculated from the expressions in Table II and it viscoelastic properties. The measurements were made three weeks apart. 

Note that VTD-variance of Loss Tangent, and that SDTD is the standard deviation of Loss Tangent with similar definitions for GSP (G or real modulus) and GDP (G or loss modulus). 

Thus, the real modulus of bulk elasticity would be higher than Ev and the ratio between the modulus of elasticity of wet and dry foam would be... 

The stress vector can be similarly considered to be the sum of two components. One component, a = ao cos is in phase with the deformation, the other component, on the other hand, a " = ao sin d, is not. A modulus can be assigned to each of these two components. The real modulus, or storage modulus, G measures the rigidity and resistance to deformation of the sample. It is given by... 

The ratio of the imaginary to the real modulus is called the loss factor ... 

The dynamic mechanical spectrum of a PAN fiber obtained by Minami [205] is shown in Figure 12.20. The real modulus at room temperature is approximately 4 x 10 MPa. The modulus begins to decrease near 75°C and drops to 2 x 10 MPa, which is a decrease of slightly over one order of magnitude, and corresponds to the onset of the glass transition. The Tg based on the midpoint of the decrease in real modulus is approximately 115°C. This is higher than the values obtained using thermal expansion coefficient (85°C) or DSC (100°C), and simply reflects the temperature shift characteristic of dynamic techniques mentioned earlier. 

Most glassy polymers such as polystyrene exhibit a much larger decrease in the real modulus upon heated through the glass transition temperature. A decrease of approximately... 

Figure 11 The real modulus and complex viscosity vs. the particle volume fraction for oxidi/.ed polyacrylonitrile/silicone oil suspensions. The applied electric field = 1.5 kV/mm. The strain amplitude = 200%, frequency = 2 Hz. Redrawn with permission from T. Hao, Y. Chen, Z. Xu, Y. Xu and Y. Huang, Chin. J. Polym. Sci., 12(1994)97...

When at least one of the refractive indices n, and n, is complex, the reflection coefficients, Eqs. (2a) and (2b), become complex and can be written in terms of real modulus and phase factor. The phase factor of a reflection coefficient amounts to the... 

An important implication finm Bq.(lO) is that one may obtain the correction for (i.e., E ) even if the system compliance cannot be obhuned fi om the materials to be tested. To do so, one needs to obtain at first the numerical expression in Eq.(lO) by testing several controlled materials with known stress-strain relationships that allow s to be derived fi om steps (1) (5) outlined above. Once s is known and relationship is established, E can be calculated using Eq.(9) for any material sample with E, obtained experimentally. E represents the correction for and its numeric value is expected to be close to that of E, the real modulus obtained fi om Eq.(l) using an extensometer. The step-by-step procedure to obtain E fium s and , is summarized as follows ... 

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... 

Storage modulus (Young s (E ), shear (G ), bulk K )) It represents the elastic part of the deformation experienced in a viscoelastic material. It is commonly referred to as the in-phase and real modulus (see Argand diagram representation ). It is proportional to the amount of energy stored and recoverable in a sample. Unit Pa (Pascals). 

Additionally, the solution path could utilize the correspondence principle discussed previously, where the elastic solution is obtained and the real modulus or stiffness is replaced with the complex modulus or stiffness. However, let us explore the classic method of directly solving the equation of motion. For the first case, consider sinusoidal vibration of the base. 

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