Elastic parameters

The mechanical properties of polymers are controlled by the elastic parameters the three moduli and the Poisson ratio these four parameters are theoretically interrelated. If two of them are known, the other two can be calculated. The moduli are also related to the different sound velocities. Since the latter are again correlated with additive molar functions (the molar elastic wave velocity functions, to be treated in Chap. 14), the elastic part of the mechanical properties can be estimated or predicted by means of the additive group contribution technique. 

There is also an empirical relationship between the shear modulus and the transition temperatures. 

Since polymers are no purely elastic materials but are viscoelastic, they exhibit time and temperature dependence these also may be estimated if the transition temperatures are known. 

In oriented polymers (e.g. stretched filaments) the tensile modulus is a function of the stretch ratio tentative expressions are provided. 

The mechanical properties of polymers are of interest, in particular in all applications where polymers are used as structural materials. Mechanical behaviour involves the deformation of a material under the influence of applied forces. 

---

Here F is the force of interaction between masses m and M, L the rod length, f the elastic parameter of the fiber, and cp the angle of the twist. 

Now assume that the spring is made in such a clever way that the restoring force is directly proportional to the original length /q and the coefficient of proportionality is exactly equal to the elastic parameter ... 

TABLE 1 Elastic Parameters for Representative Soft (SOPC) and Rigid (SOPC CHOL, 1 1) Membranes (Parameters E, K, and Hq are Taken from Refs. 117-119, and rj from Ref 53)... 

Fig. 10. Estimated viscoelatic properties in a normal human breast in vivo. (A) T2 anatomical image. (B) Shear modulus image of the same slice. (Q Young s modulus image of the same slice. Grey scale bars are in kPa. Images B and C are extracted from 3D data sets of reconstructed elasticity parameters, obtained with the subzone based method used in Fig. 8. Note the good contrast in image C, even though the mechanical parameters are not obviously correlated to the structural properties depicted in image A (reprinted with permission from Ref. 48 2000 IOP Publishing Ltd.).

Fig. 11. Tensor-valued elasticity parameters in a human breast in vivo. A dotted circle symbolizes a carcinoma previously localized using gadolinium-enhanced Ti-weighted imaging. Eigenvalues Ei, E2, and E3 of the elasticity tensor are shown in (a), (b), and (c) respectively. Also shown in (d) is the isotropic elasticity...

He and Hutchinson (1989) considered a crack approaching an interface as a continuous distribution of dislocations along a semi-infinite half space. The effect of mismatch in elastic properties on the ratio of the strain energy release rates, Gi/Gj, is related to two non-dimensional parameters, the elastic parameters of Dundurs, a and /f (Dundurs, 1968) ... 

It is noted that these parameters become unity for an isotropic material. The two elastic parameters, a and, are also modified accordingly, taking into account the anisotropy ... 

ASTM E 1426-94, Standard Test Method for Determining Effective Elastic Parameter for X-Ray Diffraction Measurements of Residual Stress, ASTM, Philadelphia, PA, 1994. 

In general, it is difficult to determine correctly the elastic parameters Ef and Vf of thin films. Hence, this translation (eq. (14a), (14b)) may introduce a large uncertainty in the A values. 

Summarizing this section, we conclude that, from an experimental point of view, the ideal network theory expressed by Eq. (III-9) seems to be obeyed rather well only in the case of swollen networks. The determination of the elasticity parameters is complicated by the presence of a free enthalpy of dilution term. In spite of this, it is possible to conclude that 1/2 is the most likely value for B in Eq. (III-9). The factor A remains undetermined because experimentally the most one can obtain is A g o 2 8. Concerning this memory parameter, a dependence on the stage of dilution during crosslinking definitely exists, (Eq. Ill-19), but its dependence upon the nature of the diluent used in swelling studies needs further investigation. 

Polymer gel Polymer cohesive energy density P (cal cm ) Deviation from geometric mean mixing rule parameter z Non-Gaussian elasticity parameter, N Curve fit crosslink density (10s mol cm-3) Experimental crosslink density (10s mol cm-3)... 

Table 18. Variation of strength and elastic parameters of epoxide resin and paraffin on filling with glass microspheres 1381...

The model predicts that for the typical elastic parameters the polarization splay is small, just a few degrees. This was also confirmed experimentally as only a small change of birefringence under the electric field is detected for the Blrev phase. The electric field eliminates the spontaneous splay by aligning the dipole moments along the electric field direction. 

Fig. 11.32 The rheological properties of the polymer resins used in the polymer mixing study at 200°C. (a) The shear viscosity, (b) The elasticity parameter C"T, )- [Proceedings of the Sixth Semiannual Meeting of the Polymer Mixing Study, Polymer Processing Institute, Hoboken, NJ (1993).]...

The most important application of the LP measurements is establishing the chemical composition of the ternary compounds. For nitrides, a standard accuracy of X-ray measurements of about 10 ppm would allow us to determine the composition with an error of about 0.1%. Unfortunately, lack of reliable data for elastic parameters of all nitrides and for lattice parameters of AIN and InN leads to a lowering of this accuracy to about 1%. 

TABLE 13.1 Survey of the elastic parameters and their definitions... 

Elementary mode of deformation Elastic parameter Symbol Definition Equations... 

The definitions of the elastic parameters are given in Table 13.1 (Eqs. (13.4)-(13.7) see also commentary to the definitions of E and S). The three elastic moduli have the dimension force per unit area, so in the S.I.-system N/m2 or Pa. For practical reasons the numerical values below the glass transition temperature are usually given in GPa. The Poisson ratio is dimensionless theoretically it varies from -1 to Vi, and in practice from 0 to Vi (incompressible rigid solids and liquids). 

TABLE 13.2 Theoretical inter-relations of elastic parameters... 

Theoretical relationship of the elastic parameters with the velocities of sound waves... 

Theoretically the elastic parameters are directly related to the speeds of the different sound waves. 

U is called Molar Elastic Wave Function it is independent of temperature or polymeric phase state and can be calculated from additive group contributions. In this way the elastic parameters can be estimated or predicted. 

Empirical relationships of elastic parameters with other physical quantities... 

The theoretical treatment of the mechanical properties of fibres is, as a matter of fact, more complicated than that of isotropic polymers. Instead of two elastic parameters, e.g. the tensile modulus and the Poisson ratio, we now need five, because of the anisotropy. 

In a subsequent investigation, with Roos and Kampschreur (1989), Northolt extended the modified series model to include viscoelasticity. For that an additional assumption was made, viz. that the relaxation process is confined solely to shear deformation of adjacent chains. The modified series model maybe applied to well-oriented fibres having a small plastic deformation (or set). In particular it explains the part of the tensile curve beyond the yield stress in which the orientation process of the fibrils takes place. The main factor governing this process is the modulus for shear, gd, between adjacent chains. At high deformation frequencies yd attains its maximum value, ydo at lower frequencies or longer times the viscoelasticity lowers the value of gd, and it becomes a function of time or frequency. Northolt s relations, that directly follow from his theoretical model for well-oriented fibres, are in perfect agreement with the experimental data if acceptable values for the elastic parameters are substituted. 

The speeds of longitudinal and transverse (shear) sonic waves can be estimated, c.q. predicted via two additive molar functions. From these sound velocities the four most important elastic parameters (the three elastic moduli and the Poisson ratio) can be estimated. 

In Chap. 13 we have already discussed the use of sound speed measurements for the derivation of elastic parameters. We shall come back on that, more elaborately, in this chapter. We have also seen that sound speeds can be expressed in terms of additive molar functions these are of course basic for estimations, as well for mechanical properties as for thermal conductivity (Chap. 17). 

When there is negligible absorption the two sound speeds are related to the elastic parameters (see Chap. 13) ... 

TABLE 14.1 Sound propagation and elastic parameter data for various polymers... 

Code Elastic parameters Secondary sonic data ... 

---