Effective modulus

Stresses are usually related to strains through an effective modulus. If the components of stress are nondimensionalized by a suitable scalar modulus c, then they are also of order c. Using (A.94), (A.lOl), and the binomial theorem in (A.39), the relation between the normalized spatial stress s = s/c and the normalized referential stress S = S/c becomes... 

Modulus Measurements Another SCC test technique is the use of changes of modulus as a measure of the damping capacity of a metal. It is known that a sample of a given test material containing cracks will have a lower effective modulus than does a sample of identical material free of cracks. The technique provides a rapid and reliable evaluation of the susceptibility of a sample material to SCC in a specific environment. The so-called internal friction test concept can also be used to detect and probe nucleation and progress of cracking and the mechanisms controlling it. 

The result is important for the discussion in Part 3. Multiplication of the v-values by RT gives the corresponding moduli. The effective modulus of the first network after removal of first network crosslinks, Gie, has been calculated for a first network modulus, G-j, of 0.75 MPa. In Figure 1, G. e is plotted against the modulus of the second network before removal of the first network cross-links, G2. It can be seen that the memory effect increases with increasing modulus or degree of cross-linking of the second network. Gx and G2 max are related to the experiment to be discussed in Part 3. 

Combining the predictions for the reptation time of the H cross-bar and the effective modulus when the arms are acting as solvent gives a prediction for the scaling of the viscosity of a melt of H-polymers on their structural parameters ... 

The microstructure of the decomposed Fe-Mo alloy, Fig. 18.136, shows strong alignment of the developing two-phase microstructure along (100) directions. Such alignment is common in cubic crystals, and it arises from the anisotropy of the effective modulus, Y, in the diffusion equation. From Eq. 18.74 it is apparent that the crystallographic directions in which Y is a minimum will correspond to the wavevector of the fastest-growing waves. 

When the shape factor is high, such that Ec/K (where K = bulk modulus) exceeds 0.1, the effective modulus will be below that expected, due to the bulk compression being appreciable. The effective modulus can then be estimated from99 ... 

The amplitude of deformation with this apparatus must change by a fairly large amount to obtain reasonable precision and, consequently, it is likely that the stress/strain curve will be non-linear over the range measured, particularly in compression. Hence, only an effective modulus ia then measured. The range of frequency obtainable is small at a level of a few hertz. 

The usual methods give not exactly the equilibrium modulus value viscoelastic effects and internal energy effects (modulus component not proportional to T), are usually not taken into account. 

If the matrix can become more graphitic, as the above examples indicate, more shear planes become available hence more microcracking can occur, resulting in greater strain at lower stress levels. Thus the apparent or effective modulus of the 2D composite materials is reduced, and more energy is required to cause failure—an outcome indicated by the difference of area under the as-received and heat-treated load-deflection curves in Figures 7 and 9 ... 

The mechanical response of composites, as shown in these exploratory studies, indicates dependence on the ease with which fracture can occur between fibers, yarns, and plies. Poorly crystallized matrices result in composites that are strong and stiff but with little yield so that failure occurs catastrophically. In contrast, more crystalline matrices seem to be not quite as strong and to have a lower effective modulus, but their increased strain capability ensures that failure is not catastrophic the composited strength decays gradually as further strain is applied. Thus, the energy required for total failure is increased, and the composite with more crystalline matrix is more tolerant of defects or stress risers. 

Two factors will determine the accuracy of the modeling. The first is the accuracy with which the dynamic mechanical properties of the constituent materials is known, and the second is the degree to which the effective modulus theory actually models the properties of the inhomogeneous material. 

The second problem facing the researcher, and the one which is the subject of this paper, is to determine which, if any, effective modulus theory accurately predicts the acoustic wave velocity and attenuation in a microscopically or macroscopically inhomogeneous material. 

Static Theories. The simplest effective modulus theory is that which assumes the components behave as an ideal liquid mixture of one substance in another (6). In this case, both the density and the compressibility are additive properties of the corresponding quantities of the two materials depending upon the proportional amount of each substance in the mixture. Thus,... 

Other investigators (1 ) have obtained results similar to those of Kerner. The results of Dewey (ll) which are valid for a dilute solution agree with the Kerner equation in the dilute solution limit. Christensen ( 12) reviews and rederives the effective modulus calculations for spherical inclusions. The three models which are... 

In this case, the dipole, quadrupole, and higher order terms may be dropped compared to the monopole term. The effective modulus when A p is given by... 

Kligman and Madigosky (j ) extended the work of Chaban to the case of a solid medium. Based on this, it can be shown that when only the lowest order terms in frequency of the monopole and quadrupole contributions are retained (the dipole contribution appears in the effective density), the effective modulus is... 

Note that the experimental data agree best with the Kerner theory and that, at high void content, this theory correctly predicts an effective modulus of zero. Up to 15 percent in void fraction, the Kerner and Chaban predictions are essentially identical. Finally, the fact that the Kerner theory agrees with the experimental data up to 45 percent void content suggests that, currently, this is the best theory for predicting the effective modulus in microscopically voided materials. 

McLeish and coworkers have published results on the rheological behavior of S2I2 miktoarm star copolymers [359]. For the temperature range between 100 and 150°C it was evident that the rheology of a polymer with 20wt%. PS was independent of temperature, implying a particular molecular mechanism for stress relaxation for this architecture. For the sample having 35 wt% PS, a failure of the superposition principle was observed, a fact that was attributed to the temperature sensitive effective modulus of the polymer. 

In spite of our reluctance to quote numerical values at this point, the effective modulus obtained from the initial portion of the tensile curve ranges from 1 to 5 X 10 dyn cm". Many individual PE crystals have moduli from 3 X 10 to 10 dyn cm" and fracture at forces of about 0.2 dyn. Orientation effects are expected to be present and are presently being investigated. There is no comparable experimental data with which we can directly relate these values. However, moduli are in the range found for bulk specimens but are considerably less than the value of 70 X 10 dyn cm reported by Perkins et al. (8) for ultra-drawn HDPE fibers. The x-ray measurements of the lattice moduli by Ito (9) using an x-ray technique for oriented sheet samples is perhaps the most relevant comparison. He found values of 240 X 10 dyn cm" in a direction parallel to the fiber axis and a value of 4 X 10 dyn cm for the perpendicular direction which would be the closest comparison with our orientation. We are not yet certain whether the initial portion of the stress-strain curve shows nonlinear viscoelastic effects such as found by Chen et al. (4) for springy polypropylene (PP) fibers. 

---