Fibrils volume fraction

Verheulpen-Heymanshas measured the fibril volume fraction profile along isolated crazes in polycarbonate using an optical technique whereas Trent, Palley and Baer have measured it in isolated polystyrene crazes in thin films by comparing craze displacements measured from the displacement of bars of an evaporated metal grid intersecting the craze thicknesses. They use TEM of the unstressed film to make the measurements. Both groups find that Vf is independent of craze thickness. 

From the surface stress profile S(x) and fibril volume fraction profile Vf(x) it is also possible to find the true stress in the craze fibrils Of(x) which is... 

As implied by the discussion above craze fibril extension ratio or its inverse the fibril volume fraction of the craze is an important parameter of the microstructure. Fibril volume fractions can be measured by several different methods. The refractive index n of the craze can be measured by measuring the critical angle for total reflection of light by the craze surface. Using the Lorentz-Lorenz equation Vf then can be computed from The method is difficult because small variations... 

To this point the craze fibril volume fraction Vf and fibril extension ratio X have discussed as if they were true constants of the craze. While this view is approximately correct, one would expect the draw ratio of the polymer fibrils to depend somewhat on the stress at which they are drawn, since the polymer in the fibrils should have a finite strain hardening rate. Experimental evidence for just such stress effects on X, is discussed below. 

Craze fibril diameters determined by TEM and SAXS are of the order of 10 nm. Craze fibril volume fractions Vf range from 0.5 to 0.1, depending on the entanglement network of the polymer and the local craze stress. 

The transmission electron microscope (TEM) image from crazes in such films, the negative from which Fig. la was printed, can be analyzed not only to reveal dimensions of the craze but also to yield the local craze fibril volume fraction v by using a microdensitometer to measure the local density of the image, its mass thickness contrast The extension ratio of the craze fibrils, X equals i. ... 

The first one is due to the fact that these profiles are optical thickness profiles, with unknown optical index in the craze. Hence, neither geometrical profiles T(x) nor fibril volume fraction Vf(x) distribution in the craze are known. Both are necessary to calculate absolute values of the craze surface stress. As discussed in reference under certain conditions, valid craze surface stresses may be... 

The constant craze fibril volume fraction assumption is rather restrictive, but seems realistic in the case of PMMA. Moreover, if the calculated craze surface stress distribution is constant, then, a posteriori, the assumption is correct, because in the case of a craze growing by means of fibrils drawing from the bulk (as it is the case for PMMA) a constant stress along the craze can hardly generate a variable craze fibril structure (i.e. a variable volume fraction) along the craze. 

As previously noted, the craze fibril volume fraction is not known from the experiment. But the craze surface stress being constant, it may be calculated by means of the Dugdale equation ... 

Fig. 46. Craze structural paraincier S. as defined by Eq. 17 and as determined by means of Eq. 19. This parameter includes craze fibril volume fraction and the tensile modulus of the bulk. The dashed i ojc encloses the values of for all the crazes in air, whereas the dots correspond to the crazes...

Figure 47 shows taken from Equation 20 versus Vj. It shows that S. is quite sensitive to Vp and is therefore a good means to evaluate v, with the numerical values of Fig. 47. It can be estimated that the tensile modulus E of the bulk PMMA is not affected by the very low pressure toluene gas environment during the short duration of the experiment. The optical craze index in PMMA in air without load is known as n = 1.32, which corresponds to v = 0.6. From the optical interferometry, it is known that the craze just before breakage is twice as thick as unloaded, (v, = 0.3) and hence using Lorentz-Lorenz equation its optical index is n = 1.15. From Figs. 46 and 47 it can be concluded that the bulk modulus around the propagating crack is about 4400 MPa, which is a somewhat high value, in view of the strain rates at a propagating crack tip (10 to s" ). Using the scatter displayed in Fig. 46, it can be concluded from Fig. 47 that the fibril volume fraction is constant, v = 0.3, within a scatter band of 0.08, and is therefore not sensitive to the toluene gas.

This result is consistent with Kramer s results showing that the fibril extension ratio (which is just the inverse of the fibril volume fraction) is equal to the bulk polymer network full extension ratio. As a matter of fact, it is unlikely that the toluene vapor changes the physical and chemical structure of the bulk it just makes the fibril drawing easier . On the other hand, it is generally admitted that the fibril diameter times the craze surface stress is constant. Therefore, the craze surface stress being lower in toluene vapor, the fibrils are probably thicker. 

The Fourier transform method has been used to calculate the craze surface stress distribution from craze shapes obtained by means of optical interferometry — the craze shapes are the same in air and in toluene gas, only their sizes vary — the craze surface stress is almost constant along the craze boundary — the craze fibril volume fraction remains constant in air and in toluene gas over the whole velocity range, despite the fact that at low velocity in toluene gas the craze length reaches 4 times the length in air — the optical interference setup may give valuable information on the variations of craze fibril volume fraction, but not on its absolute numerical value. 

In many fracture mechanics-based approaches, crack advance is taken to occur when K reaches some critical value Kc (equivalent energy-based criteria are also widely used). Kc may then be measured using a pre-cracked specimen, in which a and hence K are well defined. In some cases (PMMA, for example) crack advance is observed to proceed via breakdown of a single craze at the crack tip. By modeling a craze as an orthotropic linear elastic body it has been shown that Kc is given by Eq. (72), where v is Poisson s ratio, a is related to the craze anisotropy, C7c is the draw stress normal to the craze-bulk interface, Vf is the fibril volume fraction in the craze, and oy is the stress to break a craze fibril [33]. 

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