Craze thickening

Note that the velocity of craze tip advance is very sensitive to the energy of the surface being created (Vq decreases strongly as F increases since 10 n 20 for most polymer glasses ). This feature will be invoked later to explain the important effect of entanglements on the stresses required for crazing. 

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. 

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The craze thickening, associated with the craze growth, implies an increase of fibril length. This is achieved by pulling out polymer chains from the craze-bulk interface, according to a behaviour analogous to plastic flow within the active layer (5-10 nm thick), as shown in Fig. 5b [21]. 

Primitive and mature fibril diameter Craze thickening rate Fibril lifetime... 

Craze thickening rate along the direction normal to the craze plane... 

Material parameters for the normal craze thickening rate preexponential... 

Material parameters for the tangential craze thickening rate pre-exponential... 

Because crazing is a precursor to failure, pioneering studies on crazing have focused on the conditions for craze initiation. Later on, estimation of the toughness motivated examination of the craze thickening and the conditions for craze breakdown. 

Descriptions of craze thickening are based on the observed crazes at the tip of a stationary crack for creep tests [29,30] and on observations of crazes in thin films by transmission electron microscopy (TEM) or small-angle X-ray scattering (SAXS) [31,32],... 

Fig. 2 Description of the craze thickening process according to Kramer [31] as drawing in new polymer chains from the craze/bulk interface into the fibrils. The fibrils have a diameter D and spacing of D0...

The deformation of the polymer within a thin active zone was originally represented by a non-Newtonian fluid [31 ] from which a craze thickening rate is thought to be governed by the pressure gradient between the fibrils and the bulk [31,32], A preliminary finite element analysis of the fibrillation process, which uses a more realistic material constitutive law [36], is not fully consistent with this analysis. In particular, chain scission is more likely to occur at the top of the fibrils where the stress concentrates rather than at the top of the craze void as suggested in [32], A mechanism of local cavitation can also be invoked for cross-tie generation [37]. 

More work on a detailed description of the fibrillation process is needed to clarify the underlying mechanism and its relationship with molecular aspects, such as the entanglement density or the molecular mobility. Nevertheless, based on the observations reported by DOll [29,30] of time-dependent craze stress and Kramer s [31,32] description of fibrillation involving an active plastic zone, one can conclude that craze thickening is a viscoplastic process. 

Based on the description of craze thickening due to Kramer et al. [31,32], Schirrer [45] proposed a phenomenological viscoplastic formulation for the fibril drawing velocity similar to the Eyring model as... 

Fig. 5 a Schematic description of the craze structure, b Idealization of the craze process according to Kramer and Berger [32] for the craze thickening after initiation, c Representation of crazes by discrete cohesive surfaces... 

Since oy is the major principal stress, we have an = oy > crm = (oy + a2)/2 and the side condition that the normal stress has to exceed the hydrostatic stress for craze initiation is satisfied. Equation 21 defines a critical normal stress which appears to be hydrostatic stress dependent. As long as an < oTr(am), crazing does not occur and when an reaches o/jr(crm) crazing initiates. Once initiated, the craze thickens and the condition (Eq. 21) is no longer relevant. 

As discussed in Sect. 3.2, once a mature fibril is created, further thickening occurs by a viscoplastic drawing mechanism which involves intense plastic deformation at the craze/bulk interface [32], Instead of using a non-Newtonian formulation as in [32] or a formulation based on Eyring s model [45], but on the basis of a preliminary study of the process [36], the craze thickening is described with a similar expression as the viscoplastic strain rate for the bulk in Eq. 3 as [20]... 

After a short transition following craze initiation, the craze thickens and results in an opening An of the cohesive surface at approximately constant normal stress. 

The heat equation (Eq. 29) is coupled to the equations governing the mechanical response through the temperature dependence of the bulk viscoplastic strain rate (Eq. 3), the craze thickening rate (Eq. 22), and the thermal expansion in Eq. 1. The system of differential equations resulting from the finite element discretization of the energy balance in [9] is modified [57] to... 

To illustrate the influence of the craze thickening kinetics on fracture, two sets of craze parameters are used and listed in Table 3. The two sets are borrowed from [22] (cases 8 and 1) and named here A and B. In Fig. 8, we report the plastic strain rate distribution observed near the crack tip. This variable is suitable to track the development of plasticity and is normalized with To = Ki/soy/Fi as a reference strain rate at the tip of the notch (the radius is rt = 0.1 mm and T = 293 K). We compare the cases for which no crazing is considered (Fig. 8a) to those where crazing is accounted for (Fig. 8b), with the set A of craze parameters in Table 3. When crazing is not present and at the particular loading rate considered, plasticity develops in the form of shear bands which originate from the tip of the notch, where the stress concentrates. 

Table 3 The sets of craze parameters used in this study the only difference originates from Ac and hence the sets exhibit different craze thickening kinetics (from [22])...

For set B, craze thickening is faster and the craze critical thickness is attained at K / (so r ) 1.32, which is significantly smaller than the value K / (so rt) 1.71 for set A. During crack propagation, some plasticity confined to the craze/crack interface is observed (Fig. 12) but the bulk remains mostly elastic. Therefore, the craze parameters B of Table 3 result in a more brittle response compared to that predicted for the craze parameters A (see Fig. 8b). 

The temperature distribution during crack propagation is shown in Fig. 15. As the crack advances, the heat continues to diffuse along the normal to the craze surfaces but the size of the hot zone remains comparable to that of the craze thickness. The maximum temperature increase is located at the crack/craze interface, where the craze thickening and related heat flux into the bulk are maxima. At this location, the temperature reaches the glass transition temperature Tg but plasticity is not enhanced in the bulk, which remains primarily elastic during crack propagation. 

The temperature distribution at craze breakdown and during crack propagation is shown in Fig. 17. As the crack advances, the temperature reaches the glass transition temperature at the location of the crack-craze transition, where plastic dissipation caused by craze thickening is maximum. However, this remains confined to a small volume around the crack-craze surfaces (see Fig. 17) so that no plasticity on a larger scale is promoted. 

There are two important questions about craze growth, namely what are the mechanisms of craze tip advance (expansion of the craze periphery generating more fibrils) and craze thickening (normal separation of craze surfaces lengthening the craze fibrils). Unlike the cloudy experimental situation regarding craze nucleation, that regarding craze growth now seems quite clear. 

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