Generalized fracture energy

In Griffith s original treatment, the surface free energy per unit area of fracture plane was employed in place of the generalized fracture energy, Gc/2. His results therefore carried the implication of thermodynamic reversibility. In contrast, Gc merely represents energy dissipated during fracture. Nevertheless, provided that it is dissipated in the immediate vicinity of the crack tip and is independent of the overall shape of the test piece and the way in which forces... 

The most-often cited theoretical underpinning for a relationship between practical adhesion energy and the work of adhesion is the generalized fracture mechanics theory of Gent and coworkers [23-25] and contributed to by Andrews and Kinloch [26-29]. This defines a linear relationship between the mechanical work of separation, kj, , and the thermodynamic work of adhesion ... 

Grillet et al. (1991) studied mechanical properties of epoxy networks with various aromatic hardeners. It is possible to compare experimental results obtained for networks exhibiting similar Tg values (this eliminates the influence of the factor Tg — T). For instance, epoxy networks based on flexible BAPP (2-2 - bis 4,4-aminophenoxy phenyl propane) show similar Tg values ( 170°C) to networks based on 3-3 DDS (diamino diphenyl sulfone). However, fracture energies are nine times larger for the former. These results constitute a clear indication that the network structure does affect the proportionality constant between ay and Tg — T. Although no general conclusions may be obtained, it may be expected that the constant is affected by crosslink density, average functionality of crosslinks and chain... 

In general, an increase in conversion increases Tg and series prepared with non-stoichiometric compositions, the minimum value of fracture energy is obtained when the stoichiometric ratio of both functionalities is used (Morgan et al., 1984 Won et al., 1990), as expected (maximum values of both Tg and [Pg.385]

In general, the fracture energy of a network is correlated to its ability for plastic deformation. This means that GIc decreases with an increase of oy. The influence of test variables (T, e) on ay was analyzed before (Sec. 12.3.4), and can explain the changes observed in GIc values. 

The interphase provided by the adhesion promoter may be hard or soft and could affect the mechanical properties. A soft interphase, for example, can significantly improve fatigue and other properties. A soft interphase will reduce stress concentrations. A rigid interphase improves stress transfer of resin to the filler or adherend and improves interfacial shear strength. Adhesion promoters generally increase adhesion between the resin matrix and substrate, thus raising the fracture energy required to initiate a crack. 

The generalized theory of fracture mechanics of Andrews (35) predicts that the cohesive fracture energy per unit surface area J is given by the energy required to break the bonds crossing the fracture plane, Jq, multiplied by a loss function, 6. 

This is an important aspect and would at least partly explain in a manner somewhat different from the more accepted explanations why Go is generally 100 to 1000 times higher than the thermodynamic work of adhesion [48], It indicates that G E(v, T). Apart from this the interesting consideration still holds that the flexibility at the interface is inversely proportional to both the intrinsic fracture energy and to the peel adhesion strength at least where the effect of is minimized. 

The generalized activation energy of fracture, U(a), may now be expanded in a Taylor series and, since x at a 0, we obtain... 

Equation (10.5) is more generally applicable than Eq. (10.6) because it is not restricted to linearly elastic materials. It constitutes a criterion for tensile rupture of a highly elastic material having a cut in one edge of length, /, in terms are of the fracture energy, Gc- Two important examples of test pieces of this type are (1) the ASTM tear test piece for vulcanized rubber (ASTM D624-54) and (2) a typical tensile test piece that has accidental small nicks caused, for example, by imperfections in the surface of the mold or die used to prepare it. 

This theory relates the force needed to separate the two surfaces to the mucoadhesive strength and is widely applied in research, where force of detachment versus distance is normally measured. In general, the fracture stress (considered to be equivalent to the mucoadhesion stress) is calculated by dividing the fracture force by the area of contact in the mucoadhesive bond. In the same study, fracture energy (or work of mucoadhesion) can be obtained as the area under the curve in the force versus distance plot. Nowadays, instruments measure directly the force of mucoadhesion between two surfaces in contact as a function of distance and time. 

Fig. 5.41 Schematic presentation of general relationship fracture energy-porosity (after [94]) high strength caused by low porosity and high ratio of unhydrated cement, B higher porosity and Ca(OH)j content, C local maximum of strength caused by fracture propagation inhibition on the pores (at porosity=22.5 %), D reduced strength by increased porosity and reduced number of intergranular contacts...

Fig. 13.41 The temperature dependence of the critical fracture energy Gic in DGEBA epoxy-resin thermosets, modified either by rubber particles or by debonding glass spheres, either in tests of conventional extension rates or in Izod impact tests, compared with the generally flat behavior of unmodified epoxy resin (Kinloch (1985) courtesy of Springer).

---