The strain energy release rate

In linear elastic fracture mechanics it is useful also to consider the energy G available for unit increase in crack length, which is called the strain energy release rate . Following Equation (12.1) above, G is 

Although the Griffith and Irwin formulations of the fracture problems are equivalent, most recent studies of polymers have followed Irwin. Before discussing results for polymers, it is useful to show how Gc can be calculated. 

This quantity C is not to be confused with an elastic stiffness constant as defined 

Here can be determined directly by combining a load-extension plot from a tensile testing machine with determination of the movement of the crack across the specimen, noting the load P for given crack lengths (points 1, 2, 3, 4, 5 in 

The critical strain-energy release rate (or, in the original Griffith terminology, the fracture surface energy y) therefore can be obtained by measurements of either the load P or the deflection A for given crack lengths c. 

The exact equivalent formulation in terms of the critical stress intensity factor can be obtained from Equation (13.9), giving 

We have discussed only the calculation for a geometrically simple specimen, so that the principles involved are not obscured by complex stress analysis. For a comprehensive discussion of the calculation of the fracture toughness parameters Gc and Kc for specimens with different geometries, see standard texts [5-7]. 

---

Fracture mechanics (qv) tests are typically used for stmctural adhesives. Thus, tests such as the double cantilever beam test (Fig. 2c), in which two thick adherends joined by an adhesive are broken by cleavage, provide information relating to stmctural flaws. Results can be reported in a number of ways. The most typical uses a quantity known as the strain energy release rate, given in energy per unit area. 

For equilibrium systems with no contact hysteresis G = W, which is the classical Griffith criterion in fracture mechanics. For such a system, Eqs. 12 and 37 are the same. That is, the strain energy release rate is given by... 

Attention will be restricted to the strain-energy-release rate for the opening mode. This mode occurs for the plate with a centrally located crack of length 2a under load P in Figure 6-11. 

The strain-energy-release rate due to crack extension 28a is the shaded area in Figure 6-11 if the loading-frame head does not move during crack extension, that is,... 

Inwin [6-14] calls the strain-energy-release rate G, so... 

The strain-energy-release rate was expressed in terms of stresses around a crack tip by Inwin. He considered a crack under a plane stress loading of a , a symmetric stress relative to the crack, and x°° a skew-symmetric stress relative to the crack in Figure 6-12. The stresses have a superscript" because they are applied an infinite distance from the crack. The stress distribution very near the crack can be shown by use of classical elasticity theory to be, for example. 

One way of looking at the fracture characteristics of a ductile material is by measuring the amount of plasticity at a crack tip prior to crack propagation (Fig. 8.84). One test which measures this is the crack-tip opening displacement (CTOD), 5. Wells has found that 6 can be related to the strain energy release rate, G, by the formula ... 

Thns the strain energy release rate is effectively an instantaneous value of Dupre s energy of adhesion, with 6 = 0(0 instead of the equilibrium value. The sign reversal in the left-hand side of Eq. (18) when compared to Eq. (15) is due simply to the fact that we have a closing crack with a spreading liquid. 

The interface debond criterion used in this analysis is based on the concept of fracture mechanics where the strain energy release rate against the incremental debond length is equated to the interface fracture toughness, Gk, which is considered to be a material constant... 

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) ... 

Fig. 6.19. Ratio of the strain energy release rates, Gt/Gr, plotted as a function of crack length. After He...

The values of the stress intensity factor (Klc) and of the strain energy release rate (Glc) of both crosslinked maleic and nadic oligomers are rather low and explain the poor mechanical properties of these materials. 

This dependence is certainly different from the amplitude of the RR stress and strain-rate fields which is Kjlt. This is an illustration of why the amplitude of the RR-field, C(t), is not necessarily the crack driving force parameter. This is in contrast to the ambient temperature situation wherein the strain energy release rate correlates exactly with either G (= K2/E) or /, both of which also govern the amplitude of the appropriate elastic or elastic-plastic stress fields. 

The unstable fracture of epoxies has been shown by Mai and Atkins to be accompanied by a negative change of the strain energy release rate, G, tvith crack velocity, a. This is in contrast to the positive dG/da which they find characterizes stable fracture. Whether a negative dG/da is the cause or the consequence of unstable fracture is, however, subject to debate... 

The behavior of propagating cracks is often described in terms of the elastic strain energy required to create unit area of fracture surface the strain energy release rate G. Between stress intensity factor K and strain energy release rate Gi the following relationship exists ... 

The thickness of the TDCB specimens (S = 10 mm) is sufficient to ensure plain strain conditions. It should be noted that during the test the arms remain within their elastic limit. Therefore, from simple beam theory [7], and by the use of linear elastic fracture mechanics, the strain energy release rate of the adhesive can be obtained using Eqn. 2, where P is the load at failure and E, is the substrate modulus. The calculated adhesive fracture energy was employed in the simulation of the TDCB and impact wedge-peel (IWP) tests. 

---