# Characterization of Ductile Crack Propagation by Fractal Energy Dissipation Rate
## 2 Fractal Aspects and Energy Dissipation Rate in Fracture
Several researchers [7; 27; 28; 29; 30] have suggested a more plausible characterization of the local separation energy related to the creation of new surfaces and plastic deformation remote from the crack tip, i.e., the energy dissipation rate, \(R\), which is defined as the rate of all nonrecoverable energy dissipation with regard to the crack increment. A stable crack increment, \(da\), must satisfy the energy balance as  
\[dw=dU \tag{1}\]  
where \(dw\) is the energy input rate, and \(dU\) is the rate of internal energy dissipation from all causes, including plastic yielding remote from the crack tip. Furthermore, the dissipated work rate \(dU\) for the cracked body can be divided into the elastic (recoverable) component \(w_{el}\), Griffith energy, \(2\gamma\), and plastic component, \(w_{pl}\)[1]:  
\[U=w_{el}+\underbrace{w_{pl}+2\gamma Bda}_{U_{dis}} \tag{2}\]  
For ductile fractures involving large-scale yielding, remote and local plasticity both contribute to \(w_{pl}\), and distinguishing between these two irreversible energies is extremely difficult. Therefore, the conventional test method that characterizes the ductile fracture toughness by resistance curves embraces all unrecoverable energy dissipation together, particularly in ductile tearing and deep notch/crack extension, which coincides with the definition of hardening cohesive energy, as shown in Figure 1[31]. For a stable crack extension, the energy dissipation rate is physically meaningful, as well as more generally, in contrast to the ambiguous driving force in terms of \(J\) or \(\delta\). [PERSON] et al. [7; 8] proved that \(R\) degenerates to G for linear elastic fracture mechanics, to \(J\) for nonlinear fracture mechanics, and to the work absorption rate for the elastoplastic material. Accordingly, \(R\) is defined as [7]  
\[R=\frac{dU_{dis}}{Bda} \tag{3}\]  
The dissipation work, \(U_{dis}\), can be determined from the area under the measured load-displacement curve, as indicated in some standards. If the \(J\)-integral is used to measure the irreversible fracture energy, using Equations (3) and (4), the following formulation is obtained [32]. \[R=\ (W-a)\frac{dJ_{pl}}{da} \tag{4}\]  
where \(B\) is the specimen thickness, \(b_{0}\) is the ligament length, and \(\eta\) is the plastic factor [26]. Although the energy dissipation rate \(R\) for ductile fracture is more physical than the classical fracture resistance, the geometrical dependency, i.e., in-plane constraint caused by the crack length and loading configuration, still exists [32; 33]. Therefore, it is essential to identify geometry-invariant material parameters from \(R\). To obtain a geometrically independent material fracture property, [PERSON] et al. [32] proposed a four-parameter exponential function to fit the \(R\)-\(\Delta a\) curve to explore the transferability of different values of geometric specimen fracture toughness, where the constraint effects on the \(J_{R}\) curves have to be parameterized by four fitting parameters. Therefore, the theoretical background of the energy dissipation rate requires further study. Figure 1: Different energy dissipation regions: fracture, local plasticity, and remote plasticity.