# Characterization of Ductile Crack Propagation by Fractal Energy Dissipation Rate
## 2 Fractal Aspects and Energy Dissipation Rate in Fracture  
Following the pioneering works by [PERSON] [9; 10], the achievements of the fractal approach in fractures are fruitful and effective, especially regarding crack topologies and surfaces in a finite range of scales. [PERSON] [11] proposed a power-law relationship between the \(J\)-integral and the crack length increment assuming an invasive fractal crack tip geometry. [PERSON] and [PERSON] [12] further investigated the relationship between the fracture resistance \(J_{\text{R}}\) curve and fractal fracture surface in alloys. Especially, the fractality of the fracture surface, crack path, and related irreversible energy have been widely recognized in both brittle and ductile materials [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. Fractal patterns were experimentally observed by [PERSON] and [PERSON] [13] in metals, where strain localization led to the formation of fractal slip bands. Pioneering investigations into fractal fracture mechanics were performed by [PERSON] [14], [PERSON] and [PERSON] [15], and [PERSON] [16]. [PERSON] et al. [17] investigated the stress singularity and path dependence of \(J\)-integral on fractal cracks. The classical [PERSON]'s theory for fractal cracks has led to the conclusion that, because of the nondifferentiable fractal sets, cracking requires infinite surface energy to create an increment of fractal crack extension regardless of the nominal crack growth length [18]. In engineering, there exists upper and lower bounds in the scaling range for all fractal structures: the upper bound is represented by the macroscopic size, whereas the lower bound is related to the internal characteristic length [19; 20; 21; 22; 23]. [PERSON] [24] thus extended [PERSON]'s criterion to fractal cracks by a fractal measurement of surface energy per unit, and the generalized [PERSON]'s criterion can be stated locally as \(\Delta U_{e}=\Delta U_{s}=2\gamma_{f}(D)\Delta m_{D}\), where \(\Delta U_{e}\) is the driving force or the strain energy release rate corresponding to the crack increment \(\Delta a\), and \(\Delta U_{\text{s}}\) is the crack growth resistance. The fractal specific surface energy, \(\gamma_{t}\), is a function of material microstructure \(\Delta m_{\text{D}}\) and fractal dimension \(D\). [PERSON] and [PERSON] [18] suggested an interpretation of crack resistance behavior in terms of a self-similar fracture surface. Accordingly, the nominal fracture resistance is related to the renormalized fracture energy and fractal crack increment. As a result, the energy dissipation rate of smooth cracks is independent of the crack length and remains constant, whereas that of fractal cracks increases with crack propagation. Because of the discrepancy between ideal and natural fractal sets, crack-size scaling energy dissipation is a geometrical multifractal. The effect of fractality on the mechanical properties gradually vanishes as the scale of the crack size increases. Owing to the fractional measurement of physical fractal sets, the corresponding mechanical quantities must consider anomalous physical dimensions to obtain physically universal quantities [19; 20; 21; 22; 23]. Current work shows that a physically universal and scale-independent energy dissipation rate can be obtained by considering an anomalous physical dimension in the energy dissipation rate, abandoning the complete similarity of the Euclidean topologies. Ductile fracture tests following BSI 7448 Part 4 [25] and ASTM E1820 [26] have confirmed that the irreversible energy rate is not only loading configuration-dependent but also intimately related to the crack and specimen size. The path-independent \(J\)-integral or crack tip open displacement thus cannot represent the strain energy release rate and becomes a description of accumulated energy at a given crack extension, deviating from the true \"driving force\" for a propagating crack.