# Characterization of Ductile Crack Propagation by Fractal Energy Dissipation Rate
## 3 Fracture Resistance and Energy Dissipation Rate in Ductile Crack Extension  
[PERSON] and [PERSON] [18] established the relationship between the structural behavior and crack growth resistance curve using the energy brittleness number, where the ductile-to-brittle transition resulting from size effects can be characterized by the modified \(J\) resistance curve. Consequently, a plausible description of the topology of the fracture domain is essential for stable crack growth resistance behaviors. It is assumed that the non-integer fractal dimensions are a proper description of the fracture domains. Owing to the fractional measurability of fractal sets, the mechanical quantities defined in these domains must take non-integer physical dimensions. Meanwhile, unlike the complete similarity in the Euclidean domain, the incomplete similarity can be used to capture the geometry in the fractal domains, i.e., to characterize the energy dissipation rate using the fractal approach. Therefore, a fractal model was developed to calculate the rugged crack length based on the Voss equation [17], where the crack is compared with a self-affine noise in time as follows:  
\[\Delta a^{*}=\Delta a\sqrt{1+\left(\frac{l_{0}}{\Delta a}\right)^{2H-2}} \tag{5}\]  
where \(\Delta a^{*}\) is the increment in the rugged crack length, \(\Delta a\) is its horizontal projection, \(l_{0}\) is the unit length or scaling factor, and \(H\) is the Hurst exponent, which is usually smaller than one and related to the fractal dimension \(D\) through \(H=I-D\) (\(i=2\) or \(3\)). In the present work, it is assumed that \(\Delta a^{*}\) and \(\Delta a\) are the geometrical descriptions of the crack path and its accompanying plastic wake in the fractal and Euclidean domains, respectively. Considering energy equivalence and fractal crack, the path-independent \(J\)-integral can be modified in fractal domains as  
\[J=-\frac{d\Pi}{da^{*}}\frac{da^{*}}{da} \tag{6}\]  
where \(\Pi\) is the potential energy of an elastic body. Incorporating Equation (6) into Equation (7), the fractal crack resistance \(J_{R}\) per unit thickness is:  
\[J_{R}=\frac{dU_{\gamma}}{da^{*}}\frac{da^{*}}{da}=\left(2\gamma_{e}+\gamma_{p }\right)\frac{da^{*}}{da}=\left(2\gamma_{e}+\gamma_{p}\right)\frac{1+(2-H)( \frac{l_{0}}{\Delta a})^{2H-2}}{\left[1+\left(\frac{l_{0}}{\Delta  
and the plastic (or irreversible) component can be obtained as  
\[J_{pl}=\gamma_{p}(2-H)(\frac{l_{0}}{\Delta a})^{H-1} \tag{10}\]  
A similar result was obtained by [PERSON] and [PERSON] for crack-resistance curves [18]. Based on Equations (5) and (9), the energy dissipation rate, \(R\), can be obtained in the fractal form:  
\[\mathrm{R}=\frac{dU_{dis}}{Bda}=(W-a)\frac{dJ_{pl}}{da}=(W-a)\left[\gamma_{p} \frac{d^{2}a^{*}}{da^{2}}\right] \tag{11}\]  
Here, \(W\) is the characteristic reference size of the specimen, i.e., the height of beam. Using Equation (10), and in the self-similar limit \(a^{*}>>a\), we obtain:  
\[\mathrm{R}=\frac{\gamma_{p}}{\eta}(W-\Delta a)(2-H)(1-H)(\Delta a)^{-H}(l_{0} )^{H-1}=(W-\Delta a)(2-H)(1-H)(\Delta a)^{-H}\left[\frac{\gamma_{p}}{\eta}(l_{ 0})^{H-1}\right] \tag{12}\]  
Because the fractal set is defined in a limit sense corresponding to the observation scale \(n\) at approximately zero, Equation (12) can be rewritten as Equation (13) to obtain the scale-independent value of \(\mathrm{R}^{*}\):  
\[\mathrm{R}^{*}=(W-\Delta a)(2-H)(1-H)(\Delta a)^{-H}\lim_{n\to\infty}\left[ \frac{\gamma_{p}}{\eta}(l_{0})^{H-1}\right] \tag{13}\]  
The expression in the square brackets is indeterminate in classical meaning because, at the infinitesimal scale of observation (\(n\)\(\to\)\(\infty\)), \(\gamma_{p}(l_{0})^{H-1}\) tends to be nondifferentiable (\(l_{0}\)\(\to\)\(0\), \(\gamma_{p}\)\(\to\)\(0\)) in the form \(0\)\(\times\)\(\infty\).