## 3 Fracture Mechanics-Based Fatigue Assessment of Heterogeneous Materials  
The heterogeneous material that is illustrated in Figure 3c is comprised of homogeneous materials \(H\) and \(L\). The static mechanical properties and the fatigue crack growth resistance of \(H\) and \(L\) are shown in Figure 3a,b, respectively. Besides, the geometric portion of \(H\) and \(L\) changes along the loading direction. For a crack initialed at the edge, such a heterogeneous material's fatigue failure process can be depicted as follows. The crack expands \(a_{p}^{H}(x)\) in H and \(a_{p}^{L}(x)\) in \(L\). The stress on the material's remaining segment rises continuously, as the fatigue crack continues to expand. While the stress on the specimen's remaining segment reaches at a definite value, the instantaneous rupture happens in the heterogeneous material.  
Like homogenous materials, a fatigue failure process of heterogeneous material is likewise composed of instantaneous rupture phase and fatigue crack propagation stage. The difference is that more than one material would be penetrated by the fatigue crack while spreading in heterogeneous materials. Besides, the heterogeneous material's instantaneous rupture segment is related not only to materials properties and the fatigue load, but also to constituent materials' geometric ratios on the load-bearing cross-section.  
The instant rupture segment, \(t_{re}(x)\), of a heterogeneous material, as illustrated in Figure 3c, is a uninterrupted piecewise function that is relevant to the initial crack location \(x\) and can be depicted as follows:  
\[t_{re}(x)=\left\{\begin{array}{ll}P_{max}/\left(\sigma_{u}^{H}.l\right)+S_{L}( a,x),&x\leq x_{0}\\ P_{max}/\left(\sigma_{u}^{L}.l\right),&x\geq x_{0}\end{array}\right\} \tag{8}\]  
where \(S_{L}(a,x)\) denotes the dimension of \(L\) on the remaining bearing segment, \(P_{max}\) represents the upper limit of a fatigue load, \(\sigma_{u}^{H}\) and \(\sigma_{u}^{L}\) are the ultimate tensile strength of \(H\) and \(L\), respectively.  
The load-bearing cross-section at \(x=x_{0}\) is called the \"typical load-bearing section\". While the fatigue crack initiates at the \"typical load-bearing section\", the failure criterion would be reached by \(H\) exactly after failure of \(L\).  
The heterogeneous material's (shown in Figure 3c) fatigue life can be calculated by adding up the stress cycles experienced by \(H\) and \(L\) on the fatigue crack growth route. According to (4), the heterogeneous material's fatigue life is summarized as follows:  
\[N(x)=\frac{1}{2\sigma_{max}4}\Bigg{[}\frac{a_{p}^{H}(x)\Big{(}\sigma_{y}^{H}+ \sigma_{u}^{H}\Big{)}\epsilon_{u}^{H}\Big{(}\sigma_{u}^{H}\Big{)}^{2}\,E^{H}} {a_{i}\Big{[}a_{i}+a_{p}^{H}(x)\Big{]}}+\frac{\Big{[}a_{c}(x)-a_{i}-a_{p}^{L}( x)\Big{]}\Big{(}\sigma_{y}^{L}+\sigma_{u}^{L}\Big{)}\epsilon_{u}^{L}\Big{(} \sigma_{u}^{L}\Big{)}^{2}\,E^{L}}{\Big{[}a_{i}+a_{p}^{L}(x)\Big{]}\sigma_{c}( x)} \tag{9}\]  
Figure 3: (**a**) Static performance of \(H\) and \(L\); (**b**) Fatigue crack growth properties of \(L\) and \(H\); (**c**) Fatigue failure process of heterogeneous and homogeneous materials.  
where \(a_{c}(x)\) represents the critical fatigue crack length under \(\sigma_{max}\), \(a_{i}\) is the initial crack size, \(\sigma_{y}^{H}\) and \(\sigma_{y}^{L}\) represent the yield strength of \(H\) and \(L\), \(\epsilon_{u}^{H}\) and \(\epsilon_{u}^{L}\) represent the elongation of \(H\) and \(L\), \(E^{H}\) and \(E^{L}\) represent the Yong's Modulus of \(H\) and \(L\). \(a_{p}^{H}(x)\) and \(a_{p}^{L}(x)\), respectively, denotes the fatigue crack propagating dimension through \(H\) and \(L\) before expanding to the instantaneous rupture segment. Equation (9) reveals that a heterogeneous material's fatigue life is related to the geometrical parameters, mechanical performance of each component material and the fatigue load.