## Materials and Methods
Batch D of hybrid configuration without post-process stress-relief, due to its low ductility, was not considered for fatigue investigations. For the investigation of fatigue behavior in very high cycle fatigue region, ultrasonic fatigue testing system _Shimadzu USF-2000A_ (Shimadzu, Kyoto, Japan) resonating at a frequency of 20 kHz was employed. To avoid stress-induced heating at an ultrasonic frequency, the tests were carried out at a so-called pulse:pause ratio of 50:50 where the system resonated for 200 ms and then stopped for the next 200 ms such that the deformation-induced rise in temperature remained below 10 K. All the fatigue testing was performed under fully-reversed loading.The details of the test setup can be seen in [42]. Tests at the two frequencies resulted in Woehler curves until 1E9 cycles. These Woehler curves were obtained by testing the configurations B, C, and E, which were post-processed by stress-relief treatment. The failure mechanisms in solids are typically modeled based on sharp crack discontinuities. However, this approach suffers in situations with complex crack topologies. This drawback is eliminated by introducing a diffusive crack modeling based on a crack phase-field \(d\), which covers values in the range [0,1], where \(d\) = 0 describes the unbroken state of the material and \(d\) = 1 the fully broken state. The model starts with the assumption for the free energy in the following form  
\[\psi(\epsilon^{\mathrm{e}},\,d)=\frac{1}{2}\Big{[}(1-d)^{2}+k_{\mathrm{n}} \Big{]}\epsilon^{\mathrm{e}}:\ \mathbb{C}:\epsilon^{\mathrm{e}},\epsilon=\ \epsilon^{\mathrm{e}}+\epsilon^{ \mathrm{P}}, \tag{1}\]  
where \(\epsilon^{\mathrm{e}}\) is the elastic strain, \(\mathbb{C}\) the elasticity matrix, and \(k_{\mathrm{n}}\) a small constant required to achieve numerical stability in the case of a fully broken material. The additive decomposition of total strains \(\epsilon\) into an elastic part \(\epsilon^{\mathrm{e}}\) and an inelastic part \(\epsilon^{\mathrm{P}}\) is a typical assumption for the small strain theory. However, Equation (1) does not represent a full model, but it must be completed by an expression governing the evolution of the damage parameter \(d\). In contrast to phenomenological methods, the present contribution follows a strategy where an assumption for a physically motivated dissipation potential is met [36]. According to such an approach, the phase-field \(d\) is related to the crack surface \(A\) of a crack embedded in a body \(B\) by the relationship  
\[A(d)=\ \int_{B}\gamma(d,\
abla d)\mathrm{d}V,\gamma(d,\ \
abla d)=\ \frac{1}{2l}d^{2}+\ \frac{l}{2}\| \
abla d\|^{2}, \tag{2}\]  
where \(\gamma\) is the crack surface density, depending on the phase-field and its gradient. [PERSON] et al. [36] introduced the constitutive dissipation functional for a rate-independent fracture process as  
\[D\Big{(}\dot{d};d\Big{)}=\ \int_{B}\phi\Big{(}\dot{d},\
abla\dot{d};d,\
abla d \Big{)}\mathrm{d}V,\phi=\ g_{c}\dot{\gamma}\Big{(}\dot{d},\dot{\
abla}\dot{d};d,\
abla d\Big{)}, \tag{3}\]  
which satisfies a global irreversibility constraint of the crack evolution. In the above equation \(g_{c}\) is a threshold value that is related to the critical Griffith-type fracture and \(\phi\) is a dissipation function per unit volume. In order to cover isotropic hardening effects, an additional term \(\mathbf{k_{y}}\Big{|}\dot{\mathbf{e}}^{\mathbf{p}}\Big{|}\) is introduced, leading to the final form of the dissipation potential  
\[D=\ \frac{g_{c}}{l}\dot{d}\dot{d}+\ g_{c}l\cdot d\cdot\
abla\dot{d}+k_{y}\| \dot{\mathbf{e}}^{\mathbf{p}}\|,\,k_{y}=\ k_{y0}+k_{\mathrm{h}},\dot{k}_{\mathrm{h}}=H \|\dot{\mathbf{e}}^{\mathbf{p}}\|, \tag{4}\]  
where \(l\) is the length-scale parameter and \(k_{y}\) the current yield limit with the evolution depending on the initial yield limit \(k_{y0}\) and isotropic hardening \(H\). The full formulation additionally requires a Lagrange term stipulating the non-negativity of the damage rate.