## 1 Introduction  
The repeated passage of train wheels over the rails induces rolling contact fatigue (RCF) cracks on both the railhead and the wheel tread. Such surface RCF cracks undergo non-proportional mixed mode I/II/III loading cycles [1; 2; 3; 4]. For a period, they grow stably at a shallow angle to the surface, according to what is considered coplanar fatigue crack growth (FCG). Once they have reached a certain length, these cracks start to branch. Coplanar cracks are not a great danger to trains as they flake off, causing the train to have a rougher ride at most. Branch cracks, instead, could lead to catastrophic failure if left to grow. However, the continued growth of coplanar cracks can obscure the branch ones from detection through conventional non-destructive methods. Therefore, the accurate prediction of growth rates for both coplanar and branch cracks is essential to prevent rail and wheel failures and develop cost-effective maintenance strategies. Many studies have been conducted on the coplanar FCG of rail steel under sequentially applied mode I and II loading cycles. Compared to these mixed mode loading tests, FCG experiments under mixed mode I/III loading are hardly performed. However, several tests under proportional loading or with one mode cyclic and other static have been conducted. [PERSON] et al. [5] studied the FCG in mode III AISI 4340 steel in torsional loading, measured on circumferentially-notched specimens and the results were compared with the behavior in mode I. FCG rates in mode III were found to be slower than those in mode I, despite that they were still Paris-type law related to mode III stress intensity \(K_{\text{III}}\) range (\(\Delta K_{\text{III}}\)). They proposed a micromechanical model for mode III FCG, in which the crack advance was considered to occur via a mode II coalescence of cracks, initiated at inclusions ahead of the main crack front. [PERSON] et al. [6] found no correlation between the FCG rate and the \(\Delta K_{\text{III}}\) or the plastic strain intensity range (\(\Delta\Gamma_{\text{III}}\)) except when superimposing a static mode I loading to the mode III one on a low-strength AISI 4140 steel. In the latter case, the variability of the FCG rate decreased. [PERSON] and [PERSON] [7] investigated the effects of static mode III on mode I FCG behavior in four materials. They showed that the loading conditions caused two main effects--a considerable reduction in FCG rate and a modification in crack path. They introduced a new criterion based on two main assumptions--fatigue cracking occurred only under the effect of local mode I opening and a fatigue crack grows in a direction where the FCG rate is the maximum. [PERSON] et al. [8] applied a novel experimental method to promote mode III coplanar FCG in bearing steel. The method comprised out-of-phase multiaxial fatigue tests, adopting a stress history typical of sub-surface RCF onto specimens containing micronotches. Their analytical model has shown that the typical crack opening values determined by out-of-phase loads can prevent the crack face contact during the RCF loading cycles. [PERSON] et al. [9] investigated the FCG behavior in cruciform specimens made of Ti6246 by applying a static load along one arm of the specimen and a cyclic load along the other arm. They used an equivalent stress intensity range in the Walker crack growth law that can consider all mode I/II/III loading and determined that a change in the FCG direction occurred depending on the static load levels. Recently, mixed mode II/III experiments were also performed on a rail steel considering RCF. [PERSON] et al. [10] performed the experiments to determine the FCG thresholds and kinetics under loading conditions using asymmetric four-point bending specimens with different angles between the crack front and the shear load. They determined that a coplanar shear-mode FCG occurred with the high loading ranges.