# Influence of Lubrication Systems on the Fatigue Strength of Bolted Joints
## 1 Introduction
### Theoretical Background  
In bolted joints, the force is transmitted by contact between the bolt and the nut threads. The resulting local stress in the bolt is critical to failure and determines the time of crack initiation under fatigue loading. Fatigue failure occurs at the location of the highest stress concentration, which is in the first load-bearing thread due to the initial redirection of the force flow from the bolt into the nut. This results in an inhomogeneous load distribution over the depth of engagement, where the first load-bearing thread transmits about 40% of the total load [23]. Numerical studies comparing metric ISO, Unified and Whitworth standard threads [24] as well as simulations evaluating the effect of nut material and depth of engagement [3] are in accordance with the experimentally gathered stress distribution [23]. In contrast, under quasi-static load, the screw fails in the weakest cross-section, which according to the design principle is present in the free-loaded thread [2]. The fatigue strength of the bolt, in the form of a permanently transmissible amplitude, is determined in an axial fatigue test according to DIN 969 in the bolt-nut test [21]. For this purpose, the bolt is usually preloaded to a constant mean load level of 70% of the yield strength via the test system and a sinusoidal operating load with constant amplitude is superimposed as a fatigue load. The test is completed when the bolt reaches a limited number of cycles N\({}_{\text{G}}\) without failure (often 5 or 10 million) or failure occurs [25]. The selection of the test loads and the required minimum number of individual tests as well as the statistical evaluation are further described in the standards DIN 969 and DIN 50100 [25; 26]. According to the VDI guideline 2230, the estimation of the fatigue strength in the form of the permanently transmissible amplitude with 50% probability of failure is carried out for bolts rolled before heat treatment exclusively via the size of the screw diameter. \[\sigma_{\text{ASV, 50\%}}=\frac{150}{\text{d}}+45 \tag{1}\]  
d: nominal diameter. \(\sigma_{\text{ASV}}\): permanently transmissible stress amplitude in N/mm\({}^{2}\) for bolts rolled before heat treatment For high operational reliability, the amount of the initial preload after tightening was designed to compensate for losses in preload by setting effects and mechanical and thermal operating loads. The reliability is given as long as the residual clamping force is able to keep the clamped parts together, and thus they were acting as one part. The coefficient of friction, the tightening torque and the preload are influenced by the material and surface pairing as well as by assembly parameters such as the speed of the tightening. In order to achieve reproducible preloads independent of the tightening method, bolted joints are adjusted to a defined friction coefficient range with the aid of lubricants. According to VDI Guideline 2230, the limits for the total friction coefficient \(\mu_{\text{tot}}\) are 0.08 to 0.16 [17]. The VDA sheet 235-101 (Verband der Automobilindustrie e.V., German Association of the Automotive Industry) further restricts the permissible total friction coefficient and defines the limits in the range of \(\mu_{\text{tot}}\) = 0.09 to 0.14 [19]. Friction coefficients are determined in the torque/preload test according to DIN EN ISO 16047 [27]. In this test, a bolt-nut connection is preloaded against a reference base with a continuously applied tightening torque. The total friction coefficient and the partial friction coefficients for the thread (\(\mu_{\text{th}}\)) and under the head (\(\mu_{\text{b}}\)) are determined for a defined test force \(F_{\text{p}}\). The applied tightening torque T is subdivided into the thread pitch torque, the thread friction torque and the head friction torque.