13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 2.2. Consideration of size effect and variable stress state In principle the fatigue life depends not only on the material, the stress range, and the stress ratio, but also on the specimen length, as found for example in [4]. This means that the longer a specimen the higher the failure probability if the same stress range Δ σ is applied. This phenomenon, called statistical size effect, is due to the fact that a larger specimen is more likely to contain a large critical crack than a smaller specimen, cracks being considered to originate fatigue failure. Furthermore, specimen geometries used in fatigue experiments frequently present a cross-section with varying diameter along their lengths (see Fig. 2). The experimental results (∆σ versus lifetime N) stemming from testing these specimens are usually evaluated considering only the maximum stress range ∆σ0, acting on the smallest cross section, and the stress ratio R=σmin/σmax. While the stress ratio R is the same for all cross sections, ∆σ is varying along the specimen length. To extrapolate from those test results to structural elements or specimens of different size it is advantageous to obtain a “normalized” Wöhler field. To accomplish this task a new method is developed, based on the following assumptions: a) Fatigue failure initiates from surface flaws. Therefore the size effect is related to the stressed surface area, i.e. the larger the stressed surface the higher is the failure probability for the same combination of ∆σ and N. b) Validity of statistical independence and weakest-link principal implying that the survival probability , of a surface S = n ⋅Si composed of n surface elements of size !" is given by the product of the individual survival probabilities , # of the subelements each loaded by a stress range ∆σi, i.e. , ,Δ $ , # ,Δ " . & "'( (2) Accordingly, if all surface elements have the same size Si and are loaded by the same stress range ∆σi one gets , ,Δ ) , # ,Δ " * + # ⁄ . (3) c) For the moment, only the uni-axial load case is considered, so that Eq. (1) describes the failure probability ,∆ for a uni-axially tensioned surface element ΔS. With those assumptions and ,∆ 1 ,∆ we can combine Eqs. (1) and (3) to obtain the survival probability for a uni-axially tensioned surface element of size !" ." ∙ ∆! as , # ,Δ )1 ,∆ ,Δ * #⁄0 exp !"∆! ln lnΔ (4) Thus, for an arbitrary structure under fatigue load with tensioned surface S = n ⋅Si composed of n surface elements of size Si, each loaded by a different stress level Δ σi , combining Eqs. (2) and (4) one gets
RkJQdWJsaXNoZXIy MjM0NDE=