PSI - Issue 13

Tomasz Tomaszewski / Procedia Structural Integrity 13 (2018) 1756–1761 Author name / Structural Integrity Procedia 00 (2018) 000–000

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2. Size effect model with a critical volume Statistical model based on the critical volume (Sosnovski (1989)) can be used to analyse the size effect for the fatigue strength within the linear and complex stress range. The theory assumes, that a statistical solid body includes a set of basic elements, each to a certain degree responsible for the strength of the entire body. The model relates the fatigue properties to the material volume. The main assumption of the model is to determine the critical volume ( V P ) of a deformable solid body, delimited with an area of finite dimensions with a critical stress level within the specimen. Total volume depends on the geometrical dimensions and the specimen volume ( V ). For fatigue tests, stress at the specimen surface can be considered a deterministic value defined by the distribution of random variable of failure probability P( σ V ) . Fatigue strength ( σ V ) is determined for the critical volume ( V P ) at assumed probability. If the failure probability is interpreted in the category of confidence interval γ = 1 – α 0 ( α 0 – significance level), the critical volume ( V P γ ) can be determined for the distribution P( σ V ). Volume V P γ depends on the specimen volume ( V ), maximum stress, parameters of fatigue probability distribution function P( σ V ), confidence interval ( γ ), shape of the specimen’s surface area and load type. It is determined for the typical load types for smooth specimen with various shapes of the cross-sectional area based on the following relationship (Sosnovskii (1975)): V P γ V = λ ቀ 1 σ V min σ ቁ β ቀ 1+ σ V min σ ቁ α 1 ቀ 2+ σ V min σ ቁ α 2 (2) for 0 < V P γ /V ≤ 1, σ ≥ σ Vmin , where: V – specimen volume, σ Vmin – lower confidence interval limit for a probability distribution of the fatigue strength function P ( σ V ), σ – stress for the specimen volume, λ , β , α 1 , α 2 – coefficients depending on the type of load and shape of the cross-sectional area. Equation (2) for a linear and uniform stress is V P γ = V . Using the failure probability distribution based on the weakest link theory and the critical volume for non-uniform strain, the following relationship can be used (Sosnovskii (1975)): log σ V = B P - B log V P γ + B 0 ( v ) (3) where: B P = 1 m log ሾ -2,3log ሺ 1- P ሻሿ B = 1/ m m – shape factor of the distribution. Model parameters B 0 ( v ) λ , β , α 1 , α 2 depend on the type of load for bending of prismatic bars: B 0 ሺ v ሻ = log ൤ሺ m +1 ሻ 1 m σ w ൨ For λ = 1, β = 1, α 1 = 0, α 2 = 0, for the axial load of the smooth specimen: B 0 ሺ v ሻ = log σ w (4) where: σ w – parameter depending on the mechanical properties of the material. 3. High-cycle fatigue tests Based on the analyses of sensitivity to changing the size of the cross-section area (Tomaszewski and Strzelecki (2016)) the tested material (austenitic acid-resistant steel 1.4301) was selected. The differences in the obtained fatigue life are noticeable, which was confirmed experimentally on minispecimens. Steel 1.4301 features a significantly higher content of alloying elements than other materials (Tomaszewski and Strzelecki (2016)). The experimental tests for small specimens necessitates referring the results to the standard specimens and describing the reasons for the potential divergences of results. The specimens used in the tests are presented in Fig. 2, and the dimensions in Table 1. The specimens are of uniform shape and fixed value of theoretical stress concentration