PSI - Issue 7

Hans-Jakob Schindler / Procedia Structural Integrity 7 (2017) 383–390 H.-J. Schindler / Structural Integrity Procedia 00 (2017) 000–000

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∆ K = K max – K min ; (1) with K max and K min being the maximum and minimum SIF K Ι of a load cycle. Experimental da/dN vs. ∆ K – curves are usually evaluated at constant R, whereas, in general, a crack in a structural component is subjected to different values of R. In particular, if residual stresses are present and taken into account, R varies continuously with crack depth, as discussed by Schindler (2017). Therefore, the effect of R on da/dN should be known in mathematical terms. In the so-called “Paris-regime”, i.e. for ∆ K th < ∆ K < (1− R ) · K i , FCG follows roughly “Paris’-law”, which means = (R) ∙ ∆ = 0 · S(R) ∙ ∆ (2) The open parameters C p and n have to be determined experimentally. There are several well-accepted guidelines that provide representative values of these parameters for common structural metals. Among them are FKM (2001), International Institute of Welding (2004), British standard 7910 (2013) and ASME (2010). They agree, essentially, on the R-effect on C p for R < 0. For R > 0, however, the situation is less clear. Most of the aforementioned guidelines consider C P as a material property, unaffected by R. British standard 7910 (2013) gives two different values: one for R<0.5 and one for R>0.5. Only ASME (2010) provides a mathematical formula for S(R). Anyway, most of the published experimental data indicate a certain influence of R on C P . In the present paper, this effect is considered from a theoretical point of view, based on fundamental relations of linear elastic fracture mechanics (LEFM). The general relation (2) is derived analytically, resulting in an equation for da/dN where R appears as another loading parameter. R = K min /K max

Nomenclature C nd

non-dimensional factor to adjust analytically predicted da/dN to measured values

C P0 Paris’ constant for R = 0, as defined in eq. (2). da/dN crack growth increment during one load cycle E Young’s modulus f(R)

non-dimensional parameter to quantify crack closure; f = Kop / K max K I required for initiation of ductile tearing, quantitatively close to K Ic , i.e. K i ≈K Ic

K i

K Ic

plain strain fracture toughness minimum K I during a fatigue load cycle maximum K I during a fatigue load cycle

K min K max

K op KI required to overcome crack closure, i.e to remove the contact pressure from the crack faces K* generalized fatigue load of a crack in terms of SIF; defined as K* = K max α · ∆ K 1- α m parameter to quantify the effect of crack tip constraints on the local yield stress in the cyclic plastic zone: m ≈ 2.5 for plain strain and m ≈ 1 for generalized plain stress n Paris’ exponent, defined in eq. (2) N f number of load cycles to failure of the material at the crack tip according to eq. (7) q exponent in the modified Coffin-Mansion-relation (7) R load ratio; R = K min / K max r pR length of the cyclic (“active”) plastic zone in the direction of the crack S(R) non-dimensional factor to quantify the R-effect on Cp, as defined in (2) α exponent in the definition of K*, determined by curve fitting ∆δ fatigue load of a crack in terms of crack tip opening displacement (CTOD) ∆ K th threshold value of ∆ K = K max - K min δ i crack tip opening displacement (CTOD) at initiation of ductile tearing δ min minimum CTOD of a load cycle δ max maximum CTOD of a load cycle σ f flow stress, defined as σ f = (R p0.2 + R m )/2, with R p0.2 and R m being yield and ultimate tensile stress