PSI - Issue 68

Josef Arthur Schönherr et al. / Procedia Structural Integrity 68 (2025) 425–431 J. A. Scho¨nherr et al. / Structural Integrity Procedia 00 (2024) 000–000

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materials, Francis et al. (2006), leading to a reduced creep strength. Defects and flaws associated with welded joints potentially determine the service life. Therefore, a fracture mechanics based component assessment is reasonable.

2. The C ∗ parameter

Introduced back in 1976 by Landes et al. (1976), the C ∗ parameter is a commonly used quantity to describe the crack tip load under steady state creep conditions, Nikbin et al. (1986); Saxena (2015); BS 7910 (2019); Ewald et el. (2019). Following the concepts of Fracture Mechanics, steady state creep is reached when the stress field around the crack tip is not subjected to changes and therefore the deformation is dominated by secondary creep. The material behavior in secondary creep can be described by the Norton model, ˙ ε = A · σ n , (1) relating the creep strain rate ˙ ε to the stress σ , incorporating the factor A and exponent n as model parameters. The C ∗ parameter allows for characterizing the stress and creep strain rate fields around the crack tip and is used for correlating data of creep crack initiation and propagation tests of various geometries. It is therefore used to bridge the gap between lab tests, conducted with standardized specimen geometries, to actual applications. The original formulation of the C ∗ parameter has been developed on the basis of the J contour integral formulation of Rice (1968), where the strains and displacements were replaced by their rates and therefore C ∗ is based on the energy density rate W ∗ (which itself depends on the strain rates) and displacement rates ˙ u i as C ∗ = Γ W ∗ d y − T i ∂ ˙ u i ∂ x d s . (2) In general, the C ∗ integral is assumed to be path independent, Landes et al. (1976), such as the J integral. Note however that this is linked to several assumptions made, such as the material homogeneity, Kienzler (1993), and is therefore, strictly speaking, not applicable to the assessment of cracks in welded joints exhibiting localized changes in the material behavior. The finite element software Abaqus Standard integrates a formulation of the C t integral of Saxena (1986) with some adaptions for usage in FE, Dassault Systemes (2022). For steady-state creep, the C t integral solution translates into the C ∗ integral. Therefore, Abaqus Standard allows for evaluating the C ∗ integral numerically for materials following the Norton power-law relation under the premise of a su ffi cient long simulation time to obtain steady-state creep conditions. According to the Abaqus documentation, it is favorable to evaluate multiple contour integrals and calculate the C t value by extrapolating the contour results to the very crack tip. 2.1. Contour integral formulation

2.2. ASTM E1457 method

The standard ASTM E1457-19e1 (2019) encompasses formulae to calculate C ∗ for various standardized specimen geometries, including those manufactured from welded joints. C ∗ follows as

F ˙ v cr B N ( W − a )

C ∗ =

H η,

(3)

where F is the applied force, ˙ v cr the creep part of the displacement rate, B N the (net) specimen thickness, W the specimen width and a the crack length. Depending on the specimen type and location of the creep displacement rate measurement, H is determined as a function of the Norton exponent n . The factor η is tabulated for di ff erent com binations of specimen type and location of the creep displacement rate measurement as a function of the normalized crack length a / W and the weld mismatch ratio M . The underlying data were obtained from Zhou et al. (2014) by finite element simulations of bi-material specimens comprising the weld material (WM) and the base material (BM). Both materials were given the same n . The HAZ was not considered in these simulations.