PSI - Issue 60

N. Khandelwal et al. / Procedia Structural Integrity 60 (2024) 582–590 Author name / StructuralIntegrity Procedia 00 (2019) 000 – 000

583

2

the material is subjected to large scale yielding with plastic strains of the order of 2 – 3% in which the ductility and the crack growth resistance of the material are important challenges to manage (Berg et al., 2008). Stresses in the pipe observed to be membrane type even for pipes subjected to bending, results in the increase in the fracture toughness by lowering down the constraint near to crack-tip stress field (Chiesa et al., 2001). Apart from that, the crack propagation in gas transmission pipelines subjected to internal gas pressure also represents a very complex phenomenon and imposes a challenge concerning with the structural integrity. Gas flowing out from the fracture opening, once a crack initiates in the pipeline, leads to a decompression front formation, and the cracks start propagating in both directions from the initiation site (Xue et al., 2021). Various experiments have been performed to investigate the fatigue and fracture behaviour involving fatigue crack initiation, fatigue crack growth and fracture resistance behaviour of the components, which in turn useful for accurate prediction of the remaining life of the piping component. In this regard, Chattopadhyay et al. (2010) conducted fracture tests on three-point bend (TPB) specimens and through-wall circumferentially cracked straight pipes having outer diameters of 219 and 406 mm. Sobel and Newman (1986), Dhalla (1987) reported experimental data on the behavior and strength of steel elbows under monotonic loading conditions. Performing a number of experimental trials is always costly in terms of cost and time. Hence, researchers have worked extensively towards numerical simulation of the crack behaviour to analyse fracture of pipe specimen under diverse loading types (monotonic and cyclic loading). In recent times, phase field method based on variational principle emerged as an competitive numerical tool to simulate the fracture problem, because of its ability to identify crack initiation and propagation within a single framework for simple and complicated structures (Bourdin et al., 2008). Phase field method introduces two parameters namely; a phase field parameter ( ϕ ) that varies between 0 and 1 differentiating between broken and intact material, and a length scale parameter (l_0) that approximates the sharp crack into the diffused crack with exponential distribution in numerical formulation. The mathematical foundation of phase field was firstly introduced by Francfort and Marigo (1998), later Bourdin et al. (2000), Amor et al. (2009) and Miehe et al. (2010) presented a regularised numerical version of PFM, which was easy to implement in commercial FE software for quasi-static brittle fracture. PFM for brittle fracture has been a topic of very intense research in the past few years. However, very limited studies were reported for the extension to ductile fracture. Duda et al. (2015) proposed a phase field model for plastic solids where limited plasticity assumed near to the crack tip. Recently, Khandelwal and Murthy (2023) performed a parametric study for various phase field models comprised of various geometric functions and degraded functions in ductile fracture. In this paper, numerical fracture studies based on PFM are carried out on pipes made of SA312 Type 304LN stainless steel under four-point loading. Numerical results thus obtained are validated with the corresponding experimental results obtained in CSIR- SERC laboratory (Vishnuvardhan et al., 2010a, 2010b). 2. Mathematical formulation Phase field formulation is an energy based method originated from the variation formulation where free energy ( ), which comprised of strain energy ( ) and fracture energy ( ), is minimized w.r.t displacement field variable ( ) and phase field parameter ( ) [representing = complete broken state and = as totally intact state of the material]. For a ductile material, strain energy ( ) can be decomposed into elastic ( ) and plastic part ( ) as presented in Eq. (1). = + = + + (1) Fracture energy ( ) of thin crack ( ) as shown in Fig. 1 can be approximated into diffuse crack by regularized volume integral as per Eq. (2). ( )=∫ ≈ ∫ 2 0 [ 2 + 0 2 | | 2 ] (2)