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Evaluation of Localized Necking Models for Fracture Prediction in Punch-Loaded Steel Panels

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Evaluation of Localized Necking Models for Fracture Prediction in Punch-Loaded Steel Panels Journal of Marine Science and Engineering Article Evaluation of Localized Necking Models for Fracture Prediction in Punch-Loaded Steel Panels Burak Can Cerik 1 , Kangsu Lee 2 and Joonmo Choung 1,* 1 Department of Naval Architecture and Ocean Engineering, Inha University, Incheon 22212, Korea; [email protected] 2 Offshore Plant Research Division, Korea Research Institute of Ships and Ocean Engineering, Daejeon 34103, Korea; [email protected] * Correspondence: [email protected] Abstract: This study compared the experimental test results on punch-loaded unstiffened and stiffened panels with numerical predictions using different localized necking modeling approaches with shell elements. The analytical models that were derived by Bressan–Williams–Hill (BWH) were used in their original form and extended version, which considers non-proportional loading paths while using the forming-severity concept and bending-induced suppression of through-thickness necking. The results suggest that the mesh size sensitivity depends on the punch geometry. Moreover, the inclusion of bending effects and the use of the forming-severity concept in the BWH criterion yielded improved estimations of fracture initiation with shell elements. Keywords: ductile fracture; punching; localized necking; collision; fracture strain; shell element 1. Introduction The assessment of a ship’s hull structure to withstand collision and stranding loads has traditionally been based on the application of a limit equivalent (effective) plastic strain Citation: Cerik, B.C.; Lee, K.; in non-linear finite element analysis, without any explicit considerations of the structural Choung, J. Evaluation of Localized processes, including through-thickness necking and associated ductile fracture. This ap- Necking Models for Fracture proach, which is commonly coupled with mesh/element size-based scaling, has a certain Prediction in Punch-Loaded Steel accuracy when applied to predict plating rupture [1]. On the other hand, practitioners Panels. J. Mar. Sci. Eng. 2021, 9, 117. https://doi.org/10.3390/jmse9020117 have generally recognized the need for more accurate fracture prediction methods, which are applicable to shell elements, involving calibration that can be performed readily while Academic Editor: José A.F.O. Correia using the available hardening properties of the material and considering the likely failure Received: 24 December 2020 modes [2–4]. Recent attention has been given to characterizing the ductile fracture response Accepted: 21 January 2021 of marine structural steels [5–8] and simulating fracture with shell elements in maritime Published: 25 January 2021 crash scenarios while using advanced fracture models [9–16]. Through-thickness neck- ing and subsequent fracture is the dominant failure mechanism in punch-loaded plated Publisher’s Note: MDPI stays neu- structures, as observed in the later studies. tral with regard to jurisdictional clai- A serious obstacle to the accurate simulations of actual fracture phenomenon induced ms in published maps and institutio- by localized necking using shell elements is the plane-stress condition assumption and nal affiliations. mesh size effects in the post-necking regime. Adopting a forming limit curve (FLC) with the assumptions on the post-necking response and mesh size effect is a straightforward approach for predicting this failure mechanism with shell elements [10,17,18]. Research related to fracture prediction with shell elements [12,13,19] confirmed that the assumption Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. of incipient necking as a failure condition for plates under biaxial membrane stretching is This article is an open access article reasonable, provided that an element size that is several multiples of the plate thickness distributed under the terms and con- is used. Nevertheless, challenges remain when using FLCs, because the finite element ditions of the Creative Commons At- analysis predictions tend to be sensitive to the numerical implementation of the localized tribution (CC BY) license (https:// necking condition [13,20]. creativecommons.org/licenses/by/ This study examined the influence of different numerical implementations of FLCs 4.0/). on fracture predictions with shell elements in punch-loaded stiffened panel simulations. J. Mar. Sci. Eng. 2021, 9, 117. https://doi.org/10.3390/jmse9020117 https://www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2021, 9, 117 2 of 13 The well-known BWH (Bressan–Williams–Hill) criterion by Alsos et al. [18] was used as a reference. Punching tests on unstiffened and stiffened panels reported in the recent literature were used to evaluate the predictions of the instance of fracture. Furthermore, the influence of the bending and loading history effects are discussed. 2. Localized Necking Prediction 2.1. Preliminaries In this study, the J2 plasticity with the associative flow rule and isotropic strain hardening was assumed. The plane stress condition holds, because shell elements are used. The von Mises yield condition can be expressed as f [σ, k] = s¯ − k = 0 (1) where k is the deformation resistance and s¯ is the von Mises equivalent stress under the plane stress condition, which is expressed in terms of the principal stresses (s1, s2), as follows: q 2 2 s¯ = s1 + s2 − s1s2 (2) As the deformation resistance function, a modified version of the Swift hardening law [21] was adopted, as follows: ( s0 if #¯p ≤ #¯L k #¯p = n (3) K #0 + #¯p if #¯p > #¯L where s0 is the initial flow stress, #¯L is Lüders plateau strain, and K, n, and #0 are the Swift law parameters. The equivalent plastic strain, #¯p is defined as the work-conjugate of the equivalent stress, s¯ . Note that strain rate effect is omitted in the present study, because it has a marginal influence on global structural response and fracture initiation in low-velocity impact problems, such as ship collisions [13]. The stress state or loading path is usually described while using the ratio of principal plastic strains (#1, #2): # a = 2 (4) #1 Alternatively, the principal stress ratio may be used, which is defined as s b = 2 (5) s1 Following the assumption of J2 plasticity with the associative flow rule and propor- tional loading, it may be shown that the following relation holds between a and b: 2b − 1 a = (6) 2 − b and, inversely, 2a + 1 b = (7) 2 + a Furthermore, under plane stress condition, stress triaxiality is defined as s s + s h = m = 1 2 (8) s¯ 3s¯ where sm is the mean stress. Note that s3 = 0. J. Mar. Sci. Eng. 2021, 9, 117 3 of 13 The relationship between the stress triaxiality and other stress state parameters, a and b, can be written, as follows: a + 1 b + 1 h = p = p (9) 3(a2 + a + 1) 3 b2 − b + 1 Similarly, an inverse relationship between a and h can be written as 8 p 0, if h = 1/ 3 < p p a = − 3h2+ 3h −(3h−2(3h+2))−2 (10) : 2(3h2−1) , otherwise The principal major strain, #1, can be transformed into the equivalent plastic strain, as follows: 2#1 p 2 #¯p = p 1 + a + a (11) 3 The last useful equation is between the equivalent von Mises stress and major principal stress, s1, which is given, as follows: s¯ s1 = p (12) 1 − b + b2 2.2. Bressan-Williams-Hill Model The BWH model forwarded by Alsos et al. [18] to predict fractures with shell elements combines Hill’s localized necking criterion, which is valid in the second quadrant of the forming limit diagram (−0.5 < a ≤ 0), and Bressan and Williams’ shear instability- based localized necking criterion [3], which is used for the first quadrant (0 ≤ a < 1.0). Alsos et al. [18] presented this criterion in a rather unconventional space of the principal stress-principal strain ratio, as follows: 8 p > p2K p1+0.5a p2 #ˆ1 1 + a + a2 , for − 0.5 ≤ a ≤ 0 > 3 1+a+a2 3 1+a < n s1,n[a] = p2 #ˆ (13) 2K 3 1 > p q , for 0 ≤ a ≤ 1 :> 3 a 2 1−( 2+a ) where #ˆ1 is a major necking strain that corresponds to the plain strain tension state (a = 0). While using a power-law hardening function (Holloman type), it can be assumed to be the hardening exponent n. In the above equation, K and n are the power law hardening pa- rameters. The BWH criterion can be transformed into the spaces, (s1, s2), (#¯p, a), and (#¯p, h), while using the transformation equations presented in the previous subsection. Figure1 shows the plots of the BWH criterion for mild steel in various variable spaces. Based on the arguments of Stoughton [22,23], Alsos et al. [18] suggested that a stress-based forming limit diagram (FLSD) is less sensitive to the non-proportionality of strain (loading) paths and advocated while using the BWH criterion expressed in the space of (s1, a). If the stress state, which is characterized by either stress triaxiality (h) or principal plastic strain ratio (a), remains constant to failure. The loading path (strain path) is then said to be proportional. If the stress triaxiality changes throughout the loading, the loading path is non-proportional. Hence, a proportional loading path means that the direction of the stress vector does not change. In the original finite element numerical implementation of the BWH criterion, incipient necking is assumed to occur if the major principal stress at the mid-layer through-thickness integration point of a shell element satisfies the following condition: s 1 ≥ 1 (14) s1, n[a] J. Mar. Sci. Eng. 2021, 9, 117 4 of 13 Note that the formulation does not consider the stress and strain histories, because it only refers to the current stress state and major principal stress. The failure condition is checked at the mid-layer of a shell element in order to ensure that only membrane loading is considered. Therefore, in its original form, the BWH criterion is a local-failure criterion for shell elements.
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