A novel measure for the material resistance to ductile fracture propagation under shear-dominated deformation

Journal Pre-proofs A novel measure for the material resistance to ductile fracture propagation under shear-dominated deformation Jingsi Jiao, Alexey Gervasyev, Cheng Lu, Frank Barbaro PII: DOI: Reference:

S0167-8442(20)30421-3 https://doi.org/10.1016/j.tafmec.2020.102845 TAFMEC 102845

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Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

25 June 2020 5 October 2020 19 November 2020

Please cite this article as: J. Jiao, A. Gervasyev, C. Lu, F. Barbaro, A novel measure for the material resistance to ductile fracture propagation under shear-dominated deformation, Theoretical and Applied Fracture Mechanics (2020), doi: https://doi.org/10.1016/j.tafmec.2020.102845

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A novel measure for the material resistance to ductile fracture propagation under shear-dominated deformation Jingsi Jiao*, Alexey Gervasyev, Cheng Lu, Frank Barbaro School of Mechanical, Materials, Mechatronic and Biomedical Engineering, Faculty of Engineering and Information Sciences, University of Wollongong, Australia Corresponding author email: [email protected]; [email protected]

Abstract. Our engineered metallic materials are being designed to be more and more ductile with an increase in strength. Consequently, the failure mode has changed – from brittle to ductile fracture, which leads to a limited applicability of the previously used parameters that quantify the material resistance developed for brittle fracture. This is owing to that a considerable amount of plastic strain is introduced to the fracture process. Shear-dominated ductile fracture failure is commonly observed in many of the structural metals. In the current work, a novel measure Omega 𝛺 is proposed, that fundamentally describes the material’s ability to motivate mass relative motion in response to Mode I shear-dominated fracture propagation. The effects of specimen geometry, fracture tip constraint and deformation modes on 𝛺 were discussed and analysed, which inspired the ideas of normalisation approaches that lead to the development of a dimensionless parameter Normalised Omega 𝛺. Accompanying with this concept, our systematically conducted experimental results (DWTT and SENT) indicated, that 𝛺 is independent to both in-plane and out-plane (specimen thickness) specimen configurations. So that it possesses the characteristic of being an intrinsic material constant for ductile fracture under shear mode. Keywords: DWTT; SENT; Ductile Fracture; 𝛺; 𝛺; Intrinsic Material Value.

1. Introduction Objects with mass in our world exist in space. This means that objects’ geometry is 3 dimensional. Material mechanics in general studies an object’s response to a force or load. Objects with different geometrical configurations have different responses to loading conditions. The concepts of strain and stress have been introduced aiming to eliminate or minimise the geometrical effect of the targeted object. So that the intrinsic material properties can be subtracted for a specific application. Taking the most commonly implemented uniaxial tensile test as an example, whose specimen is schematically shown in Fig. 1(a), the narrow region is set to be the gauge area so that most of the mechanical energy inputted from the testing machine is consumed at this volume of the material. In order to obtain its stress-strain property, the deformation along z-direction is captured and normalised by its original gauge length (the geometrical property in z-direction) for

strain; the uniaxial load is normalised by the gauge volume’s other geometrical features in x and y-directions (the cross-sectional area) so that the stress is obtained.

(a) (b) Fig. 1: Schematics of (a) a uniaxial tensile test specimen in xyz coordinate system; (b) three characteristics for a successful test approach – Effectiveness, Descriptiveness and Consistency. The stress-strain property of a uniaxial tensile test would be one of the most important macroscopic properties to evaluate the material’s mechanical performance. It could be considered as a material intrinsic value (at least close to) because, that the geometrical effect of the given test piece have been normalised in all three dimensions (particularly for true stressstrain concept). From this well-established testing method, three critical characteristics can be subtracted: 1. Effectiveness – majority of the input energy for testing is consumed by the nominated (gauge volume) mass – the narrower part of the specimen; 2. Descriptiveness – the nominated mass is deformed by a specified deformation mode – the uniform cylindrical shape enables an uniform tension; 3. Consistency – the nominated area can be determined and measured in a consistent fashion – tensile test is one of the most cost efficient test methods with well-established testing procedure. Fig. 1(b) aims to provide an opportunity to visualise the idea for better understanding. It would be evident that, the determination of the gauge volume is of essence for establishing a welldesigned testing approach. In order to appropriately determine the gauge area, all the three characteristics (labelled as 1, 2, and 3 in Fig. 1(b)) listed above need to be achieved simultaneously. This is illustrated by the overlapped area of 1, 2 and 3. This discussion will be continued when introducing the author’s approach to determine the gauge volume for the fracture tests in the current work.

Foregoing discussion leads a general topic, that is, to recover the intrinsic material properties for a given deformation mode. With the development of metallic materials, the ductile fracture becomes one of the major failure modes in many engineering applications (Xue and Wierzbicki 2009, Besson, McCowan et al. 2013, Lu and Wang 2018, Kõrgesaar 2019). In past decades, the fracture propagation in metallic materials under Mode I has been studied extensively, different fracture criteria have been proposed (Newman Jr, James et al. 2003). The critical crack-tipopening angle (CTOA) or displacement (CTOD) at a specified distance from the fracture tip was found to be the most suited for analysing the stable fracture propagation (Shih, DeLorenzi et al. 1979, Newman Jr, James et al. 2003). It is a directly measurable quantity (Lam, Kim et al. 2006) and has been standardised as a formal material parameter in the field of fracture mechanics (Lam, Kim et al. 2006). CTOA/D was experimentally measured and studied in numerous studies with different approaches under various testing conditions. Using a high-resolution photographic camera with a video system, Dawicke et al. (Dawicke and Sutton 1994, Dawicke, Newman et al. 1995) observed that in a thin-sheet aluminium alloy, the critical CTOA values were nearly constant during stable fracture propagation after initiation (see Fig. 2(a)), which went through a transition from tensile mode at fracture initiation stage to slant shear deformation mode at fracture propagation stage (Newman Jr 1998, Xue and Wierzbicki 2009), as illustrated in Fig. 2(b). In their studies, two different specimen configurations have been investigated – middle-crack tension that is primarily a tension specimen and compact tension that is primarily a bend specimen. It was shown that the critical CTOA was not sensitive to the specimen configurations. Moreover, in their work, it was shown that the critical CTOA obtained from middle-crack tension specimens with different sample widths remained constant. So that it was concluded that the measured critical CTOA values for a given material is independent of in-plane configuration and loading types (Newman Jr, James et al. 2003).

(a)

(b)

(c) Fig. 2: (a) CTOA remains constant during stable fracture propagation after a tearing at the fracture initiation stage (Dawicke, Newman et al. 1995); (b) the tensile mode for initiation, transition region, and the slant shear mode for fracture propagation (Newman Jr 1998); (c) CTOA decreases with increasing specimen thickness (Mahmoud and Lease 2003). Furthermore, several studies have been carried out to investigate the influence of ‘out-plane’ configuration – the thickness 𝐵 of the specimen – on the critical CTOA. In the work of (Mahmoud and Lease 2003) the sensitive study of critical CTOA with thickness range between 2 to 25 mm was conducted for the material of 2024-T351, using optical and digital image correlation system (DIC). It was found that the critical CTOA values decrease with increasing thickness as shown in Fig. 2(c), which is also in line with results from the works of (Newman Jr, Dawicke et al. 1992, Dawicke, Sutton et al. 1995). This effect was considered to relate to the constraint on the plastic deformation near the fracture. It was stated that the fracture tip constraint effect is one of the main influencing parameters in ductile fracture; the other is the amount of the plastic strain (Henry and Luxmoore 1997). The constraint effect can be literally understood as a structural or material obstacle against plastic deformation (Zhang, Xu et al. 2010), which is induced mainly by geometrical and physical boundary conditions (Yuan and Brocks 1998). The constraint effect is zero at the free surfaces so that the material deforms under plane stress condition; and it increases to its maximum at the centre to deform under plane strain condition (as schematically shown in Fig. 3(a)). As shown in Fig. 3(b), that different deformation conditions (plane stress/strain) result in various level of plastic straining that leads to different fracture speed (Zerbst, Heinimann et al. 2009). So that the tunnelling of the fracture is expected in thicker specimens. This geometrical constraint effect on CTOA/D was discussed by Newman Jr. et al (Newman Jr, James et al. 2003) who stated that, it is one of the two critical parameters for the function of CTOA/D – the other one is the absolute specimen thickness. The severity level of this constraint results in different fracture surface appearance of test specimens, which are schematically illustrated in Fig. 3(c). Thin specimens have low constraint levels, so that the failure is dominated by 45-degree slant shear deformation through the thickness (Newman Jr, James et al. 2003, Nahshon and Hutchinson 2008). Increasing

the thickness will increase the constraint level in the centre of the specimen that results in the plane strain deformation mode, which causes the flat fracture surface appearance shown in Fig. 3(c) of dotted areas in the medium and thick specimens.

(a)

(b)

(c) Fig. 3: (a) The schematic of the plastic deformation zone around the fracture tip and it varies along thickness owing to the geometrical constraint effect (Ewalds and Wanhill 1984); (b) The tunneling effect in a thicker specimen caused by the geometrical constraint (Hayes, Edwards et al. 2015); (c) the evaluation of fracture surface appearance with increasing the specimen thickness from plane stress to plane strain conditions (Mouritz 2012). Based on the foregoing discussions, the well-recognised characteristics of the CTOA/D approach can be summarised as below: 1. The CTOA/D at the fracture initiation stage is large due to tear deformation mode, and reduces dramatically to a much smaller constant value when the fracture propagates stably in the shear mode; 2. CTOA/D is independent to the in-plane configuration of specimen and is a function of the absolute specimen thickness and the crack-front constraint; 3. It is an in-situ measure that can describes the fracture propagation process, so that it requires well-calibrated testing devices such as high-resolution photographic camera, a video system, or a DIC system, which would greatly raise the testing cost and require very skilled operators.

Arguably, critical CTOA/D is a successful measure for ductile fracture propagation. It can be intuitively understood as the measure that relates to the quantification of the strain – for a given thickness more CTOA/D means that more deformation is required for separating the specimen. However, with a high testing cost and lack of experienced operators, CTOA/D approach would not be an applicable approach for all the research institutes. Current work proposes a potentially more cost-effective approach to measure the material properties with respect to the ductile fracture propagation in shear mode. A series of systematically conducted testing results from drop weight tear test (DWTT) and dynamic single edge notched tensile (SENT) test were used to develop and verify our proposed material measure 𝛺 [mm-1]. The concept 𝛺 allows a direct quantification of the geometrical effects from different specimen thicknesses, which leads to the development of a normalised dimensionless quantity 𝛺 that was observed to be independent to the specimen thickness. This approach would lead a small step closer to appreciate the material’s intrinsic performance when resisting ductile fracture propagation in shear dominated mode.

2. The introduction of Omega concept 2.1.

Physical foundation of Omega

The 𝛺 concept was initially inspired by the phenomena that if plastically stretch a cylindrical metallic solid, the more severe the deformation is, the more the surface area (of the cylindrical shape) it will generate. This is illustrated with Fig. 4, the surface area (excluding the ends) of the left cylindrical solid can be calculated with 𝑆1 = 2𝜋ℎ1𝑟1, in where ℎ and 𝑟 are the height and radius respectively. Under a constant volume assumption, the surface area of the right volume 𝑟1

can be calculated with 𝑆2 = 2𝜋ℎ1𝑟1 ∙ 𝑟2. Because that 𝑟1 > 𝑟2, 𝑆2 is larger than 𝑆1. This increase of area is quantified without including any material properties. It is critical to note that this property is geometrically necessary for adopting the shape change. The ratio of the surface area to the affected volume might be therefore used as an alternative indicator to describe the severity of the deformation for this particular case. The increase of the cylindrical surface area is essentially related to the permanent relative mass motion within the material of the specimen, which might be categorised as plastic strain. Plastic strain is realised by the motion of the dislocations (Cotterell 1953) that in microscopic scale also generate extra surface areas for the grain volume once they are released when reaching the grain boundary during plastic flow.

Fig. 4: Stretching a cylindrical volume under idealized geometrical assumptions. One critical advantage of implementing the concept of the surface area to volume ratio for the fracture process is that, the concept is potentially a 3-dimensional measure of the plastic deformation. So that it might be more suitable to describe the fracture phenomena that possesses complex strain history with irregular 3-dimensional specimen geometry (after broken). The current work is orientated based on seeking a suitable quantification approach using this surface area to volume ratio concept for ductile fracture process, which fundamentally quantifies the relative mass motion of the material. In the following section, a series of systematically conducted experimental trials including DWTT (drop weight tear test) and dynamic SENT (singe edged notch test) was introduced, with the aims of assisting and verifying the development of this novel measure. 2.2.

Experiment setups and data processing approach

2.2.1. Drop-weight tear test The experiments were conducted at the Tubular Goods Research Institute in China using a pendulum-type DWTT machine. Its appearance and testing schematic are shown in Fig. 5(a) and (b) respectively. The impact capacity of the machine can achieve up to 50 kJ, which is stored in form of the potential energy of a punch, and inputted to the testing material with a transit impact at the back the specimen while the two ends of the specimen were supported (see Fig. 5(b)). In order to effectively initiate the fracture, chevron notch is implemented for all the specimens as shown in Fig. 5(c) with the specimen thickness 𝐵 and the width 𝑊. In the current work nine different pipeline steels have been tested. Specimens were machined from industrially produced line pipes. DWTT specimens either were full-thickness, i.e. their thickness was equal to the thickness of corresponding pipe, or were reduced to half-thickness. The specimens are labelled from Steel No.1 to Steel No. 9, and their main dimensions are listed in Table 1. All of the specimen’s fracture surfaces appear to have the characteristics of ductile fracture. Some of the broken specimen’s surfaces are shown in Fig. 5(d).

(a)

(b)

(c) (d) Fig. 5: (a) 50 kJ special-designed DWTT machine in the testing laboratory of TGRI China; (b) the schematic of the machine; (c) configurations of the specimen and notch information; (d) the broken specimen samples after DWTT. Different from CTOA tests, the experimental data for the current study was collected after testing. Using a high precision 3 dimensional laser scanner, the surface profiles of the broken samples can be acquired (demonstrated in Fig. 6(a)). The accuracy of the device was up to 50 micrometres. Detailed information of scanned broken parts is demonstrated in Fig. 6(b) and (c) from different viewing directions. In line with the description for the fracture process (Newman Jr 1998), who pointed out the “flat-to-slant” transition – for a facture developed from the initiation stage that was in tensile mode, to the stable propagation stage that is in the 45-degree shear mode. Our scanned data correlates with this description very well: after the impact at the back of the specimen, the fracture is initiated at the tip of the chevron notch under tensile deformation mode and this is reflected by the appearance of “Tensile mode fracture surface” indicated in Fig. 6(b) and (c); after a period of transition, the tensile mode transferred to a shear mode reflected by the approx. 45 degrees shear surface labelled as “Shear mode fracture surface” in Fig. 6(b) and (c); finally, while the fracture tip approaching to the back impacted area where the specimen is thickened, the fracture path was deviated. This impact affected area is not considered in the analysis owing to the distortion of the materials and the loss of the material’s structural integrity.

This qualitative description for the fracture surface appearances leads to an idea, that the fracture process might be quantified by properly studying the 3D scanning data. Table 1: The configurations of the DWTT specimens for nine different pipeline steels. Steel code

Thickness 𝑩 [mm]

Width 𝑾 [mm]

Steel No. 1

20.6/10.3

72

Steel No. 2

20.6/10.3

72

Steel No. 3

20.6/10.3

72

Steel No. 4

20.6/10.3

72

Steel No. 5

20.6/10.3

72

Steel No. 6

27.7/13.8

72

Steel No. 7

27.7/13.8

72

Steel No. 8

27.7/13.8

72

Steel No. 9

18.4/10

72

(a)

(b) (c) Fig. 6: (a) Using high resolution 3D laser scanner to collect the geometrical information of broken specimens; (b) and (c) are different views of a broken specimen – a description of the fracture process from initiation to stable propagation and to the deviation of the fracture path – based on the characteristics of the fracture surfaces. We have chosen a 3D graphics software Autodesk NetFabb to process our 3D scanning data owing to its computational effectiveness and its applicability to our analysis requirements. The software provided the function of slicing the scanned parts based on a pre-defined incremental length 𝑛. It outputted the contour profiles of the sliced cross sections, as well as the quantitative information of total contour length and contour area with a 100 micrometre precision. An example of the slicing profiles can be demonstrated in Fig. 7 – for illustrative purpose the two broken part 1 and 2 are aligned to each other in Fig. 7(a) – using the slicing function, the broken specimens were incrementally sliced from the bottom (Section 1-1) to the top (Section n-n) in direction z. So that the parts’ cross sectional information on the x-y plane can be obtained; and they are shown in Fig. 7(b). In Fig. 7(b) different fracture stages can be observed; and it is corresponding to the previous qualitative description: for Section 1-1 a flat surface is observed owing to the position is at the notch surface that was not deformed as also shown in Fig. 6(b) and (c); with slicing proceeds (so does the fracture process), a tensile-mode fracture surface appears; then swiftly transfers to 45-degree shear fracture mode. It can be seen that this shear mode remained a consistent appearance until the slicing approached to the back of the specimen at Section n-n. This slicing approach showed its potential to describe the entire fracture process after testing.

(a)

(b) Fig. 7: (a) Using Autodesk NetFabb to slice of the broken specimens along z-direction for collecting the cross-sectional information on x-y plane; (b) the sliced cross-sectional information that describes the entire fracture process. (figure a and b are in the different dimensional scales) It can be observed in Fig. 7(b), that majority of the material is not associated with the fracture process and only has a role of structural support, a rational strategy for the determination of the gauge area is therefore needed. The author’s proposed approach for the gauge area can be demonstrated in Fig. 8(a), by sectioning the contour profiles with the plane that is coincident to the notch surface, the portions on the left side were removed from the analysis because their irrelevance to the fracture deformation. So that the gauge area was determined as shown in Fig. 8(b), it was the material between the plane of the notch surface and the end tip of the broken specimen.

(a) (b) Fig. 8: The determination of the gauge area for fracture analysis (a) sectioning the fracture slicing profile with the notch surface; (b) the remaining fracture profiles are the nominated gauge areas. The current gauge area determination is in line with the three characteristics subtracted from well-established testing methods in the introduction section: firstly, the gauge area consumes the most of the inputted energy for the test – satisfies the requirement for Effectiveness. Secondly, the gauge area captures the characteristics of the targeted testing phenomenon – from fracture initiation to “flat-to-slant” transition then reaches stable propagation stage – it is therefore satisfies the requirement of Descriptiveness. Finally, the notch surface is used for determining the gauge area, whose feature is consistent and easily detectable among different specimens – the requirement of Consistency is therefore satisfied. The magnitude of omega 𝛺 (concept introduction is in Section 2.1) can be obtained with Equation 1, in where 𝐿𝑐𝑜𝑛 and 𝐴𝑐𝑜𝑛 are the contour length and area of the gauge area respectively as shown in Fig. 8(b); and they are automatically calculated and outputted by Autodesk NetFaab. As mentioned before, 𝑛 is the slicing step length and set to be 0.1 mm. 𝛺=

𝐿𝑐𝑜𝑛𝑛 𝐴𝑐𝑜𝑛𝑛

Equation 1

Fig. 9 shows a typical results of 𝛺 for from the DWTT specimens with respect to the length of the fracture propagation ∆𝑎: from 0 mm being at the tip of the chevron notch, to approx. 68 mm at the back of the specimen. There are three characteristics observed from the 𝛺-∆𝑎 diagram in Fig. 9. First, 𝛺 is outstandingly high at the beginning of the fracture process at 45 mm-1, which rapidly decreases to approx. 1 mm-1 when ∆𝑎 reaches around 15 mm. This behaviour of 𝛺 corresponds to the gauge area profile of “Initiation stage” in Fig. 8(b). Second, the value of 𝛺 remains constant

for ∆𝑎 between approx. 15 and 35 mm, which indicates a “Stable fracture propagation stage” in Fig. 8(b). Third, a range of fluctuated 𝛺 values are observed after ∆𝑎 reaches 40 mm, which indicates the fracture propagates into the back impacted affected zone as shown in Fig. 8(b). The result shown in 𝛺-∆𝑎 diagram clearly demonstrates its critical shared characteristics/connections with the well-established CTOA approach as introduced in the introduction section, which would indicate the sound physical basis of this novel concept. Similarly to the CTOA approach, the magnitude of 𝛺 is set to be the average 𝛺 value during the stable propagation – for the case in Fig. 9 is the average 𝛺 of ∆𝑎 ranging from 15 to 35 mm.

Fig. 9: The proposed 𝜴 approach quantitatively describes the fracture process of DWTT. 2.2.2. Dynamic single-edge notched tensile test The SENT tests were conducted using Instron Dynamic Testing Machine with the cross-head speed range up to 20 meter per second. The configuration of the specimen is demonstrated in Fig. 10. The specimens were cut by spark erosion from Steel No. 1 and Steel No. 2 (6 specimens from each).

Fig. 10: Specimen configuration for dynamic SENT tests of Steel No. 1 and Steel No. 2. During the SENT testing, a high-speed camera system was coupled to the machine during the dynamic tearing tests. It was observed that before the fracture propagated, the specimen was under tension for a short period of time. As loading continued the fracture initiated at the notched side, and this further increased the unsymmetrical condition of the specimen between the notched and un-notched sides, which resulted a torsion-like deformation on the specimen that caused a rotation of the specimen’s broken ends as shown in Fig. 11(a). After the completion of the tests, same specimen scanning and slicing approach of DWTT was carried out for SENT. The rotation of the specimen’s ends led to a misalignment of the fracture propagation path to the slicing direction-z. So that as shown in Fig. 11(a), a rigid body rotation was applied to the broken parts to ensure the fracture propagation path is coincident to the slicing z-direction as shown in Fig. 11(b).

Fig. 11: (a) The 3D scanned parts with unadjusted relative position; (b) with adjusted position corresponding to its vertical slicing path in z-direction. In line with the reasoning discussed for DWTT, identical gauge area determination and slicing approach were applied to the SENT tests. As shown in Fig. 12(a) that, the gauge area was determined by separating each of the broken parts with the notch surface plane that is parallel to the y-z plane and remove the undeformed portions. The section cuts from ① to ⑥ were marked at the positions shown in Fig. 12(a) and their profiles in x-y plane are demonstrated in

Fig. 12(b). It provided a comprehensive yet qualitative description about the fracture process from initiation stage of ① to ③ as tensile mode, then to previously discussed “flat-to-slant” (tensile to 45-degree-shear mode) transition of ④ and ⑤, finally reaches to ⑥ that appears to be consistence for a reasonably long propagation distance relative to the specimen’s configuration. Same as DWTT approach, this description can be also quantified with a 𝛺-∆𝑎 diagram.

(a) (b) Fig. 12: (a) The determination of the gauge area for SENT test – sectional slicing profiles from ① to ⑥ are shown in (b) that illustrates the fracture appearances from initiation stage (①②③) to “tensile-to-slant transition” stage (④⑤), finally reaches the stable fracture propagation stage ⑥. Fig. 13 shows a typical results of 𝛺 against ∆𝑎 for SENT. Besides the deformation mode difference of SENT (tension) to DWTT (bend), there was no back impact affected zone for the SENT specimens. Two characteristics can be observed in Fig. 13. Similarly to the results from DWTT, first, 𝛺 is also significantly higher at the beginning of the fracture process for approx. 75 mm-1; and also undergoes a rapidly decrease to approx. 5 mm-1 when ∆𝑎 is 2 mm. Second, the value of 𝛺 remains constant for the propagation length between approx. 2 and 7 mm, which corresponds to the “Stable fracture propagation stage” illustrated in Fig. 12 (b). The results related to the back impact affected area are deviated owing to the loss of the specimen structural integrity that is out the scope of our investigation. The 𝛺 value determination for SENT is identical to DWTT, for the case in Fig. 13 is the average 𝛺 of the ∆𝑎 ranging from 2 to 7 mm.

Fig. 13: The proposed 𝜴 approach quantitatively characterizes the fracture process of SENT.

3. Results and discussions In this section, firstly the 𝛺 results from DWTT and SENT tests were presented and discussed, which followed by the normalisation of 𝛺 with the purpose of normalising the effects of specimen geometry and low-level fracture tip constraint. So that finally, the normalised omega 𝛺 that is potentially an intrinsic material parameter to quantify the resistance of ductile fracture propagation was obtained. 3.1.

Absolute omega values

For DWTT and SENT tests, each of the specimens broke into two parts, which means that each test has two 𝛺 results. We propose two possible approaches to determine a single-valued 𝛺 for each test with corresponding reasoning as follows: 1. Using the average value of the two 𝛺 results from the two broken parts (the average approach). This approach could provide mean-valued results and reduce possible measurement variations; 2. Using the maximum value of the two 𝛺 results (the maximum approach). This approach could possibly detect the best performance of the specimen with respect to create relative mass motions (strains). The 𝛺 results from both of the approaches the average approach and the maximum approach are shown in Fig. 14(a) and (b) respectively. A nonlinear decrease trend of 𝛺 is observed with increasing specimen thickness 𝐵; as in Fig. 14(a) (the average value approach) from approx. 5.7 ± 0.7 mm-1 when 𝐵=2 mm reduces to approx. 0.5 mm-1 when 𝐵 is 27.7 mm. Similar trend was observed when studying the effect of thickness on CTOA, as shown in Fig. 2(b). Almost identical trend can be observed in Fig. 14(b) for the maximum value approach, except that this approach

leads to larger result variations across the tested thickness range; and this effect is the most severe at 𝐵=2 mm. With respect to CTOA concept it was explained as that, increasing thickness leads to the increase of the fracture tip constraint that impedes the development of plastic straining and finally leads to the reduction of CTOA. Presumably, similar interpretation could be applied to the current 𝛺 concept. However, unlike the CTOA/D approach, 𝛺 has its intrinsic/pure geometrical values for different specimen thicknesses. A proper interpretation cannot be conducted until Omega’s pure-geometry related value is carefully normalised.

(a)

(b) Fig. 14: The tested values of 𝜴 from different steels and testing methods (a) the average value approach and (b) the maximum value approach. 3.2.

Normalisation of omega

3.2.1. Pure geometrical value of 𝜴 and its normalisations The geometry of the 45-degree shear fracture process can be schematically illustrated with Fig. 15(a), in where 𝑐 is the maximum length of the shear lip during the propagation, 𝑎 is the partial

thickness corresponding to 𝑐. So that the geometrical 𝛺𝐺𝑒𝑜 can be calculated with Equation 2. In an idealised case the broken Part 1 and 2 are reversely symmetrical with 𝜃=45 degrees, so 𝑎 = 𝑐 = 𝐵/2. Then Equation 2 becomes Equation 3 that is used to calculate the ideal geometrical omega 𝛺𝐺𝑒𝑜_𝑖𝑑𝑒𝑎𝑙 that is a pure geometrical property. In Fig. 14(a) and (b), this ideal geometrical omega is plotted as solid lines. It can be observed that it decreases with increasing 𝐵. In order to reveal the material related properties, this geometrical related effect needs to be normalised.

(a) (b) Fig. 15: (a) Schematic of the shear surface appearance on the testing specimens; (b) real situation of the shear appearance. 𝛺𝐺𝑒𝑜 =

(𝑎 + 𝑐 +

𝑎2 + 𝑏2)𝑛

0.5𝑎𝑐𝑛

𝛺𝐺𝑒𝑜_𝑖𝑑𝑒𝑎𝑙 =

8(1 + 2) 𝐵

Equation 2

Equation 3

The first normalised results can be seen in Fig. 16(a) and (b) for the average value approach and the maximum value approach respectively. The vertical axis is the normalised value of 𝛺/ 𝛺𝐺𝑒𝑜_𝑖𝑑𝑒𝑎𝑙 (a dimensionless quantity), which is designed to remove the idealised geometrical effect of 𝛺 obtained across a wide range of thickness. The steep slope of the absolute-valued 𝛺 against thickness observed in Fig. 14 is now significantly smoothed in Fig. 16 with a linear manner. Both Fig. 16(a) and (b) show an increase trend of 𝛺/𝛺𝐺𝑒𝑜_𝑖𝑑𝑒𝑎𝑙 with 𝐵 – the only major difference between the two approaches is that for all given thicknesses, the maximum value approach in Fig. 16(b) shows a wider result variation range comparing to the average value approach in Fig. 16(a).

(a)

(b) Fig. 16: The results of first normalization attempt 𝜴/𝜴𝑮𝒆𝒐_𝒊𝒅𝒆𝒂𝒍 from different steels and testing methods (a) the average value approach and (b) the maximum value approach. It is not a straight forward task to interpret the trend showed in Fig. 16. As for the CTOA/D concept, it decreases with increasing thickness increases owing to that thicker specimen is subject to more severe fracture tip constraint effect. In other words, the thicker the specimen is, the higher level of impediment of plastic straining could be. As a measure that fundamentally quantifies the deformation, 𝛺 would process a similar physical meaning to CTOA/D. So that the increase trend shown in Fig. 16 cannot be explained as per fracture tip constraint. In fact, the 45-degree shear lip appearance (as a necessary condition in all of our analyses) indicates that the geometrical constraint effect during the fracture process is at low level. The normalisation component 𝛺𝐺𝑒𝑜_𝑖𝑑𝑒𝑎𝑙 is a function of a pure geometrical parameter 𝐵 (Equation 3) out of an idealised case as illustrated in Fig. 15(a). The results in Fig. 16 implied that the idealised geometrical property may not be a proper assumption to calculate the geometrical effect for the 𝛺 concept. Measurements about 𝑎 and 𝑐 in all the specimens (see Fig. 15(b)) were

therefore conducted to check the reliability of the ideal fracture surface appearance assumption. Based on the measured 𝑎 and 𝑐, 𝜃 can be obtained with considering a right angle geometry. The results in Fig. 17(a) and (b) are the measured values of 𝑎 and 𝑐 respectively with respect to the specimen thickness. Both 𝑎 and 𝑐 increase linearly with 𝐵 as expected with a significant amount of result deviation. Once converted 𝑎 and 𝑐 into 𝜃 as shown in Fig. 17(c), the deviation reduces to approx. 5 ~ 10% from mean values across the whole range of 𝐵. It is shown that with increasing the specimen thickness 𝐵 from 2 to 27.7 mm, the fracture shear lip angle 𝜃 reduces from 50 degrees to approx. 40 degrees. A second order polynomial fitting equation is applied to describe to this behaviour. It is therefore understood that previous idealised geometrical assumption of the 45-degree shear lip angle is not sufficient to fully quantify the geometrical properties of the 𝛺 concept for varying thicknesses.

(a)

(b)

(c) Fig. 17: The geometrical properties of the tested specimens for (a) 𝑎; (b) 𝑐; and (c) 𝜃. 𝜃 = 0.0453𝐵2 ― 1.5454𝐵 + 53.803

Equation 4

3.2.2. Normalisations of 𝜴 based on real geometrical properties As forgoing discussions, with Equation 2, the absolute-valued 𝛺 is normalised with its corresponding real shear lip geometrical properties (𝑎 and 𝑏 in Fig. 15(b)). As can be observed in Fig. 18(a) and (b) for the average and maximum approach respectively, a nearly horizontal trend (at 𝛺/𝛺𝐺𝑒𝑜=1) is constructed by the results from the whole thickness range and from different test methods (DWTT for bend and SENT for tensile). Similar to previous results, the major different between the average and maximum approach is that the maximum approach leads to more result deviations and an outstanding outlier at 𝐵=13.8 mm.

(a)

(b) Fig. 18: the normalized 𝜴 (𝜴/𝜴𝑮𝒆𝒐) from different steels and testing methods (a) the average value approach and (b) the maximum value approach. This horizontal trend leads to an important indication that, with the current approach the geometrical effect from specimen thickness is fully normalised. Furthermore, as discussed that our study scope has been focused on the specimens that have the slant shear lip appearance,

which results in plane stress condition. So that the fracture tip constraint effect is contained at a low level, and could be neglected for our current work. Even if the effect could not be neglected, we consider this low-level fracture constraint effect is included in the current normalisation approach with integrating the real geometrical properties (𝑎 and 𝑐 or 𝜃). In other words, 𝜃 is considered to be the geometrical representative of the shear lip appearance, which is an important characteristic to quantify the fracture tip constraint effect in the current work. So that an easily accessible material resistance parameter to ductile fracture propagation under sheardominated fracture mode is proposed as normalised Omega 𝛺 (𝛺/𝛺𝐺𝑒𝑜). It is a dimensionless quantity, which quantifies the material’s ability to activate its relative mass motions with respect to shear facture. The higher the value is, the better the performance of the metallic material will have.

4. Conclusions and Future works The engineering performance of a material is a function of its intrinsic material properties as well as its geometrical properties. The material properties are related to the geometrical properties because that, different geometries can trigger different deformation modes for the material to deform. On one hand, for engineering applications, it is necessary to design geometries for the small scale tests that are close to their practical application with certain deformation modes. On the other hand, it is also important to normalise the geometrical effect from different specimen configurations to reveal its intrinsic material properties on this application. Therefore a successful testing approach needs: a) A suitable specimen geometrical configurations that enables it to deform in a desired deformation mode; b) An appropriate approach to normalise/minimize the geometrical effect that may overshadow the intrinsic material properties to this specific deformation mode. The current work is constructed based on above logics and can be summarised as follow: 1. Three characteristics from well-established uniaxial tensile test were subtracted as: Effectiveness, Descriptiveness, and Consistency, which served as guidelines to determine the gauge area for our proposed 𝛺 approach; 2. With the assistance of a series of systematically conducted dynamic tests for DWTT and SENT, the concept of 𝛺 (Equation 1) was proposed that fundamentally describes materials’ ability of activating relative mass motion with respect to the shear dominated fracture propagation; 3. The 𝛺 concept possesses important characteristics that were found in the well-established CTOA/D approach, which would present a solid case for this concept to be physically sound;

4. It was shown that the pure/intrinsic geometrical effect on 𝛺 (Equation 3) with respect to different specimen thicknesses can be normalised, however, it provided unsatisfying results owing to that the assumption for the slant shear to be perfectly 45-degree was too coarse; and this was verified with the measured angle 𝜃 for varying specimen thicknesses (Equation 4); 5. Finally, with including the measured geometrical properties of the shear lip (Equation 2), the normalised Omega 𝛺 (𝛺/𝛺𝐺𝑒𝑜) was proposed as a dimensionless quantity that is potentially an intrinsic material property – it was observed from our experimental results that 𝛺 is independent to both in-plane and out-plane configurations of the specimens. Future works originated from the current proposed concept are listed as follows: 1. The connections between microscopic features of the fracture surfaces to the normalised Omega is to be analysed and quantified; 2. The relationship between ductility and the normalised Omega is to be investigated; 3. A systematic parametric study of normalised Omega to a wide range of strain rate (fracture speed) is to be conducted; 4. Numerical studies for implementing Omega concept as a criteria for thin-sheet (low triaxiality) fracture initiation and propagation are to be conducted.

Acknowledgements The authors would like to express their gratitude for the financial support provided by CBMM Technology Suisse. The kind assistances to conduct DWTT experiments from Dr. Chunying Huo, Dr. He Li, Dr. Xiongxiong Gao, and Ms. Na Li at CNPC Tubular Goods Research Institute are greatly appreciated.

Conflict of interest statement The authors declare that they have no conflict of interest.

Authors contributions Jiao: Conceptualization, Methodology, Formal Analysis, Resources, Investigation, Writing – Original draft preparation, Review & Editing. Gervasyev: Resources, Writing – Review & Editing, Investigation. Lu: Project administration. Barbaro: Resources, Funding acquisition.

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Highlights 1. A novel parameter of 𝛺 [mm-1] was proposed that fundamentally describes materials’ ability of activating relative mass motion with respect to the shear-dominated fracture propagation; 2. The 𝛺 concept possesses important characteristics that were found in the well-established CTOA/D approach; 3. With a proper normalisation strategy, a dimensionless quantity – Normalised Omega 𝛺 was proposed that is independent to both in-plane and out-plane configurations of the specimens.

Author statement

Jiao: Conceptualization, Methodology, Formal Analysis, Resources, Writing – Original draft preparation, Review & Editing, Investigation. Gervasyev: Resources, Writing – Review & Editing, Investigation. Lu: Project administration. Barbaro: Resources, Funding acquisition.