Validation of the Two-Parameter Fracture Criterion using finite-element analyses with the critical CTOA fracture criterion

Engineering Fracture Mechanics 136 (2015) 131–141

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Validation of the Two-Parameter Fracture Criterion using finite-element analyses with the critical CTOA fracture criterion J.C. Newman Jr. a,⇑, J.C. Newman III b a b

Department of Aerospace Engineering, Mississippi State University, MS 39762, United States Department of Computational Engineering, University of Tennessee-Chattanooga, Chattanooga, TN 37403, United States

a r t i c l e

i n f o

Article history: Received 26 May 2014 Received in revised form 20 January 2015 Accepted 26 January 2015 Available online 7 February 2015 Keywords: Fracture Aluminum alloys Plasticity Cracks Stress-intensity factor Finite elements CTOA

a b s t r a c t Elastic–plastic finite-element analyses employing the critical crack-tip-opening-angle (CTOA) fracture criterion were used to simulate fracture on thin-sheet middle-crack-tension, M(T), specimens made of several aluminum alloys over an extremely large range of widths (18–5700-mm) for a wide range in crack-length-to-width ratios. A linear relation between elastic stress-intensity factor (KIe) and net-section stress (Sn) at failure was observed for Sn less than yield stress of the material. Test data on two aluminum alloys (2024-T3 and 2219-T87) over a wide range of M(T) specimen widths (76–600-mm and 76–1200-mm, respectively) supported the linear relationship that validated the Two-Parameter Fracture Criterion. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The concepts of linear-elastic fracture mechanics (LEFM) have been very useful in correlating fracture data for cracked plates and structural components in which the crack-tip plastic deformations are constrained to small regions (plane-strain fracture [1]). However, for high-toughness sheet materials where stable tearing of the crack and large amounts of plastic deformation occur near the crack tip at fracture, the elastic stress-intensity factor at failure (KIe) varies with planar dimensions, such as crack length, specimen width, and specimen type [2–5]. To account for variation in KIe with various crack-configuration parameters, the elastic–plastic stress–strain behavior near the crack-tip region must be considered. Several equations for calculating the elastic–plastic stress–strain behavior at notches or cracks have been proposed. Among these are equations derived for notches by Hardrath and Ohman [6], and by Neuber [7]. For cracks, equations have been derived by Rice and Rosengren [8] and by Hutchinson [9] for non-linear elastic materials. The Hardrath–Ohman equation was later generalized for a cracked plate and was applied as a fracture criterion by Kuhn and Figge [10]. In a similar way, Newman [4,5], using the Neuber relation and the elastic-stress distribution in the crack-tip region, derived a fracture criterion for a cracked plate, which related the elastic stress-intensity factor at failure (KIe), the elastic nominal (net-section) failure stress (Sn), and two material fracture parameters. The Two-Parameter Fracture Criterion (TPFC) has been used to analyze failure of surface- and through-cracked sheet and plate specimens under tensile loading [4]; and was also used to analyze failure of compact and notch-bend sheet specimens for a wide range of materials [5].

⇑ Corresponding author. Tel.: +1 901 734 6642; fax: +1 662 325 7730. E-mail address: [email protected] (J.C. Newman Jr.). http://dx.doi.org/10.1016/j.engfracmech.2015.01.021 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.

132

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

Nomenclature B c E hc KF KIe KT Ke Kr m Pf S Sf Sn Su w

q q⁄ ru rys wc

specimen thickness, mm half-length of crack, mm modulus of elasticity, GPa one-half height of plane-strain core, mm elastic–plastic fracture toughness, MPa m1/2 elastic stress-intensity factor at failure, MPa m1/2 elastic stress-concentration factor plastic-strain-concentration factor plastic-stress-concentration factor fracture-toughness parameter failure (maximum) load under displacement control, kN gross-section (remote) stress, MPa remote failure (maximum) stress, MPa net-section stress at failure, MPa plastic-hinge stress, MPa one-half width of M(T) specimen, mm notch-root radius, mm critical length parameter for a crack, mm uniaxial ultimate tensile strength, MPa uniaxial yield stress, MPa critical crack-tip-opening-angle (CTOA), deg.

The phenomenon of stable crack growth and fracture in metallic materials under tensile and bending loads has been studied extensively using elastic–plastic finite-element methods [11–19]. These studies were conducted to develop efficient techniques to simulate crack extension and to study various local and global fracture criteria. Some of these criteria were based on crack-tip stress or strain, crack-tip-opening displacement or angle, crack-tip force, energy-release rates, J-integral, and the tearing modulus. Of these, either the critical crack-tip-opening displacement (CTOD, dc) at a specified distance from the crack tip or the critical crack-tip-opening angle (CTOA, wc) were shown to be the most suited for modeling stable crack growth and instability during the fracture process. In the present paper, the elastic–plastic finite-element method employing the critical CTOA fracture criterion was used to study the relationship between the elastic stress-intensity factor (KIe) and the net-section stress (Sn) at failure for middlecrack-tension, M(T), specimens made of three materials over an extremely wide range in widths (18–5700-mm). Fig. 1 shows the largest M(T) specimen modeled and analyzed in this study in relation to an average size man! This specimen would be very difficult to test in the laboratory, but could be virtually tested on the computer. One material was a hypothetical 2000series aluminum alloy with an elastic-perfectly-plastic stress–strain behavior under plane-stress conditions; while the other two materials, 2024-T3 and 2219-T87, were modeled with the two-dimensional finite-element method using the planestrain core concept [20]. For the latter two materials, comparisons are made using three-dimensional finite-element analyses with a constant critical CTOA that did not require the plane-strain core and also on experimental test data for a fairly wide range in M(T) specimen widths (76–1200-mm).

2. Two-Parameter Fracture Criterion In the early 1970’s, the Two-Parameter Fracture Criterion (TPFC) was derived [4] from the Inglis elastic stress-concentrap tion equation, KT = 1 + 2 (c/q), for an elliptical hole under remote uniform stress [21] and Neuber’s equation that relates the plastic-stress and plastic-strain-concentration factors to the square of the elastic stress-concentration factor. Neuber [7] considered the problem of anti-plane shear-strained bodies containing notches with various non-linear stress–strain laws and concluded that

K r K e ¼ K 2T

ð1Þ

where Kr and Ke are the plastic stress- and strain-concentration factors, respectively, and KT is the elastic stress-concentration factor. Crews [22] had shown by experiments that Neuber’s equation represents the local stress–strain behavior of mildly notched sheet specimens better than the Hardrath–Ohman notch equation [6]. Later, Rice–Rosengren [8] and Hutchinson [9] derived the local stress and strain fields for a crack in a non-linear elastic material. Using these results showed that the product of the plastic stresses, plastic strains and modulus of elasticity was equivalent to the square of Irwin’s [23] elastic stress field. Thus, these results showed that Neuber’s equation applied for a crack in a non-linear elastic material under tensile-mode loading.

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

133

S

2L 2ci

2w

S Fig. 1. Largest M(T) specimen analyzed using the finite-element method and an average size man to denote scale.

Because of the ‘‘1’’ in the Inglis elastic stress-concentration equation, the resulting fracture criterion produced two groups of material constants involving fracture strength (rf), fracture strain (ef), modulus of elasticity (E), and a critical length parameter (q⁄). Thus, two parameters (KF and m) had to be used. For net-section stress less than the yield stress of the material, the elastic–plastic fracture toughness was

K F ¼ K Ie =½1  mðSn =Su Þ for Sn < rys

ð2Þ

where KIe is the elastic stress-intensity factor at failure, Sn is the net-section (or nominal) stress at failure, and Su is the plastic-hinge stress. For the M(T) specimen, Su is equal to the tensile strength, ru; but for a pure bend specimen, Su = 1.5 ru. The first parameter, KF, had units of stress-intensity factor, and the second term, m, was non-dimensional. There was a linear relationship between the elastic stress-intensity factor, KIe, and net-section stress, Sn, at failure for Sn less than the yield stress of the material, and the slope was related to the m-parameter. If m = 0, the fracture criterion was linear-elastic fracture mechanics (LEFM), but if m = 1 and KF was very large, then a plastic-hinge analysis controlled the failure stress. The fracture parameters KF and m are constant only in the same limited sense as the ultimate tensile strength, that is, the parameters vary with material thickness, state of stress, temperature, and rate of loading. To obtain fracture constants, which are representative for the material and test conditions, the fracture data should be from a single batch of material and should be over the widest range of crack and specimen (component) size. At least two fracture tests are required, of course, because two constants must be evaluated. But a relationship was found between KF/E against m for a wide range of materials [4]. This relation could be used if limited fracture data was available. Once the fracture parameters are known, the failure stress, Sf, or KIe, can be calculated [4,5] knowing the stress-intensity factor solution, the initial crack length, and the plastic-hinge stress for the particular crack configuration. Most investigators do not record critical crack lengths (stably tearing crack length at the maximum load) because of the difficulties involved in measuring stably tearing cracks. Therefore, the initial crack length and maximum (failure) loads were used in the fracture calculations. Thus, the m-parameter accounts for both plasticity and stable tearing of the crack. (Orange [24] has shown that the TPFC gives a two-parameter KR-curve description, KF is the maximum asymptote and m relates to the shape of the resistance curve, which appears to fit experimental resistance curve test data quite well.)

134

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

3. Critical crack-tip-opening-angle fracture criterion The early CTOA applications were restricted to using two-dimensional, elastic–plastic, finite-element analyses, assuming either plane-stress or plane-strain behavior, which led to generally non-constant values of CTOA, especially in the early stages of crack extension [15,16,18,19]. Later, the non-constant CTOA values were traced to inappropriate state-of-stress (or constraint) assumptions in the crack-front region and severe crack tunneling in thin-sheet materials [20,25]. In the past two decades, the CTOA fracture criterion has been used with two- and three-dimensional, elastic–plastic, finite-element analyses to study constraint effects, crack tunneling, and the fracture process [26,27]. The constant CTOA criterion (from crack initiation to fracture) has been successfully applied to numerous structural applications, such as stiffened panels with multiple-site damage (MSD) crack configurations [28], buckling behavior of thin-sheet materials [29], an aircraft fuselage [30] and pipeline materials [31]. The fracture criterion has been able to link laboratory fracture coupons to structural applications. This paper uses the two-dimensional (2D) finite-element method with the plane-strain core concept [20,32] to model the fracture process in very small and very large width M(T) specimens. (The plane-strain core concept was used with the STAGS shell code to ‘‘predict’’ the failure of a damaged aircraft fuselage within 5% of the failure pressure, before the test was ever conducted [30].) Some comparisons are also made using the three-dimensional (3D) finite-element method, which naturally accounts for constraint around the crack front and does not need the plane-strain core. 4. Materials Three aluminum alloy materials have been selected to evaluate the relationship between the stress-intensity factor and the net-section stress at failure. First, a hypothetical material with an elastic-perfectly-plastic stress–strain behavior under plane-stress conditions was analyzed with two critical CTOA (wc) values. One wc value was selected to exhibit low-toughness behavior, while the other exhibited high toughness. The other two materials, 2024-T3 and 2219-T87, were selected because test data in the literature covered a fairly wide range in M(T) specimen widths (76 to 1,200-mm). For the latter two materials, the two-dimensional elastic–plastic finite-element analysis with the plane-strain core concept [20] was used to fit the test data. The critical CTOA (wc) and the plane-strain core height (hc) were found by trial-and-error. A summary of the materials, tensile and fracture properties are shown in Table 1. 5. Finite-element method Two elastic–plastic finite-element codes for crack problems have been developed at NASA Langley. The ZIP2D code, using constant-strain triangles under small-strain conditions, was initially developed to simulate fatigue-crack growth and closure [33], but was modified to simulate the fracture process using various fracture criteria [14]. Of course, the two-dimensional code was developed to apply either plane-stress or plane-strain conditions, but the stress state around a crack even in a thinsheet material develops nearly plane-strain conditions [34]; thus, the plane-strain core concept [20] was developed to more accurately model the fracture process. Here, the core was a constant height of elements that had plane-strain conditions, while all other elements were plane-stress conditions. Later, Xu and Sun [32] showed that a moving plane-strain core (circular region) around the crack tip produced essentially the same load-against-crack-extension results as the stationary core. Similarly, the ZIP3D code [35] used the 8-noded isoparametric element with the small-strain assumption; and was developed to simulate fatigue-crack-growth with crack closure [36] and the fracture process, again, using several fracture criteria. However, the 3-D code naturally accounts for the constraint variations around the crack front and, thus, only the critical CTOA is needed to model the fracture process. 6. Fracture simulations and test data The two-dimensional finite-element code, ZIP2D, was used either under plane-stress conditions for one aluminum alloy or the plane-strain-core concept in the other two aluminum alloys to simulate the fracture process. Seven models of M(T) specimens were generated with 0.5-mm size elements all along the crack line; thus, a wide range in initial crack-lengthto-width (ci/w) ratios may be analyzed. These models are summarized in Table 2. The largest mesh (#9), a one-quarter model, is shown in Fig. 2. The insert shows the local pattern of elements that are common to all models, which has

Table 1 Materials, tensile and fracture properties. Material 2000

a

2024-T3 2219-T87 a

rys (MPa)

ru (MPa)

E (GPa)

B (mm)

wc (°)

hc (mm)

KF (MPa m1/2)

m

400 400 355 395

400 400 490 470

70 70 71.4 72.1

– – 2.3 2.54

3.0 4.5 5.35 4.7

– – 1.5 1.9

147 810 545 200

0.98 1.1 1.28 0.9

Hypothetical material under plane-stress conditions.

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

135

Table 2 Finite-element models and dimensions used in the fracture analyses.

a b c d

Model

Width, 2w mma

Nodes

Elementsb

1c 2c 3c 4d 5c 6d 7d 8d 9d

18 36 90 180 355 710 1425 2850 5700

182 371 1035 1035 3439 3439 7179 15,621 31,161

303 639 1841 1841 6161 6161 12,960 28,480 56,960

Smallest element size was 0.5-mm and L/w = 2. Constant-strain triangular elements. One-half model (crack-line symmetry). One-quarter model (crack-line and centerline symmetry).

Fig. 2. Finite-element model of the largest M(T) specimen analyzed using the critical CTOA fracture criterion (units are millimeters).

0.5-mm size elements all along the crack path. For some cases, two meshes were used to model two different specimen widths by: (1) modeling a crack in the center of the model (two crack tips) or (2) modeling an edge crack with centerline boundary conditions (one crack tip). Model 1 and 2 (smallest models) would have only about 5 or 10 elements, respectively, to model the crack or uncracked ligament for ci/w = 0.3 or 0.7, so the results from Model 1 may be suspect for shallow or deep cracks. Model 3–9 would have more than 10 elements to model the crack or uncracked ligament for ci/w ratios of 0.1–0.9. The reason for the ‘‘uniform size elements and mesh pattern along the crack path’’ in the widely different sheet widths is based on the Fracture Mechanics principle of similitude. When the stress and strain distributions reach a critical condition for one crack configuration during stable tearing at a constant CTOA, to predict the failure of another crack configuration, the ‘‘same’’ stress and strain distributions have to develop at the critical condition. The same size elements and mesh pattern along the crack path helps to insure similitude. In all cases, the critical CTOA value was held constant for each alloy, and the specimens were subjected to uniform displacement along the grip line. First, a displacement (Do) was applied to cause the highest stressed element at the crack tip to

136

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

reach the yield stress of the material (elastic conditions). Then small displacement steps (1.5% of Do) were incrementally applied until the maximum load had been achieved. The critical crack-tip-opening displacement (CTOD) to produce the desired critical CTOA was monitored at the second node from the crack tip; and the crack would extend (crack-tip node severed and stiffness matrix was updated) when the applied CTOD reached or exceeded the critical value. Very stiff springs were used to model boundary conditions all along the anticipated crack path [33,35]. When the crack-tip spring was broken at the critical CTOA, the spring force was re-applied and then slowly released in five (5) equal load steps. Comparisons were made between ZIP2D calculations and the TPFC analyses for failure stress (Sf), or KIe at failure, against crack length, specimen width, or the net-section stress. Experimental test data were also compared for the latter two materials. The stress–strain behavior for each material was modeled with a multi-linear tabular input, as shown in Table 3. 6.1. Aluminum alloy 2000-Series The 2000-series alloy was a hypothetical material with an elastic-perfectly-plastic stress–strain behavior and two different critical CTOA values under plane-stress conditions. ZIP2D fracture simulations were made on various width panels for several initial crack-length-to-width (ci/w) ratios. Fig. 3(a) shows KIe values plotted against the ratio of net-section-stressto-flow-stress, Sn/ru. The open symbols show KIe values calculated from finite-element analyses using the failure load (Pf) and initial crack length (ci). The low wc (3-deg.) material produced a nearly linear relation for specimens widths (2w) larger than about 32-mm, while the high wc (4.5-deg.) material exhibited a linear relation for 2w greater than about 180-mm or in both cases Sn < 0.85 ru. The slope of the KIe–Sn plot was not greatly influenced by the ci/w ratio, but the magnitude was about ±5% (dashed curves) about the mean for the range of ci/w from 0.2 to 0.6 on both the low- and high-toughness materials. For the smaller width specimens, the region of the non-linear relation (KIe against Sn) occurred under large-scale plastic deformations; and, in some cases, exhibited failures with net-section stresses greater than the flow stress of the material (crack or notch strengthening due to constraint elevations of the local flow stress). The solid straight lines were calculated from the TPFC using the finite-element results for ci/w = 0.4; and the values of the two parameters are as shown. The elastic–plastic fracture toughness, KF, is the largest value of stress-intensity factor at failure possible. This value would occur at Sn = 0 for an extremely large M(T) panel, as shown by the asymptote (dotted line) in Fig. 3(b) for the low-toughness material. This is the reason for generating the very large M(T) models that would be very difficult to test in the laboratory, but could be virtually tested on the computer. The TPFC was able to fit the calculated results well (±5%) for Sn less than 0.85 ru. For Sn > 0.85 ru, the current TPFC equation would be conservative. But, for the high-toughness material the m-value was greater than unity. This was not expected, and appears to be related to constraint effects under large-scale plastic deformations, and this is a subject for future research. Because there appeared to be a slight effect of the initial-crack-length-to-width ratio on the KIe against Sn relation, the results over a very wide range in ci/w ratios are shown in Fig. 3(c) for a standard width specimen (2w = 710 mm) that is commonly tested in the laboratory. Solid symbols show finite-element analyses (FEA) for the two critical CTOA values for 0.05 6 ci/w 6 0.95. Solid curves are the KIe values calculated from the TPFC that was fitted to the results at ci/w = 0.4. The TPFC predicted failure values on the low-toughness material were within ±5%, and the high-toughness material were within ±8% over the extremely wide range in crack lengths. The results shown in Fig. 3(c) are showing a general trend in the comparisons between the FEA and the TPFC. For shallow cracks, the TPFC is generally under predicting the failure stress; while for deep cracks, the TPFC is over predicting the failure stress. As will be shown later on test data for 2024-T3 aluminum alloy specimens, the same trend was observed between test data and the TPFC. This is a subject for future research. However, the FEA and TPFC are matching within ±5% over an extremely wide range in specimen widths for 0.2 6 ci/w 6 0.6. 6.2. Aluminum alloy 2024-T3 The 2024-T3 aluminum alloy (B = 2.3 mm thick [37]) was extensively tested at NASA Langley during the NASA Aging Aircraft Program. All panels were tested with anti-buckling guides with TeflonÒ sheets between the specimen and guides. The

Table 3 Stress–strain tables used in the finite-element fracture analysesa. 2024-T3

2219-T87

r (MPa)

e

r (MPa)

e

345 390 431 472 490 490

.00483 .015 .04 .1 .16 .2

379 407 430 451 465 470

.005263 .009 .018 .03 .05 .12

a 2000-series aluminum alloy was elastic-perfectly-plastic stress–strain behavior with E = 70 GPa and a yield stress of 400 MPa.

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

137

Fig. 3. (a) – Elastic stress-intensity factor at failure against the net-section stress for 2000-series aluminum alloy for two critical CTOA values and TPFC analyses. (b) – Elastic stress-intensity factor at failure against specimen width for 2000-series aluminum alloy for two critical CTOA values and corresponding TPFC analyses. (c) – Elastic stress-intensity factor at failure against crack-length-to-width ratio for 2000-series aluminum alloy for two critical CTOA values with a total width of 640-mm.

138

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

symbols in Fig. 4(a) show the failure stress against specimen width for panels tested with an initial crack-length-to-width ratio of 1/3. First, ZIP3D was used to find a critical CTOA (wc = 5.25°) to fit the results fairly well. Then the ZIP2D code was used under either plane-stress or plane-strain conditions with the same critical angle. The plane-stress results over estimated the failure stress on the largest width panel, while plane-strain conditions under estimated the failure stress. In the current paper, the ZIP2D code was again used with a slightly different critical angle (5.35°) and a plane-strain core to account for constraint (hc = 1.5 mm) that fit the test results very well. (Note: A scale factor of 0.857 was applied to the models in Table 2 to match the widths tested.) Then the critical CTOA and plane-strain-core size was used in ZIP2D fracture simulations on the nine models (Table 2). Fig. 4(b) shows the KIe values plotted against the ratio of the net-section stress to the ultimate tensile strength, ru, for ci/w = 1/3. The square symbols show the test data, while the open and solid circular symbols are the ZIP3D and ZIP2D analysis results, respectively. The results from ZIP2D and ZIP3D were both linear for Sn less than the yield stress. But here again, the ZIP2D results produced an m-value much greater than unity. From previous TPFC analyses [4,5], it was expected that m = 1 would be the upper limiting value. But these results clearly show that the fracture simulations and test data support an m-value greater than unity. The solid line (denoted m = 1) and dashed curve shows the original TPFC analyses on the 2024-T3. However, only the 305-mm wide test results were available when KF and m were originally evaluated. Fig. 4(c) shows KIe against specimen width that clearly illustrates why testing large panels are necessary to establish appropriate fracture properties—size does matter. The elastic–plastic fracture toughness, KF, was 545 MPa m1/2 and m = 1.28. Although testing an M(T) panel that is 5120 mm wide may be difficult, these dimensions are very reasonable

Fig. 4. (a) – Measured and calculated failure stress against specimen width for 2024-T3 aluminum alloy. (b) – Measured and calculated KIe against netsection stress for 2024-T3 aluminum alloy. (c) – Measured and calculated KIe against specimen width for 2024-T3 aluminum alloy. (d) – Measured and calculated failure stress against crack-length-to-width ratio for 305-mm wide 2024-T3 aluminum alloy specimens.

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

139

for the side of a jumbo-jet! Of course, the fuselage has windows, frames, stiffeners, and riveted-lap-splice joints that complicate the configuration. But, Hsu et al. [30] used the STAGS finite-element shell code with a critical CTOA and the planestrain core on 2014-T6 aluminum alloy to accurately predict the critical pressure on a DC-9 damaged fuselage within 5%, before the test was ever conducted at Wright-Patterson Air Force Base.

Fig. 5. (a) – Measured and calculated failure stress against specimen width for 2219-T87 aluminum alloy. (b) – Measured and calculated KIe against netsection stress for 2219-T87 aluminum alloy. (c) – Measured and calculated KIe against specimen width for 2219-T87 aluminum alloy.

140

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

Fig. 4(d) shows the results on 305-mm wide M(T) panels tested over a wide range in initial crack-length-to-width ratios. These tests were conducted many years before the wide panel tests on the same material batch. Again, the ZIP2D calculations fit the test data extremely well; and the original TPFC analysis with 267 MPa m1/2 and m = 1 also fit the test data very well. But the higher KF value with m = 1.28 did not do as well, but the maximum error was only about 5%. However, the trend in KIe with crack-length-to-width ratio was modeled fairly well by all methods. 6.3. Aluminum alloy 2219-T87 The 2219 tests were of interest because Boeing [38] had tested much larger width panels than the NASA Langley tests and they also used anti-buckling guides. The symbols in Fig. 5(a) show the failure stress against specimen width for panels tested with an initial crack-length-to-width ratio of 1/3. Here the ZIP2D code was used with trial-and-error to find the critical CTOA and plane-strain core height. (Again, a scale factor of 0.857 was applied to the models in Table 2 to match the widths tested.) Past experience had indicated that the core half-height was less than the sheet thickness (B). The critical CTOA was selected to match the failure stress on the largest width panel. The solid curve shows the calculations with the plane-strain core that fit most of the tests quite well, but under predicted the smallest width specimen by about 6%. Then ZIP2D was used to calculate the failure stresses under either plane-stress or plane-strain conditions using the same angle, but these calculations did not do well on the largest width specimens. The plot of KIe against Sn/ru, again, showed that the relationship was linear for the larger width panels, but the calculations showed some slight discrepancies for the smaller width specimens (see Fig. 5(b)). Apparently, the KIe–Sn plot amplifies the differences between the tests and analyses because the 150-mm test was only about 6% from the ZIP2D calculation. Thus, the differences between the ZIP2D and TPFC results, in term of failure load, are within a few percent. Again, the smallest width (15-mm) specimen had a net-section stress greater than the ultimate tensile strength. Fig. 5(c) shows KIe against specimen width. The open symbols are the test data, while the solid symbols are the ZIP2D calculations. And the curve is the TPFC results (KF = 200 MPa m1/2; m = 0.98). 7. Concluding remarks The Two-Parameter Fracture Criterion (TPFC) had previously been derived from the Inglis elastic stress-concentration p equation, KT = 1 + 2 (c/q), for an elliptical hole under remote uniform stress and Neuber’s equation that relates the plastic-stress and plastic-strain-concentration factors to the square of the elastic stress-concentration factor. Because of the ‘‘1’’ in the Inglis elastic stress-concentration equation, the resulting fracture criterion produced two groups of material constants involving fracture strength, fracture strain, modulus of elasticity, and a critical length parameter (q⁄). Thus, two parameters (KF and m) had to be used. The first parameter, KF, had units of stress-intensity factor, and the second term, m, was non-dimensional; and the relation between the elastic stress-intensity factor at failure, KIe, and the net-section stress, Sn, at failure was ‘‘linear’’ for net-section stresses less than the yield stress (rys). The slope was related to the m-parameter. If m = 0, the fracture criterion was LEFM, but if m = 1 and KF was very large, then a plastic-hinge analysis controlled the failure stress. This study was undertaken to investigate the linear relationship. The linear relationship was validated using elastic–plastic finite-element analyses employing the critical crack-tip-opening-angle (CTOA) fracture criterion to simulate fracture tests on several thin-sheet aluminum alloys over an extremely large range in middle-crack-tension, M(T), specimen widths (18–5700-mm). One hypothetical material, under plane-stress conditions (ru = 400 MPa), was analyzed with either a low or high critical CTOA (wc). The low wc (3-deg.) material produced a linear relation for specimens widths (2w) larger than about 36-mm, while the high wc (4.5-deg.) material exhibited a linear relation for 2w greater than about 180-mm or in both cases Sn < 0.85 ru. The slope of the KIe–Sn plot was not greatly influenced by the initial crack-length-to-width (ci/w) ratio, and the magnitude was within about ±5% about the mean curve for the range of ci/w from 0.2 to 0.6 on both the low- and high-toughness materials. For the smaller width specimens, the region of the non-linear relation (KIe against Sn) occurred under large-scale plastic deformations; and, in some cases, exhibited failures with net-section stresses greater than the uniaxial tensile strength (crack or notch strengthening due to constraint elevations of the local flow stress). The other two materials, 2024-T3 and 2219-T87, were analyzed with two-dimensional finite-element analyses with the plane-strain core to account for the level of constraint at the crack tip. Again, fracture simulations showed the linear relation between KIe and Sn at failure for net-section stresses less than the yield stress of the material. These results validated the original Two-Parameter Fracture Criterion. For net-section stresses greater than the yield stress, a more complicated equation is required; and constraint variations not considered in the original derivation of the TPFC must be evaluated. Test data on these two aluminum alloys over a wide range of M(T) specimen widths (76–1200-mm) supported the linear relationship for Sn < rys. References [1] Plane Strain Crack Toughness Testing of High Strength Metallic Materials, ASTM STP 410, American Society for Testing and Materials; 1966. [2] Fracture Toughness Testing, ASTM STP 381, American Society for Testing and Materials; 1964.

J.C. Newman Jr., J.C. Newman III / Engineering Fracture Mechanics 136 (2015) 131–141

141

[3] Kuhn P. Residual tensile strength in the presence of through cracks or surface cracks. NASA TN D-5432. National Aeronautics and Space Administration. Washington, D.C.; 1970. [4] Newman Jr JC. Fracture analysis of surface- and through-cracked sheets and plates. Engng Fract Mechs J 1973;5:667–89. [5] Newman Jr JC. Fracture analysis of various cracked configurations in sheet and plate materials. ASTM STP 1976;605:104–23. [6] Hardrath HF, Ohman L. A study of elastic and plastic stress concentration factors due to notches and fillets in flat plates. NACA Rep. 1117; 1953. (supersedes NACA TN 2566). [7] Neuber H. Theory of stress concentration for shear-strained prismatical bodies with arbitrary non-linear stress-strain law. Trans ASME Ser E, J Appl Mechs 1961;28:544–50. [8] Rice JR, Rosengren GF. Plane strain deformation near a crack tip in a power-law hardening material. Mech Phys Solids 1968;16:1–12. [9] Hutchinson JW. Singular behavior at the end of a tensile crack in a hardening material. J Mech Phy Solids 1968;16:13–31. [10] Kuhn P, Figge IE. Unified notch-strength analysis for wrought aluminum alloys. NASA TN D-1259, National Aeronautics and Space Administration. Washington, D.C.; 1962. [11] Kobayashi A, Chiu S, Beeuwkes R. A numerical and experimental investigation on the use of the J-Integral. Engng Frac Mech 1973;5(2):293–305. [12] Andersson H. A finite-element representation of stable crack growth. J Mech Phys Solids 1973;21:337–56. [13] de Koning A. A contribution to the analysis of slow stable crack growth, National Aerospace Laboratory. Report NLR MP 75035U; 1975. [14] Newman Jr JC. Finite-element analysis of crack growth under monotonic and cyclic loading. ASTM STP 1976;637:56–80. [15] Shih C, de Lorenzi H, Andrews W. Studies on crack initiation and stable crack growth. ASTM STP 1979;668:65–120. [16] Kanninen M, Rybicki E, Stonesifer R, Broek D, Rosenfield A, Nalin G. Elastic–plastic fracture mechanics for two-dimensional stable crack growth and instability problems. ASTM STP 1979;668:121–50. [17] Newman Jr JC. An elastic–plastic finite element analysis of crack initiation, stable crack growth and instability. ASTM STP 1984;833:93–117. [18] Brocks W, Yuan H. Numerical studies on stable crack growth. ESIS Pub 1991;9:19–33. [19] Newman Jr JC, Shivakumar K, McCabe D. Finite element fracture simulation of A533B steel sheet specimens. ESIS Pub 1991;9:117–26. [20] Newman Jr JC, Dawicke DS, Sutton MA, Bigelow CA. A fracture criterion for wide-spread cracking in thin-sheet materials. Seventeenth Symposium. Int. Comm. Aeronautical Fatigue, Stockholm, Sweden; 1993. [21] Inglis CE. Stresses in a plate due to presence of cracks and sharp corners. Trans Naval Architects 1913;55:219–42. [22] Crews Jr JH. Elastoplastic stress–strain behavior at notch roots in sheet specimens under constant-amplitude loading. NASA TN D-5253; 1969. [23] Irwin GR. Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mechs 1957;24:361–4. [24] Orange TW. A relation between semi-empirical fracture analyses and R-curves. NASA TP 1600; January 1980. [25] Newman Jr JC, Dawicke D, Bigelow C. Finite-element analysis and measurement of CTOA during stable tearing in a thin-sheet aluminum alloy. Durability of Metal Aircraft Structures. S. Atluri et al., Eds.; 1992. [26] Dawicke D, Newman Jr JC, Bigelow C. Three-dimensional CTOA and constraint effects during stable tearing in a thin-sheet material. ASTM STP 1995;1256:223–42. [27] James M, Newman Jr JC, Johnston W. Three-dimensional analyses of crack-tip-opening angles and d5-resistance curves for 2024–T351 aluminum alloy. ASTM STP 2002;1406:279–97. [28] Seshadri B, Newman Jr JC, Dawicke D, Young R. Fracture analysis of the FAA/NASA wide stiffened panels, NASA CP-208982. 1999; 513–24. [29] Seshadri B, Newman Jr JC. Analysis of buckling and stable tearing in thin-sheet materials. ASTM STP 1998;1332:114–34. [30] Hsu C, Lo J, Yu J, Lee X, Tan P. Residual strength analysis using CTOA criteria for fuselage structures containing multiple site damage. Engng Frac Mech 2003;70(3–4):525–45. [31] Darcis P, McCowan C, Drexler E, McColskey J, Shtechman A, Siewert T. Fracture toughness through a welded pipeline section – Crack tip opening criterion. Welding in the World 51(Special Issue); 2007. p. 225–34. [32] Su XM, Sun CT. A plane-strain core model for crack growth in ductile materials. Proc. ASME Aerospace Division. 1996; AD-52: 217–226. [33] Newman Jr JC. Finite-element analysis of fatigue crack propagation including the effects of crack closure. Ph.D. Thesis. Virginia Polytechnic Institute and State University; 1974. [34] Hom CL, McMeeking RM. Large crack tip opening in thin elastic–plastic sheets. Int J Fract 1990;45:103–22. [35] Shivakumar KN, Newman Jr JC. ZIP3D – An elastic and elastic–plastic finite-element analysis program for cracked bodies. NASA TM-102753; November 1990. [36] Chermahini RG, Shivakumar KN, Newman Jr JC. Three dimensional finite-element simulation of fatigue crack growth and closure. ASTM STP 982. 1988; 319–341. [37] Dawicke DS, Newman Jr JC., Starnes Jr JH., Rose CA., Young RD, Seshadri BR. Residual strength analysis methodology: laboratory coupons to structural components. In: Third Joint FAA/DoD/NASA Conference on Aging Aircraft. Albuquerque, NM; 1999. [38] Eichenberger TW. Fracture resistance data summary. Report No. DA-20947. The Boeing Airplane Company. Seattle, Washington; 1962.