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CC(T) Specimen Load-Bearing Capacity Related to Yield Strength and Upper-Shelf Charpy-V Energy

CC(T) Specimen Load-Bearing Capacity Related to Yield Strength and Upper-Shelf Charpy-V Energy Article CC(T) Specimen Load-Bearing Capacity Related to Yield Strength and Upper-Shelf Charpy-V Energy Kim Wallin K -solutions, Ltd., Heinäkuja 4A, 02760 Espoo, Finland; Kim.Wallin@kwsolutions.fi Abstract: The load-bearing capacity of a CC(T) specimen (Center-Cracked Tension) in the ductile fracture regime is usually controlled by plastic collapse. If the material’s tearing resistance is suffi- ciently low, the load-bearing capacity can drop below the plastic collapse value. Here, a recently developed simple fracture mechanics-based Charpy-V impact energy criterion for plastic collapse was used to provide a best estimate assessment of the CC(T) specimen load-bearing capacity. Keywords: structural assessment; fracture mechanics; impact fracture; limit load; CC(T) specimen 1. Introduction Fracture mechanics is usually used mainly to ensure that a structure will not fail due to possible flaws. Normal engineering methods such as the one in Eurocode 3 focus mainly on avoidance of brittle fracture. If brittle fracture can be ruled out, it is commonly assumed that the structure will fail by plastic collapse and a limit load dimensioning is considered sufficient. However, if the material’s ductile tearing resistance is sufficiently low, it is possible that ductile crack growth lowers the maximum load-bearing capacity below the plastic limit load. To safeguard against such events, a criterion for the ductile tearing resistance is also needed. Citation: Wallin, K. CC(T) Specimen In a recent publication, a simple criterion for plastic collapse was developed based on Load-Bearing Capacity Related to the K /L ratio [1]. It postulates that if the K /L ratio is less than 0.4, the failure will be r r r r Yield Strength and Upper-Shelf controlled by plastic collapse. K represents the ratio between the elastic crack driving force Charpy-V Energy. Appl. Mech. 2021, 2, and the material’s fracture toughness, corresponding to a specific fracture definition [1]. L 666–680. https://doi.org/10.3390/ represents the ratio between primary load and plastic limit load corresponding to yield applmech2040038 strength [1]. The function relating K to L represents the reciprocal of a plasticity correction r r Received: 14 July 2021 on the elastic crack driving force. The criterion results in a simple Charpy-V requirement Accepted: 26 September 2021 to ensure that the maximum load will be controlled by the limit load. Published: 28 September 2021 The Charpy-V impact energy requirement is conservative as it is based on a conserva- tive estimate of the crack driving force and it is based on a lower bound-type correlation Publisher’s Note: MDPI stays neutral between J (J-integral value corresponding to 1 mm crack extension) and KVus (Charpy- 1mm with regard to jurisdictional claims in V impact energy at upper shelf), and the correlations have been developed for highly published maps and institutional affil- constrained bend specimens (SE(B) (single-edged bend) and C(T) (compact tension)). The iations. requirement is not intended to provide a best estimate of the Charpy-V impact energy required to ensure plastic collapse. It is a general screening criterion for plastic collapse. When the requirement is fulfilled, the likelihood of reaching the plastic limit load is high. In the original publication [1], there was a lack of experimental validation of the developed Copyright: © 2021 by the author. requirement. The goal of this work was to validate the requirement using actual test Licensee MDPI, Basel, Switzerland. data. The CC(T) specimen geometry was chosen because it mimics structures with mainly This article is an open access article tension loading as is the basic assumption in Eurocode 3 (EN 1993-1-10). distributed under the terms and Here, the method was used to assess the maximum load values of CC(T) speci- conditions of the Creative Commons mens. When the method is used to make a best estimate assessment of lowly constrained Attribution (CC BY) license (https:// CC(T) specimens, many of the aspects of the requirement need to be considered creativecommons.org/licenses/by/ and modified. 4.0/). Appl. Mech. 2021, 2, 666–680. https://doi.org/10.3390/applmech2040038 https://www.mdpi.com/journal/applmech Appl. Mech. 2021, 2 667 1.1. The Basis of the Criteria The K /L = 0.4 criterion allows the definition of a simple fracture toughness criterion r r for plastic collapse. The stress intensity factor in terms of load P can be expressed in the form of Equation (1) and the plastic limit load can be expressed in the form of Equation (2) [1], K =  f(a/W) (1) 1/2 B W P =   B W F a/W (2) ( ) L Y where K is the stress intensity factor, P is the limit load, P is the load, B is the specimen I L thickness, W is the specimen height, and  is the yield stress. The functions f(a/W) and F(a/W) are application-specific geometry functions that can be found from various handbooks [2,3]. The ratio K /L corresponding to the limit load (L = 1) can then be written in the r r r form of Equation (3) by inserting P from Equation (2) into Equation (1) [1]. The fracture toughness K represents the true material’s fracture toughness, not the linear-elastic IC fracture toughness. This means that K can correspond to J-integral-based K or to a IC JC point on the tearing resistance curve, expressed in the same units as the stress intensity factor, K. Equation (3) can be used for any geometry. 1/2 W  f(a/W) F(a/W) r(L =1) r Y = (3) L K r IC Applying simplified equations for the stress intensity factor and the limit load and replacing the yield strength with the material’s flow strength ( ), an upper bound estimate of the required fracture toughness to achieve plastic collapse is given by Equation (4) [1]. It is based on the maximum of Equation (3) for several different crack and loading geometries and contains a plasticity correction equal to 2.5, which corresponds to the typical level of plasticity at L = 1 [1]. IC p  2 ) plastic collapse (4) Equation (4) gives a conservative estimate of the crack driving force close to the plastic limit load. This needs to be matched with the material’s tearing resistance. The publica- tion [1] favored the use of the tearing resistance corresponding to 1 mm ductile crack growth (J ). The use of an initiation definition, such as J , is considered overly conservative. 1mm Ic The value of J was estimated from a well validated J KV correlation [1,4]. The 1mm 1mm us previous work [1] made use of the 5% lower bound correlation in the form of Equation (5). The J-integral values need to be written in the form of K , Equation (6), where E is the IC modulus of elasticity and  is Poisson’s ratio. 1.28 KV US J = 193 (5) 1mm5% J E K = (6) IC (1 ) The publication [1] ended up with a simple Charpy-V impact energy requirement in the form of Equation (7). KV(J )  exp 0.59 + 0.78 ln(W) + (7) 1mm 1.2. Modifications Specific to CC(T) Specimen The CC(T) specimen has a very simple geometry (Figure 1). There are several stress intensity factor solutions for the CC(T) specimen [2]. Equation (8) is supposed to have Appl. Mech. 2021, 2, FOR PEER REVIEW 3   KV J ≥+ exp 0.59 0.78⋅ ln W+ () ( )  (7) 1mm   Appl. Mech. 2021, 2 668 Appl. Mech. 2021, 2, FOR PEER REVIEW 3 1.2. Modifications Specific to CC(T) Specimen The CC(T) specimen has a very simple geometry (Figure 1). There are several stress intensity factor solutions for the CC(T) specimen [2]. Equation (8) is supposed to have an accur an accuracy acy of 0.of 1% for 0.1%any p for any roppr ort oportional ional crack crack lengtlength h a/W [2 a/W ]. The co [2]. The rrespondin corresponding g limit lo limit ad   equation load equation is very simple is very simple, , due to the dueextre to the mely extr low emely con low straint constraint of the Y specimen. It c of the specimen. an simply It can KV J ≥+ exp 0.59 0.78⋅ ln W+ () ( ) (7)  1mm simply be expressed in terms of a constant net section stress, Equation 343 (9) [3]. The crack be expressed in terms of a constant net section stress, Equation (9) [3]. The crack driving   driving force equation corresponding to limit load thus becomes Equation (10). It includes force equation corresponding to limit load thus becomes Equation (10). It includes the the plasticity correction of approximately 2.5, as used in the original simplified method plasticity correction of approximately 2.5, as used in the original simplified method de- 1.2. Modifications Specific to CC(T) Specimen described above. Equation (10) is shown graphically in Figure 2. As seen from the figure, scribed above. Equation (10) is shown graphically in Figure 2. As seen from the figure, the the maximum crack driving force is close to 2, as assumed in Equation (4). maximum The CC(T) sp crack driv ecimen has a ing force is clo very simple g se to 2, as assum eometry (F ed in Equatio igure 1). There n (4). are several stress intensity factor solutions for the CC(T) specimen [2]. Equation (8) r is supposed to have an p 2 4 a a  a accuracy of 0.1% for any proportional crack length a/W [2]. The corresponding limit load K =   a 1 0.025 + 0.06  sec  (8) W W 2 W equation is very simple, due to the extremely low constraint of the specimen. It can simply be expressed in terms of a constant net section stress, Equation (9) [3]. The crack driving =   1 (9) L f force equation corresponding to limit load thus becomes Equation (10). It includes the r  r plasticity correction of approximately 2.5, as used in the original simplified method de- K 2 4 a a a  a a I(L =1) scribed above. Equation (10) is shown graphically in Figure 2. As seen from the figure, the p =   1 0.025 + 0.06  sec   1  2.5 (10) W W W 2 W W maximum crack driving force is close to 2, as assumed in Equation (4). Figure 1. Definition of CC(T) specimen dimensions. The dimensions 2a and 2W are the crack length and specimen width, respectively.  aa  πa  K =σ ⋅ π⋅ a ⋅ 1 − 0.025 ⋅ + 0.06 ⋅ ⋅ sec ⋅ (8) I      WW 2W       a σ=σ ⋅ 1− (9)  Lf Figure 1. Definition of CC(T) specimen dimensions. TheW dimensions 2a and 2W are the crack length Figure 1. Definition of CC(T) specimen dimensions. The dimensions 2a and 2W are the crack length  and specimen width, respectively. and specimen width, respectively.  aa  πa  K =σ ⋅ π⋅ a ⋅ 1 − 0.025 ⋅ + 0.06 ⋅ ⋅ sec ⋅ (8) I      WW 2W       a σ=σ ⋅ 1− (9)  Lf  I(Lr=1) Figure 2. Plasticity-corrected crack driving force at limit load for a CC(T) specimen, as a function of relative crack length a/W. The J –CVN correlation that is based on the 5% lower bound is not applicable us 1mm for a best estimate. Instead, the median correlation should be used. This has the form of Equation (11) [1,4]. 1.28 KV US J = 269 (11) 1mm Appl. Mech. 2021, 2 669 The above-described energy requirement does not account for constraint differences. Deeply cracked bend specimens experience a high stress triaxiality or constraint, because the yielding is contained due to the compressive stresses opposite of the crack. Single- cracked-tension-loaded specimens experience an uncontained yielding because the whole ligament is in tension and, thus, the stress triaxiality or constraint is much lower. The CC(T) specimen thus has a much lower constraint than the bend specimens used for the Charpy-V correlation, and this has a pronounced effect on the tearing resistance curve. The T-stress, which is an elastic parameter, corresponds to the higher-order nonsin- gular term in the series expansion of the stress field equation. Even though the T-stress describes the nonsingular stress component in the direction of the crack plane, it also gives an approximation of the opening stress in the loading direction. Figure 3 shows the effect of constraint, expressed in the form of T-stress at limit load, on the J-integral for a specific crack growth compared to a high-constraint configuration [5]. Deeply cracked bend specimens have positive T-stresses, whereas the CC(T) specimens have strongly negative T-stresses, as seen from Figure 4 [5]. The T-stress effect on the tearing resistance can be approximated by Equation (12) [5]. The accuracy of the constraint Appl. Mech. 2021, 2, FOR PEER REVIEW 5 effect description is of the order of 15%. J T-stress 100 MPa exp . . . T-stress  100 MPa (12) J 1500 MPa T>0 3.0 2¼Cr1Mo σ = 260 MPa 2.8 Y StE460 σ = 460 MPa 2.6 Carbon steel σ = 252 MPa Q&T steel σ = 700 MPa 2.4 X80 σ = 570 MPa 2.2 HY-80 σ = 610 MPa A533B Cl.1 σ = 397 MPa 2.0 HY-100 σ = 749 MPa 1.8 1.6 1.4 1.2 1.0 0.8 -600 -500 -400 -300 -200 -100 0 100 200 T-stress (limit load) [MPa] Figure 3. Effect of constraint expressed in the form of T-stress at limit load, on the J-integral for a Figure 3. Effect of constraint expressed in the form of T-stress at limit load, on the J-integral for a specific crack growth compared to a high-constraint configuration (J/J ) [5]. Dashed lines indicate T>0 specific crack growth compared to a high-constraint configuration (J/JT>0) [5]. Dashed lines indicate 15% uncertainty. ±15% uncertainty. Figure 4 contains a fit to the individual T-stress data. It has the form of Equation (13) [5]. 2 3 T-stress a a a 1.14 + 0.89 + 0.152 0.495 (13) W W W It is important to note that Equation (13) corresponds to limit load. If the net sec- tion stress ( ) is below the flow stress, Equation (13) needs to be multiplied by the net ratio  / . net The Q parameter is an elastic–plastic constraint parameter. A definition, in line with the T-stress, refers to the normalized distance between the actual tensile stress  at a yy Figure 4. T-stress at limit load, normalized by yield strength, for a CC(T) specimen, as a function of relative crack length a/W [5]. The Q parameter is an elastic–plastic constraint parameter. A definition, in line with the T-stress, refers to the normalized distance between the actual tensile stress σyy at a specified location in front of the crack and the matching value corresponding to the small- scale yielding solution for T = 0 [6]. It can be expressed in the form of Equation (14) [6]. σ− σ () yy yy r ⋅σ T0 = Y (14) Qa≡θt=0, =2 σ J Figure 5 shows a comparison of the yield stress-normalized elastic T-stress corre- sponding to limit load and the Q value corresponding to the load at JIC, where the Q values have been collected from the references leading to Figure 3. Overall, the two parameters follow an approximate offset of 0.15. The T-stress underestimates very low Q values and overestimates very high values, but in the range Q = −1 to 0, the estimates generally differ less than 0.2, which is of the same order as the uncertainty in the parameter estimates. This provides validation for the use of Equation (12) to describe the constraint effect for ductile fracture. J/J T>0 Appl. Mech. 2021, 2, FOR PEER REVIEW 5 3.0 2¼Cr1Mo σ = 260 MPa 2.8 StE460 σ = 460 MPa 2.6 Carbon steel σ = 252 MPa Q&T steel σ = 700 MPa 2.4 Y X80 σ = 570 MPa 2.2 HY-80 σ = 610 MPa A533B Cl.1 σ = 397 MPa 2.0 HY-100 σ = 749 MPa 1.8 1.6 1.4 1.2 Appl. Mech. 2021, 2 670 1.0 0.8 -600 -500 -400 -300 -200 -100 0 100 200 T-stress (limit load) [MPa] specified location in front of the crack and the matching value corresponding to the small- scale yielding solution for T = 0 [6]. It can be expressed in the form of Equation (14) [6]. Figure 3. Effect of constraint expressed  in the form of T-stress at limit load, on the J-integral for a yy yy r T=0 specific crack growth compared to a high-constraint configuration (J/JT>0) [5]. Dashed lines indicate Q  at  = 0, = 2 (14) ±15% uncertainty. Figure 4. T-stress at limit load, normalized by yield strength, for a CC(T) specimen, as a function of Figure 4. T-stress at limit load, normalized by yield strength, for a CC(T) specimen, as a function of relative crack length a/W [5]. relative crack length a/W [5]. Figure 5 shows a comparison of the yield stress-normalized elastic T-stress correspond- The Q parameter is an elastic–plastic constraint parameter. A definition, in line with ing to limit load and the Q value corresponding to the load at J , where the Q values IC have been collected from the references leading to Figure 3. Overall, the two parameters the T-stress, refers to the normalized distance between the actual tensile stress σyy at a follow an approximate offset of 0.15. The T-stress underestimates very low Q values and specified location in front of the crack and the matching value corresponding to the small- overestimates very high values, but in the range Q = 1 to 0, the estimates generally differ scale yielding solution for T = 0 [6]. It can be expressed in the form of Equation (14) [6]. Appl. Mech. 2021, 2, FOR PEER REVIEW 6 less than 0.2, which is of the same order as the uncertainty in the parameter estimates. This provides validation for the use of Equation (12) to describe the constraint effect for σ− σ () yy yy r ⋅σ ductile fracture. T0 = Y (14) Qa≡θt=0, =2 σ J Figure 5 shows a comparison of the yield stress-normalized elastic T-stress corre- sponding to limit load and the Q value corresponding to the load at JIC, where the Q values have been collected from the references leading to Figure 3. Overall, the two parameters follow an approximate offset of 0.15. The T-stress underestimates very low Q values and overestimates very high values, but in the range Q = −1 to 0, the estimates generally differ less than 0.2, which is of the same order as the uncertainty in the parameter estimates. This provides validation for the use of Equation (12) to describe the constraint effect for ductile fracture. Figure 5. Relation between elastic T-stress at limit load (T/ ) and elastic–plastic Q parameter Figure 5. Relation between elastic T-stress at limit load (T/σY) and elastic–plastic Q parameter at JIc at J [5]. Ic [5]. The constraint effect data in Figure 3 are verified only to a J/JT>0 ratio of 2. Due to this, the constraint correction for the CC(T) specimens is limited to 2. This is in line with actual tearing resistance data for CC(T) specimens, as compared to C(T) specimen tearing re- sistance data shown in Figure 6 [7]. Figure 6. Comparison of CC(T) and C(T) specimen tearing resistance (J-integral versus crack exten- sion Δa) for a medium-strength steel [7]. 2. Materials and Methods The data for the CC(T) tests were taken from [8–10]. Nine different steels with differ- ent mechanical properties were covered. Their mechanical properties are given in Table 1. The strength levels varied from moderate (325 MPa) to extra high (988 MPa). The Charpy-V upper-shelf energies, measured at ambient temperature, were in the range 85 J to 250 J. The specimens were CC(T) specimens with W = 150 mm and having varying crack lengths. The plate thickness varied between 20 and 40 mm, being in most cases 30 mm. The thickness dimension was not significant for the CC(T) specimens, when assessed in terms of nominal and net section stress. The first four steels had specimens with pre-fa- tigue cracks, whereas the other steels had specimens with narrow Electric Discharge Ma- chining (EDM) notches. All tests were performed at room temperature. The specimens were loaded in tension under displacement control and, for this work, the maximum load J/J T>0 Appl. Mech. 2021, 2, FOR PEER REVIEW 6 Figure 5. Relation between elastic T-stress at limit load (T/σY) and elastic–plastic Q parameter at JIc Appl. Mech. 2021, 2 [5]. 671 The constraint effect data in Figure 3 are verified only to a J/JT>0 ratio of 2. Due to this, the constraint correction for the CC(T) specimens is limited to 2. This is in line with actual The constraint effect data in Figure 3 are verified only to a J/J ratio of 2. Due to T>0 this, the constraint correction for the CC(T) specimens is limited to 2. This is in line with tearing resistance data for CC(T) specimens, as compared to C(T) specimen tearing re- actual tearing resistance data for CC(T) specimens, as compared to C(T) specimen tearing sistance data shown in Figure 6 [7]. resistance data shown in Figure 6 [7]. Figure 6. Comparison of CC(T) and C(T) specimen tearing resistance (J-integral versus crack exten- Figure 6. Comparison of CC(T) and C(T) specimen tearing resistance (J-integral versus crack exten- sion Da) for a medium-strength steel [7]. sion Δa) for a medium-strength steel [7]. 2. Materials and Methods The data for the CC(T) tests were taken from [8–10]. Nine different steels with different 2. Materials and Methods mechanical properties were covered. Their mechanical properties are given in Table 1. The strength The da levels tavaried for thfr e CC( om moderate T) tests were taken from [ (325 MPa) to extra high 8–1 (988 0]. Ni MPa). ne dif The ferent steel Charpy-V s with differ- upper-shelf energies, measured at ambient temperature, were in the range 85 J to 250 J. The ent mechanical properties were covered. Their mechanical properties are given in Table specimens were CC(T) specimens with W = 150 mm and having varying crack lengths. The 1. The strength levels varied from moderate (325 MPa) to extra high (988 MPa). The plate thickness varied between 20 and 40 mm, being in most cases 30 mm. The thickness Charpy-V upper-shelf energies, measured at ambient temperature, were in the range 85 J dimension was not significant for the CC(T) specimens, when assessed in terms of nominal to 250 J. The specimens were CC(T) specimens with W = 150 mm and having varying crack and net section stress. The first four steels had specimens with pre-fatigue cracks, whereas the other steels had specimens with narrow Electric Discharge Machining (EDM) notches. lengths. The plate thickness varied between 20 and 40 mm, being in most cases 30 mm. All tests were performed at room temperature. The specimens were loaded in tension The thickness dimension was not significant for the CC(T) specimens, when assessed in under displacement control and, for this work, the maximum load during the test was used terms of nominal and net section stress. The first four steels had specimens with pre-fa- in the analysis. None of the specimens experienced unstable fracture prior to maximum tigue cracks, whereas the other steels had specimens with narrow Electric Discharge Ma- load during the tests. chining (EDM) notches. All tests were performed at room temperature. The specimens Table 1. Mechanical properties of the materials [8–10]. were loaded in tension under displacement control and, for this work, the maximum load Material  MPa  MPa KV J Ref. U us S355N 364 526 250 [8] S500M 584 639 227 [8] S690QL 845 884 192 [8] S960QL 988 1055 125 [8] St52-3 403 594 200 [9] St52-3 325 490 85 [9] Steel A 420 640 200 [10] Steel B 565 715 200 [10] Steel C 755 835 120 [10] Appl. Mech. 2021, 2, FOR PEER REVIEW 7 during the test was used in the analysis. None of the specimens experienced unstable frac- ture prior to maximum load during the tests. The CC(T) test results for the three sets of steels are shown compiled in Figures 7–9, in the form of nominal stress at load maximum. Table 1. Mechanical properties of the materials [8–10]. Material σY MPa σU MPa KVus J Ref. S355N 364 526 250 [8] S500M 584 639 227 [8] S690QL 845 884 192 [8] S960QL 988 1055 125 [8] St52-3 403 594 200 [9] Appl. Mech. 2021, 2 672 St52-3 325 490 85 [9] Steel A 420 640 200 [10] Steel B 565 715 200 [10] The CC(T) test results for the three sets of steels are shown compiled in Figures 7–9, in Steel C 755 835 120 [10] the form of nominal stress at load maximum. Appl. Mech. 2021, 2, FOR PEER REVIEW 8 Figure 7. CC(T) test maximum load (nominal stress) values divided by flow stress for four different Figure 7. CC(T) test maximum load (nominal stress) values divided by flow stress for four different steels, as a function of relative crack length a/W [8]. steels, as a function of relative crack length a/W [8]. Figure 8. CC(T) test maximum load (nominal stress) values divided by flow stress for two different Figure 8. CC(T) test maximum load (nominal stress) values divided by flow stress for two different St52-3 steels, as a function of relative crack length a/W [9]. St52-3 steels, as a function of relative crack length a/W [9]. Figure 9. CC(T) test maximum load (nominal stress) values divided by flow stress for three different steels, as a function of relative crack length a/W [10]. 3. Results The individual test results were analyzed with Equations (10)–(13), to estimate the required Charpy-V impact energy to achieve the plastic limit load in terms of Equation (9). In cases when Equation (12) indicated a larger constraint effect on J than 2, the adjust- ment was limited to 2. This only occurred with the smallest crack length and higher- strength steels. Figures 10–12 show the measured proportional maximum net section stresses as a function of relative crack length. The results are comparable as all specimens had a width (2W) of 300 mm. The crack length does seem to have a pronounced effect on the maximum net section stress. The figures include the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. With some exceptions, there is a Appl. Mech. 2021, 2, FOR PEER REVIEW 8 Appl. Mech. 2021, 2 673 Figure 8. CC(T) test maximum load (nominal stress) values divided by flow stress for two different St52-3 steels, as a function of relative crack length a/W [9]. Appl. Mech. 2021, 2, FOR PEER REVIEW 9 Figure 9. CC(T) test maximum load (nominal stress) values divided by flow stress for three different Figure 9. CC(T) test maximum load (nominal stress) values divided by flow stress for three different trend that a low maximum net section stress is related to a low ratio between the Charpy- steels, as a function of relative crack length a/W [10]. steels, as a function of relative crack length a/W [10]. V impact energy and estimated required impact energy. 3. Results This effect is examined more closely in Figure 13. With one exception, the results 3. Results The individual test results were analyzed with Equations (10)–(13), to estimate the show a positive trend between proportional maximum net section stress and the ratio be- The individual test results were analyzed with Equations (10)–(13), to estimate the required Charpy-V impact energy to achieve the plastic limit load in terms of Equation (9). tween real and required impact energy. The second St52-3 steel appears to differ from the required Charpy-V impact energy to achieve the plastic limit load in terms of Equation In cases when Equation (12) indicated a larger constraint effect on J than 2, the adjustment general trend. Generally, considering that the estimates are average best estimates, the (9). In cases when Equation (12) indicated a larger constraint effect on J than 2, the adjust- was limited to 2. This only occurred with the smallest crack length and higher-strength data tend to confirm that when the material’s impact energy matches or is above the re- ment was steels. Figurli es mi 10 ted to 2. T –12 show the his onl measur y occurred ed proportional with the sm maximum allestnet crack section length a stresses nd higher- as a quired value, the maximum net section stress equals or is above the plastic limit load function of relative crack length. The results are comparable as all specimens had a width strength steels. Figures 10–12 show the measured proportional maximum net section value, based on flow stress. The variation in the results can be attributed to the scatter in (2W) of 300 mm. The crack length does seem to have a pronounced effect on the maximum stresses as a function of relative crack length. The results are comparable as all specimens the J-integral—Charpy-V correlation and to small (± 5%) deviations in the flow strength. net section stress. had a width (2W) of 300 mm. The crack length does seem to have a pronounced effect on the maximum net section stress. The figures include the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. With some exceptions, there is a Figure 10. Figure 10. Net Net section section stre stress ssvalues values d divided ivided by by flow flow strstres ess for s fo the r the da data in ta Figur in Fig eu 7re 7 as a as a function function of of relative relative crack crack length and related to the rat length and related to the ratioibetween o between the Charpy-V impac the Charpy-V impact ener t energy and estimated gy and estimated required impact energy to obtain the plastic limit load. required impact energy to obtain the plastic limit load. Figure 11. Proportional net section stress values divided by flow stress for the data in Figure 8 as a function of relative crack length and related to the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. Appl. Mech. 2021, 2, FOR PEER REVIEW 9 trend that a low maximum net section stress is related to a low ratio between the Charpy- V impact energy and estimated required impact energy. This effect is examined more closely in Figure 13. With one exception, the results show a positive trend between proportional maximum net section stress and the ratio be- tween real and required impact energy. The second St52-3 steel appears to differ from the general trend. Generally, considering that the estimates are average best estimates, the data tend to confirm that when the material’s impact energy matches or is above the re- quired value, the maximum net section stress equals or is above the plastic limit load value, based on flow stress. The variation in the results can be attributed to the scatter in the J-integral—Charpy-V correlation and to small (± 5%) deviations in the flow strength. Figure 10. Net section stress values divided by flow stress for the data in Figure 7 as a function of Appl. Mech. 2021, 2 674 relative crack length and related to the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. Figure 11. Proportional net section stress values divided by flow stress for the data in Figure 8 as a Figure 11. Proportional net section stress values divided by flow stress for the data in Figure 8 as a Appl. Mech. 2021, 2, FOR PEER REVIEW 10 function of relative crack length and related to the ratio between the Charpy-V impact energy and function of relative crack length and related to the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. estimated required impact energy to obtain the plastic limit load. Figure 12. Figure 12. Proportional net Proportional net section section stress va stress values lues div divided ided bby y flow flow stress for the stress for the data in Figure 9 as a data in Figure 9 as function of relative crack length and related to ratio between the Charpy-V impact energy and esti- a function of relative crack length and related to ratio between the Charpy-V impact energy and mated required impact energy to obtain the plastic limit load. estimated required impact energy to obtain the plastic limit load. The figures include the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. With some exceptions, there is a trend that a low maximum net section stress is related to a low ratio between the Charpy-V impact energy and estimated required impact energy. This effect is examined more closely in Figure 13. With one exception, the results show a positive trend between proportional maximum net section stress and the ratio between real and required impact energy. The second St52-3 steel appears to differ from the general trend. Generally, considering that the estimates are average best estimates, the data tend to confirm that when the material’s impact energy matches or is above the required value, the maximum net section stress equals or is above the plastic limit load Figure 13. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. 4. Discussion The load-bearing capacity of CC(T) specimens is seen to be controlled by the mate- rial’s flow stress, based on the net section stress in the ligament, provided the material’s upper-shelf impact energy is sufficient. The second St52-3 steel seems to deviate from the general trend. For this steel, there are more test results from different-size CC(T) speci- mens [11]. The results refer to specimen widths in the range 2W = 120–600 mm and a va- riety of relative crack lengths. The material and plate thickness (30 mm) are the same as for the second St52-3. Most 300 mm test results are the same as in Figure 8. The nominal stress at load maximum, as a function of relative crack length for the different specimen Appl. Mech. 2021, 2, FOR PEER REVIEW 10 Appl. Mech. 2021, 2 675 Figure 12. Proportional net section stress values divided by flow stress for the data in Figure 9 as a function of relative crack length and related to ratio between the Charpy-V impact energy and esti- value, based on flow stress. The variation in the results can be attributed to the scatter in mated required impact energy to obtain the plastic limit load. the J-integral—Charpy-V correlation and to small (5%) deviations in the flow strength. Figure 13. Compilation of proportional net section stress divided by flow stress as a function of ratio Figure 13. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. limit load. 4. Discussion 4. Discussion The load-bearing capacity of CC(T) specimens is seen to be controlled by the material’s flow stress, based on the net section stress in the ligament, provided the material’s upper- The load-bearing capacity of CC(T) specimens is seen to be controlled by the mate- shelf impact energy is sufficient. The second St52-3 steel seems to deviate from the general rial’s flow stress, based on the net section stress in the ligament, provided the material’s trend. For this steel, there are more test results from different-size CC(T) specimens [11]. upper-shelf impact energy is sufficient. The second St52-3 steel seems to deviate from the The results refer to specimen widths in the range 2W = 120–600 mm and a variety of relative general trend. For this steel, there are more test results from different-size CC(T) speci- crack lengths. The material and plate thickness (30 mm) are the same as for the second mens [11]. The results refer to specimen widths in the range 2W = 120–600 mm and a va- St52-3. Most 300 mm test results are the same as in Figure 8. The nominal stress at load riety of relative crack lengths. The material and plate thickness (30 mm) are the same as maximum, as a function of relative crack length for the different specimen widths, is shown in Figure 14 [11]. The maximum nominal stress is clearly both dependent on the relative for the second St52-3. Most 300 mm test results are the same as in Figure 8. The nominal crack length and the specimen width. stress at load maximum, as a function of relative crack length for the different specimen The data in Figure 14 were analyzed similarly to the data for a constant specimen width. The result, in the form of the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load, is shown in Figure 15. The results confirm that the crack driving force directly affects the maximum net section stress. It is unlikely that the Charpy-V energy would be much lower than 85 J. Even though the correlation between J-integral and Charpy-V energy is based on the median correlation, the trend with low net section stress values in Figure 15 cannot be explained by the uncertainty in the correlation alone, as seen from Figure 16. The calculation in this case is performed using the 5% lower-bound Charpy-V–J correlation, as given in Equation (5). The constraint correction is also conservatively taken as 1. Even for this conservative estimate, the net section stress values are low with respect to the expectations. One possible explanation is that the true flow stress of this St52-3 is 10–15% lower than assumed based on the reported tensile test results. Appl. Mech. 2021, 2, FOR PEER REVIEW 11 widths, is shown in Figure 14 [11]. The maximum nominal stress is clearly both dependent on the relative crack length and the specimen width. The data in Figure 14 were analyzed similarly to the data for a constant specimen width. The result, in the form of the ratio between the Charpy-V impact energy and esti- mated required impact energy to obtain the plastic limit load, is shown in Figure 15. The results confirm that the crack driving force directly affects the maximum net section stress. It is unlikely that the Charpy-V energy would be much lower than 85 J. Even though the correlation between J-integral and Charpy-V energy is based on the median correlation, the trend with low net section stress values in Figure 15 cannot be explained by the un- certainty in the correlation alone, as seen from Figure 16. The calculation in this case is performed using the 5% lower-bound Charpy-V–J correlation, as given in Equation (5). The constraint correction is also conservatively taken as 1. Even for this conservative esti- mate, the net section stress values are low with respect to the expectations. One possible Appl. Mech. 2021, 2 676 explanation is that the true flow stress of this St52-3 is 10–15% lower than assumed based on the reported tensile test results. Appl. Mech. 2021, 2, FOR PEER REVIEW 12 Figure 14. CC(T) test maximum load (nominal stress) values for different specimen widths, as a Figure 14. CC(T) test maximum load (nominal stress) values for different specimen widths, as a function of relative crack length a/W [11]. function of relative crack length a/W [11]. The assumption of a lower-than-assumed flow stress is strengthened by a second set of test results for the same St52-3 steel [11]. These test results correspond to a 15 mm plate thickness. The mechanical properties are higher than for the 30 mm plate, but the steel composition is identical. The material’s impact energy is not known, but from experience, it is expected to be higher than for the 30 mm plate. The data are comprised of a set of different-width CC(T) specimens with a constant initial flaw with 2a = 15 mm. The data, in the form of a maximum net section stress in relation to the flow stress, are reproduced in Figure 17, as a function of specimen width [11]. In addition, in this case, the maximum net section stress is lower than expected, similar to the 30 mm-thick plate data. For com- parison, a test result for a similar-strength Fe510 steel with identical flaw size (2a = 15 mm) and plate thickness (15 mm) [12] is included in Figure 17. The behavior of the Fe510 steel is similar to those of all the other steels in the study. This is an additional indication that the true flow stress of this St52-3 is 10–15% lower than assumed based on the reported tensile test results. Overall, the present work provides an experimental validation of the criterion and methodology developed previously [1]. Figure 15. Compilation of proportional net section stress divided by flow stress as a function of ratio Figure 15. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load for the second St52-3 steel. limit load for the second St52-3 steel. The assumption of a lower-than-assumed flow stress is strengthened by a second set of test results for the same St52-3 steel [11]. These test results correspond to a 15 mm plate thickness. The mechanical properties are higher than for the 30 mm plate, but the steel composition is identical. The material’s impact energy is not known, but from experience, it is expected to be higher than for the 30 mm plate. The data are comprised of a set of different-width CC(T) specimens with a constant initial flaw with 2a = 15 mm. The data, in the form of a maximum net section stress in relation to the flow stress, are reproduced in Figure 17, as a function of specimen width [11]. In addition, in this case, the maximum net Figure 16. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load for the second St52-3 steel. Calculation based on conservative correlation, Equation (5), and omitting constraint correction. Appl. Mech. 2021, 2, FOR PEER REVIEW 12 Appl. Mech. 2021, 2 677 section stress is lower than expected, similar to the 30 mm-thick plate data. For comparison, a test result for a similar-strength Fe510 steel with identical flaw size (2a = 15 mm) and plate thickness (15 mm) [12] is included in Figure 17. The behavior of the Fe510 steel is Figure 15. Compilation of proportional net section stress divided by flow stress as a function of ratio similar to those of all the other steels in the study. This is an additional indication that the between the Charpy-V impact energy and estimated required impact energy to obtain the plastic true flow stress of this St52-3 is 10–15% lower than assumed based on the reported tensile limit load for the second St52-3 steel. test results. Figure 16. Compilation of proportional net section stress divided by flow stress as a function of ratio Figure 16. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic between the Charpy-V impact energy and estimated required impact energy to obtain the plastic Appl. Mech. 2021, 2, FOR PEER REVIEW 13 limit load for the second St52-3 steel. Calculation based on conservative correlation, Equation (5), limit load for the second St52-3 steel. Calculation based on conservative correlation, Equation (5), and omitting constraint correction. and omitting constraint correction. Figure 17. Proportional net section stress as a function of specimen width for 15 mm thick specimens Figure 17. Proportional net section stress as a function of specimen width for 15 mm thick specimens with fixed flaw size, 2a = 15 mm [11,12]. with fixed flaw size, 2a = 15 mm [11,12]. Overall, the present work provides an experimental validation of the criterion and The original Charpy-V criterion for the limit load [1] assumes a high-constraint ge- methodology developed previously [1]. ometry. If the principle is applied to simple tension-loaded surface-cracked structures, the criterion can be relaxed. Figure 18 shows the Q-parameter values for shallow surface- cracked tension specimens [5,13,14]. Except for the crack ends, the stress-normalized Q is close to −0.5. This enables a constraint correction to the criterion given in [1]. Following the same calculations as in [1] but correcting the crack driving force for constraint leads to a new criterion for the Charpy-V energy. The J1mm criterion was used because it proved to describe the CC(T) specimens well. The resulting criterion is shown in Figure 19. Figure 18. Comparison of angular dependence of elastic–plastic Q parameter normalized by relative stress and elastic T-stress for shallow uniaxial and biaxial tensile-loaded surface flaws [5,13,14]. Appl. Mech. 2021, 2, FOR PEER REVIEW 13 Figure 17. Proportional net section stress as a function of specimen width for 15 mm thick specimens Appl. Mech. 2021, 2 678 with fixed flaw size, 2a = 15 mm [11,12]. The original Charpy-V criterion for the limit load [1] assumes a high-constraint ge- ometry. If the principle is applied to simple tension-loaded surface-cracked structures, the The original Charpy-V criterion for the limit load [1] assumes a high-constraint ge- criterion can be relaxed. Figure 18 shows the Q-parameter values for shallow surface- ometry. If the principle is applied to simple tension-loaded surface-cracked structures, cracke the criterion d tensi can on sp be r ecim elaxed. ensFigur [5,13, e 1 18 4]. shows Excep the t fo Q-parameter r the crack end values s, the for stress-normalize shallow surface- d Q is cracked tension specimens [5,13,14]. Except for the crack ends, the stress-normalized Q is close to −0.5. This enables a constraint correction to the criterion given in [1]. Following close to 0.5. This enables a constraint correction to the criterion given in [1]. Following the same calculations as in [1] but correcting the crack driving force for constraint leads to the same calculations as in [1] but correcting the crack driving force for constraint leads to a new criterion for the Charpy-V energy. The J1mm criterion was used because it proved to a new criterion for the Charpy-V energy. The J criterion was used because it proved to 1mm describe the CC(T) specimens well. The resulting criterion is shown in Figure 19. describe the CC(T) specimens well. The resulting criterion is shown in Figure 19. Appl. Mech. 2021, 2, FOR PEER REVIEW 14 Figure 18. Comparison of angular dependence of elastic–plastic Q parameter normalized by relative Figure 18. Comparison of angular dependence of elastic–plastic Q parameter normalized by relative stress and elastic T-stress for shallow uniaxial and biaxial tensile-loaded surface flaws [5,13,14]. stress and elastic T-stress for shallow uniaxial and biaxial tensile-loaded surface flaws [5,13,14]. Figure 19. Charpy-V impact energy requirements to ensure plastic collapse for J , as a function 1mm Figure 19. Charpy-V impact energy requirements to ensure plastic collapse for J1mm, as a function of of section thickness for tension-loaded surface flaws. The estimate is based on the 5% lower-bound section thickness for tension-loaded surface flaws. The estimate is based on the 5% lower-bound Charpy-V correlation and accounting for the lower constraint. Charpy-V correlation and accounting for the lower constraint. The curves in Figure 19 can be approximated by Equation (15), which is a fit to the data in the figure.   KV J ≥+ exp 2.69 0.78⋅ ln W− () ( ) (15)  1mm  y  The previous criterion [1] also has a requirement for temperature. Equation (16). This too can be adjusted for constraint, simply by adjusting the temperature by Equation (17) [5], where ΔTPC corresponds to the temperature shift due to T-stress. 9742 71452 T ≥− T 536+ + 40.31⋅ ln W+ 65.28⋅ lnσ − () ( ) (16) PC KV27 J Y σ⋅ Wσ YY Ts − tress Δ= T (17) PC 12 MPa / °C The temperature TPC gives the lowest temperature where plastic collapse is predicted, provided that the Charpy-V impact energy at TPC (or lower) fulfils Equation (15). The Charpy-V data do not need to correspond to the upper shelf, as the plastic collapse crite- rion does not intend to guarantee upper-shelf behavior of the structure. It only guarantees plastic collapse prior to any possible brittle fracture. If the low-constraint condition of the application cannot be guaranteed, the criterion in [1] should be used. The above criteria regarding impact energy and temperature are valid for structures in quasistatic loading, even though based on a correlation between impact energy from a dynamic test. The correlation is only used to estimate the quasistatic tearing resistance of the material. Dynamic loading as in the case of an earthquake requires additional correc- tions both for impact energy and temperature. It should be noted that the present criterion is aimed at low-constraint steel structures covered by Eurocode 3 and is not intended for pressure equipment, such as piping or pressure vessels, or steel structures where large thermal or residual stresses are present. It can only be used if residual stresses and constraints can be ruled out. Appl. Mech. 2021, 2 679 The curves in Figure 19 can be approximated by Equation (15), which is a fit to the data in the figure. ( s ) KV(J )  exp 2.69 + 0.78 ln(W) (15) 1mm The previous criterion [1] also has a requirement for temperature. Equation (16). This too can be adjusted for constraint, simply by adjusting the temperature by Equation (17) [5], where DT corresponds to the temperature shift due to T-stress. PC 9742 71452 T  T 536 + + 40.31 ln(W) + 65.28 ln( ) (16) PC KV27J Y Y Y T-stress DT = (17) PC 12 MPa/ C The temperature T gives the lowest temperature where plastic collapse is predicted, PC provided that the Charpy-V impact energy at T (or lower) fulfils Equation (15). The PC Charpy-V data do not need to correspond to the upper shelf, as the plastic collapse criterion does not intend to guarantee upper-shelf behavior of the structure. It only guarantees plastic collapse prior to any possible brittle fracture. If the low-constraint condition of the application cannot be guaranteed, the criterion in [1] should be used. The above criteria regarding impact energy and temperature are valid for structures in quasistatic loading, even though based on a correlation between impact energy from a dynamic test. The correlation is only used to estimate the quasistatic tearing resistance of the material. Dynamic loading as in the case of an earthquake requires additional corrections both for impact energy and temperature. It should be noted that the present criterion is aimed at low-constraint steel structures covered by Eurocode 3 and is not intended for pressure equipment, such as piping or pressure vessels, or steel structures where large thermal or residual stresses are present. It can only be used if residual stresses and constraints can be ruled out. 5. Conclusions The load-bearing capacity of a CC(T) specimen in the ductile fracture regime is usually controlled by plastic collapse. If the material’s tearing resistance is sufficiently low, the load-bearing capacity can drop below the plastic collapse value. Here, a recently developed simple fracture mechanics-based Charpy-V impact energy criterion for plastic collapse was used to provide a best estimate assessment of CC(T) specimen load-bearing capacity. The previous criterion was modified to specifically address the CC(T) specimen geometry with respect to crack driving force and limit load. In addition, the effect of the low constraint connected to the CC(T) specimen was accounted for with respect to ductile tearing. Because the goal was a best estimate, the mean J –CVNus correlation was used. It was shown 1mm that if the material’s Charpy-V impact energy in relation to the material’s yield strength is below a specific value, the load-bearing capacity will be less than the plastic limit load. This result verifies the previously developed simple fracture mechanics-based Charpy-V impact energy criterion for plastic collapse. A new criterion, specifically for tension-loaded surface flaws, was developed, in line with the previous high-constraint criterion. The present criterion is limited to steel structures covered by Eurocode 3. It is not intended for pressure equipment such as piping or pressure vessels, or steel structures where large thermal or residual stresses are present. It cannot be used if residual stresses and constraints may be present in the structure. If the low-constraint condition of the application cannot be guaranteed, the high-constraint criterion should be used. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Appl. Mech. 2021, 2 680 Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The author declares no conflict of interest. References 1. Wallin, K. A simple fracture mechanics based Charpy-V impact energy criterion for plastic collapse. Eng. Fract. Mech. 2020, 237, e107247. [CrossRef] 2. Tada, H.; Paris, P.C.; Irwin, G.R. The Stress Analysis of Cracks Handbook, 2nd ed.; Paris Productions Incorporated: St. Louis, MO, USA, 1985. 3. Miller, A.G. Review of limit loads of structures containing defects. Int. J. PVP 1988, 32, 197–327. [CrossRef] 4. Wallin, K. Low-cost J-R curve estimation based on CVN upper shelf energy. Fatigue Fract. Eng. Mater. Struct. 2001, 24, 537–549. [CrossRef] 5. Wallin, K. Fracture Toughness of Engineering Materials—Estimation and Application; EMAS Publishing: Warrington, UK, 2011. 6. O’Dowd, N.P.; Shih, C.F. Family of Crack-Tip Fields Characterized by a Triaxiality Parameter. II. Fracture applications. J. Mech. Phys. Solids 1992, 40, 939–963. [CrossRef] 7. Brocks, W.; Schmitt, W. Quantitative Assessment of the Role of Crack Tip Constraint on Ductile Tearing; ASTM International: West Con- shohocken, PA, USA, 1993; pp. 64–78. 8. Feldmann, M.; Schaffrath, S.; Citarelli, S. Approach for Upper-Shelf Toughness Requirements for the Design of Steel Structures Based on Damage Mechanics; CEN/TC 250/SC 3 N 3162; RWTH Aachen University: Aachen, Germany, July 2020. 9. Dahl, W.; Hesse, W. Auswirkung der Beurteilung von Stählen auf dei Anwndung im Hoch- und Anlagenbau. Stahl Eisen 1986, 106, 695–702. 10. Hubo, R.; Twickler, W.; Hesse, W.; Dahl, W. Failure behaviour of steels with different levels of strength and toughness. Nucl. Eng. Des. 1988, 108, 457–466. [CrossRef] 11. Rosezin, H.-J.; Dahl, W. Einfluss von Temperatur und Probengeometrie auf die Ergebnisse von Grosssugversuchen. Stahl Eisen 1985, 105, 523–529. 12. Dormagen, D.; Halim, A.; Ehrhardt, H.; Dahl, W. On the toughness evaluation of sulphide-controlled Fe510 steel by Charp- V-impact, fracture mechanics and wide plate tests. In The Crack Tip Opening Displacement in Elastic-Plastic Fracture Mechanics; Schwalbe, K.H., Ed.; Springer: Berlin/Heidelberg, Germany, 1986. [CrossRef] 13. Wang, X. Two-Parameter Characterization of Elastic-Plastic crack Front Fields: Surface Cracked Plates under Tensile Loading. Eng. Fract. Mech. 2009, 76, 958–982. [CrossRef] 14. Wang, X. Elastic T-stress Solutions for Semi-Elliptical Surface Cracks in Finite Thickness Plates. Eng. Fract. Mech. 2003, 70, 731–756. [CrossRef] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mechanics Multidisciplinary Digital Publishing Institute

CC(T) Specimen Load-Bearing Capacity Related to Yield Strength and Upper-Shelf Charpy-V Energy

Wallin, Kim
Applied Mechanics , Volume 2 (4) – Sep 28, 2021
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Abstract

Article CC(T) Specimen Load-Bearing Capacity Related to Yield Strength and Upper-Shelf Charpy-V Energy Kim Wallin K -solutions, Ltd., Heinäkuja 4A, 02760 Espoo, Finland; Kim.Wallin@kwsolutions.fi Abstract: The load-bearing capacity of a CC(T) specimen (Center-Cracked Tension) in the ductile fracture regime is usually controlled by plastic collapse. If the material’s tearing resistance is suffi- ciently low, the load-bearing capacity can drop below the plastic collapse value. Here, a recently developed simple fracture mechanics-based Charpy-V impact energy criterion for plastic collapse was used to provide a best estimate assessment of the CC(T) specimen load-bearing capacity. Keywords: structural assessment; fracture mechanics; impact fracture; limit load; CC(T) specimen 1. Introduction Fracture mechanics is usually used mainly to ensure that a structure will not fail due to possible flaws. Normal engineering methods such as the one in Eurocode 3 focus mainly on avoidance of brittle fracture. If brittle fracture can be ruled out, it is commonly assumed that the structure will fail by plastic collapse and a limit load dimensioning is considered sufficient. However, if the material’s ductile tearing resistance is sufficiently low, it is possible that ductile crack growth lowers the maximum load-bearing capacity below the plastic limit load. To safeguard against such events, a criterion for the ductile tearing resistance is also needed. Citation: Wallin, K. CC(T) Specimen In a recent publication, a simple criterion for plastic collapse was developed based on Load-Bearing Capacity Related to the K /L ratio [1]. It postulates that if the K /L ratio is less than 0.4, the failure will be r r r r Yield Strength and Upper-Shelf controlled by plastic collapse. K represents the ratio between the elastic crack driving force Charpy-V Energy. Appl. Mech. 2021, 2, and the material’s fracture toughness, corresponding to a specific fracture definition [1]. L 666–680. https://doi.org/10.3390/ represents the ratio between primary load and plastic limit load corresponding to yield applmech2040038 strength [1]. The function relating K to L represents the reciprocal of a plasticity correction r r Received: 14 July 2021 on the elastic crack driving force. The criterion results in a simple Charpy-V requirement Accepted: 26 September 2021 to ensure that the maximum load will be controlled by the limit load. Published: 28 September 2021 The Charpy-V impact energy requirement is conservative as it is based on a conserva- tive estimate of the crack driving force and it is based on a lower bound-type correlation Publisher’s Note: MDPI stays neutral between J (J-integral value corresponding to 1 mm crack extension) and KVus (Charpy- 1mm with regard to jurisdictional claims in V impact energy at upper shelf), and the correlations have been developed for highly published maps and institutional affil- constrained bend specimens (SE(B) (single-edged bend) and C(T) (compact tension)). The iations. requirement is not intended to provide a best estimate of the Charpy-V impact energy required to ensure plastic collapse. It is a general screening criterion for plastic collapse. When the requirement is fulfilled, the likelihood of reaching the plastic limit load is high. In the original publication [1], there was a lack of experimental validation of the developed Copyright: © 2021 by the author. requirement. The goal of this work was to validate the requirement using actual test Licensee MDPI, Basel, Switzerland. data. The CC(T) specimen geometry was chosen because it mimics structures with mainly This article is an open access article tension loading as is the basic assumption in Eurocode 3 (EN 1993-1-10). distributed under the terms and Here, the method was used to assess the maximum load values of CC(T) speci- conditions of the Creative Commons mens. When the method is used to make a best estimate assessment of lowly constrained Attribution (CC BY) license (https:// CC(T) specimens, many of the aspects of the requirement need to be considered creativecommons.org/licenses/by/ and modified. 4.0/). Appl. Mech. 2021, 2, 666–680. https://doi.org/10.3390/applmech2040038 https://www.mdpi.com/journal/applmech Appl. Mech. 2021, 2 667 1.1. The Basis of the Criteria The K /L = 0.4 criterion allows the definition of a simple fracture toughness criterion r r for plastic collapse. The stress intensity factor in terms of load P can be expressed in the form of Equation (1) and the plastic limit load can be expressed in the form of Equation (2) [1], K =  f(a/W) (1) 1/2 B W P =   B W F a/W (2) ( ) L Y where K is the stress intensity factor, P is the limit load, P is the load, B is the specimen I L thickness, W is the specimen height, and  is the yield stress. The functions f(a/W) and F(a/W) are application-specific geometry functions that can be found from various handbooks [2,3]. The ratio K /L corresponding to the limit load (L = 1) can then be written in the r r r form of Equation (3) by inserting P from Equation (2) into Equation (1) [1]. The fracture toughness K represents the true material’s fracture toughness, not the linear-elastic IC fracture toughness. This means that K can correspond to J-integral-based K or to a IC JC point on the tearing resistance curve, expressed in the same units as the stress intensity factor, K. Equation (3) can be used for any geometry. 1/2 W  f(a/W) F(a/W) r(L =1) r Y = (3) L K r IC Applying simplified equations for the stress intensity factor and the limit load and replacing the yield strength with the material’s flow strength ( ), an upper bound estimate of the required fracture toughness to achieve plastic collapse is given by Equation (4) [1]. It is based on the maximum of Equation (3) for several different crack and loading geometries and contains a plasticity correction equal to 2.5, which corresponds to the typical level of plasticity at L = 1 [1]. IC p  2 ) plastic collapse (4) Equation (4) gives a conservative estimate of the crack driving force close to the plastic limit load. This needs to be matched with the material’s tearing resistance. The publica- tion [1] favored the use of the tearing resistance corresponding to 1 mm ductile crack growth (J ). The use of an initiation definition, such as J , is considered overly conservative. 1mm Ic The value of J was estimated from a well validated J KV correlation [1,4]. The 1mm 1mm us previous work [1] made use of the 5% lower bound correlation in the form of Equation (5). The J-integral values need to be written in the form of K , Equation (6), where E is the IC modulus of elasticity and  is Poisson’s ratio. 1.28 KV US J = 193 (5) 1mm5% J E K = (6) IC (1 ) The publication [1] ended up with a simple Charpy-V impact energy requirement in the form of Equation (7). KV(J )  exp 0.59 + 0.78 ln(W) + (7) 1mm 1.2. Modifications Specific to CC(T) Specimen The CC(T) specimen has a very simple geometry (Figure 1). There are several stress intensity factor solutions for the CC(T) specimen [2]. Equation (8) is supposed to have Appl. Mech. 2021, 2, FOR PEER REVIEW 3   KV J ≥+ exp 0.59 0.78⋅ ln W+ () ( )  (7) 1mm   Appl. Mech. 2021, 2 668 Appl. Mech. 2021, 2, FOR PEER REVIEW 3 1.2. Modifications Specific to CC(T) Specimen The CC(T) specimen has a very simple geometry (Figure 1). There are several stress intensity factor solutions for the CC(T) specimen [2]. Equation (8) is supposed to have an accur an accuracy acy of 0.of 1% for 0.1%any p for any roppr ort oportional ional crack crack lengtlength h a/W [2 a/W ]. The co [2]. The rrespondin corresponding g limit lo limit ad   equation load equation is very simple is very simple, , due to the dueextre to the mely extr low emely con low straint constraint of the Y specimen. It c of the specimen. an simply It can KV J ≥+ exp 0.59 0.78⋅ ln W+ () ( ) (7)  1mm simply be expressed in terms of a constant net section stress, Equation 343 (9) [3]. The crack be expressed in terms of a constant net section stress, Equation (9) [3]. The crack driving   driving force equation corresponding to limit load thus becomes Equation (10). It includes force equation corresponding to limit load thus becomes Equation (10). It includes the the plasticity correction of approximately 2.5, as used in the original simplified method plasticity correction of approximately 2.5, as used in the original simplified method de- 1.2. Modifications Specific to CC(T) Specimen described above. Equation (10) is shown graphically in Figure 2. As seen from the figure, scribed above. Equation (10) is shown graphically in Figure 2. As seen from the figure, the the maximum crack driving force is close to 2, as assumed in Equation (4). maximum The CC(T) sp crack driv ecimen has a ing force is clo very simple g se to 2, as assum eometry (F ed in Equatio igure 1). There n (4). are several stress intensity factor solutions for the CC(T) specimen [2]. Equation (8) r is supposed to have an p 2 4 a a  a accuracy of 0.1% for any proportional crack length a/W [2]. The corresponding limit load K =   a 1 0.025 + 0.06  sec  (8) W W 2 W equation is very simple, due to the extremely low constraint of the specimen. It can simply be expressed in terms of a constant net section stress, Equation (9) [3]. The crack driving =   1 (9) L f force equation corresponding to limit load thus becomes Equation (10). It includes the r  r plasticity correction of approximately 2.5, as used in the original simplified method de- K 2 4 a a a  a a I(L =1) scribed above. Equation (10) is shown graphically in Figure 2. As seen from the figure, the p =   1 0.025 + 0.06  sec   1  2.5 (10) W W W 2 W W maximum crack driving force is close to 2, as assumed in Equation (4). Figure 1. Definition of CC(T) specimen dimensions. The dimensions 2a and 2W are the crack length and specimen width, respectively.  aa  πa  K =σ ⋅ π⋅ a ⋅ 1 − 0.025 ⋅ + 0.06 ⋅ ⋅ sec ⋅ (8) I      WW 2W       a σ=σ ⋅ 1− (9)  Lf Figure 1. Definition of CC(T) specimen dimensions. TheW dimensions 2a and 2W are the crack length Figure 1. Definition of CC(T) specimen dimensions. The dimensions 2a and 2W are the crack length  and specimen width, respectively. and specimen width, respectively.  aa  πa  K =σ ⋅ π⋅ a ⋅ 1 − 0.025 ⋅ + 0.06 ⋅ ⋅ sec ⋅ (8) I      WW 2W       a σ=σ ⋅ 1− (9)  Lf  I(Lr=1) Figure 2. Plasticity-corrected crack driving force at limit load for a CC(T) specimen, as a function of relative crack length a/W. The J –CVN correlation that is based on the 5% lower bound is not applicable us 1mm for a best estimate. Instead, the median correlation should be used. This has the form of Equation (11) [1,4]. 1.28 KV US J = 269 (11) 1mm Appl. Mech. 2021, 2 669 The above-described energy requirement does not account for constraint differences. Deeply cracked bend specimens experience a high stress triaxiality or constraint, because the yielding is contained due to the compressive stresses opposite of the crack. Single- cracked-tension-loaded specimens experience an uncontained yielding because the whole ligament is in tension and, thus, the stress triaxiality or constraint is much lower. The CC(T) specimen thus has a much lower constraint than the bend specimens used for the Charpy-V correlation, and this has a pronounced effect on the tearing resistance curve. The T-stress, which is an elastic parameter, corresponds to the higher-order nonsin- gular term in the series expansion of the stress field equation. Even though the T-stress describes the nonsingular stress component in the direction of the crack plane, it also gives an approximation of the opening stress in the loading direction. Figure 3 shows the effect of constraint, expressed in the form of T-stress at limit load, on the J-integral for a specific crack growth compared to a high-constraint configuration [5]. Deeply cracked bend specimens have positive T-stresses, whereas the CC(T) specimens have strongly negative T-stresses, as seen from Figure 4 [5]. The T-stress effect on the tearing resistance can be approximated by Equation (12) [5]. The accuracy of the constraint Appl. Mech. 2021, 2, FOR PEER REVIEW 5 effect description is of the order of 15%. J T-stress 100 MPa exp . . . T-stress  100 MPa (12) J 1500 MPa T>0 3.0 2¼Cr1Mo σ = 260 MPa 2.8 Y StE460 σ = 460 MPa 2.6 Carbon steel σ = 252 MPa Q&T steel σ = 700 MPa 2.4 X80 σ = 570 MPa 2.2 HY-80 σ = 610 MPa A533B Cl.1 σ = 397 MPa 2.0 HY-100 σ = 749 MPa 1.8 1.6 1.4 1.2 1.0 0.8 -600 -500 -400 -300 -200 -100 0 100 200 T-stress (limit load) [MPa] Figure 3. Effect of constraint expressed in the form of T-stress at limit load, on the J-integral for a Figure 3. Effect of constraint expressed in the form of T-stress at limit load, on the J-integral for a specific crack growth compared to a high-constraint configuration (J/J ) [5]. Dashed lines indicate T>0 specific crack growth compared to a high-constraint configuration (J/JT>0) [5]. Dashed lines indicate 15% uncertainty. ±15% uncertainty. Figure 4 contains a fit to the individual T-stress data. It has the form of Equation (13) [5]. 2 3 T-stress a a a 1.14 + 0.89 + 0.152 0.495 (13) W W W It is important to note that Equation (13) corresponds to limit load. If the net sec- tion stress ( ) is below the flow stress, Equation (13) needs to be multiplied by the net ratio  / . net The Q parameter is an elastic–plastic constraint parameter. A definition, in line with the T-stress, refers to the normalized distance between the actual tensile stress  at a yy Figure 4. T-stress at limit load, normalized by yield strength, for a CC(T) specimen, as a function of relative crack length a/W [5]. The Q parameter is an elastic–plastic constraint parameter. A definition, in line with the T-stress, refers to the normalized distance between the actual tensile stress σyy at a specified location in front of the crack and the matching value corresponding to the small- scale yielding solution for T = 0 [6]. It can be expressed in the form of Equation (14) [6]. σ− σ () yy yy r ⋅σ T0 = Y (14) Qa≡θt=0, =2 σ J Figure 5 shows a comparison of the yield stress-normalized elastic T-stress corre- sponding to limit load and the Q value corresponding to the load at JIC, where the Q values have been collected from the references leading to Figure 3. Overall, the two parameters follow an approximate offset of 0.15. The T-stress underestimates very low Q values and overestimates very high values, but in the range Q = −1 to 0, the estimates generally differ less than 0.2, which is of the same order as the uncertainty in the parameter estimates. This provides validation for the use of Equation (12) to describe the constraint effect for ductile fracture. J/J T>0 Appl. Mech. 2021, 2, FOR PEER REVIEW 5 3.0 2¼Cr1Mo σ = 260 MPa 2.8 StE460 σ = 460 MPa 2.6 Carbon steel σ = 252 MPa Q&T steel σ = 700 MPa 2.4 Y X80 σ = 570 MPa 2.2 HY-80 σ = 610 MPa A533B Cl.1 σ = 397 MPa 2.0 HY-100 σ = 749 MPa 1.8 1.6 1.4 1.2 Appl. Mech. 2021, 2 670 1.0 0.8 -600 -500 -400 -300 -200 -100 0 100 200 T-stress (limit load) [MPa] specified location in front of the crack and the matching value corresponding to the small- scale yielding solution for T = 0 [6]. It can be expressed in the form of Equation (14) [6]. Figure 3. Effect of constraint expressed  in the form of T-stress at limit load, on the J-integral for a yy yy r T=0 specific crack growth compared to a high-constraint configuration (J/JT>0) [5]. Dashed lines indicate Q  at  = 0, = 2 (14) ±15% uncertainty. Figure 4. T-stress at limit load, normalized by yield strength, for a CC(T) specimen, as a function of Figure 4. T-stress at limit load, normalized by yield strength, for a CC(T) specimen, as a function of relative crack length a/W [5]. relative crack length a/W [5]. Figure 5 shows a comparison of the yield stress-normalized elastic T-stress correspond- The Q parameter is an elastic–plastic constraint parameter. A definition, in line with ing to limit load and the Q value corresponding to the load at J , where the Q values IC have been collected from the references leading to Figure 3. Overall, the two parameters the T-stress, refers to the normalized distance between the actual tensile stress σyy at a follow an approximate offset of 0.15. The T-stress underestimates very low Q values and specified location in front of the crack and the matching value corresponding to the small- overestimates very high values, but in the range Q = 1 to 0, the estimates generally differ scale yielding solution for T = 0 [6]. It can be expressed in the form of Equation (14) [6]. Appl. Mech. 2021, 2, FOR PEER REVIEW 6 less than 0.2, which is of the same order as the uncertainty in the parameter estimates. This provides validation for the use of Equation (12) to describe the constraint effect for σ− σ () yy yy r ⋅σ ductile fracture. T0 = Y (14) Qa≡θt=0, =2 σ J Figure 5 shows a comparison of the yield stress-normalized elastic T-stress corre- sponding to limit load and the Q value corresponding to the load at JIC, where the Q values have been collected from the references leading to Figure 3. Overall, the two parameters follow an approximate offset of 0.15. The T-stress underestimates very low Q values and overestimates very high values, but in the range Q = −1 to 0, the estimates generally differ less than 0.2, which is of the same order as the uncertainty in the parameter estimates. This provides validation for the use of Equation (12) to describe the constraint effect for ductile fracture. Figure 5. Relation between elastic T-stress at limit load (T/ ) and elastic–plastic Q parameter Figure 5. Relation between elastic T-stress at limit load (T/σY) and elastic–plastic Q parameter at JIc at J [5]. Ic [5]. The constraint effect data in Figure 3 are verified only to a J/JT>0 ratio of 2. Due to this, the constraint correction for the CC(T) specimens is limited to 2. This is in line with actual tearing resistance data for CC(T) specimens, as compared to C(T) specimen tearing re- sistance data shown in Figure 6 [7]. Figure 6. Comparison of CC(T) and C(T) specimen tearing resistance (J-integral versus crack exten- sion Δa) for a medium-strength steel [7]. 2. Materials and Methods The data for the CC(T) tests were taken from [8–10]. Nine different steels with differ- ent mechanical properties were covered. Their mechanical properties are given in Table 1. The strength levels varied from moderate (325 MPa) to extra high (988 MPa). The Charpy-V upper-shelf energies, measured at ambient temperature, were in the range 85 J to 250 J. The specimens were CC(T) specimens with W = 150 mm and having varying crack lengths. The plate thickness varied between 20 and 40 mm, being in most cases 30 mm. The thickness dimension was not significant for the CC(T) specimens, when assessed in terms of nominal and net section stress. The first four steels had specimens with pre-fa- tigue cracks, whereas the other steels had specimens with narrow Electric Discharge Ma- chining (EDM) notches. All tests were performed at room temperature. The specimens were loaded in tension under displacement control and, for this work, the maximum load J/J T>0 Appl. Mech. 2021, 2, FOR PEER REVIEW 6 Figure 5. Relation between elastic T-stress at limit load (T/σY) and elastic–plastic Q parameter at JIc Appl. Mech. 2021, 2 [5]. 671 The constraint effect data in Figure 3 are verified only to a J/JT>0 ratio of 2. Due to this, the constraint correction for the CC(T) specimens is limited to 2. This is in line with actual The constraint effect data in Figure 3 are verified only to a J/J ratio of 2. Due to T>0 this, the constraint correction for the CC(T) specimens is limited to 2. This is in line with tearing resistance data for CC(T) specimens, as compared to C(T) specimen tearing re- actual tearing resistance data for CC(T) specimens, as compared to C(T) specimen tearing sistance data shown in Figure 6 [7]. resistance data shown in Figure 6 [7]. Figure 6. Comparison of CC(T) and C(T) specimen tearing resistance (J-integral versus crack exten- Figure 6. Comparison of CC(T) and C(T) specimen tearing resistance (J-integral versus crack exten- sion Da) for a medium-strength steel [7]. sion Δa) for a medium-strength steel [7]. 2. Materials and Methods The data for the CC(T) tests were taken from [8–10]. Nine different steels with different 2. Materials and Methods mechanical properties were covered. Their mechanical properties are given in Table 1. The strength The da levels tavaried for thfr e CC( om moderate T) tests were taken from [ (325 MPa) to extra high 8–1 (988 0]. Ni MPa). ne dif The ferent steel Charpy-V s with differ- upper-shelf energies, measured at ambient temperature, were in the range 85 J to 250 J. The ent mechanical properties were covered. Their mechanical properties are given in Table specimens were CC(T) specimens with W = 150 mm and having varying crack lengths. The 1. The strength levels varied from moderate (325 MPa) to extra high (988 MPa). The plate thickness varied between 20 and 40 mm, being in most cases 30 mm. The thickness Charpy-V upper-shelf energies, measured at ambient temperature, were in the range 85 J dimension was not significant for the CC(T) specimens, when assessed in terms of nominal to 250 J. The specimens were CC(T) specimens with W = 150 mm and having varying crack and net section stress. The first four steels had specimens with pre-fatigue cracks, whereas the other steels had specimens with narrow Electric Discharge Machining (EDM) notches. lengths. The plate thickness varied between 20 and 40 mm, being in most cases 30 mm. All tests were performed at room temperature. The specimens were loaded in tension The thickness dimension was not significant for the CC(T) specimens, when assessed in under displacement control and, for this work, the maximum load during the test was used terms of nominal and net section stress. The first four steels had specimens with pre-fa- in the analysis. None of the specimens experienced unstable fracture prior to maximum tigue cracks, whereas the other steels had specimens with narrow Electric Discharge Ma- load during the tests. chining (EDM) notches. All tests were performed at room temperature. The specimens Table 1. Mechanical properties of the materials [8–10]. were loaded in tension under displacement control and, for this work, the maximum load Material  MPa  MPa KV J Ref. U us S355N 364 526 250 [8] S500M 584 639 227 [8] S690QL 845 884 192 [8] S960QL 988 1055 125 [8] St52-3 403 594 200 [9] St52-3 325 490 85 [9] Steel A 420 640 200 [10] Steel B 565 715 200 [10] Steel C 755 835 120 [10] Appl. Mech. 2021, 2, FOR PEER REVIEW 7 during the test was used in the analysis. None of the specimens experienced unstable frac- ture prior to maximum load during the tests. The CC(T) test results for the three sets of steels are shown compiled in Figures 7–9, in the form of nominal stress at load maximum. Table 1. Mechanical properties of the materials [8–10]. Material σY MPa σU MPa KVus J Ref. S355N 364 526 250 [8] S500M 584 639 227 [8] S690QL 845 884 192 [8] S960QL 988 1055 125 [8] St52-3 403 594 200 [9] Appl. Mech. 2021, 2 672 St52-3 325 490 85 [9] Steel A 420 640 200 [10] Steel B 565 715 200 [10] The CC(T) test results for the three sets of steels are shown compiled in Figures 7–9, in Steel C 755 835 120 [10] the form of nominal stress at load maximum. Appl. Mech. 2021, 2, FOR PEER REVIEW 8 Figure 7. CC(T) test maximum load (nominal stress) values divided by flow stress for four different Figure 7. CC(T) test maximum load (nominal stress) values divided by flow stress for four different steels, as a function of relative crack length a/W [8]. steels, as a function of relative crack length a/W [8]. Figure 8. CC(T) test maximum load (nominal stress) values divided by flow stress for two different Figure 8. CC(T) test maximum load (nominal stress) values divided by flow stress for two different St52-3 steels, as a function of relative crack length a/W [9]. St52-3 steels, as a function of relative crack length a/W [9]. Figure 9. CC(T) test maximum load (nominal stress) values divided by flow stress for three different steels, as a function of relative crack length a/W [10]. 3. Results The individual test results were analyzed with Equations (10)–(13), to estimate the required Charpy-V impact energy to achieve the plastic limit load in terms of Equation (9). In cases when Equation (12) indicated a larger constraint effect on J than 2, the adjust- ment was limited to 2. This only occurred with the smallest crack length and higher- strength steels. Figures 10–12 show the measured proportional maximum net section stresses as a function of relative crack length. The results are comparable as all specimens had a width (2W) of 300 mm. The crack length does seem to have a pronounced effect on the maximum net section stress. The figures include the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. With some exceptions, there is a Appl. Mech. 2021, 2, FOR PEER REVIEW 8 Appl. Mech. 2021, 2 673 Figure 8. CC(T) test maximum load (nominal stress) values divided by flow stress for two different St52-3 steels, as a function of relative crack length a/W [9]. Appl. Mech. 2021, 2, FOR PEER REVIEW 9 Figure 9. CC(T) test maximum load (nominal stress) values divided by flow stress for three different Figure 9. CC(T) test maximum load (nominal stress) values divided by flow stress for three different trend that a low maximum net section stress is related to a low ratio between the Charpy- steels, as a function of relative crack length a/W [10]. steels, as a function of relative crack length a/W [10]. V impact energy and estimated required impact energy. 3. Results This effect is examined more closely in Figure 13. With one exception, the results 3. Results The individual test results were analyzed with Equations (10)–(13), to estimate the show a positive trend between proportional maximum net section stress and the ratio be- The individual test results were analyzed with Equations (10)–(13), to estimate the required Charpy-V impact energy to achieve the plastic limit load in terms of Equation (9). tween real and required impact energy. The second St52-3 steel appears to differ from the required Charpy-V impact energy to achieve the plastic limit load in terms of Equation In cases when Equation (12) indicated a larger constraint effect on J than 2, the adjustment general trend. Generally, considering that the estimates are average best estimates, the (9). In cases when Equation (12) indicated a larger constraint effect on J than 2, the adjust- was limited to 2. This only occurred with the smallest crack length and higher-strength data tend to confirm that when the material’s impact energy matches or is above the re- ment was steels. Figurli es mi 10 ted to 2. T –12 show the his onl measur y occurred ed proportional with the sm maximum allestnet crack section length a stresses nd higher- as a quired value, the maximum net section stress equals or is above the plastic limit load function of relative crack length. The results are comparable as all specimens had a width strength steels. Figures 10–12 show the measured proportional maximum net section value, based on flow stress. The variation in the results can be attributed to the scatter in (2W) of 300 mm. The crack length does seem to have a pronounced effect on the maximum stresses as a function of relative crack length. The results are comparable as all specimens the J-integral—Charpy-V correlation and to small (± 5%) deviations in the flow strength. net section stress. had a width (2W) of 300 mm. The crack length does seem to have a pronounced effect on the maximum net section stress. The figures include the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. With some exceptions, there is a Figure 10. Figure 10. Net Net section section stre stress ssvalues values d divided ivided by by flow flow strstres ess for s fo the r the da data in ta Figur in Fig eu 7re 7 as a as a function function of of relative relative crack crack length and related to the rat length and related to the ratioibetween o between the Charpy-V impac the Charpy-V impact ener t energy and estimated gy and estimated required impact energy to obtain the plastic limit load. required impact energy to obtain the plastic limit load. Figure 11. Proportional net section stress values divided by flow stress for the data in Figure 8 as a function of relative crack length and related to the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. Appl. Mech. 2021, 2, FOR PEER REVIEW 9 trend that a low maximum net section stress is related to a low ratio between the Charpy- V impact energy and estimated required impact energy. This effect is examined more closely in Figure 13. With one exception, the results show a positive trend between proportional maximum net section stress and the ratio be- tween real and required impact energy. The second St52-3 steel appears to differ from the general trend. Generally, considering that the estimates are average best estimates, the data tend to confirm that when the material’s impact energy matches or is above the re- quired value, the maximum net section stress equals or is above the plastic limit load value, based on flow stress. The variation in the results can be attributed to the scatter in the J-integral—Charpy-V correlation and to small (± 5%) deviations in the flow strength. Figure 10. Net section stress values divided by flow stress for the data in Figure 7 as a function of Appl. Mech. 2021, 2 674 relative crack length and related to the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. Figure 11. Proportional net section stress values divided by flow stress for the data in Figure 8 as a Figure 11. Proportional net section stress values divided by flow stress for the data in Figure 8 as a Appl. Mech. 2021, 2, FOR PEER REVIEW 10 function of relative crack length and related to the ratio between the Charpy-V impact energy and function of relative crack length and related to the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. estimated required impact energy to obtain the plastic limit load. Figure 12. Figure 12. Proportional net Proportional net section section stress va stress values lues div divided ided bby y flow flow stress for the stress for the data in Figure 9 as a data in Figure 9 as function of relative crack length and related to ratio between the Charpy-V impact energy and esti- a function of relative crack length and related to ratio between the Charpy-V impact energy and mated required impact energy to obtain the plastic limit load. estimated required impact energy to obtain the plastic limit load. The figures include the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. With some exceptions, there is a trend that a low maximum net section stress is related to a low ratio between the Charpy-V impact energy and estimated required impact energy. This effect is examined more closely in Figure 13. With one exception, the results show a positive trend between proportional maximum net section stress and the ratio between real and required impact energy. The second St52-3 steel appears to differ from the general trend. Generally, considering that the estimates are average best estimates, the data tend to confirm that when the material’s impact energy matches or is above the required value, the maximum net section stress equals or is above the plastic limit load Figure 13. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. 4. Discussion The load-bearing capacity of CC(T) specimens is seen to be controlled by the mate- rial’s flow stress, based on the net section stress in the ligament, provided the material’s upper-shelf impact energy is sufficient. The second St52-3 steel seems to deviate from the general trend. For this steel, there are more test results from different-size CC(T) speci- mens [11]. The results refer to specimen widths in the range 2W = 120–600 mm and a va- riety of relative crack lengths. The material and plate thickness (30 mm) are the same as for the second St52-3. Most 300 mm test results are the same as in Figure 8. The nominal stress at load maximum, as a function of relative crack length for the different specimen Appl. Mech. 2021, 2, FOR PEER REVIEW 10 Appl. Mech. 2021, 2 675 Figure 12. Proportional net section stress values divided by flow stress for the data in Figure 9 as a function of relative crack length and related to ratio between the Charpy-V impact energy and esti- value, based on flow stress. The variation in the results can be attributed to the scatter in mated required impact energy to obtain the plastic limit load. the J-integral—Charpy-V correlation and to small (5%) deviations in the flow strength. Figure 13. Compilation of proportional net section stress divided by flow stress as a function of ratio Figure 13. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load. limit load. 4. Discussion 4. Discussion The load-bearing capacity of CC(T) specimens is seen to be controlled by the material’s flow stress, based on the net section stress in the ligament, provided the material’s upper- The load-bearing capacity of CC(T) specimens is seen to be controlled by the mate- shelf impact energy is sufficient. The second St52-3 steel seems to deviate from the general rial’s flow stress, based on the net section stress in the ligament, provided the material’s trend. For this steel, there are more test results from different-size CC(T) specimens [11]. upper-shelf impact energy is sufficient. The second St52-3 steel seems to deviate from the The results refer to specimen widths in the range 2W = 120–600 mm and a variety of relative general trend. For this steel, there are more test results from different-size CC(T) speci- crack lengths. The material and plate thickness (30 mm) are the same as for the second mens [11]. The results refer to specimen widths in the range 2W = 120–600 mm and a va- St52-3. Most 300 mm test results are the same as in Figure 8. The nominal stress at load riety of relative crack lengths. The material and plate thickness (30 mm) are the same as maximum, as a function of relative crack length for the different specimen widths, is shown in Figure 14 [11]. The maximum nominal stress is clearly both dependent on the relative for the second St52-3. Most 300 mm test results are the same as in Figure 8. The nominal crack length and the specimen width. stress at load maximum, as a function of relative crack length for the different specimen The data in Figure 14 were analyzed similarly to the data for a constant specimen width. The result, in the form of the ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load, is shown in Figure 15. The results confirm that the crack driving force directly affects the maximum net section stress. It is unlikely that the Charpy-V energy would be much lower than 85 J. Even though the correlation between J-integral and Charpy-V energy is based on the median correlation, the trend with low net section stress values in Figure 15 cannot be explained by the uncertainty in the correlation alone, as seen from Figure 16. The calculation in this case is performed using the 5% lower-bound Charpy-V–J correlation, as given in Equation (5). The constraint correction is also conservatively taken as 1. Even for this conservative estimate, the net section stress values are low with respect to the expectations. One possible explanation is that the true flow stress of this St52-3 is 10–15% lower than assumed based on the reported tensile test results. Appl. Mech. 2021, 2, FOR PEER REVIEW 11 widths, is shown in Figure 14 [11]. The maximum nominal stress is clearly both dependent on the relative crack length and the specimen width. The data in Figure 14 were analyzed similarly to the data for a constant specimen width. The result, in the form of the ratio between the Charpy-V impact energy and esti- mated required impact energy to obtain the plastic limit load, is shown in Figure 15. The results confirm that the crack driving force directly affects the maximum net section stress. It is unlikely that the Charpy-V energy would be much lower than 85 J. Even though the correlation between J-integral and Charpy-V energy is based on the median correlation, the trend with low net section stress values in Figure 15 cannot be explained by the un- certainty in the correlation alone, as seen from Figure 16. The calculation in this case is performed using the 5% lower-bound Charpy-V–J correlation, as given in Equation (5). The constraint correction is also conservatively taken as 1. Even for this conservative esti- mate, the net section stress values are low with respect to the expectations. One possible Appl. Mech. 2021, 2 676 explanation is that the true flow stress of this St52-3 is 10–15% lower than assumed based on the reported tensile test results. Appl. Mech. 2021, 2, FOR PEER REVIEW 12 Figure 14. CC(T) test maximum load (nominal stress) values for different specimen widths, as a Figure 14. CC(T) test maximum load (nominal stress) values for different specimen widths, as a function of relative crack length a/W [11]. function of relative crack length a/W [11]. The assumption of a lower-than-assumed flow stress is strengthened by a second set of test results for the same St52-3 steel [11]. These test results correspond to a 15 mm plate thickness. The mechanical properties are higher than for the 30 mm plate, but the steel composition is identical. The material’s impact energy is not known, but from experience, it is expected to be higher than for the 30 mm plate. The data are comprised of a set of different-width CC(T) specimens with a constant initial flaw with 2a = 15 mm. The data, in the form of a maximum net section stress in relation to the flow stress, are reproduced in Figure 17, as a function of specimen width [11]. In addition, in this case, the maximum net section stress is lower than expected, similar to the 30 mm-thick plate data. For com- parison, a test result for a similar-strength Fe510 steel with identical flaw size (2a = 15 mm) and plate thickness (15 mm) [12] is included in Figure 17. The behavior of the Fe510 steel is similar to those of all the other steels in the study. This is an additional indication that the true flow stress of this St52-3 is 10–15% lower than assumed based on the reported tensile test results. Overall, the present work provides an experimental validation of the criterion and methodology developed previously [1]. Figure 15. Compilation of proportional net section stress divided by flow stress as a function of ratio Figure 15. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load for the second St52-3 steel. limit load for the second St52-3 steel. The assumption of a lower-than-assumed flow stress is strengthened by a second set of test results for the same St52-3 steel [11]. These test results correspond to a 15 mm plate thickness. The mechanical properties are higher than for the 30 mm plate, but the steel composition is identical. The material’s impact energy is not known, but from experience, it is expected to be higher than for the 30 mm plate. The data are comprised of a set of different-width CC(T) specimens with a constant initial flaw with 2a = 15 mm. The data, in the form of a maximum net section stress in relation to the flow stress, are reproduced in Figure 17, as a function of specimen width [11]. In addition, in this case, the maximum net Figure 16. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic limit load for the second St52-3 steel. Calculation based on conservative correlation, Equation (5), and omitting constraint correction. Appl. Mech. 2021, 2, FOR PEER REVIEW 12 Appl. Mech. 2021, 2 677 section stress is lower than expected, similar to the 30 mm-thick plate data. For comparison, a test result for a similar-strength Fe510 steel with identical flaw size (2a = 15 mm) and plate thickness (15 mm) [12] is included in Figure 17. The behavior of the Fe510 steel is Figure 15. Compilation of proportional net section stress divided by flow stress as a function of ratio similar to those of all the other steels in the study. This is an additional indication that the between the Charpy-V impact energy and estimated required impact energy to obtain the plastic true flow stress of this St52-3 is 10–15% lower than assumed based on the reported tensile limit load for the second St52-3 steel. test results. Figure 16. Compilation of proportional net section stress divided by flow stress as a function of ratio Figure 16. Compilation of proportional net section stress divided by flow stress as a function of ratio between the Charpy-V impact energy and estimated required impact energy to obtain the plastic between the Charpy-V impact energy and estimated required impact energy to obtain the plastic Appl. Mech. 2021, 2, FOR PEER REVIEW 13 limit load for the second St52-3 steel. Calculation based on conservative correlation, Equation (5), limit load for the second St52-3 steel. Calculation based on conservative correlation, Equation (5), and omitting constraint correction. and omitting constraint correction. Figure 17. Proportional net section stress as a function of specimen width for 15 mm thick specimens Figure 17. Proportional net section stress as a function of specimen width for 15 mm thick specimens with fixed flaw size, 2a = 15 mm [11,12]. with fixed flaw size, 2a = 15 mm [11,12]. Overall, the present work provides an experimental validation of the criterion and The original Charpy-V criterion for the limit load [1] assumes a high-constraint ge- methodology developed previously [1]. ometry. If the principle is applied to simple tension-loaded surface-cracked structures, the criterion can be relaxed. Figure 18 shows the Q-parameter values for shallow surface- cracked tension specimens [5,13,14]. Except for the crack ends, the stress-normalized Q is close to −0.5. This enables a constraint correction to the criterion given in [1]. Following the same calculations as in [1] but correcting the crack driving force for constraint leads to a new criterion for the Charpy-V energy. The J1mm criterion was used because it proved to describe the CC(T) specimens well. The resulting criterion is shown in Figure 19. Figure 18. Comparison of angular dependence of elastic–plastic Q parameter normalized by relative stress and elastic T-stress for shallow uniaxial and biaxial tensile-loaded surface flaws [5,13,14]. Appl. Mech. 2021, 2, FOR PEER REVIEW 13 Figure 17. Proportional net section stress as a function of specimen width for 15 mm thick specimens Appl. Mech. 2021, 2 678 with fixed flaw size, 2a = 15 mm [11,12]. The original Charpy-V criterion for the limit load [1] assumes a high-constraint ge- ometry. If the principle is applied to simple tension-loaded surface-cracked structures, the The original Charpy-V criterion for the limit load [1] assumes a high-constraint ge- criterion can be relaxed. Figure 18 shows the Q-parameter values for shallow surface- ometry. If the principle is applied to simple tension-loaded surface-cracked structures, cracke the criterion d tensi can on sp be r ecim elaxed. ensFigur [5,13, e 1 18 4]. shows Excep the t fo Q-parameter r the crack end values s, the for stress-normalize shallow surface- d Q is cracked tension specimens [5,13,14]. Except for the crack ends, the stress-normalized Q is close to −0.5. This enables a constraint correction to the criterion given in [1]. Following close to 0.5. This enables a constraint correction to the criterion given in [1]. Following the same calculations as in [1] but correcting the crack driving force for constraint leads to the same calculations as in [1] but correcting the crack driving force for constraint leads to a new criterion for the Charpy-V energy. The J1mm criterion was used because it proved to a new criterion for the Charpy-V energy. The J criterion was used because it proved to 1mm describe the CC(T) specimens well. The resulting criterion is shown in Figure 19. describe the CC(T) specimens well. The resulting criterion is shown in Figure 19. Appl. Mech. 2021, 2, FOR PEER REVIEW 14 Figure 18. Comparison of angular dependence of elastic–plastic Q parameter normalized by relative Figure 18. Comparison of angular dependence of elastic–plastic Q parameter normalized by relative stress and elastic T-stress for shallow uniaxial and biaxial tensile-loaded surface flaws [5,13,14]. stress and elastic T-stress for shallow uniaxial and biaxial tensile-loaded surface flaws [5,13,14]. Figure 19. Charpy-V impact energy requirements to ensure plastic collapse for J , as a function 1mm Figure 19. Charpy-V impact energy requirements to ensure plastic collapse for J1mm, as a function of of section thickness for tension-loaded surface flaws. The estimate is based on the 5% lower-bound section thickness for tension-loaded surface flaws. The estimate is based on the 5% lower-bound Charpy-V correlation and accounting for the lower constraint. Charpy-V correlation and accounting for the lower constraint. The curves in Figure 19 can be approximated by Equation (15), which is a fit to the data in the figure.   KV J ≥+ exp 2.69 0.78⋅ ln W− () ( ) (15)  1mm  y  The previous criterion [1] also has a requirement for temperature. Equation (16). This too can be adjusted for constraint, simply by adjusting the temperature by Equation (17) [5], where ΔTPC corresponds to the temperature shift due to T-stress. 9742 71452 T ≥− T 536+ + 40.31⋅ ln W+ 65.28⋅ lnσ − () ( ) (16) PC KV27 J Y σ⋅ Wσ YY Ts − tress Δ= T (17) PC 12 MPa / °C The temperature TPC gives the lowest temperature where plastic collapse is predicted, provided that the Charpy-V impact energy at TPC (or lower) fulfils Equation (15). The Charpy-V data do not need to correspond to the upper shelf, as the plastic collapse crite- rion does not intend to guarantee upper-shelf behavior of the structure. It only guarantees plastic collapse prior to any possible brittle fracture. If the low-constraint condition of the application cannot be guaranteed, the criterion in [1] should be used. The above criteria regarding impact energy and temperature are valid for structures in quasistatic loading, even though based on a correlation between impact energy from a dynamic test. The correlation is only used to estimate the quasistatic tearing resistance of the material. Dynamic loading as in the case of an earthquake requires additional correc- tions both for impact energy and temperature. It should be noted that the present criterion is aimed at low-constraint steel structures covered by Eurocode 3 and is not intended for pressure equipment, such as piping or pressure vessels, or steel structures where large thermal or residual stresses are present. It can only be used if residual stresses and constraints can be ruled out. Appl. Mech. 2021, 2 679 The curves in Figure 19 can be approximated by Equation (15), which is a fit to the data in the figure. ( s ) KV(J )  exp 2.69 + 0.78 ln(W) (15) 1mm The previous criterion [1] also has a requirement for temperature. Equation (16). This too can be adjusted for constraint, simply by adjusting the temperature by Equation (17) [5], where DT corresponds to the temperature shift due to T-stress. PC 9742 71452 T  T 536 + + 40.31 ln(W) + 65.28 ln( ) (16) PC KV27J Y Y Y T-stress DT = (17) PC 12 MPa/ C The temperature T gives the lowest temperature where plastic collapse is predicted, PC provided that the Charpy-V impact energy at T (or lower) fulfils Equation (15). The PC Charpy-V data do not need to correspond to the upper shelf, as the plastic collapse criterion does not intend to guarantee upper-shelf behavior of the structure. It only guarantees plastic collapse prior to any possible brittle fracture. If the low-constraint condition of the application cannot be guaranteed, the criterion in [1] should be used. The above criteria regarding impact energy and temperature are valid for structures in quasistatic loading, even though based on a correlation between impact energy from a dynamic test. The correlation is only used to estimate the quasistatic tearing resistance of the material. Dynamic loading as in the case of an earthquake requires additional corrections both for impact energy and temperature. It should be noted that the present criterion is aimed at low-constraint steel structures covered by Eurocode 3 and is not intended for pressure equipment, such as piping or pressure vessels, or steel structures where large thermal or residual stresses are present. It can only be used if residual stresses and constraints can be ruled out. 5. Conclusions The load-bearing capacity of a CC(T) specimen in the ductile fracture regime is usually controlled by plastic collapse. If the material’s tearing resistance is sufficiently low, the load-bearing capacity can drop below the plastic collapse value. Here, a recently developed simple fracture mechanics-based Charpy-V impact energy criterion for plastic collapse was used to provide a best estimate assessment of CC(T) specimen load-bearing capacity. The previous criterion was modified to specifically address the CC(T) specimen geometry with respect to crack driving force and limit load. In addition, the effect of the low constraint connected to the CC(T) specimen was accounted for with respect to ductile tearing. Because the goal was a best estimate, the mean J –CVNus correlation was used. It was shown 1mm that if the material’s Charpy-V impact energy in relation to the material’s yield strength is below a specific value, the load-bearing capacity will be less than the plastic limit load. This result verifies the previously developed simple fracture mechanics-based Charpy-V impact energy criterion for plastic collapse. A new criterion, specifically for tension-loaded surface flaws, was developed, in line with the previous high-constraint criterion. The present criterion is limited to steel structures covered by Eurocode 3. It is not intended for pressure equipment such as piping or pressure vessels, or steel structures where large thermal or residual stresses are present. It cannot be used if residual stresses and constraints may be present in the structure. If the low-constraint condition of the application cannot be guaranteed, the high-constraint criterion should be used. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Appl. Mech. 2021, 2 680 Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The author declares no conflict of interest. References 1. Wallin, K. A simple fracture mechanics based Charpy-V impact energy criterion for plastic collapse. Eng. Fract. Mech. 2020, 237, e107247. [CrossRef] 2. Tada, H.; Paris, P.C.; Irwin, G.R. The Stress Analysis of Cracks Handbook, 2nd ed.; Paris Productions Incorporated: St. Louis, MO, USA, 1985. 3. Miller, A.G. 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Auswirkung der Beurteilung von Stählen auf dei Anwndung im Hoch- und Anlagenbau. Stahl Eisen 1986, 106, 695–702. 10. Hubo, R.; Twickler, W.; Hesse, W.; Dahl, W. Failure behaviour of steels with different levels of strength and toughness. Nucl. Eng. Des. 1988, 108, 457–466. [CrossRef] 11. Rosezin, H.-J.; Dahl, W. Einfluss von Temperatur und Probengeometrie auf die Ergebnisse von Grosssugversuchen. Stahl Eisen 1985, 105, 523–529. 12. Dormagen, D.; Halim, A.; Ehrhardt, H.; Dahl, W. On the toughness evaluation of sulphide-controlled Fe510 steel by Charp- V-impact, fracture mechanics and wide plate tests. In The Crack Tip Opening Displacement in Elastic-Plastic Fracture Mechanics; Schwalbe, K.H., Ed.; Springer: Berlin/Heidelberg, Germany, 1986. [CrossRef] 13. Wang, X. Two-Parameter Characterization of Elastic-Plastic crack Front Fields: Surface Cracked Plates under Tensile Loading. Eng. Fract. Mech. 2009, 76, 958–982. [CrossRef] 14. Wang, X. 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Journal

Applied Mechanics – Multidisciplinary Digital Publishing Institute

Published: Sep 28, 2021

Keywords: structural assessment; fracture mechanics; impact fracture; limit load; CC(T) specimen

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