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Experimental and numerical investigation of thickness effect on ductile fracture toughness of steel alloy sheets
Engineering Fracture Mechanics 77 (2010) 646–659
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Experimental and numerical investigation of thickness effect on ductile fracture toughness of steel alloy sheets A.R. Shahani *, M. Rastegar, M. Botshekanan Dehkordi, H. Moayeri Kashani Faculty of Mechanical Engineering, K.N. Toosi University of Technology, P.O. Box 19395-1999, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 20 January 2009 Received in revised form 2 December 2009 Accepted 29 December 2009 Available online 4 January 2010 Keywords: Ductile fracture toughness Thickness effect Finite element analysis
a b s t r a c t Effect of thickness on ductile fracture toughness of plates made of steel alloy GOST 08Ch22N6T is investigated experimentally. Multiple specimen tests for determining fracture toughness have been conducted using compact tension (CT) specimens with thicknesses of 1.25, 1.64 and 4.06 mm according to standard test method ASTM E813. The results show the significant effect of thickness on fracture toughness. It is observed that in low thickness, Jc increases with the thickness increase until it reaches a maximum; however, further increase in the thickness causes the Jc-value to decrease. Two-dimensional finite element analysis is also performed to reproduce the experimental results. The comparison shows a very good agreement. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Sheet metals have wide applications such as pressure vessels, vehicles and aircrafts. Due to size effects, the fracture behavior of such plates is different from bulk materials. Fracture toughness which is an explanatory of cracked specimen resistance to stable crack growth, is extremely affected by specimen size. Other parameters such as loading mode, material micro-structure, chemical environment and temperature affect on fracture resistance. In thin plates or very ductile materials, where there is a large plastic zone near the crack tip, fracture toughness is characterized with the critical J-integral near the onset of slow stable crack growth. The measurement of the critical value of J-integral, JIc, is performed according to the standards of ASTM E813 [1] and ASTM E1820 [2]. Specimen thickness as the most effective parameter affecting fracture toughness was considered by many researchers. In fact, it is insufficient to present a single value for fracture toughness for a particular sheet material. In such cases, the variation of the fracture resistance with respect to the plate thickness is given in a graph. As the plate thickness intended for a particular problem is made smaller, the fracture toughness increases to reflect the nature of plane stress behavior. A design based solely on the JIc parameter in this situation would be too conservative and hence, Jc as the thickness dependent fracture toughness should be used in place of JIc. In a tough material having fine structure and good ductility or in thin plates, the work of fracture and the degree of thickness necking can be very high, so that the material is able to absorb energy and resist fracture [3]. A practical application of thin members with greater Jc-value than JIc is the framework of cars where high fracture resistance might lead to a very beneficial effect on crashworthiness. The effect of other parameters like crack length, specimen width and initial ligament are also noticed by some investigators [4–6]. Pardoen et al. [7,8] conducted some experiments using DENT1 specimens to study the effect of thickness on critical values of J and CTOD (Jc and dc) near the stable crack growth in aluminum plates 6082T0. They reported that increasing thickness from 1 * Corresponding author. Tel.: +98 21 84063221; fax: +98 21 88677273. E-mail address: [email protected] (A.R. Shahani). 1 Double edge notched tension. 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.12.017
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Nomenclature a0 ai ap Apl b0 B Be Bn B0 Ci E J Jel/Jpl Jc, JQ JIc K KIc Pi rp Sut ULL v W
m rys ry dc Da
g h CMOD CTOA CTOD
initial crack length crack length at the end of an unloading/reloading sequence with the slope of 1/Ci final physical crack length plastic component of the area under load–displacement curve initial ligament of CT specimen which is equal to Wa0 thickness of CT specimen effective thickness of side grooved CT specimen net thickness of side grooved CT specimen thickness corresponding to maximum value of fracture toughness crack opening compliance (Dv/DP) of an unloading/reloading sequence Young modulus J-integral elastic/plastic component of J-integral critical J-integral near the onset of slow stable crack growth ductile fracture toughness stress intensity factor brittle fracture toughness load at the unloading point in single specimen test method or final load in multiple specimen test method plastic zone radius measured from the crack tip ultimate strength a parameter in Eq. (8) used for crack length calculation displacement measured using clip gage at the end of the specimen notch width of CT specimen Poisson’s ratio yield stress average of yield and ultimate stresses critical CTOD near the onset of slow stable crack growth crack growth a parameter in Eq. (6) used for calculation of Jpl polar coordinate based on the right-hand crack tip used for plastic zone radius calculation crack mouth opening displacement crack tip opening angle crack tip opening displacement
to 6 mm, Jc and dc increase linearly for thinner specimens and nonlinearly for larger thicknesses. They attained the same results through testing sixteen different alloys including stainless steel, aluminum, brass, bronze, zinc and lead for specimens with a variety of thickness ranges prepared in rolling and transverse directions [9]. Mahmoud and Lease [10] considered the critical value of CTOA (related to stable crack growth) as ductile fracture toughness of aluminum alloy 2024-T351. Their experimental results showed that the critical CTOA decreases as the thickness of CT specimens increases from 2.3 to 25.4 mm. Kang et al. [11] investigated the effect of thickness and rolling direction on ductile fracture toughness of T2 copper foils with thickness ranging between 0.02 and 1 mm using DENT specimens. For both specimens prepared in rolling and transverse directions, Jc increases for thicknesses up to 0.3 mm and then decreases. Fracture toughness was found to be higher for specimens prepared normal to rolling direction. Some more different results have been reported by Jitsukawa et al. [12] who studied the effect of specimen thickness, width, ligament and span using three-point bend specimens made of aluminum alloy 7075-T6. They found that the mentioned parameters have negligible effect on critical values of J and CTOD for specimens with thicknesses ranged from 1.25 to 25 mm. Finite element analysis (FEA) in fracture mechanics problems is mainly performed to verify the experimental results. The finite element analysis can also lead to predictive schemes in such cases where there is lack of experimental results. The most important point in stable crack growth problems is the crack growth criterion used. Dawicke and Newman [13] introduced seven different criteria for crack growth in two general groups: constant critical parameters and resistance curve parameters. The former assumes that crack growth occurs when a particular parameter (especially J-integral or CTOD/CTOA) reaches its critical value; while, the second group of the criteria can be used when the resistance curve is accessible. Both single critical parameters and resistance curve parameters are determined experimentally. Constant critical parameters, especially critical CTOD/CTOA, as a crack growth criterion are widely used in two and three-dimensional FEA by many investigators [14–18]. This is because of independence of the critical parameters from specimen type and crack shape; while, the resistance curve parameters are strongly dependent on specimen type. The present study investigates the effect of thickness on ductile fracture toughness of thin specimens made of steel alloy GOST 08Ch22N6T. CT specimens are used with thicknesses of 1.25, 1.64 and 4.06 mm. To prevent buckling of thin specimens,
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multiple specimen tests using anti-buckling plates are carried out. The experiments were conducted according to standard test method ASTM E813. Section 2 of the paper deals with the experimental set-up and procedures followed. Section 3 of the paper is devoted to the FEA of the problem. Finite element analysis using ABAQUS [19] is performed to investigate the coherency of the experimental approach and to consider some more thicknesses. Section 4, results and discussion, includes three parts of experimental results, numerical results and effect of thickness. In the numerical analysis, the curve of crack mouth opening displacement (CMOD) vs. crack length (CMOD-a curve) is used as an input material property. Some other thicknesses besides the tested specimens are considered in FEA. For these thicknesses, CMOD-a curves are obtained by linear interpolation between the experimental graphs of the three thicknesses available for the tests. J-integral vs. crack growth (J–Da) or resistance curve, which is used to measure Jc-values, is estimated from FEA for the tested specimens and shows good agreement with the experimental results. It is then concluded that the developed finite element model is applicable for other additional thicknesses. Jc-value is found to increase up to the thickness 1.64 mm and then decreases to the thickness 4.06 mm.
2. Experimental set-up and procedures 2.1. Material properties The specimens employed in this study are made of steel alloy GOST 08Ch22N6T with thicknesses of 1.25, 1.64 and 4.06 mm. All the plates have been machined from the same thick plate. So, the specimens have the same chemical composition and mechanical properties as shown in Tables 1 and 2, respectively. All the specimens are prepared in the rolling direction of the sheet using CNC machine with wire-cut equipments. 2.2. Experiment performance Fracture toughness tests are conducted according to standard test method ASTM E813. All the experiments have been performed using a 100 KN Zwick/Roell servo-hydraulic test machine. Three different thicknesses of 1.25, 1.64 and 4.06 mm are tested using multiple specimen test method and compact tension (CT) specimen with dimensions indicated in Fig. 1. In each multiple test series, the specimens are fatigue pre-cracked to a length of 2 mm. They are loaded to a specified extension and load–displacement curve is plotted. The extension is measured at the end of the specimen notch (crack mouth) with a clip gage. Four anti-buckling plates are used to prevent buckling of the specimen during the loading. Previously, Castrodeza et al. [20] applied the same method for preventing the specimen buckling when testing on 1.42 mm thick CT specimens made of fiber metal composites. Fig. 2 shows the application of these plates. It should be pointed out that the single specimen test method is applied only to the thickest specimen. This technique is not applied to the thin specimens because of buckling problems. J-integral is calculated according to ASTM E813 in single or multiple specimen test method. In the single specimen test method, sequences of unloading/reloading are applied to the specimen and the following equations are used to calculate Jintegral.
J ¼ J el þ J pl
ð1Þ
K i ð1 m2 Þ J elðiÞ ¼ E K i ¼ ½Pi =ðBBn WÞ1=2 f ða0 =WÞ
ð2Þ ð3Þ 2
f ða0 =WÞ ¼ J plðiÞ ¼
3
4
ð2 þ a0 =WÞð0:886 þ 4:64a0 =W 13:32ða0 =WÞ þ 14:72ða0 =WÞ 5:6ða0 =WÞ Þ
ð4Þ
ð1 a0 =WÞ3=2
gAplðiÞ
ð5Þ
Bn b0 g ¼ 2 þ 0:522b0 =W
ð6Þ
Elastic component of J-integral can be also calculated by replacing Apl in Eq. (5) with elastic component of area under load–displacement curve. Therefore, using total area under load–displacement curve in Eq. (5) instead of Apl, total J-integral is obtained.
Table 1 Chemical composition of tested specimens. Material
C
Mn
Si
Cr
Ni
P
S
Ti
Fe
Percent
60.08
60.8
60.8
21–23
5.3–6.3
60.035
60.025
0.3–0.6
Remainder
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Table 2 Mechanical properties of tested specimens. Young’s modulus (GPa) Poisson’s ratio Yield stress (MPa) Ultimate strength (MPa)
206 0.3 520 750
Fig. 1. Dimensions of CT specimen in mm.
Fig. 2. Anti-buckling plates around CT specimen.
According to ASTM E813, crack length is measured using compliance method in single specimen test or visually in multiple specimen test method. In the compliance method, crack length is estimated using the following equation:
ai =W ¼ 1:000196 4:06319U LL þ 11:242U 2LL 106:043U 3LL þ 464:335U 4LL 650:677U 5LL
ð7Þ
where
U LL ¼
1 ½Be EC i 1=2 þ 1
;
Be ¼ B ðB Bn Þ2 =B
Ci ¼
Dv DP
ð8Þ ð9Þ
In multiple specimen test method, the specimens are finally exposed to fatigue cycling and broken to measure the stable crack growth size, i.e., the distance between the fatigue pre-cracked and the fatigue cycling regions. These three regions are discerned by taking high resolution pictures from the fracture surface employing a Canon IXUS-850-IS camera with 7-Mega Pixel resolution. Fig. 3 shows these zones for one of the specimens with 1.64 mm thickness. Along the front of the fatigue crack and the front of the region of slow stable crack extension, the crack size is measured at nine equally spaced points centered about the specimen centerline and extending to 0.005 W from the surface of the specimen. The average of two near-surface measurements is combined with the remaining seven crack length measurements to determine initial and final crack sizes (a0 and ap). Crack growth is then calculated by Da = ap – a0. It should be pointed out that all the crack length measurements are performed using the fracture surface photographs and the digitizer software.
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Fig. 3. Stable crack growth zone between two fatigue regions.
After determination of J and crack extension, resistance curve for each particular thickness can be plotted. The related critical J-value near the onset of slow stable crack growth (JQ) is calculated from the intersection point of the resistance curve and a line parallel to blunting line (the equation of which is J = 2ryDa) with an offset of 0.2 mm. 3. Finite element simulation Two-dimensional finite element analysis (FEA) of the problem is performed using ABAQUS standard code. Since the main concentration of the research is on the experimental investigation of the thickness effect on the fracture resistance, finite element analysis is mostly accomplished to show the coherency of the experimental approach. Therefore, 2D analysis is preferred to the more complicated 3D solution for the sake of comparing experimental and numerical results. However, threedimensional analysis is a beneficial approach in problems with numerical aspects through which more accurate and useful
Fig. 4. (a) FE model and the paths of J-integral calculation. (b) Fine elements near the crack tip.
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651
4
CMOD (mm)
3.5 3 2.5 2
B=1.25 mm
1.5
B=1.64 mm B=4.06 mm
1 30
30.5
31
31.5
32
32.5
33
Crack Length (mm) Fig. 5. Finite element analysis input curves for different thicknesses.
consequences may be achieved. Due to symmetry of the geometry and loading, only the upper half of the specimen is modeled and symmetry condition is applied to the ligament. The center of the specimen hole is defined as a reference point and coupled to its upper half. The load is then applied to the center of the hole in the form of a displacement boundary condition. The J-integral is calculated for different crack lengths and the resistance curve is plotted. Crack mouth opening displacement (CMOD) vs. crack length curve (CMOD-a curve) obtained from experimental results is used as the input material fracture property and linear hardening elastic–plastic behavior is considered. Second-order shell elements (S8R) are used to discretize the model. The model containing 4192 nodes and 1572 elements is shown in Fig. 4a. Near the crack tip elements are also fine enough to give a proper precision as shown in Fig. 4b. The behavior of pre-cracked model for each special thickness is defined by CMOD vs. crack length curve obtained from the experiments. These curves are shown in Fig. 5. In FEA, for each amount of crack growth which is equal to a fine element length in the ligament direction, one half of the corresponding CMOD (due to symmetry) is applied as displacement boundary condition and J-integral is calculated on 20 different paths. On the near crack tip paths, J-value changes extremely due to the presence of plastic zone and may not be a correct estimation of J-integral. On the paths outside of the plastic region, J remains nearly constant. The integral on 20th path is used as the constant J-value to plot resistance curve. 4. Results and discussion 4.1. Experimental results Specification and number of tested specimens in multiple specimen test series are shown in Table 3. According to ASTM E813, at least five specimens are needed for multiple specimen test method. For each specimen with a particular thickness, a proper extension value is applied in order to obtain valid points on the resistance curve. The extension values are indicated in Table 4. The resistance curves obtained for different thicknesses via the multiple specimen test method are shown in Appendix A. Specimen D as the thickest specimen was tested successfully using multiple specimen test method without using anti-buckling plates and no buckling was observed. Therefore, it was possible to test this specimen according to single specimen test procedure. Load–displacement curve for this single specimen test method and the corresponding resistance curve are indicated in Appendix A. The critical value of J-integral near the onset of slow stable crack growth (JQ) is calculated as described in Section 2.2. According to ASTM E813, JQ should satisfy some requirements in order to be ductile fracture toughness (JIc) of the tested material. The obtained critical values from the resistance curves (JQ) satisfy all the requirements mentioned in this standard test method except the thickness requirement:
B > 25J Q =ry
ð10Þ
Therefore, the values of JQ may be known as Jc which is authentic for just the respective thickness. The experimental values of Jc and the corresponding K Jc from Eq. (11) are indicated in Table 5.
K Jc ¼
pffiffiffiffiffiffi Jc E
ð11Þ
Table 3 Specification and number of tested specimen. Specimen code
Thickness (mm)
Number of tested specimens in multiple specimen test method
A B D
1.25 1.64 4.06
5 6 7
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Table 4 Extension values for tested specimens. Specimen code Extension (mm)
A1 1.6
A2 2
A3 2.4
A4 2.8
A5 3
Specimen code Extension (mm)
B1 1.3
B2 2
B3 2.4
B4 3
B5 3.4
B6 3.8
Specimen code Extension (mm)
D1 1.6
D2 1.8
D3 2
D4 2.4
D5 3
D6 3.4
D7 3.8
Table 5 Jc-values for different thicknesses. Thickness (mm)
Jc (kJ/m2)
K Jc ðMPa
1.25 1.64 4.06 (Multiple test) 4.06 (Single test)
141.5 247.2 153.4 173.3
170.7 225.7 177.8 188.9
pffiffiffiffiffi mÞ
Jc-value obtained from the single specimen test method is nearly 13% different from the multiple specimen test result. Since load–displacement curves from both test methods coincide with each other as shown in Fig. A4, this difference cannot be due to J-integral calculation. Such a difference for Jc-values is attributed to different crack length measurement techniques used in this study. The resistance curve and fracture toughness of materials depend on some parameters like material sub-structure and grain size, loading history, temperature and working environment. Due to three-dimensional stress field in plates, the critical J-integral near the onset of stable crack growth (Jc) is highly dependent on geometry and especially specimen thickness. So, the measured fracture toughness is valid only for the corresponding specimens. In thin specimens for which the state of plane stress is nearly dominant due to large plastic region, the resistance curve is steeper and Jc-value is greater than that of thicker specimens. The plastic zone size may be estimated for all the specimens using standard solutions for plane stress and plane strain. The following equations give the plastic zone radius related to both Von Mises’ and Tresca’s criteria [21]:
8 h i 2 K2 > 1 þ 32 sin h þ cos h ; Plane stress < r p ðhÞ ¼ 4pr 2 ys 0 h i Von Mises criterion : > : r p ðhÞ ¼ K 2 2 32 sin2 h þ ð1 2mÞ2 ð1 þ cos hÞ ; Plane strain 4pr
ð12Þ
8 h h2 K2 > ; Plane stress > > r p ðhÞ ¼ 2pr2ys cos 2 1 þ sin 2 > < h 2 9 0 K2 2 h Tresca s criterion : r p ðhÞ ¼ 2pr2 cos 2 1 2m þ sin 2 = ys > > whichever is lorger; Plane strain > > ; : r p ðhÞ ¼ K 2 2 cos2 h 2 2pr
ð13Þ
ys
ys
where the stress intensity factor is calculated from Eq. (3). The radius of the plastic region may be compared with the specimen thickness using rp/B ratio. Minimum and maximum values of rp/B and also the ratios corresponding to h = 0° and h = 90° for the extension of 2 mm as an instance have been calculated for different specimens as shown in Table 6. The values have been computed both by the Von Mises’ and Tresca’s Criteria. It is seen that the values of rp/B for the two thinnest specimens are much higher than that of the thickest one. This Table 6 The ratio of rp/B from Eqs. (12) and (13) for extension of 2 mm. Specimen code
Plane stress equations
Plane stress equations
h = 0°
h = 90°
Minimum
Maximum
h = 0°
h = 90°
Minimum
Maximum
Von Mises’ criterion A B D
4.11 3.49 1.52
5.14 4.37 1.90
4.11 3.49 1.52
5.49 4.66 2.02
0.66 0.56 0.24
3.41 2.90 1.26
0.66 0.56 0.24
3.42 2.91 1.26
Tresca’s criterion A B D
4.11 3.49 1.52
5.99 5.09 2.21
4.11 3.49 1.52
6.93 5.90 2.56
4.11 3.49 1.52
2.52 2.14 0.93
2.06 1.75 0.76
4.11 3.49 1.52
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means that the specimens A and B are close to plane stress state; while, the plastic zone size in the specimen D is in the order of the plate thickness and plane strain state prevails. Similar results can be obtained for all of the other extension values. Fig. 6 demonstrates the presence of considerable plastic zone in the specimen A with the extension of 2.8 mm. In Fig. 7, all of the resistance curves (R-curve) are shown in a single graph. The resistance curve for the specimen B has greater slope than that of the specimen D. This means that the less the specimen thickness is, the more energy is required for crack growth. This is because in the thinner specimen B, that is closer to plane stress, the size of plastic region at the crack tip is greater and more energy is required for the formation of new plastic region with greater size corresponding to the longer crack. So, in the thinner specimen B, stable crack growth occurs more lately and the R-curve is more sloping. On the other hand, Jc-value is larger for the specimen B in which the crack front is closer to the free surface. As the thickness increases from the specimen A to B the state of plane strain prevails. This results in the delay in stable crack growth. Fracture toughness of thin plates is mainly associated with the amount of energy required to reduce the initial cracked specimen thickness which is frequently higher than the energy needed for crack growth. The former depends on the initial specimen thickness [21]. In the other words, fracture resistance of thin metallic sheets is significantly dependent on the energy dissipated in crack tip necking. Thus, the less the amount of necking is, the less energy is needed for crack extension. The percentage values of thickness necking with respect to the initial specimen thickness are given in Table 7 for different extensions. More considerable amount of necking in specimen B than specimen A indicates larger Jc-value for the thicker one. In the other words, as the state of stress is close to plane stress (specimens A and B), the energy which causes thickness reduction and thus, the fracture toughness is higher for thicker specimens. As the thickness increases, the state of stress in crack tip field gradually changes to plane strain and less amount of necking is observed as shown in Table 7. This implies lower Jcvalue for specimen D than specimen B. Stress triaxiality tends to increase at the crack tip for the thicker specimen leading to higher rate of void growth and lower fracture strain and crack tip necking degree. It causes the fracture toughness to decrease [3]. Resistance curves from single and multiple specimen tests for the specimen D can be compared in Fig. 7. The resistance to crack growth is estimated higher from single specimen test. This is attributed to the different methods for crack length measurement. However, the single specimen method is more economic and takes more little time to carry out if it is performable. Fig. 7 also reveals nearly the same resistance to crack growth for the specimens A and D from multiple specimen tests. Critical J-integrals and necking degree for these specimens are also very close to each other as shown in Tables 5 and 7, respectively. 4.2. Numerical results Resistance curves from the finite element analysis are obtained as described in Section 3 and compared with the experimental results as illustrated in Fig. 8. It is seen that the FE results underestimate the experimental graphs, especially for the
Fig. 6. Large plastic zone on the fracture surface of specimen A4.
700
specimen A, B=1.25 mm
J(KJ/m2)
600
specimen B, B=1.64 mm specimen D, B=4.06 mm- multiple
500
specimen D, B=4.06 mm-single
400 300 200 100 0
0.5
1
1.5
Crack Extension (mm) Fig. 7. Comparison between resistance curves.
2
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Table 7 Percentage values of thickness necking for different specimens and extensions. Specimen code
Extension (mm) 1.3
A B D
1.6
1.8
2
2.4
2.8
3
3.4
3.8
– 9.1 –
7.9 – 6.3
– – 6.8
8.2 17.7 6.5
8.7 19.0 7.7
11.7 – –
12.5 20.0 8.2
– 24.2 10.7
specimen A
500
J (KJ/m2)
– 36.2 11.8
400 300 EXP
200 100
FEA
0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
2.2
Crack Extension (mm)
J (KJ/m2)
700
specimen B
600 500 400 EXP
300 200
FEA
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
Crack Extension (mm)
J (KJ/m2)
600
specimen D
500 400 300
EXP
200 100
FEA
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
Crack Extension (mm) Fig. 8. Comparison between experimental and FEA resistance curves.
specimens A and D; however they are in good agreement. The maximum differences for the specimens A, B and D are 5.8%, 4.3% and 8.7%, respectively. These negligible errors show the coherency of the experimental approach and acceptable precision of the finite element model. Thus, FE simulation can be applied for other models like different thicknesses as it will be discussed afterwards. 4.3. Effect of thickness More complete investigation of thickness effect on the fracture toughness needs considering some other thicknesses in addition to the thicknesses of previously tested specimens. Therefore, five other thicknesses were considered beside the first three thicknesses mentioned in Table 3. The developed finite element model is employed for the thicknesses of 1.45, 2, 2.5, 3 and 3.5 mm. Finite element input behavior for new thicknesses (i.e., CMOD-a curve) is estimated by linear interpolation between the experimental curves obtained for 1.25, 1.64 and 4.06 mm thick specimens (Fig. 5). As indicated in Fig. 5, the CMOD corresponding to the specimen B is greater than that of the two other specimens for all crack lengths. Also, the material
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behavior curve (i.e. CMOD vs. crack length curves shown in Fig. 5) corresponding to specimen A lies between the graphs related to the thicknesses of 1.64 and 4.06 mm. On the other hand, the CMOD rises up by increasing the thickness from 1.25 to 1.64 mm and then decreases by further increase of the thickness to 4.06 mm for all crack lengths. Furthermore, the material behavior for the intermediate thicknesses (i.e. thicknesses between 1.25 and 1.64 mm and also between 1.64 and 4.06 mm) is unknown due to limited number of thicknesses available for the experiments. In this situation, linear interpolation method seems reasonable for estimation of the behavior of the intermediate thicknesses. All the input curves for FEA are indicated in Fig. 9. Crack resistance curves obtained from the FEA for all thicknesses are given in Figs. 10 and 11. Fig. 10 shows increasing slope as the thickness increases. While, as indicated in Fig. 11, the slope of the R-curve decreases with increasing thickness. In Table 8, the R-curve power law equation, i.e., J ¼ C 1 ðDaÞC 2 , the intersection point of the R-curve with 0.2 mm offset line (DaQ), critical J-integral (Jc-value) and its corresponding K Jc (from Eq. (11)) are shown for different thicknesses. All the data in Table 8 are the FEA results. It is seen that for thicknesses 1.25–1.64 mm, Jc rises up following the increase of the slope of the resistance curve. However, for the thicknesses greater than 1.64 mm the Jcvalue decreases with the increase of thickness.
4 B=1.25 mm B=1.45 mm B=1.64 mm
3
B=2 mm B=2.5 mm
2.5
B=3 mm B=3.5 mm
2
B=4.06 mm
1.5 30
31
32
33
Crack Length (mm) Fig. 9. Experimental and interpolated input curves for FEA.
800
J (KJ/m2)
700 600 500 400 B=1.25 mm
300
B=1.45 mm
200
B=1.64 mm
100 0
0.5
1
1.5
2
2.5
Crack Extension (mm) Fig. 10. Resistance curves for first three thicknesses.
700 B=2 mm B=2.5 mm B=3 mm B=3.5 mm B=4.06 mm
600
J (KJ/m2)
CMOD (mm)
3.5
500 400 300 200 100 0
0.5
1
1.5
2
Crack Extension (mm) Fig. 11. Resistance curves for thicknesses 2–4.06 mm.
2.5
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Table 8 FEA results for different thicknesses. B (mm)
C1 (kJ/m2)
C2
DaQ (mm)
Jc (kJ/m2)
K Jc ðMPa
1.25 1.45 1.64 2 2.5 3 3.5 4.06
275.88 346.07 437.35 401.63 363.19 336.02 305.21 275.69
0.6676 0.6105 0.6854 0.7299 0.7232 0.723 0.6839 0.6269
0.2965 0.3414 0.3763 0.3488 0.3276 0.3147 0.3073 0.3027
122.6 179.6 223.9 189 162.1 145.7 136.3 130.4
158.9 192.3 214.8 197.3 182.7 173.2 167.6 163.9
pffiffiffiffiffi mÞ
Fig. 12 plots variations of critical J-integral against specimen thickness. Experimental results are also shown in this figure. A five-order polynomial curve is fitted to the FEA results according to Eq. (14).
J c ¼ 28:149B5 410:24B4 þ 2314:4B3 6351:1B2 þ 8401:5B 4063:7 ðkJ=m2 Þ for 1:25 < B < 4:06 mm
ð14Þ
Considering the experimental results, the specimen B with thickness of 1.64 mm has the maximum Jc-value (equal to 247.2 KJ/m2) among the tested specimens which may not be the maximum value of fracture toughness for the material under consideration. The low variety of thicknesses available for the experiments and having no approximation of ductile fracture toughness for this particular steel alloy makes it difficult to estimate Jc at different thicknesses and also the thickness corresponding to the maximum Jc-value using the models proposed in the literature [21]. Nevertheless, the present investigation provides more or less proper and reliable estimations for the thickness dependent fracture toughness in the range of 1.25–4.06 mm. The thickness at which Jc reaches its maximum value (B0) depends on the material type. This thickness is usually considered to correspond to the real plane stress fracture toughness. It means that B0 must be equal to the plastic zone size in plane strain which is approximated by [21]:
B0 ffi
K 2Ic
ð15Þ
3pr2ys
Jc (KJ/m2)
Assuming that the thickness B0 for this special material is equal to 1.64 mm, the values of KIc and its equivalent JIc are obpffiffiffiffiffi tained from Eqs. (15) and (11) as 64:6 MPa m and 20.3 kJ/m2, respectively. This implies that the minimum thickness required for a valid JIc test should be about 0.8 mm according to Eq. (10). All the tested specimens satisfy this requirement; while, their estimated fracture toughness values are significantly different. Therefore, the thickness related to the maximum Jc-value should be between 1.64 and 4.06 mm due to the higher value of KIc expected for this low-strength steel alloy. In the other words, the curve of Jc vs. specimen thickness might be higher than that obtained in Fig. 12 for the thicknesses 1.64– 4.06 mm if some more thicknesses were provided. However, the experimental and numerical values in Tables 5 and 8 could be used as the conservative estimations of the thickness dependent fracture toughness for many design purposes involving sheet metals. For very thin specimens (B = 1.25 and 1.45 mm), due to the low thickness and large plastic region at the crack tip, fracture occurs at lower load levels. Hence, Jc-values for these thicknesses are lower than that of the specimen B. While, in thicker specimens, as the thickness increases, the input energy is dominantly spent on forming new surfaces, and so growing the crack. Therefore, the crack growth occurs earlier than the thinner specimens for which the major proportion of the input energy causes plastic zone formation.
260 240 220 200 180 160 140 120 100
Experimental results FEA results
0
1
2
3
4
Thickness (mm) Fig. 12. Variations of Jc-value vs. thickness.
5
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657
Since the thickness B0 was discussed to be higher than 1.64 mm, the fracture toughness values for thicknesses B = 1.25 and 1.45 mm do not experience changes like the thicker specimens do when some more thicknesses are tested. On the other hand, Jc-values have been estimated less conservatively for specimens thinner than the specimen B.
5. Conclusions An experimental and numerical study of the effect of thickness on the fracture toughness has been carried out. Jc-values for plates with thicknesses of 1.25, 1.64 and 4.06 mm made of steel alloy GOST 08Ch22N6T have been obtained experimentally using CT specimens and according to the standard test method ASTM E813. Then, some other thicknesses of 1.45, 2, 2.5, 3 and 3.5 mm have been examined through the use of the finite element method to study the thickness effect more comprehensively. Input material property in the FEA was CMOD vs. crack length curve. For the latter group of thicknesses, the mentioned input curve was estimated by linear interpolation between the experimental graphs obtained for the former thicknesses. Following conclusions are drawn from the present investigation: 1. Due to very low thickness of the specimens, single specimen test method was faced to buckling problem. To prevent this problem, multiple specimen tests were conducted and also anti-buckling plates were used. However, the specimen D (B = 4.06 mm) did not show any buckling and single specimen test was also conducted for this specimen. The resistance curves corresponding to the single and multiple specimen testing methods performed for the specimen D are in good agreement.
specimen A
1000
J(KJ/m2)
800
Jmax=847 KJ/m2
600 400 200 0 0 0.244 0.5
1
1.5
1.8
2
2.5
Crack Extension (mm) 1000
Jmax=847 KJ/m2
2
J(KJ/m )
800
specimen B
600 400 200 0 0 0.32 0.5
1
1.5
2 2.021 2.5
Crack Extension (mm)
specimen D
1000 Jmax=847 KJ/m2
J(KJ/m2)
800 600 400 200 0 0 0.241 0.5
1
1.5 1.873 2
2.5
Crack Extension (mm) Fig. A1. Resistance curves obtained from multiple specimen test method.
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2. Resistance curves for the first three thicknesses were plotted. Critical J-integral values near the onset of slow stable crack growth (JQ) for these thicknesses satisfied all the requirements mentioned in ASTM E813 except the thickness requirement (Eq. (10)). So, they may be considered as Jc. 3. Two-dimensional finite element analysis was performed using ABAQUS standard code to investigate the coherency of the experimental approach and considering some other thicknesses. Resistance curves obtained from the FEA showed a very good agreement with the experimental results. Insignificant differences of 5.8%, 4.3% and 8.7% exist between the FEA resistance curves compared with the experimental graphs for 1.25, 1.64 and 4.06 mm thick specimens, respectively. This good precision showed the acceptable accuracy of the developed finite element model. 4. Jc-value increases for the thicknesses from 1.25 to 1.64 mm and then decreases for thicker specimens up to B = 4.06 mm. The ratio of the plastic zone size to the specimen thickness (rp/B) was found to be higher for the thinnest specimens than that of the specimen D. It means that the state of plane stress prevails in the specimens A and B unlike the thickest sample where the plane strain is more dominant. The large degree of thickness necking observed in the specimen B showed the considerable amount of energy required for crack growth indicating the greater value of the fracture toughness corresponding to the specimen B than that of the other specimens. However, the low Jc-value for the thickest specimen could be attributed to the increase of stresses in the crack tip field. 5. Determination of the thickness related to the maximum value of the fracture toughness needs more variety of the thicknesses for the experiments. However, it was concluded that the thickness B0 is between 1.64 and 4.06 mm for the material under consideration. On the other hand, the curve of the fracture toughness vs. specimen thickness shown in Fig. 12 suggests conservative Jc-values at different thicknesses, especially in the range of 1.64–4.06 mm. Having approximated values of the thickness dependent fracture toughness is beneficial to many design purposes.
Fig. A2. Load–displacement curve from single specimen test for specimen D.
specimen D
1000 Jmax=847 KJ/m
2
J (KJ/m2)
800 600 400 200 0 0 0.276 0.5
1
1.5 1.926 2
2.5
3
Crack Extension (mm) Fig. A3. Resistance curve obtained from single specimen test method for specimen D.
Load (KN)
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8 7 6 5 4 3 2 1 0
D5-multiple test
0
0.5
1
1.5
2
659
D8-single test
2.5
3
3.5
Extension (mm) Fig. A4. Load–displacement curves from single and multiple specimen test methods for specimen D.
Appendix A Three different CT specimens with the thicknesses indicated in Table 3 have been tested using multiple specimen test method. Crack growth and J-integral are calculated for each test series and the resistance curves are plotted as shown in Fig. A1. For each graph in Fig. A1, blunting line and its parallel exclusion lines with 0.15, 0.2, 0.5, 1 and 1.5 mm offsets are shown according to ASTM E813. The valid region is shown by dashed lines. Maximum J-integral (Jmax = b0ry/15) is the same for all the specimens. Vertical dashed lines indicate minimum and maximum crack extension (Damin and Damax) corresponding to the intersection of the initial power law curve (which is estimated using all the initial points) with the 0.15 mm and 1.5 mm exclusion lines, respectively. Final power law curve is achieved after neglecting invalid points out of this region. Load–displacement curve related to single specimen test method for the specimen D, with 12 unloading/reloading sequences and final extension of 3 mm, was plotted as shown in Fig. A2. The corresponding resistance curve is illustrated in Fig. A3. Load–displacement curves from single and multiple specimen test methods for specimen D are compared in Fig. A4. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
ASTM code E813: standard test method for JIc. A measure of fracture toughness 1991. ASTM code E1820: standard test method for measurement of fracture toughness 2001. Shukla A. Practical fracture mechanics in design. 2nd ed. New York: Marcel Dekker; 2005. Qingfen L, Limin Z, Shouren L. The effect of a/W ratio on crack initiation values of cod and J-integral. Eng Fract Mech 1986;23(5):925–8. Masanori K. Study on the effect of the crack length on the JIc value. Nucl Eng Des 1997;174(1):41–9. Jinkook K, Guihua Z, Xiaosheng G. Modeling of ductile fracture: application of the mechanism-based concepts. Int J Solids Struct 2007;44:1844–62. Pardoen T, Marchal Y, Delannay F. Thickness dependence of cracking resistance in thin aluminum plates. J Mech Phys Solids 1999;47:2093–123. Pardoen T, Marchal Y, Delannay F. Essential work of fracture compared to fracture mechanics-towards a thickness independent plane stress toughness. Eng Fract Mech 2002;69:617–31. Pardoen T, Hacheza F, Marchionia B, Blythb PH, Atkinsb AG. Mode I fracture of sheet metal. J Mech Phys Solids 2004;52:423–52. Mahmoud S, Lease K. The effect of specimen thickness on the experimental characterization of critical crack-tip-opening angle in 2024-T351 aluminum alloy. Eng Fract Mech 2003;70:443–56. Kang YL, Zhang ZF, Wang HW, Qin QH. Investigation of the effect of thickness on fracture toughness of metallic foils. Mater Sci Eng A 2005;394:312–9. Jitsukawa S, Naito A, Segawa J. Effect of size and configuration of 3-point bend bar specimens on J–R curves. J Nucl Mater 1999;271–272:87–91. Dawicke DS, Newman Jr JC. Evaluation of fracture parameters for predicting residual strength of multi-site damage. In: Proceeding of 1st joint DoD/ FAA/NASA conference on aging aircraft, Ogden, Utah; 1997. Newman Jr JC, James MA, Zerbst U. A review of the CTOA/CTOD fracture criterion. Eng Fract Mech 2003;70(3–4):371–85. Newman JC, Dawicke DS, Bigelow CA. Finite-element analyses and fracture simulation in thin-sheet aluminum alloy. In: Proceeding of the international workshop on structural integrity of aging airplanes, Atlanta, Georgia; 1992. Shivakumar KN, Newman Jr JC. Numerical fracture simulation of bend specimens using a CTOD criterion. Eng Fract Mech 1989;32(2):203–10. Sutton MA, Deng X, Fashang M, James C, Newman Jr JC, James M. Development and application of a crack tip opening displacement-based mixed mode fracture criterion. Int J Solids Struct 2000;37:3591–618. Sutton MA, Helm JD, Boone ML. Experimental study of crack growth in thin sheet 2024-T3 aluminum under tension-torsion loading. Int J Fract 2001;109(3):285–301. Hibbit, Karlsson, Sorensen, HKS Inc. ABAQUS finite element code v. 6.7.1; 2007. Castrodeza EM, Schneider Abdala MRW, Bastian FL. Crack resistance curves of GLARE laminates by elastic compliance. Eng Fract Mech 2006;73:2292–303. Broek D. Elementary engineering fracture mechanics. 3rd ed. Boston: Martinus Nijhoff Publishers; 1982.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Experimental and numerical investigation of thickness effect on ductile fracture toughness of steel alloy sheets A.R. Shahani *, M. Rastegar, M. Botshekanan Dehkordi, H. Moayeri Kashani Faculty of Mechanical Engineering, K.N. Toosi University of Technology, P.O. Box 19395-1999, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 20 January 2009 Received in revised form 2 December 2009 Accepted 29 December 2009 Available online 4 January 2010 Keywords: Ductile fracture toughness Thickness effect Finite element analysis
a b s t r a c t Effect of thickness on ductile fracture toughness of plates made of steel alloy GOST 08Ch22N6T is investigated experimentally. Multiple specimen tests for determining fracture toughness have been conducted using compact tension (CT) specimens with thicknesses of 1.25, 1.64 and 4.06 mm according to standard test method ASTM E813. The results show the significant effect of thickness on fracture toughness. It is observed that in low thickness, Jc increases with the thickness increase until it reaches a maximum; however, further increase in the thickness causes the Jc-value to decrease. Two-dimensional finite element analysis is also performed to reproduce the experimental results. The comparison shows a very good agreement. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Sheet metals have wide applications such as pressure vessels, vehicles and aircrafts. Due to size effects, the fracture behavior of such plates is different from bulk materials. Fracture toughness which is an explanatory of cracked specimen resistance to stable crack growth, is extremely affected by specimen size. Other parameters such as loading mode, material micro-structure, chemical environment and temperature affect on fracture resistance. In thin plates or very ductile materials, where there is a large plastic zone near the crack tip, fracture toughness is characterized with the critical J-integral near the onset of slow stable crack growth. The measurement of the critical value of J-integral, JIc, is performed according to the standards of ASTM E813 [1] and ASTM E1820 [2]. Specimen thickness as the most effective parameter affecting fracture toughness was considered by many researchers. In fact, it is insufficient to present a single value for fracture toughness for a particular sheet material. In such cases, the variation of the fracture resistance with respect to the plate thickness is given in a graph. As the plate thickness intended for a particular problem is made smaller, the fracture toughness increases to reflect the nature of plane stress behavior. A design based solely on the JIc parameter in this situation would be too conservative and hence, Jc as the thickness dependent fracture toughness should be used in place of JIc. In a tough material having fine structure and good ductility or in thin plates, the work of fracture and the degree of thickness necking can be very high, so that the material is able to absorb energy and resist fracture [3]. A practical application of thin members with greater Jc-value than JIc is the framework of cars where high fracture resistance might lead to a very beneficial effect on crashworthiness. The effect of other parameters like crack length, specimen width and initial ligament are also noticed by some investigators [4–6]. Pardoen et al. [7,8] conducted some experiments using DENT1 specimens to study the effect of thickness on critical values of J and CTOD (Jc and dc) near the stable crack growth in aluminum plates 6082T0. They reported that increasing thickness from 1 * Corresponding author. Tel.: +98 21 84063221; fax: +98 21 88677273. E-mail address: [email protected] (A.R. Shahani). 1 Double edge notched tension. 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.12.017
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Nomenclature a0 ai ap Apl b0 B Be Bn B0 Ci E J Jel/Jpl Jc, JQ JIc K KIc Pi rp Sut ULL v W
m rys ry dc Da
g h CMOD CTOA CTOD
initial crack length crack length at the end of an unloading/reloading sequence with the slope of 1/Ci final physical crack length plastic component of the area under load–displacement curve initial ligament of CT specimen which is equal to Wa0 thickness of CT specimen effective thickness of side grooved CT specimen net thickness of side grooved CT specimen thickness corresponding to maximum value of fracture toughness crack opening compliance (Dv/DP) of an unloading/reloading sequence Young modulus J-integral elastic/plastic component of J-integral critical J-integral near the onset of slow stable crack growth ductile fracture toughness stress intensity factor brittle fracture toughness load at the unloading point in single specimen test method or final load in multiple specimen test method plastic zone radius measured from the crack tip ultimate strength a parameter in Eq. (8) used for crack length calculation displacement measured using clip gage at the end of the specimen notch width of CT specimen Poisson’s ratio yield stress average of yield and ultimate stresses critical CTOD near the onset of slow stable crack growth crack growth a parameter in Eq. (6) used for calculation of Jpl polar coordinate based on the right-hand crack tip used for plastic zone radius calculation crack mouth opening displacement crack tip opening angle crack tip opening displacement
to 6 mm, Jc and dc increase linearly for thinner specimens and nonlinearly for larger thicknesses. They attained the same results through testing sixteen different alloys including stainless steel, aluminum, brass, bronze, zinc and lead for specimens with a variety of thickness ranges prepared in rolling and transverse directions [9]. Mahmoud and Lease [10] considered the critical value of CTOA (related to stable crack growth) as ductile fracture toughness of aluminum alloy 2024-T351. Their experimental results showed that the critical CTOA decreases as the thickness of CT specimens increases from 2.3 to 25.4 mm. Kang et al. [11] investigated the effect of thickness and rolling direction on ductile fracture toughness of T2 copper foils with thickness ranging between 0.02 and 1 mm using DENT specimens. For both specimens prepared in rolling and transverse directions, Jc increases for thicknesses up to 0.3 mm and then decreases. Fracture toughness was found to be higher for specimens prepared normal to rolling direction. Some more different results have been reported by Jitsukawa et al. [12] who studied the effect of specimen thickness, width, ligament and span using three-point bend specimens made of aluminum alloy 7075-T6. They found that the mentioned parameters have negligible effect on critical values of J and CTOD for specimens with thicknesses ranged from 1.25 to 25 mm. Finite element analysis (FEA) in fracture mechanics problems is mainly performed to verify the experimental results. The finite element analysis can also lead to predictive schemes in such cases where there is lack of experimental results. The most important point in stable crack growth problems is the crack growth criterion used. Dawicke and Newman [13] introduced seven different criteria for crack growth in two general groups: constant critical parameters and resistance curve parameters. The former assumes that crack growth occurs when a particular parameter (especially J-integral or CTOD/CTOA) reaches its critical value; while, the second group of the criteria can be used when the resistance curve is accessible. Both single critical parameters and resistance curve parameters are determined experimentally. Constant critical parameters, especially critical CTOD/CTOA, as a crack growth criterion are widely used in two and three-dimensional FEA by many investigators [14–18]. This is because of independence of the critical parameters from specimen type and crack shape; while, the resistance curve parameters are strongly dependent on specimen type. The present study investigates the effect of thickness on ductile fracture toughness of thin specimens made of steel alloy GOST 08Ch22N6T. CT specimens are used with thicknesses of 1.25, 1.64 and 4.06 mm. To prevent buckling of thin specimens,
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multiple specimen tests using anti-buckling plates are carried out. The experiments were conducted according to standard test method ASTM E813. Section 2 of the paper deals with the experimental set-up and procedures followed. Section 3 of the paper is devoted to the FEA of the problem. Finite element analysis using ABAQUS [19] is performed to investigate the coherency of the experimental approach and to consider some more thicknesses. Section 4, results and discussion, includes three parts of experimental results, numerical results and effect of thickness. In the numerical analysis, the curve of crack mouth opening displacement (CMOD) vs. crack length (CMOD-a curve) is used as an input material property. Some other thicknesses besides the tested specimens are considered in FEA. For these thicknesses, CMOD-a curves are obtained by linear interpolation between the experimental graphs of the three thicknesses available for the tests. J-integral vs. crack growth (J–Da) or resistance curve, which is used to measure Jc-values, is estimated from FEA for the tested specimens and shows good agreement with the experimental results. It is then concluded that the developed finite element model is applicable for other additional thicknesses. Jc-value is found to increase up to the thickness 1.64 mm and then decreases to the thickness 4.06 mm.
2. Experimental set-up and procedures 2.1. Material properties The specimens employed in this study are made of steel alloy GOST 08Ch22N6T with thicknesses of 1.25, 1.64 and 4.06 mm. All the plates have been machined from the same thick plate. So, the specimens have the same chemical composition and mechanical properties as shown in Tables 1 and 2, respectively. All the specimens are prepared in the rolling direction of the sheet using CNC machine with wire-cut equipments. 2.2. Experiment performance Fracture toughness tests are conducted according to standard test method ASTM E813. All the experiments have been performed using a 100 KN Zwick/Roell servo-hydraulic test machine. Three different thicknesses of 1.25, 1.64 and 4.06 mm are tested using multiple specimen test method and compact tension (CT) specimen with dimensions indicated in Fig. 1. In each multiple test series, the specimens are fatigue pre-cracked to a length of 2 mm. They are loaded to a specified extension and load–displacement curve is plotted. The extension is measured at the end of the specimen notch (crack mouth) with a clip gage. Four anti-buckling plates are used to prevent buckling of the specimen during the loading. Previously, Castrodeza et al. [20] applied the same method for preventing the specimen buckling when testing on 1.42 mm thick CT specimens made of fiber metal composites. Fig. 2 shows the application of these plates. It should be pointed out that the single specimen test method is applied only to the thickest specimen. This technique is not applied to the thin specimens because of buckling problems. J-integral is calculated according to ASTM E813 in single or multiple specimen test method. In the single specimen test method, sequences of unloading/reloading are applied to the specimen and the following equations are used to calculate Jintegral.
J ¼ J el þ J pl
ð1Þ
K i ð1 m2 Þ J elðiÞ ¼ E K i ¼ ½Pi =ðBBn WÞ1=2 f ða0 =WÞ
ð2Þ ð3Þ 2
f ða0 =WÞ ¼ J plðiÞ ¼
3
4
ð2 þ a0 =WÞð0:886 þ 4:64a0 =W 13:32ða0 =WÞ þ 14:72ða0 =WÞ 5:6ða0 =WÞ Þ
ð4Þ
ð1 a0 =WÞ3=2
gAplðiÞ
ð5Þ
Bn b0 g ¼ 2 þ 0:522b0 =W
ð6Þ
Elastic component of J-integral can be also calculated by replacing Apl in Eq. (5) with elastic component of area under load–displacement curve. Therefore, using total area under load–displacement curve in Eq. (5) instead of Apl, total J-integral is obtained.
Table 1 Chemical composition of tested specimens. Material
C
Mn
Si
Cr
Ni
P
S
Ti
Fe
Percent
60.08
60.8
60.8
21–23
5.3–6.3
60.035
60.025
0.3–0.6
Remainder
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Table 2 Mechanical properties of tested specimens. Young’s modulus (GPa) Poisson’s ratio Yield stress (MPa) Ultimate strength (MPa)
206 0.3 520 750
Fig. 1. Dimensions of CT specimen in mm.
Fig. 2. Anti-buckling plates around CT specimen.
According to ASTM E813, crack length is measured using compliance method in single specimen test or visually in multiple specimen test method. In the compliance method, crack length is estimated using the following equation:
ai =W ¼ 1:000196 4:06319U LL þ 11:242U 2LL 106:043U 3LL þ 464:335U 4LL 650:677U 5LL
ð7Þ
where
U LL ¼
1 ½Be EC i 1=2 þ 1
;
Be ¼ B ðB Bn Þ2 =B
Ci ¼
Dv DP
ð8Þ ð9Þ
In multiple specimen test method, the specimens are finally exposed to fatigue cycling and broken to measure the stable crack growth size, i.e., the distance between the fatigue pre-cracked and the fatigue cycling regions. These three regions are discerned by taking high resolution pictures from the fracture surface employing a Canon IXUS-850-IS camera with 7-Mega Pixel resolution. Fig. 3 shows these zones for one of the specimens with 1.64 mm thickness. Along the front of the fatigue crack and the front of the region of slow stable crack extension, the crack size is measured at nine equally spaced points centered about the specimen centerline and extending to 0.005 W from the surface of the specimen. The average of two near-surface measurements is combined with the remaining seven crack length measurements to determine initial and final crack sizes (a0 and ap). Crack growth is then calculated by Da = ap – a0. It should be pointed out that all the crack length measurements are performed using the fracture surface photographs and the digitizer software.
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Fig. 3. Stable crack growth zone between two fatigue regions.
After determination of J and crack extension, resistance curve for each particular thickness can be plotted. The related critical J-value near the onset of slow stable crack growth (JQ) is calculated from the intersection point of the resistance curve and a line parallel to blunting line (the equation of which is J = 2ryDa) with an offset of 0.2 mm. 3. Finite element simulation Two-dimensional finite element analysis (FEA) of the problem is performed using ABAQUS standard code. Since the main concentration of the research is on the experimental investigation of the thickness effect on the fracture resistance, finite element analysis is mostly accomplished to show the coherency of the experimental approach. Therefore, 2D analysis is preferred to the more complicated 3D solution for the sake of comparing experimental and numerical results. However, threedimensional analysis is a beneficial approach in problems with numerical aspects through which more accurate and useful
Fig. 4. (a) FE model and the paths of J-integral calculation. (b) Fine elements near the crack tip.
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651
4
CMOD (mm)
3.5 3 2.5 2
B=1.25 mm
1.5
B=1.64 mm B=4.06 mm
1 30
30.5
31
31.5
32
32.5
33
Crack Length (mm) Fig. 5. Finite element analysis input curves for different thicknesses.
consequences may be achieved. Due to symmetry of the geometry and loading, only the upper half of the specimen is modeled and symmetry condition is applied to the ligament. The center of the specimen hole is defined as a reference point and coupled to its upper half. The load is then applied to the center of the hole in the form of a displacement boundary condition. The J-integral is calculated for different crack lengths and the resistance curve is plotted. Crack mouth opening displacement (CMOD) vs. crack length curve (CMOD-a curve) obtained from experimental results is used as the input material fracture property and linear hardening elastic–plastic behavior is considered. Second-order shell elements (S8R) are used to discretize the model. The model containing 4192 nodes and 1572 elements is shown in Fig. 4a. Near the crack tip elements are also fine enough to give a proper precision as shown in Fig. 4b. The behavior of pre-cracked model for each special thickness is defined by CMOD vs. crack length curve obtained from the experiments. These curves are shown in Fig. 5. In FEA, for each amount of crack growth which is equal to a fine element length in the ligament direction, one half of the corresponding CMOD (due to symmetry) is applied as displacement boundary condition and J-integral is calculated on 20 different paths. On the near crack tip paths, J-value changes extremely due to the presence of plastic zone and may not be a correct estimation of J-integral. On the paths outside of the plastic region, J remains nearly constant. The integral on 20th path is used as the constant J-value to plot resistance curve. 4. Results and discussion 4.1. Experimental results Specification and number of tested specimens in multiple specimen test series are shown in Table 3. According to ASTM E813, at least five specimens are needed for multiple specimen test method. For each specimen with a particular thickness, a proper extension value is applied in order to obtain valid points on the resistance curve. The extension values are indicated in Table 4. The resistance curves obtained for different thicknesses via the multiple specimen test method are shown in Appendix A. Specimen D as the thickest specimen was tested successfully using multiple specimen test method without using anti-buckling plates and no buckling was observed. Therefore, it was possible to test this specimen according to single specimen test procedure. Load–displacement curve for this single specimen test method and the corresponding resistance curve are indicated in Appendix A. The critical value of J-integral near the onset of slow stable crack growth (JQ) is calculated as described in Section 2.2. According to ASTM E813, JQ should satisfy some requirements in order to be ductile fracture toughness (JIc) of the tested material. The obtained critical values from the resistance curves (JQ) satisfy all the requirements mentioned in this standard test method except the thickness requirement:
B > 25J Q =ry
ð10Þ
Therefore, the values of JQ may be known as Jc which is authentic for just the respective thickness. The experimental values of Jc and the corresponding K Jc from Eq. (11) are indicated in Table 5.
K Jc ¼
pffiffiffiffiffiffi Jc E
ð11Þ
Table 3 Specification and number of tested specimen. Specimen code
Thickness (mm)
Number of tested specimens in multiple specimen test method
A B D
1.25 1.64 4.06
5 6 7
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Table 4 Extension values for tested specimens. Specimen code Extension (mm)
A1 1.6
A2 2
A3 2.4
A4 2.8
A5 3
Specimen code Extension (mm)
B1 1.3
B2 2
B3 2.4
B4 3
B5 3.4
B6 3.8
Specimen code Extension (mm)
D1 1.6
D2 1.8
D3 2
D4 2.4
D5 3
D6 3.4
D7 3.8
Table 5 Jc-values for different thicknesses. Thickness (mm)
Jc (kJ/m2)
K Jc ðMPa
1.25 1.64 4.06 (Multiple test) 4.06 (Single test)
141.5 247.2 153.4 173.3
170.7 225.7 177.8 188.9
pffiffiffiffiffi mÞ
Jc-value obtained from the single specimen test method is nearly 13% different from the multiple specimen test result. Since load–displacement curves from both test methods coincide with each other as shown in Fig. A4, this difference cannot be due to J-integral calculation. Such a difference for Jc-values is attributed to different crack length measurement techniques used in this study. The resistance curve and fracture toughness of materials depend on some parameters like material sub-structure and grain size, loading history, temperature and working environment. Due to three-dimensional stress field in plates, the critical J-integral near the onset of stable crack growth (Jc) is highly dependent on geometry and especially specimen thickness. So, the measured fracture toughness is valid only for the corresponding specimens. In thin specimens for which the state of plane stress is nearly dominant due to large plastic region, the resistance curve is steeper and Jc-value is greater than that of thicker specimens. The plastic zone size may be estimated for all the specimens using standard solutions for plane stress and plane strain. The following equations give the plastic zone radius related to both Von Mises’ and Tresca’s criteria [21]:
8 h i 2 K2 > 1 þ 32 sin h þ cos h ; Plane stress < r p ðhÞ ¼ 4pr 2 ys 0 h i Von Mises criterion : > : r p ðhÞ ¼ K 2 2 32 sin2 h þ ð1 2mÞ2 ð1 þ cos hÞ ; Plane strain 4pr
ð12Þ
8 h h2 K2 > ; Plane stress > > r p ðhÞ ¼ 2pr2ys cos 2 1 þ sin 2 > < h 2 9 0 K2 2 h Tresca s criterion : r p ðhÞ ¼ 2pr2 cos 2 1 2m þ sin 2 = ys > > whichever is lorger; Plane strain > > ; : r p ðhÞ ¼ K 2 2 cos2 h 2 2pr
ð13Þ
ys
ys
where the stress intensity factor is calculated from Eq. (3). The radius of the plastic region may be compared with the specimen thickness using rp/B ratio. Minimum and maximum values of rp/B and also the ratios corresponding to h = 0° and h = 90° for the extension of 2 mm as an instance have been calculated for different specimens as shown in Table 6. The values have been computed both by the Von Mises’ and Tresca’s Criteria. It is seen that the values of rp/B for the two thinnest specimens are much higher than that of the thickest one. This Table 6 The ratio of rp/B from Eqs. (12) and (13) for extension of 2 mm. Specimen code
Plane stress equations
Plane stress equations
h = 0°
h = 90°
Minimum
Maximum
h = 0°
h = 90°
Minimum
Maximum
Von Mises’ criterion A B D
4.11 3.49 1.52
5.14 4.37 1.90
4.11 3.49 1.52
5.49 4.66 2.02
0.66 0.56 0.24
3.41 2.90 1.26
0.66 0.56 0.24
3.42 2.91 1.26
Tresca’s criterion A B D
4.11 3.49 1.52
5.99 5.09 2.21
4.11 3.49 1.52
6.93 5.90 2.56
4.11 3.49 1.52
2.52 2.14 0.93
2.06 1.75 0.76
4.11 3.49 1.52
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means that the specimens A and B are close to plane stress state; while, the plastic zone size in the specimen D is in the order of the plate thickness and plane strain state prevails. Similar results can be obtained for all of the other extension values. Fig. 6 demonstrates the presence of considerable plastic zone in the specimen A with the extension of 2.8 mm. In Fig. 7, all of the resistance curves (R-curve) are shown in a single graph. The resistance curve for the specimen B has greater slope than that of the specimen D. This means that the less the specimen thickness is, the more energy is required for crack growth. This is because in the thinner specimen B, that is closer to plane stress, the size of plastic region at the crack tip is greater and more energy is required for the formation of new plastic region with greater size corresponding to the longer crack. So, in the thinner specimen B, stable crack growth occurs more lately and the R-curve is more sloping. On the other hand, Jc-value is larger for the specimen B in which the crack front is closer to the free surface. As the thickness increases from the specimen A to B the state of plane strain prevails. This results in the delay in stable crack growth. Fracture toughness of thin plates is mainly associated with the amount of energy required to reduce the initial cracked specimen thickness which is frequently higher than the energy needed for crack growth. The former depends on the initial specimen thickness [21]. In the other words, fracture resistance of thin metallic sheets is significantly dependent on the energy dissipated in crack tip necking. Thus, the less the amount of necking is, the less energy is needed for crack extension. The percentage values of thickness necking with respect to the initial specimen thickness are given in Table 7 for different extensions. More considerable amount of necking in specimen B than specimen A indicates larger Jc-value for the thicker one. In the other words, as the state of stress is close to plane stress (specimens A and B), the energy which causes thickness reduction and thus, the fracture toughness is higher for thicker specimens. As the thickness increases, the state of stress in crack tip field gradually changes to plane strain and less amount of necking is observed as shown in Table 7. This implies lower Jcvalue for specimen D than specimen B. Stress triaxiality tends to increase at the crack tip for the thicker specimen leading to higher rate of void growth and lower fracture strain and crack tip necking degree. It causes the fracture toughness to decrease [3]. Resistance curves from single and multiple specimen tests for the specimen D can be compared in Fig. 7. The resistance to crack growth is estimated higher from single specimen test. This is attributed to the different methods for crack length measurement. However, the single specimen method is more economic and takes more little time to carry out if it is performable. Fig. 7 also reveals nearly the same resistance to crack growth for the specimens A and D from multiple specimen tests. Critical J-integrals and necking degree for these specimens are also very close to each other as shown in Tables 5 and 7, respectively. 4.2. Numerical results Resistance curves from the finite element analysis are obtained as described in Section 3 and compared with the experimental results as illustrated in Fig. 8. It is seen that the FE results underestimate the experimental graphs, especially for the
Fig. 6. Large plastic zone on the fracture surface of specimen A4.
700
specimen A, B=1.25 mm
J(KJ/m2)
600
specimen B, B=1.64 mm specimen D, B=4.06 mm- multiple
500
specimen D, B=4.06 mm-single
400 300 200 100 0
0.5
1
1.5
Crack Extension (mm) Fig. 7. Comparison between resistance curves.
2
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Table 7 Percentage values of thickness necking for different specimens and extensions. Specimen code
Extension (mm) 1.3
A B D
1.6
1.8
2
2.4
2.8
3
3.4
3.8
– 9.1 –
7.9 – 6.3
– – 6.8
8.2 17.7 6.5
8.7 19.0 7.7
11.7 – –
12.5 20.0 8.2
– 24.2 10.7
specimen A
500
J (KJ/m2)
– 36.2 11.8
400 300 EXP
200 100
FEA
0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
2.2
Crack Extension (mm)
J (KJ/m2)
700
specimen B
600 500 400 EXP
300 200
FEA
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
Crack Extension (mm)
J (KJ/m2)
600
specimen D
500 400 300
EXP
200 100
FEA
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
Crack Extension (mm) Fig. 8. Comparison between experimental and FEA resistance curves.
specimens A and D; however they are in good agreement. The maximum differences for the specimens A, B and D are 5.8%, 4.3% and 8.7%, respectively. These negligible errors show the coherency of the experimental approach and acceptable precision of the finite element model. Thus, FE simulation can be applied for other models like different thicknesses as it will be discussed afterwards. 4.3. Effect of thickness More complete investigation of thickness effect on the fracture toughness needs considering some other thicknesses in addition to the thicknesses of previously tested specimens. Therefore, five other thicknesses were considered beside the first three thicknesses mentioned in Table 3. The developed finite element model is employed for the thicknesses of 1.45, 2, 2.5, 3 and 3.5 mm. Finite element input behavior for new thicknesses (i.e., CMOD-a curve) is estimated by linear interpolation between the experimental curves obtained for 1.25, 1.64 and 4.06 mm thick specimens (Fig. 5). As indicated in Fig. 5, the CMOD corresponding to the specimen B is greater than that of the two other specimens for all crack lengths. Also, the material
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behavior curve (i.e. CMOD vs. crack length curves shown in Fig. 5) corresponding to specimen A lies between the graphs related to the thicknesses of 1.64 and 4.06 mm. On the other hand, the CMOD rises up by increasing the thickness from 1.25 to 1.64 mm and then decreases by further increase of the thickness to 4.06 mm for all crack lengths. Furthermore, the material behavior for the intermediate thicknesses (i.e. thicknesses between 1.25 and 1.64 mm and also between 1.64 and 4.06 mm) is unknown due to limited number of thicknesses available for the experiments. In this situation, linear interpolation method seems reasonable for estimation of the behavior of the intermediate thicknesses. All the input curves for FEA are indicated in Fig. 9. Crack resistance curves obtained from the FEA for all thicknesses are given in Figs. 10 and 11. Fig. 10 shows increasing slope as the thickness increases. While, as indicated in Fig. 11, the slope of the R-curve decreases with increasing thickness. In Table 8, the R-curve power law equation, i.e., J ¼ C 1 ðDaÞC 2 , the intersection point of the R-curve with 0.2 mm offset line (DaQ), critical J-integral (Jc-value) and its corresponding K Jc (from Eq. (11)) are shown for different thicknesses. All the data in Table 8 are the FEA results. It is seen that for thicknesses 1.25–1.64 mm, Jc rises up following the increase of the slope of the resistance curve. However, for the thicknesses greater than 1.64 mm the Jcvalue decreases with the increase of thickness.
4 B=1.25 mm B=1.45 mm B=1.64 mm
3
B=2 mm B=2.5 mm
2.5
B=3 mm B=3.5 mm
2
B=4.06 mm
1.5 30
31
32
33
Crack Length (mm) Fig. 9. Experimental and interpolated input curves for FEA.
800
J (KJ/m2)
700 600 500 400 B=1.25 mm
300
B=1.45 mm
200
B=1.64 mm
100 0
0.5
1
1.5
2
2.5
Crack Extension (mm) Fig. 10. Resistance curves for first three thicknesses.
700 B=2 mm B=2.5 mm B=3 mm B=3.5 mm B=4.06 mm
600
J (KJ/m2)
CMOD (mm)
3.5
500 400 300 200 100 0
0.5
1
1.5
2
Crack Extension (mm) Fig. 11. Resistance curves for thicknesses 2–4.06 mm.
2.5
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Table 8 FEA results for different thicknesses. B (mm)
C1 (kJ/m2)
C2
DaQ (mm)
Jc (kJ/m2)
K Jc ðMPa
1.25 1.45 1.64 2 2.5 3 3.5 4.06
275.88 346.07 437.35 401.63 363.19 336.02 305.21 275.69
0.6676 0.6105 0.6854 0.7299 0.7232 0.723 0.6839 0.6269
0.2965 0.3414 0.3763 0.3488 0.3276 0.3147 0.3073 0.3027
122.6 179.6 223.9 189 162.1 145.7 136.3 130.4
158.9 192.3 214.8 197.3 182.7 173.2 167.6 163.9
pffiffiffiffiffi mÞ
Fig. 12 plots variations of critical J-integral against specimen thickness. Experimental results are also shown in this figure. A five-order polynomial curve is fitted to the FEA results according to Eq. (14).
J c ¼ 28:149B5 410:24B4 þ 2314:4B3 6351:1B2 þ 8401:5B 4063:7 ðkJ=m2 Þ for 1:25 < B < 4:06 mm
ð14Þ
Considering the experimental results, the specimen B with thickness of 1.64 mm has the maximum Jc-value (equal to 247.2 KJ/m2) among the tested specimens which may not be the maximum value of fracture toughness for the material under consideration. The low variety of thicknesses available for the experiments and having no approximation of ductile fracture toughness for this particular steel alloy makes it difficult to estimate Jc at different thicknesses and also the thickness corresponding to the maximum Jc-value using the models proposed in the literature [21]. Nevertheless, the present investigation provides more or less proper and reliable estimations for the thickness dependent fracture toughness in the range of 1.25–4.06 mm. The thickness at which Jc reaches its maximum value (B0) depends on the material type. This thickness is usually considered to correspond to the real plane stress fracture toughness. It means that B0 must be equal to the plastic zone size in plane strain which is approximated by [21]:
B0 ffi
K 2Ic
ð15Þ
3pr2ys
Jc (KJ/m2)
Assuming that the thickness B0 for this special material is equal to 1.64 mm, the values of KIc and its equivalent JIc are obpffiffiffiffiffi tained from Eqs. (15) and (11) as 64:6 MPa m and 20.3 kJ/m2, respectively. This implies that the minimum thickness required for a valid JIc test should be about 0.8 mm according to Eq. (10). All the tested specimens satisfy this requirement; while, their estimated fracture toughness values are significantly different. Therefore, the thickness related to the maximum Jc-value should be between 1.64 and 4.06 mm due to the higher value of KIc expected for this low-strength steel alloy. In the other words, the curve of Jc vs. specimen thickness might be higher than that obtained in Fig. 12 for the thicknesses 1.64– 4.06 mm if some more thicknesses were provided. However, the experimental and numerical values in Tables 5 and 8 could be used as the conservative estimations of the thickness dependent fracture toughness for many design purposes involving sheet metals. For very thin specimens (B = 1.25 and 1.45 mm), due to the low thickness and large plastic region at the crack tip, fracture occurs at lower load levels. Hence, Jc-values for these thicknesses are lower than that of the specimen B. While, in thicker specimens, as the thickness increases, the input energy is dominantly spent on forming new surfaces, and so growing the crack. Therefore, the crack growth occurs earlier than the thinner specimens for which the major proportion of the input energy causes plastic zone formation.
260 240 220 200 180 160 140 120 100
Experimental results FEA results
0
1
2
3
4
Thickness (mm) Fig. 12. Variations of Jc-value vs. thickness.
5
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Since the thickness B0 was discussed to be higher than 1.64 mm, the fracture toughness values for thicknesses B = 1.25 and 1.45 mm do not experience changes like the thicker specimens do when some more thicknesses are tested. On the other hand, Jc-values have been estimated less conservatively for specimens thinner than the specimen B.
5. Conclusions An experimental and numerical study of the effect of thickness on the fracture toughness has been carried out. Jc-values for plates with thicknesses of 1.25, 1.64 and 4.06 mm made of steel alloy GOST 08Ch22N6T have been obtained experimentally using CT specimens and according to the standard test method ASTM E813. Then, some other thicknesses of 1.45, 2, 2.5, 3 and 3.5 mm have been examined through the use of the finite element method to study the thickness effect more comprehensively. Input material property in the FEA was CMOD vs. crack length curve. For the latter group of thicknesses, the mentioned input curve was estimated by linear interpolation between the experimental graphs obtained for the former thicknesses. Following conclusions are drawn from the present investigation: 1. Due to very low thickness of the specimens, single specimen test method was faced to buckling problem. To prevent this problem, multiple specimen tests were conducted and also anti-buckling plates were used. However, the specimen D (B = 4.06 mm) did not show any buckling and single specimen test was also conducted for this specimen. The resistance curves corresponding to the single and multiple specimen testing methods performed for the specimen D are in good agreement.
specimen A
1000
J(KJ/m2)
800
Jmax=847 KJ/m2
600 400 200 0 0 0.244 0.5
1
1.5
1.8
2
2.5
Crack Extension (mm) 1000
Jmax=847 KJ/m2
2
J(KJ/m )
800
specimen B
600 400 200 0 0 0.32 0.5
1
1.5
2 2.021 2.5
Crack Extension (mm)
specimen D
1000 Jmax=847 KJ/m2
J(KJ/m2)
800 600 400 200 0 0 0.241 0.5
1
1.5 1.873 2
2.5
Crack Extension (mm) Fig. A1. Resistance curves obtained from multiple specimen test method.
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2. Resistance curves for the first three thicknesses were plotted. Critical J-integral values near the onset of slow stable crack growth (JQ) for these thicknesses satisfied all the requirements mentioned in ASTM E813 except the thickness requirement (Eq. (10)). So, they may be considered as Jc. 3. Two-dimensional finite element analysis was performed using ABAQUS standard code to investigate the coherency of the experimental approach and considering some other thicknesses. Resistance curves obtained from the FEA showed a very good agreement with the experimental results. Insignificant differences of 5.8%, 4.3% and 8.7% exist between the FEA resistance curves compared with the experimental graphs for 1.25, 1.64 and 4.06 mm thick specimens, respectively. This good precision showed the acceptable accuracy of the developed finite element model. 4. Jc-value increases for the thicknesses from 1.25 to 1.64 mm and then decreases for thicker specimens up to B = 4.06 mm. The ratio of the plastic zone size to the specimen thickness (rp/B) was found to be higher for the thinnest specimens than that of the specimen D. It means that the state of plane stress prevails in the specimens A and B unlike the thickest sample where the plane strain is more dominant. The large degree of thickness necking observed in the specimen B showed the considerable amount of energy required for crack growth indicating the greater value of the fracture toughness corresponding to the specimen B than that of the other specimens. However, the low Jc-value for the thickest specimen could be attributed to the increase of stresses in the crack tip field. 5. Determination of the thickness related to the maximum value of the fracture toughness needs more variety of the thicknesses for the experiments. However, it was concluded that the thickness B0 is between 1.64 and 4.06 mm for the material under consideration. On the other hand, the curve of the fracture toughness vs. specimen thickness shown in Fig. 12 suggests conservative Jc-values at different thicknesses, especially in the range of 1.64–4.06 mm. Having approximated values of the thickness dependent fracture toughness is beneficial to many design purposes.
Fig. A2. Load–displacement curve from single specimen test for specimen D.
specimen D
1000 Jmax=847 KJ/m
2
J (KJ/m2)
800 600 400 200 0 0 0.276 0.5
1
1.5 1.926 2
2.5
3
Crack Extension (mm) Fig. A3. Resistance curve obtained from single specimen test method for specimen D.
Load (KN)
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8 7 6 5 4 3 2 1 0
D5-multiple test
0
0.5
1
1.5
2
659
D8-single test
2.5
3
3.5
Extension (mm) Fig. A4. Load–displacement curves from single and multiple specimen test methods for specimen D.
Appendix A Three different CT specimens with the thicknesses indicated in Table 3 have been tested using multiple specimen test method. Crack growth and J-integral are calculated for each test series and the resistance curves are plotted as shown in Fig. A1. For each graph in Fig. A1, blunting line and its parallel exclusion lines with 0.15, 0.2, 0.5, 1 and 1.5 mm offsets are shown according to ASTM E813. The valid region is shown by dashed lines. Maximum J-integral (Jmax = b0ry/15) is the same for all the specimens. Vertical dashed lines indicate minimum and maximum crack extension (Damin and Damax) corresponding to the intersection of the initial power law curve (which is estimated using all the initial points) with the 0.15 mm and 1.5 mm exclusion lines, respectively. Final power law curve is achieved after neglecting invalid points out of this region. Load–displacement curve related to single specimen test method for the specimen D, with 12 unloading/reloading sequences and final extension of 3 mm, was plotted as shown in Fig. A2. The corresponding resistance curve is illustrated in Fig. A3. Load–displacement curves from single and multiple specimen test methods for specimen D are compared in Fig. A4. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
ASTM code E813: standard test method for JIc. A measure of fracture toughness 1991. ASTM code E1820: standard test method for measurement of fracture toughness 2001. Shukla A. Practical fracture mechanics in design. 2nd ed. New York: Marcel Dekker; 2005. Qingfen L, Limin Z, Shouren L. The effect of a/W ratio on crack initiation values of cod and J-integral. Eng Fract Mech 1986;23(5):925–8. Masanori K. Study on the effect of the crack length on the JIc value. Nucl Eng Des 1997;174(1):41–9. Jinkook K, Guihua Z, Xiaosheng G. Modeling of ductile fracture: application of the mechanism-based concepts. Int J Solids Struct 2007;44:1844–62. Pardoen T, Marchal Y, Delannay F. Thickness dependence of cracking resistance in thin aluminum plates. J Mech Phys Solids 1999;47:2093–123. Pardoen T, Marchal Y, Delannay F. Essential work of fracture compared to fracture mechanics-towards a thickness independent plane stress toughness. Eng Fract Mech 2002;69:617–31. Pardoen T, Hacheza F, Marchionia B, Blythb PH, Atkinsb AG. Mode I fracture of sheet metal. J Mech Phys Solids 2004;52:423–52. Mahmoud S, Lease K. The effect of specimen thickness on the experimental characterization of critical crack-tip-opening angle in 2024-T351 aluminum alloy. Eng Fract Mech 2003;70:443–56. Kang YL, Zhang ZF, Wang HW, Qin QH. Investigation of the effect of thickness on fracture toughness of metallic foils. Mater Sci Eng A 2005;394:312–9. Jitsukawa S, Naito A, Segawa J. Effect of size and configuration of 3-point bend bar specimens on J–R curves. J Nucl Mater 1999;271–272:87–91. Dawicke DS, Newman Jr JC. Evaluation of fracture parameters for predicting residual strength of multi-site damage. In: Proceeding of 1st joint DoD/ FAA/NASA conference on aging aircraft, Ogden, Utah; 1997. Newman Jr JC, James MA, Zerbst U. A review of the CTOA/CTOD fracture criterion. Eng Fract Mech 2003;70(3–4):371–85. Newman JC, Dawicke DS, Bigelow CA. Finite-element analyses and fracture simulation in thin-sheet aluminum alloy. In: Proceeding of the international workshop on structural integrity of aging airplanes, Atlanta, Georgia; 1992. Shivakumar KN, Newman Jr JC. Numerical fracture simulation of bend specimens using a CTOD criterion. Eng Fract Mech 1989;32(2):203–10. Sutton MA, Deng X, Fashang M, James C, Newman Jr JC, James M. Development and application of a crack tip opening displacement-based mixed mode fracture criterion. Int J Solids Struct 2000;37:3591–618. Sutton MA, Helm JD, Boone ML. Experimental study of crack growth in thin sheet 2024-T3 aluminum under tension-torsion loading. Int J Fract 2001;109(3):285–301. Hibbit, Karlsson, Sorensen, HKS Inc. ABAQUS finite element code v. 6.7.1; 2007. Castrodeza EM, Schneider Abdala MRW, Bastian FL. Crack resistance curves of GLARE laminates by elastic compliance. Eng Fract Mech 2006;73:2292–303. Broek D. Elementary engineering fracture mechanics. 3rd ed. Boston: Martinus Nijhoff Publishers; 1982.