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Stress triaxiality based transferability of cohesive zone parameters
Journal Pre-proofs Stress triaxiality based transferability of cohesive zone parameters Viswa Teja Vanapalli, B.K. Dutta, J. Chattopadhyay, Nevil Martin Jose PII: DOI: Reference:
S0013-7944(19)30859-8 https://doi.org/10.1016/j.engfracmech.2019.106789 EFM 106789
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
10 July 2019 19 November 2019 19 November 2019
Please cite this article as: Teja Vanapalli, V., Dutta, B.K., Chattopadhyay, J., Martin Jose, N., Stress triaxiality based transferability of cohesive zone parameters, Engineering Fracture Mechanics (2019), doi: https://doi.org/ 10.1016/j.engfracmech.2019.106789
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Stress triaxiality based transferability of cohesive zone parameters Viswa Teja Vanapalli1*, B.K. Dutta1**, J. Chattopadhyay1, 2, Nevil Martin Jose2 1Homi
2 Reactor
Bhabha National Institute, Anushaktinagar, Mumbai 400094, India Safety Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India *[email protected], **[email protected]
Abstract Present study is to investigate the dependency of cohesive zone parameters on crack tip stress triaxiality for SA333 Grade 6 steel. An exponential cohesive law is used to simulate ductile fracture behavior of 14 three-point bend specimens made of SA333 Gr. 6 steel. Cohesive parameters are determined by varying peak stress to match experimental results with the computed data. To extend the validity of parameter peak stress as a function of triaxiality, experimental results of six piping components with through-wall circumferential crack made up of SA333 Grade 6 steel are used. A confidence interval band is then plotted to study the transferability of cohesive parameters. The variation in peak stress for a single value of multiaxiality quotient is attributed to the micro structural variation in material during manufacturing near the crack tip, which is statistical in nature. A normal variation of peak stress is assumed for a given value of multiaxiality quotient. To test the accuracy of the cohesive parameters with such normal variation, experimental results of six piping components are again used. Significant match of computed results with the measured values shows transferability of the present material parameters. The methodology presented in this work can be applied to other structural steels to find out cohesive zone parameters. Such material parameters will be useful to the designers to carry out safety analysis of piping components of nuclear and other installations. Keywords: Cohesive zone model; multi-axiality quotient; Transferability of cohesive parameters
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Nomenclature a0 A1 A2 B BN da dx G h J JSZW n q q0 π Tp Ti Tl Tu W Z
Initial crack size Lower transition value in Boltzmann sigmoidal function [Eq. (9)] Upper transition value in Boltzmann sigmoidal function [Eq. (9)] Thickness of TPBB specimen Thickness of TPBB specimen after side grooving is done Crack growth Transition slope coefficient in Boltzmann sigmoidal function [Eq. (9)] Cohesive energy [Eq. (4)] Triaxiality factor [Eq. (5)] J-integral Initiation toughness at stretch zone width Ramberg-Osgood exponent Multi axiality quotient [Eq. (8)] Inflection point in Boltzmann sigmoidal function [Eq. (9)] Effective traction [Eq. (3)] Peak stress / cohesive strength [Eq. (2)] Typical peak stress value for a value of Z score [Eq. (12)] The value of peak stress at -3Ο limit The value of peak stress at +3Ο limit Width of TPBB specimen Z score of standard normal distribution [Eq. (10)]
Greek symbols Ξ± Ramberg-Osgood coefficient Ξ² Mode mixity parameter [Eq. (1)] Ξ΄ Effective opening displacement [Eq. (1)] Ξ΄n Normal/opening displacement [Eq. (1)] Ξ΄p Effective displacement at peak stress [Eq. (2)] Ξ΄t Tangential/sliding displacement [Eq. (1)] Ο Free energy potential [Eq. (2)] Οsd Standard deviation of the normal variation [Eq. (11)] Οh Hydrostatic stress [Eq. (6)] Οv von Mises equivalent stress [Eq. (7)] Οy Yield strength Οu Ultimate strength ΟZ Probability density function [Eq. (13)] ΞΌ Value of peak stress obtained from sigmoidal curve fit. Abbreviations ASTM American Society of Testing and Materials CMOD Crack Mouth Opening Displacement EPFM Elastic plastic fracture mechanics NB Nominal Bore TPBB Three-Point Bend Bar specimen TSL Traction Separation Law
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1. Introduction: Various techniques are available for numerical simulation of crack growth in a component used in any power plant. Some of these are node-release-technique [1], Gurson-Tvergaard-Needleman damage model [2], Cohesive-zone-model, etc. The cohesive zone model [3, 4] is a damage mechanics methodology used to simulate the initiation and propagation of cracks in solids using finite element technique. In this model, a narrow band of vanishing thickness lies ahead of crack tip to represent the fracture process zone. The upper and lower surfaces, as shown in Fig. 1(a), are called the cohesive surfaces. Both the surfaces are subjected to cohesive tractions during the loading of the component. The cohesive tractions follow a constitutive law called the Traction Separation Law (TSL), which relates the cohesive traction (T) with the effective opening displacement (Ξ΄). A typical TSL is shown in Fig. 1(b). There are various types of TSL available in the literature developed primarily for different engineering materials such as trapezoidal, exponential and polynomial TSLs [5, 6, 7, 8]. Three parameters which are generally used to define a TSL are cohesive energy, peak stress and the effective opening displacement. It can be shown that any two out of these three parameters are independent. The philosophy behind TSL is that crack growth will occur when the separation distance between the two adjoining points on the opposite cohesive surfaces reaches a critical value (Ξ΄c). The amount of energy required for separation of surfaces is known as cohesive energy. The various cohesive zone models vary according to the shape of traction separation laws and corresponding governing equations.
Fig. 1 (a) Cohesive zone model (b) Exponential Traction Separation Law (TSL).
In the present work, an exponential TSL is used to simulate the crack growth in Three-Point Bend Bar (TPBB) specimens made up of SA333 Gr.6 material. Finite element code WARP3D available in open literature is used for this purpose [9]. An exponential TSL combining normal and shear tractions into a single effective traction (T) is available in WARP3D. The model thus has a single traction separation curve and a single energy of separation (G). The shape of the exponential law is dependent on userspecified values of peak stress (Tp) and displacement at peak stress during the analysis (Ξ΄p). The effective opening displacement is calculated using the following equation (1). πΏ = π½2πΏ2π‘ + πΏ2π (1) Here Ξ΄t and Ξ΄n are tangential and normal displacements and Ξ² is a mode-mixity parameter which assigns different weights to tangential and normal displacements. For a mode I fracture problem, the value of Ξ² is taken as zero. 3|Page
The constitutive relation for the cohesive surface is derived from an exponential form of a free energy potential Ξ¨ [9] as shown below.
[ (
πΏ
) (
πΏ
)]
(2)
πΉ = ππ₯π(1)πππΏπ 1 β 1 + πΏπ exp β πΏπ The relationship T- Ξ΄ follows from π=
βπΉ βπΏ
πΏ
(
πΏ
)
= ππ₯π(1)πππΏπππ₯π β πΏπ
(3)
The work of separation per unit area of the cohesive surface is given for an exponential TSL as β
πΊ = β«0 πππΏ = exp (1)πππΏπ
(4)
The infinite value of distance of separation to calculate cohesive energy has been taken [9] to address exponential decrease in the value of traction with the distance of separation. However, in actual practice, while computing the value of G, a finite value of Ξ΄ is taken as a cut off distance. In the present case, the cut off distance is so taken that the current value of traction is less than 5% of peak value of traction. As, T decreases with Ξ΄ in an exponential manner, such cut off distance comes out to be 5.74 times of the Ξ΄p. The crack propagation in a component (pipe, elbow, t-joint, etc.) depends on the material J-R curve, which in turn depends on the stress triaxiality. Stress triaxiality changes with the crack propagation due to loss of constraint. Hence, to carry out crack propagation analysis by EPFM, one should consider change in J-R curve with crack propagation. This is generally not done. It is anticipated that the cohesive parameters are not only material constants but vary with the state of stress triaxiality at the crack tip. Previous works from authors like C. Chen et al [10] and A. Banerjee [11] investigated the interrelations between cohesive parameters and crack tip triaxiality. It is observed that with decrease of cohesive strength, triaxiality decreases while cohesive energy increases with triaxiality. While Chen et al. [10, 12] studied the variation of parameters along crack front and crack extension for a Compact Tension specimen made of 20MnMoNi55 & St37 materials using a polynomial TSL, A. Banerjee et al [11] incorporated the triaxiality parameter directly into the equation of TSL for aluminum and steel materials. I. Schieder et al. [13] studied on the transferability of cohesive parameters based on shape of TSL for different materials and constraint conditions. In many cases, constant cohesive parameters showed dependence as long as cracked models are investigated. In present work, the βJ-hβ approach is used to calculate the βqβ parameter. Where βhβ is the triaxiality factor defined as ratio between hydrostatic stress (Οh) and the von Mises equivalent stress (Οv). πβ
(5)
β = ππ£ where πβ =
(π1 + π2 + π3) 3
(6) ππ£ = { (π1 β π2)2 + (π2 β π3)2 + (π3 β π1)2}/ 2
(7)
where, Ο1, Ο2, Ο3, are the principal stresses. The multi-axiality quotient βqβ is given by, π = 1/( 3β) (8) The βqβ parameter varies with the increase in applied load. In order to compare βqβ for different cracked specimens, the variation of q is plotted with the increase in applied J integral. The applied J-integral value at any displacement increment may be calculated using the closed form formula given in ASTM E 1820 [14] or numerically using domain integral technique [9]. The βqβ is calculated in the adjacent elements using a distance from the initial crack tip. The stress information is taken at the integral points (Gauss points) at the load step when the applied J-integral is equal to the J-initiation of the material. The objective of present work is to ascertain the transferability of cohesive prameters as a function of crack tip stress triaxiality made of SA333 Grade 6 steel. In this paper, section 2 & 3 deal with procedure 4|Page
adopted to determine cohesive parameters and triaxiality parameters of three-point bend (TPBB) specimens and pipe components respectively using an exponential TSL; Section 4 describes the development of a correlation between cohesive parameter peak stress with stress triaxiality. Results of a normal statistical variation in peak stress for constant multiaxiality quotient have been presented in Section 5 along with concluding remarks in Section 6. 2. Fracture simulation analysis of TPBB specimens 2.1 Material properties & Experimental data The experimental results of TPBB specimens made up of SA333 Grade 6 steel used for the present analyses are taken from the reported results [15]. This material is widely used in Indian nuclear reactors for fabricating straight pipes & elbows. The material properties of SA333 Grade 6 steel are given in Table 1. J-initiation value for this material is found to be close to 220 kJ/m2 determined using stretch zone width at crack initiation (JSZW) [15, 16, 17, 18]. The experimental results of TPBB specimens having 8 mm, 12 mm and 25 mm thicknesses, have been used in the present analysis. The 8mm thick specimens are prepared from 200 mm NB (8-inch diameter) pipes, while 12mm and 25mm thick specimens are prepared from 400 mm NB (16-inch diameter) pipes. Table 1 Mechanical properties of SA333 Grade-6 steel [15] 200 mm NB pipe
400 mm NB pipe
Youngβs Modulus, E
203000 MPa
203000 MPa
Yield Strength, Οy
285 MPa
307 MPa
Ultimate Strength, Οu
420 MPa
463 MPa
Ramberg Osgood parameters Ξ± n
10.759 4.301
10.249 4.23
2.2 Finite element model of TPBB specimen Fig. 2 shows the schematic diagram of a three-point bend bar (TPBB) specimen with side grooves. In the present analysis, 3D finite element model of a quarter specimen is used making use of symmetric boundary conditions. The finite element (FE) model of the specimen is prepared using 8-noded solid brick elements while cohesive zone is modeled using 8-noded cohesive interface elements. The true stress-strain curve as reported in [16, 17] is used in the analysis. The finite element mesh for TPBB specimen consists of 1719 nodes and 1180 elements. Aspect ratio of elements present near the crack tip is 1 in the direction of crack propagation. Cohesive elements are located at the crack plane from the initial crack tip up to a distance of 4 mm in the direction of crack growth as shown in Fig. 3(c). The cohesive elements initially have zero thickness. The size of the cohesive elements in the direction of crack propagation is taken as 0.2 mm which is the least crack growth measured. The details of boundary conditions and loading are shown in Fig. 3(a) and Fig. 3(b).
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Fig. 2 Geometric details of a TPBB specimen with side grooves
Fig. 3 Quarter 3D FE mesh with symmetry boundary conditions in (a) z-direction, (b) x & y-direction and loading nodes in x-direction and (c) enlarged view showing cohesive elements near the crack tip.
2.3 Computed cohesive parameters for TPBB specimens The cohesive parameters for the TPBB specimens have been found by comparing the simulated results with the experimental results. During the analysis, the value of cohesive energy (G) is taken as the value of initiation fracture toughness for SA333 Grade 6 steel, which is 220 kJ/m2. It has been pointed out in reference [7] that setting βGβ with the initiation fracture toughness does not necessarily cause substantial errors. This is due to the fact that the total energy absorbed in ductile failure is much greater than initiation fracture toughness so that an inaccuracy in cohesive energy does not substantially affect the result of the simulation of a crack growth resistance curve. Thus, the pragmatic view of setting cohesive zone energy equal to initiation fracture toughness is justified. In the present work, the initiation fracture toughness (JSZW) is determined using a stretch zone width methodology (SZW) [18]. It may be mentioned here that initiation fracture toughness determined by SZW method is independent of stress triaxiality. 6|Page
To match computed results with the experimental results, sensitivity analysis of peak stress (Tp) on the load-CMOD curve is done in the present analysis. An algorithm to determine peak stress (Tp) is given in Fig. 4. This helped to obtain suitable cohesive parameter Tp for a given specimen. The cohesive element gets deleted sequentially when the separation distance between the two adjacent points on the opposite cohesive surfaces reaches a critical value (Ξ΄c) and crack growth is calculated by multiplying the size of cohesive element with the number of elements βkilledβ at that particular displacement increment. The word βkilledβ refers to those cohesive elements which are no longer able to transmit stresses during further displacement increments. The load v/s CMOD plot can be obtained from the output file. The J-R curves are calculated using the closed form formula given in ASTME1820 [14]. Table 2 shows geometrical details of a typical TPBB specimen used in the analysis having a0/W ratio of 0.305.
Fig. 4 Algorithm to compute the optimized cohesive parameters and 'q' parameter for TPBB specimens.
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Fig. 5 Comparison of (a) Load v/s CMOD, (b) Load v/s crack growth and (c) J-Resistance curve with experimental result for specimen T082C
Fig. 6 Comparison of (a) Load v/s CMOD, (b) Load v/s crack growth and (c) J-Resistance curves with experimental results for TPBB specimens having thickness of 8 mm.
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Fig. 7 Comparison of (a) Load v/s CMOD, (b) Load v/s crack growth and (c) J-Resistance curves with experimental results for TPBB specimens having 12 mm thickness
Fig. 8 Comparison of (a) Load v/s CMOD, (b) Load v/s crack growth and (c) J-Resistance curves with experimental results for TPBB specimens having 25 mm thickness
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Table 2 Geometrical details & cohesive parameters of a typical TPBB specimen Dimensions Thickness (B) 8.14 mm Thickness after side grooving (BN) 6.76 mm Width (W) 25.08 mm a0/W ratio 0.305 Cohesive parameters determined through present analysis Cohesive Energy (G) 220 KJ/m2 Peak Stress (T) 1300 MPa Table 3 Analyses of TPBB specimens: geometrical details and cohesive parameters Specimen T082B T082C T083A T083B T085B T122A T123B T123A T125C T125B T252B T252A T253A T253B
B (mm) 8.18 8.14 8.12 7.90 8.06 12.68 12.58 12.72 12.64 12.48 25.32 25.18 25.24 25.20
BN (mm) 6.98 6.76 6.7 6.46 7.00 10.14 10.12 10.12 10.14 10.10 20.80 20.50 20.52 20.46
a0/W 0.244 0.305 0.347 0.356 0.513 0.275 0.354 0.362 0.430 0.505 0.189 0.200 0.342 0.354
W (mm) 25.10 25.08 25.06 25.06 25.06 25.08 25.08 25.08 25.08 25.02 50.18 50.14 50.20 50.28
Peak stress, Tp (MPa) 1275 1300 1300 1275 1275 1700 1600 1625 1825 1700 1700 1750 1825 1775
Ξ΄p/2 using Eqn. (4) (mm) 0.031738 0.031128 0.031128 0.031738 0.031738 0.023803 0.025291 0.024902 0.022173 0.023803 0.023803 0.023123 0.022173 0.022798
Finite element analyses are conducted on fourteen TPBB specimens using the cohesive zone elements along crack front. Fig. 5 shows computed (a) Load v/s CMOD, (b) Load v/s crack growth (da)and (c) J-Resistance curve in comparison to experimental results for a TPBB specimen T082C for which dimensions are shown in Table 2. Such results have been obtained by varying the value peak stress βTpβ. The results shown in Fig. 5 are for the most optimal value of βTpβ for which best comparison between computed and experimental results have been obtained. Fig. 6, Fig. 7 and Fig. 8 show the best fitted results of TPBB specimens with the experimental results having thickness of 8 mm, 12 mm and 25 mm respectively. The values of best fitted cohesive parameters for all the specimens along with geometric details are tabulated in Table 3. Table 4. Geometrical details & cohesive parameters of a typical TPBB specimen Dimensions Thickness (B) 8.14 mm Thickness after side grooving (BN) 6.76 mm Width (W) 25.08 mm a0/W ratio 0.305 Cohesive parameters determined through present analysis Cohesive Energy (G) 220 KJ/m2 Peak Stress (T) 1300 MPa
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2.4 Determination of βqβ parameter of TPBB specimens by elastic-plastic FEA Standard procedure, cited in the literature, is to first calculate multi-axiality quotient βqβ at the crack tip by carrying out elastic-plastic analysis of the component with initial crack. Based on the calculated value of βqβ, a J-R curve is selected and assumed to be applicable throughout the crack propagation analysis. Similar approach has been adopted in the present work. In place of J-R curve, cohesive parameter βTpβ is used. The value of βTpβ is selected for a particular value of multi-axiality quotient βqβ, which is calculated by carrying out an elastic-plastic analysis of the cracked component. The value of βqβ has been taken to be constant during the crack propagation as is done in case of J-R curve. The stress triaxiality parameters βqβ for TPBB specimens have been determined for initial crack length. For this purpose, elastic-plastic finite element analyses of all fourteen TPBB specimens have been carried out using von Mises yield criterion, Prandtl Reuss flow rule, isotropic hardening law and large strain formulation. True stress-strain data mentioned in section 2.2 is employed yet again. The finite element mesh used in elastic-plastic analysis of TPBB specimen consists of 1656 nodes and 1140 elements. The stress information is taken at the integral points (Gauss points) at the load step when the applied J-integral is equal to the J-initiation of the material. An algorithm to find out βqβ parameter is also shown in Fig. 4. It is observed that in all the cases, the minimum value of βqβ, and hence the maximum constraint, has been obtained at the mid-section of the model across the thickness. The βqβ parameter for all specimens are calculated when the applied J-integral in the specimen reaches a value of 220 kJ/m2. A plot of βqβ v/s distance from the crack tip for TPBB specimen at mid-section is shown in Fig. 9 for a J-integral value close to 220 KJ/m2. The minimum values of βqβ ahead of the crack tip from such plots for all the specimens are shown in Table 5.
Fig. 9 'q' v/s distance from crack tip for T08_2C. Minimum value of 'q' is obtained in Layer 1 is equal to 0.33388 at J ~221.9472 kJ/m2.
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Table 5 The minimum value of 'q' parameter ahead of the crack tip at an applied J~220 kJ/m2 determined for the TPBB specimens Minimum value of βqβ parameter 0.32697 0.33388 0.31896 0.32574 0.30957 0.30772 0.30764 0.30341 0.28148 0.26774 0.27268 0.26977 0.24349 0.23761
Specimen T082B T082C T083A T083B T085B T122A T123B T123A T125C T125B T252B T252A T253A T253B
3. Fracture simulation analysis of Straight pipes with through-wall flaws Using the results of peak stress βTpβ and minimum value of βqβ parameters for TPBB specimens, it is possible to generate correlation between them for βqβ values ranging from 0.237 to 0.333, which is the range shown in Table 5. But the constraint level in real life components is generally lower. Therefore, it is necessary to include the βqβ parameters in the database from large components to increase the range of validity of such correlation. In the present work, experimental data for straight pipes with through wall cracks are used for this purpose. 3.1 Material properties and Experimental data The experimental results of straight pipes used for such purpose are taken from the experiments conducted under the project, named, βPiping component integrity test programβ, carried out by Chattopadhyay et al [16]. Under this project, fracture experiments are carried out on piping components made up of SA333 Grade 6 steel. Table 6 shows geometric details of six pipes selected here for the determination of cohesive parameters. The straight pipes were subjected to four-point bending load. Table 6 Geometrical Details of the Pipes [16] Pipe component
SPBMTWC8-1 SPBMTWC8-2 SPBMTWC8-3 SPBMTWC16-1 SPBMTWC16-2 SPBMTWC16-3
Outer Diameter, OD (mm) 219 219 219 406 406 406
Wall Thickness, t (mm)
Outer span (mm)
Inner span (mm)
15.15 15.10 15.29 32.38 32.15 32.36
4000 4000 4000 5820 5820 5820
1480 1480 1480 1480 1480 1480
Through wall circumferential crack angle, 2ΞΈ (degrees) 65.6 93.9 126.4 96.0 126.3 157.8
3.2 Finite element model Fig. 10 shows schematic diagram of the straight pipe having through-wall circumferential crack subjected to four-point bending load. Due to such arrangement, inner span experiences pure bending moment and hence the crack plane. Elastic-plastic finite element analyses of all pipe components having through wall circumferential cracks have been carried out using von Mises yield criterion, Prandtl Reuss 12 | P a g e
flow rule, isotropic hardening law and large strain formulation A 3D finite element model using 8-node solid brick elements is adopted using quarter symmetricity. The finite element mesh used for cracked pipe consists of 35292 nodes and 28401 elements. The stress information is taken at the integral points (Gauss points) at the load step when the applied J-integral is equal to the J-initiation of the material. A spider web type mesh is employed near the crack tip. It is well known that one needs to have very fine mesh near the crack tip to get correct value of stress triaxiality parameter βqβ. However, such fine mesh is difficult to employ in case of cohesive zone analysis due to exorbitant requirement of computational time. Hence, in the present research work, two different models have been used separately for calculating βqβ and cohesive zone analysis. The mesh used for calculating βqβ, we have used a very small hole (100-micron radius) to model the blunting of crack tip during yielding as the present material is highly ductile (of the order of 40%). Such methodology has been also suggested by many authors in the past such as [17, 19]. The aspect ratio of elements is 1 in the direction of crack propagation. Six layers of elements are taken along the thickness. For cohesive zone analysis, a transition mesh is employed. The finite element mesh used for cracked pipe with cohesive elements consists of 56077 nodes and 46290 elements. Crack growth is simulated by placing cohesive elements along the crack plane. Displacement controlled four-point bending load is applied to simulate the experimental conditions.
Fig. 10 Geometry of straight pipe with through wall crack subjected to four-point bending load.
3.3 Computation of βqβ and cohesive parameters for straight pipe specimens The multi-axiality quotient βqβ for all six straight pipes are determined by carrying out elastic-plastic finite element analyses. Finite element computer code WARP3D [9] available in public domain is again used for this purpose. J-integral along the crack front is also calculated by domain integral method for increasing load. Fig. 11 shows variation of triaxiality parameter βqβ as a function of distance from crack tip for straight pipe SPBMTWC8-1 for all the five layers of elements along thickness. Layer 1 to layer 5 corresponding to outer diameter to inner diameter of the pipe in a sequential manner. These values are plotted for a J-integral value of 220 kJ/m2. The minimum value of βqβ is for layer 4 and is equal to 0.31878. This value is shown in Table 7. This procedure is repeated for all the six straight pipes to compute minimum values of βqβ ahead of the crack front. As followed earlier in case of TPBB specimens, cohesive zone analyses are then carried out for all the six straight pipes for various values of βTpβ with an aim to find out best values of βTpβ which compute load-displacement and load-crack growth curves closest to the measured values. Fig. 12 and Fig. 13 show load-displacement and crack-growth curves for all the pipe specimens. For pipe SPBMTWC8-1, after carrying out analyses for various values of βTpβ, it is seen that a value of Tp = 1200MPa produces computed data which have best comparison with the measured values. Table 7 shows the values of βqβ and corresponding βTpβ parameters for all the six straight pipes.
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Table 7 Stress triaxiality 'q' at an applied J integral of ~220 kJ/m2 and optimized cohesive parameters for straight pipes Piping Component
Minimum value of βqβ parameter
SPBMTWC8-1 SPBMTWC8-2 SPBMTWC8-3 SPBMTWC16-1 SPBMTWC16-2 SPBMTWC16-3
0.31878 0.328761 0.351526 0.332117 0.330341 0.349647
Peak stress Tp (MPa) 1200 1154 900 1166 1133 1077
Ξ΄p/2 using Eqn. (4) (mm) 0.033722 0.035066 0.044963 0.034706 0.035716 0.037574
Fig. 11 Triaxiality parameter 'q' v/s distance from crack tip for pipe specimen SPBMTWC8-1 (The minimum value of 'q' is obtained in Layer 4 and is equal to 0.31878. Layer 1 to Layer 5 is from outer diameter to inner diameter side of the pipe).
Fig. 12 Comparison of Load v/s CMOD with experimental results of pipe components.
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Fig. 13 Comparison of Load v/s crack growth with experimental results of pipe components.
4. Variation of cohesive zone parameter βTpβ with crack tip stress triaxiality 4.1 An Empirical Relation By using the values of peak stress βTpβ and stress triaxiality parameter βqβ (minimum value ahead of crack tip) for all the fourteen TPBB specimens and six piping components, a plot between these two parameters is shown Fig. 14. The equation of best fit sigmoidal curve between these two parameters is as shown below [20] . π(πππ) =
π΄1 β π΄2 1+π
(π β π0)/ππ₯
+ π΄2, where 0.24 < q < 0.35
(9)
Table 8 A1, A2, q0 and dx values of Eq. (9) for different sigmoidal curve fits Upper bound Tl (+3Ο) Best fit ΞΌ Lower bound Tu (-3Ο)
A1 (MPa) 1710.74005 1781.9732 1853.20635
A2 (MPa) 908.59687 979.83002 1051.06317
q0 0.31655 0.31655 0.31655
dx 0.01205 0.01205 0.01205
A significant dependency of peak stress βTpβ on the stress triaxiality is observed in Fig. 14. It is also observed that βTpβ increases with the decrease in the value of βqβ. This means that the value of peak stress βTpβ increases with the increase in constraint value. The value of βTpβ saturates close to a value of 1800 MPa for SA333 Gr. 6 material for a value of βqβ less than 0.26. This is interpreted as material is close to plain strain condition for a value of βqβ less than 0.26. Similarly, the value of βTpβ saturates close to value of 1000 MPa for increase in the value of βqβ value more than 0.36, which may be treated as close to plane stress condition.
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Fig. 14 Plot of parameter 'q' v/s peak stress (Tp) for SA333 Grade 6 steel obtained from the cohesive analysis of TPBB and straight pipe specimens.
4.2 A proposed normal variation of peak stress over upper to lower bounds It is also observed that there is a range of variation of peak stress for a constant value of βqβ. Such variation is attributed to the inhomogeneity in the material microstructure over geometry. This may lead to different microstructures at the crack tip of various TPBB specimens during fabrication with respect to a non-homogenous medium. A range of upper and lower bounds of peak stress with respect to mean value is also shown in Fig. 14. The sigmoidal curve fit parameters for upper and lower bounds are given in Table 8. The maximum probability of having the value of βTpβ for any cracked component is the mean value (Β΅) given by equation (9). Similarly, the minimum probability is at the upper and lower bound values. A normal variation of parameter βTpβ is proposed here over maximum and minimum limits. By considering that the upper and lower bounds lie within Β±3Ο limit, the value of Z score is varied between -3 to +3. Z score for a standard normal distribution is given by Eqn. (10). π=
ππ β π
(10)
ππ π
Where the standard deviation is obtained by using the following equation ππ π =
ππ’ β π
(
3
)
(11)
Here Xu is the upper limit of peak stress (Tp) and ΞΌ is the mean value. The peak stress values of intermediate Z scores can be obtained by using following equation ππ = π + ππ πππ
(12)
As an illustration, the typical values of βTpβ for a constant value of q = 0.31878 for seven values of Z scores are shown in Fig. 15. The values of corresponding probabilities are given in Table 9. The probability density function in terms of Z score is given by
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ππ =
1
π 2π
β π2 2
(13)
Table 9 Values of Z score with corresponding probability of peak stress less than Z score value Z score -3 -1.5 -0.67 0 0.68 1.5 3
π·(π»π β€ π»πππ) 0.135% 6.681% 25.143% 50.000% 75.175% 93.319% 99.869%
Fig. 15 Typical peak stress values at different Z scores for 'q' value of 0.31878.
5. Analysis of straight pipes with through-wall circumferential cracks making use of variation of cohesive parameters 5.1
Determination of cohesive zone parameters.
The cohesive parameters are determined above as a function of stress triaxiality with a normal variation over a bound for SA333 Gr. 6 material. In the present section, the usefulness of these parameters is verified by re-analyzing six straight pipes mentioned above. The 3D finite element models described in Section 3 are used for this purpose. The crack growth is simulated by placing cohesive elements along the crack plane. Displacement controlled four-point bending load is applied to simulate the experimental conditions. Each component is analyzed by using cohesive energy (G) as 220 kJ/m2. The analyses are carried out for seven values of peak stress βTpβ corresponding to probability of occurrence. These values are computed using Equation (12) and are shown in Table 10 for respective values of Z-score. These seven values of βTpβ for each pipe are also shown in Fig. 16 for all six pipes as per their stress triaxiality βqβ values.
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Table 10 Computed values of peak stress (Tp) at different Z scores for all pipes Pipe component
SPBMTWC8-1 SPBMTWC 8-2 SPBMTWC 8-3 SPBMTWC 16-1 SPBMTWC 16-2 SPBMTWC 16-3
Z = -3 (Tl) 1272.6 6 1122.2 3 950.31 6 1081.4 5 1102.3 2 956.93 6
Z = -1.5 1308.27 9 1157.85 6 985.933 3 1117.07 2 1137.94 2 992.552 6
Peak stress, Tp (MPa) Z = -0.67 Z=0 Z =+0.68 (ΞΌ) 1327.987 1343.896 1360.042
1379.513
Z = +3 (Tu) 1415.129
1177.563
1193.472
1209.618
1229.089
1264.705
1005.641
1021.55
1037.696
1057.167
1092.783
1136.779
1152.688
1168.834
1188.305
1223.921
1157.65
1173.559
1189.705
1209.176
1244.792
1012.26
1028.169
1044.315
1063.786
1099.402
Z =+1.5
Fig. 16 Z-score intervals taken for different pipes within the limits.
5.2
Results and discussions
The results are shown in Fig. 17 and Fig. 18 along with the experimental results. It may be seen that the experimental values lie reasonably well within the computed results for all straight pipes except for an 8β pipe having 60Β° crack angle denoted by SPBMTWC8-1.
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Fig. 17 Comparison of Load v/s CMOD for simulation results of different Z scores with experimental result for pipes (a) SPBMTWC8-1, (b) SPBMTWC8-2, (c) SPBMTWC8-3, (d) SPBMTWC16-1, (e) SPBMTWC16-2 and (f) SPBMTWC16-3.
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Fig. 18 Comparison of Load v/s crack growth (da) for simulation results of different Z scores with experimental result for pipes (a) SPBMTWC8-1, (b) SPBMTWC8-2, (c) SPBMTWC8-3, (d) SPBMTWC16-1, (e) SPBMTWC16-2 and (f) SPBMTWC16-3.
6. Conclusions In this work, fracture behavior of a nuclear pressure vessel material SA333 Grade 6 steel was studied using cohesive zone model. A step by step procedure was proposed to predict cohesive zone parameter peak stress βTpβ as a function of stress triaxiality βqβ, in addition to its material dependency. The variation of βTpβ and βqβ as well as normal variation within two bounds have been quantified by numerical analyses employing experimental results of TPBB and pipe specimens. Parameter βTpβ increases with decrease in value of βqβ and reaches a constant value asymptotically which may be represented by a sigmoidal curve. For a constant value of βqβ, there may be statistical variation in the 20 | P a g e
value of peak stress βTpβ due to microstructural changes over the geometry. Such uncertainty may be represented by a normal variation within two bounds. It is shown that experimental load-displacement and load-crack growth data of cracked pipe made up of SA333 Grade 6 steel fall within the values calculated numerically assuming normal variation of βTpβ for a given value of βqβ. It is expected that the methodology presented in this work can be extended to other structural steels to find out cohesive parameter βTpβ. Such material parameters will be useful to the designers to carry out safety analysis of piping components of nuclear and other installations. For this purpose, one needs to carry out elastic-plastic analysis of the flawed component to determine the multiaxiality quotient βqβ at the crack tip. The value of peak stress βTpβ can be determined using the correlation suggested in this work. Peak stress βTpβ then can be used to carry out cohesive zone analysis of the flawed component. References [1] W. Brocks, Plasticity and Fracture, Springer, 2017. [2] A. Needleman, Tvergaard V. and J.W. Hutchinson, "Void growth in Plastic solids.," Topics in Fracture and Fatigue, pp. 145-178, 1992. [3] G. Barenblatt, "The mathematical theory of equilibrium cracks in brittle fracture," Advanced Applied Mechanics, vol. 7, pp. 55-129, 1962. [4] D. Dugdale, "Yielding of steel sheets containing slits," Journal of Mechanics and Physics of solids, vol. 8, pp. 100-104, 1960. [5] A. Cornec, Ingo Schieder and Karl-Heinz Schwalbe, "On the practical application of the cohesive model," Engineering fracture mechanics, vol. 70, pp. 1963-1987, 2002. [6] M. Elices, G.V. Guinea, J. Gomez and J. Planas, "The cohesive zone model: advantages, limitations and challenges," Engineering Fracture Mechanics, vol. 69, pp. 137-163, 2002. [7] K.-H. Schwalbe, Ingo Scheider and Alfred Cornec, "Guidelines for applying cohesive models to the damage behaviorr of engineering materials and structures," Springer, Germany, 2013. [8] Y. Shao, Hong-Ping Zhao, Xi-Qiao Feng and Huajian Gao, "Discontinuous crack-bridging model for fracture toughness analysis of nacre," Journal of the Mechanics and Physics of Solids, vol. 60, pp. 1400-1419, 2012. [9] B. Healy, Arne Gullerud, Kyle Koppenhoefer, Arun Roy, Sushovan Roy Chowdhary, Jason Petti, Matt Walters, Barron Bichon, Kristine Cochhran and et al, "Warp3D-Release 17.7.0 User's Guide," 2016. [10] C. Chen, O. Kolednik, J. Heerens and F.D. Fischer, "Three-dimensional modeling of ductile crack growth: Cohesive zone parameters and crack tip triaxiality," Engineering Fracture Mechanics, vol. 72, pp. 2072-2094, 2005. [11] A. banerjee and R. Manivasagam, "Triaxiality dependent cohesive zone model," Engineering Fracture Mechanics, vol. 76, pp. 1761-1770, 2009. [12] C. R. Chen, O. Kolednik, I. Scheider, T. Seigmund, A. Tatschl and F.D. Fischer, "On the determination of the cohesive zone parameters for the modeling of micro-ductile crack growth in thick specimens," International Journal of Fracture, vol. 120, pp. 517-536, 2003.
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[13] I. Scheider and W. Brocks, "Effect of cohesive law and triaxiality dependence of cohesive parameters in ductile tearing," Fracture of Nano and Engineering Materials and Structures, no. Springer, Dordrecht, 2006. [14] A. E11820--15a, "Standard Test Method for Measurement of Fracture Toughness," ASTM International, 2015. [15] Tarafder,S., Sivaprasad, S., Tarafder, M., Prasad, P., Ranganath, V. R. and Das, S., "Specimen size and constraint effects on J-R curves of SA333 Gr.6 Steel, TPB and C(T) geometries," National Metallurgical Laboratory, Jamshedpur, India, 2000. [16] J. Chattopadhyay, T.V. Pavankumar, B.K. Dutta and H.S. Kushwaha, "Experimental and Analytical study of three point bend specimenn and throughwall circumferentially crack straight pipe," International Journal of Pressure Vessels and Piping, vol. 77, pp. 455-471, 2000. [17] T. V. Pavankumar, J. Chattopadhyay, B.K. Dutta and H.S. Kushwaha, "Transferability of specimen J-R curve to straight pipes with throughwall circumferential flaws," International Journal of Pressure Vessels and Piping, vol. 79, pp. 127-134, 2002. [18] J. Chattopadhyay, "Improved J and COD estimation by GE/EPRI method in elastic to fully plastic transition zone," Engineering Fracture Mechanics, vol. 73, pp. 1959-1979, 2006. [19] Kumar, V., German, M. and Shih, C., "Elastic-Plastic and Fully Plastic Analysis of Crack Initiation, Stable Growth, and Instability in Flawed Cylinders," Elastic-Plastic Fracture: Second Symposium, Vols. 1-Inelastic Crack analysis, no. ASTM International, pp. I-306 to I-353, 1983. [20] J. ResenDiz-MuNoZ, M A Corona-Rivera, J L Fernandez-Munoz, M Zapata-Torres, A MarquezHerrera and V M Ovando-Medina, "Mathematical model of Boltzmann's sigmoidal equation applicabble to the set-up of the RF-magnetron co-sputtering in thin films deposition of BaxSr1xTi03," Bull. Mater. Sci., vol. 40, no. 5, pp. 1043-1047, 2017. [21] M. Mahler and Jarir Aktaa, "Approach for J-R curve determination based on sub-size specimens using a triaxiality dependent cohesive model on a (ferritic martensitic) steel," Engineering Fracture Mechanics, vol. 1144, pp. 222-237, 2015. [22] V. Shanmugam, Ravi Penmetsa, Eric Tuegel and Stephen Clay, "Stochastic modeling of delamination growth in unidirectional composite DCB specimens using cohesive zone models," Composite structures, vol. 102, pp. 38-60, 2013. [23] R. M. Siddiqui, Dattatray, N. Jadhav and Nilesh. R. Raykar, "Prediction of crack initiation load in Throughwall circumferentially cracked pipes using 3D cohesive zone model," International journal of materials science and engineering, vol. 3, no. 3, pp. 183-192, 2015. [24] CSIR, "Characterizing numerical SZW evaluation for determining material fracture toughness (Jszw)," Board of Research on Nuclear Sciences (BRNS), Mumbai, Bhopal, 2014. [25] M. K. Sahu, J. Chattopadhyay and B. K. Dutta, "Fracture studies of straight pipes subjected to internal pressure and bending moment," International Journal of Pressure Vessels and piping, vol. 134, pp. 56-71, 2015. [26] T. V. Pavankumar, M. K. Samal, J. Chattopadhyay, B.K. Dutta, H.S. Kushwaha, E. Roos and M. Seidenfuss, "Transferability of fracture parameters from specimens to component level," International Journal of Pressure Vessels and Piping, vol. 82, no. 5, pp. 386-399, 2005.
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HIGHLIGHTS ο· ο· ο· ο·
Prediction of the load-deflection and load-crack growth variation of cracked pipes employing cohesive zone parameters and stress triaxiality. Peak stress cohesive zone parameter is a strong function of stress triaxiality with a sigmoidal variation. A normal statistical variation in the value of peak stress within two bounds for a constant value of multiaxiality quotient is suggested. Present methodology is useful to predict maximum load carrying capacity of a cracked component for safety analysis.
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S0013-7944(19)30859-8 https://doi.org/10.1016/j.engfracmech.2019.106789 EFM 106789
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
10 July 2019 19 November 2019 19 November 2019
Please cite this article as: Teja Vanapalli, V., Dutta, B.K., Chattopadhyay, J., Martin Jose, N., Stress triaxiality based transferability of cohesive zone parameters, Engineering Fracture Mechanics (2019), doi: https://doi.org/ 10.1016/j.engfracmech.2019.106789
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Β© 2019 Published by Elsevier Ltd.
Stress triaxiality based transferability of cohesive zone parameters Viswa Teja Vanapalli1*, B.K. Dutta1**, J. Chattopadhyay1, 2, Nevil Martin Jose2 1Homi
2 Reactor
Bhabha National Institute, Anushaktinagar, Mumbai 400094, India Safety Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India *[email protected], **[email protected]
Abstract Present study is to investigate the dependency of cohesive zone parameters on crack tip stress triaxiality for SA333 Grade 6 steel. An exponential cohesive law is used to simulate ductile fracture behavior of 14 three-point bend specimens made of SA333 Gr. 6 steel. Cohesive parameters are determined by varying peak stress to match experimental results with the computed data. To extend the validity of parameter peak stress as a function of triaxiality, experimental results of six piping components with through-wall circumferential crack made up of SA333 Grade 6 steel are used. A confidence interval band is then plotted to study the transferability of cohesive parameters. The variation in peak stress for a single value of multiaxiality quotient is attributed to the micro structural variation in material during manufacturing near the crack tip, which is statistical in nature. A normal variation of peak stress is assumed for a given value of multiaxiality quotient. To test the accuracy of the cohesive parameters with such normal variation, experimental results of six piping components are again used. Significant match of computed results with the measured values shows transferability of the present material parameters. The methodology presented in this work can be applied to other structural steels to find out cohesive zone parameters. Such material parameters will be useful to the designers to carry out safety analysis of piping components of nuclear and other installations. Keywords: Cohesive zone model; multi-axiality quotient; Transferability of cohesive parameters
1|Page
Nomenclature a0 A1 A2 B BN da dx G h J JSZW n q q0 π Tp Ti Tl Tu W Z
Initial crack size Lower transition value in Boltzmann sigmoidal function [Eq. (9)] Upper transition value in Boltzmann sigmoidal function [Eq. (9)] Thickness of TPBB specimen Thickness of TPBB specimen after side grooving is done Crack growth Transition slope coefficient in Boltzmann sigmoidal function [Eq. (9)] Cohesive energy [Eq. (4)] Triaxiality factor [Eq. (5)] J-integral Initiation toughness at stretch zone width Ramberg-Osgood exponent Multi axiality quotient [Eq. (8)] Inflection point in Boltzmann sigmoidal function [Eq. (9)] Effective traction [Eq. (3)] Peak stress / cohesive strength [Eq. (2)] Typical peak stress value for a value of Z score [Eq. (12)] The value of peak stress at -3Ο limit The value of peak stress at +3Ο limit Width of TPBB specimen Z score of standard normal distribution [Eq. (10)]
Greek symbols Ξ± Ramberg-Osgood coefficient Ξ² Mode mixity parameter [Eq. (1)] Ξ΄ Effective opening displacement [Eq. (1)] Ξ΄n Normal/opening displacement [Eq. (1)] Ξ΄p Effective displacement at peak stress [Eq. (2)] Ξ΄t Tangential/sliding displacement [Eq. (1)] Ο Free energy potential [Eq. (2)] Οsd Standard deviation of the normal variation [Eq. (11)] Οh Hydrostatic stress [Eq. (6)] Οv von Mises equivalent stress [Eq. (7)] Οy Yield strength Οu Ultimate strength ΟZ Probability density function [Eq. (13)] ΞΌ Value of peak stress obtained from sigmoidal curve fit. Abbreviations ASTM American Society of Testing and Materials CMOD Crack Mouth Opening Displacement EPFM Elastic plastic fracture mechanics NB Nominal Bore TPBB Three-Point Bend Bar specimen TSL Traction Separation Law
2|Page
1. Introduction: Various techniques are available for numerical simulation of crack growth in a component used in any power plant. Some of these are node-release-technique [1], Gurson-Tvergaard-Needleman damage model [2], Cohesive-zone-model, etc. The cohesive zone model [3, 4] is a damage mechanics methodology used to simulate the initiation and propagation of cracks in solids using finite element technique. In this model, a narrow band of vanishing thickness lies ahead of crack tip to represent the fracture process zone. The upper and lower surfaces, as shown in Fig. 1(a), are called the cohesive surfaces. Both the surfaces are subjected to cohesive tractions during the loading of the component. The cohesive tractions follow a constitutive law called the Traction Separation Law (TSL), which relates the cohesive traction (T) with the effective opening displacement (Ξ΄). A typical TSL is shown in Fig. 1(b). There are various types of TSL available in the literature developed primarily for different engineering materials such as trapezoidal, exponential and polynomial TSLs [5, 6, 7, 8]. Three parameters which are generally used to define a TSL are cohesive energy, peak stress and the effective opening displacement. It can be shown that any two out of these three parameters are independent. The philosophy behind TSL is that crack growth will occur when the separation distance between the two adjoining points on the opposite cohesive surfaces reaches a critical value (Ξ΄c). The amount of energy required for separation of surfaces is known as cohesive energy. The various cohesive zone models vary according to the shape of traction separation laws and corresponding governing equations.
Fig. 1 (a) Cohesive zone model (b) Exponential Traction Separation Law (TSL).
In the present work, an exponential TSL is used to simulate the crack growth in Three-Point Bend Bar (TPBB) specimens made up of SA333 Gr.6 material. Finite element code WARP3D available in open literature is used for this purpose [9]. An exponential TSL combining normal and shear tractions into a single effective traction (T) is available in WARP3D. The model thus has a single traction separation curve and a single energy of separation (G). The shape of the exponential law is dependent on userspecified values of peak stress (Tp) and displacement at peak stress during the analysis (Ξ΄p). The effective opening displacement is calculated using the following equation (1). πΏ = π½2πΏ2π‘ + πΏ2π (1) Here Ξ΄t and Ξ΄n are tangential and normal displacements and Ξ² is a mode-mixity parameter which assigns different weights to tangential and normal displacements. For a mode I fracture problem, the value of Ξ² is taken as zero. 3|Page
The constitutive relation for the cohesive surface is derived from an exponential form of a free energy potential Ξ¨ [9] as shown below.
[ (
πΏ
) (
πΏ
)]
(2)
πΉ = ππ₯π(1)πππΏπ 1 β 1 + πΏπ exp β πΏπ The relationship T- Ξ΄ follows from π=
βπΉ βπΏ
πΏ
(
πΏ
)
= ππ₯π(1)πππΏπππ₯π β πΏπ
(3)
The work of separation per unit area of the cohesive surface is given for an exponential TSL as β
πΊ = β«0 πππΏ = exp (1)πππΏπ
(4)
The infinite value of distance of separation to calculate cohesive energy has been taken [9] to address exponential decrease in the value of traction with the distance of separation. However, in actual practice, while computing the value of G, a finite value of Ξ΄ is taken as a cut off distance. In the present case, the cut off distance is so taken that the current value of traction is less than 5% of peak value of traction. As, T decreases with Ξ΄ in an exponential manner, such cut off distance comes out to be 5.74 times of the Ξ΄p. The crack propagation in a component (pipe, elbow, t-joint, etc.) depends on the material J-R curve, which in turn depends on the stress triaxiality. Stress triaxiality changes with the crack propagation due to loss of constraint. Hence, to carry out crack propagation analysis by EPFM, one should consider change in J-R curve with crack propagation. This is generally not done. It is anticipated that the cohesive parameters are not only material constants but vary with the state of stress triaxiality at the crack tip. Previous works from authors like C. Chen et al [10] and A. Banerjee [11] investigated the interrelations between cohesive parameters and crack tip triaxiality. It is observed that with decrease of cohesive strength, triaxiality decreases while cohesive energy increases with triaxiality. While Chen et al. [10, 12] studied the variation of parameters along crack front and crack extension for a Compact Tension specimen made of 20MnMoNi55 & St37 materials using a polynomial TSL, A. Banerjee et al [11] incorporated the triaxiality parameter directly into the equation of TSL for aluminum and steel materials. I. Schieder et al. [13] studied on the transferability of cohesive parameters based on shape of TSL for different materials and constraint conditions. In many cases, constant cohesive parameters showed dependence as long as cracked models are investigated. In present work, the βJ-hβ approach is used to calculate the βqβ parameter. Where βhβ is the triaxiality factor defined as ratio between hydrostatic stress (Οh) and the von Mises equivalent stress (Οv). πβ
(5)
β = ππ£ where πβ =
(π1 + π2 + π3) 3
(6) ππ£ = { (π1 β π2)2 + (π2 β π3)2 + (π3 β π1)2}/ 2
(7)
where, Ο1, Ο2, Ο3, are the principal stresses. The multi-axiality quotient βqβ is given by, π = 1/( 3β) (8) The βqβ parameter varies with the increase in applied load. In order to compare βqβ for different cracked specimens, the variation of q is plotted with the increase in applied J integral. The applied J-integral value at any displacement increment may be calculated using the closed form formula given in ASTM E 1820 [14] or numerically using domain integral technique [9]. The βqβ is calculated in the adjacent elements using a distance from the initial crack tip. The stress information is taken at the integral points (Gauss points) at the load step when the applied J-integral is equal to the J-initiation of the material. The objective of present work is to ascertain the transferability of cohesive prameters as a function of crack tip stress triaxiality made of SA333 Grade 6 steel. In this paper, section 2 & 3 deal with procedure 4|Page
adopted to determine cohesive parameters and triaxiality parameters of three-point bend (TPBB) specimens and pipe components respectively using an exponential TSL; Section 4 describes the development of a correlation between cohesive parameter peak stress with stress triaxiality. Results of a normal statistical variation in peak stress for constant multiaxiality quotient have been presented in Section 5 along with concluding remarks in Section 6. 2. Fracture simulation analysis of TPBB specimens 2.1 Material properties & Experimental data The experimental results of TPBB specimens made up of SA333 Grade 6 steel used for the present analyses are taken from the reported results [15]. This material is widely used in Indian nuclear reactors for fabricating straight pipes & elbows. The material properties of SA333 Grade 6 steel are given in Table 1. J-initiation value for this material is found to be close to 220 kJ/m2 determined using stretch zone width at crack initiation (JSZW) [15, 16, 17, 18]. The experimental results of TPBB specimens having 8 mm, 12 mm and 25 mm thicknesses, have been used in the present analysis. The 8mm thick specimens are prepared from 200 mm NB (8-inch diameter) pipes, while 12mm and 25mm thick specimens are prepared from 400 mm NB (16-inch diameter) pipes. Table 1 Mechanical properties of SA333 Grade-6 steel [15] 200 mm NB pipe
400 mm NB pipe
Youngβs Modulus, E
203000 MPa
203000 MPa
Yield Strength, Οy
285 MPa
307 MPa
Ultimate Strength, Οu
420 MPa
463 MPa
Ramberg Osgood parameters Ξ± n
10.759 4.301
10.249 4.23
2.2 Finite element model of TPBB specimen Fig. 2 shows the schematic diagram of a three-point bend bar (TPBB) specimen with side grooves. In the present analysis, 3D finite element model of a quarter specimen is used making use of symmetric boundary conditions. The finite element (FE) model of the specimen is prepared using 8-noded solid brick elements while cohesive zone is modeled using 8-noded cohesive interface elements. The true stress-strain curve as reported in [16, 17] is used in the analysis. The finite element mesh for TPBB specimen consists of 1719 nodes and 1180 elements. Aspect ratio of elements present near the crack tip is 1 in the direction of crack propagation. Cohesive elements are located at the crack plane from the initial crack tip up to a distance of 4 mm in the direction of crack growth as shown in Fig. 3(c). The cohesive elements initially have zero thickness. The size of the cohesive elements in the direction of crack propagation is taken as 0.2 mm which is the least crack growth measured. The details of boundary conditions and loading are shown in Fig. 3(a) and Fig. 3(b).
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Fig. 2 Geometric details of a TPBB specimen with side grooves
Fig. 3 Quarter 3D FE mesh with symmetry boundary conditions in (a) z-direction, (b) x & y-direction and loading nodes in x-direction and (c) enlarged view showing cohesive elements near the crack tip.
2.3 Computed cohesive parameters for TPBB specimens The cohesive parameters for the TPBB specimens have been found by comparing the simulated results with the experimental results. During the analysis, the value of cohesive energy (G) is taken as the value of initiation fracture toughness for SA333 Grade 6 steel, which is 220 kJ/m2. It has been pointed out in reference [7] that setting βGβ with the initiation fracture toughness does not necessarily cause substantial errors. This is due to the fact that the total energy absorbed in ductile failure is much greater than initiation fracture toughness so that an inaccuracy in cohesive energy does not substantially affect the result of the simulation of a crack growth resistance curve. Thus, the pragmatic view of setting cohesive zone energy equal to initiation fracture toughness is justified. In the present work, the initiation fracture toughness (JSZW) is determined using a stretch zone width methodology (SZW) [18]. It may be mentioned here that initiation fracture toughness determined by SZW method is independent of stress triaxiality. 6|Page
To match computed results with the experimental results, sensitivity analysis of peak stress (Tp) on the load-CMOD curve is done in the present analysis. An algorithm to determine peak stress (Tp) is given in Fig. 4. This helped to obtain suitable cohesive parameter Tp for a given specimen. The cohesive element gets deleted sequentially when the separation distance between the two adjacent points on the opposite cohesive surfaces reaches a critical value (Ξ΄c) and crack growth is calculated by multiplying the size of cohesive element with the number of elements βkilledβ at that particular displacement increment. The word βkilledβ refers to those cohesive elements which are no longer able to transmit stresses during further displacement increments. The load v/s CMOD plot can be obtained from the output file. The J-R curves are calculated using the closed form formula given in ASTME1820 [14]. Table 2 shows geometrical details of a typical TPBB specimen used in the analysis having a0/W ratio of 0.305.
Fig. 4 Algorithm to compute the optimized cohesive parameters and 'q' parameter for TPBB specimens.
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Fig. 5 Comparison of (a) Load v/s CMOD, (b) Load v/s crack growth and (c) J-Resistance curve with experimental result for specimen T082C
Fig. 6 Comparison of (a) Load v/s CMOD, (b) Load v/s crack growth and (c) J-Resistance curves with experimental results for TPBB specimens having thickness of 8 mm.
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Fig. 7 Comparison of (a) Load v/s CMOD, (b) Load v/s crack growth and (c) J-Resistance curves with experimental results for TPBB specimens having 12 mm thickness
Fig. 8 Comparison of (a) Load v/s CMOD, (b) Load v/s crack growth and (c) J-Resistance curves with experimental results for TPBB specimens having 25 mm thickness
9|Page
Table 2 Geometrical details & cohesive parameters of a typical TPBB specimen Dimensions Thickness (B) 8.14 mm Thickness after side grooving (BN) 6.76 mm Width (W) 25.08 mm a0/W ratio 0.305 Cohesive parameters determined through present analysis Cohesive Energy (G) 220 KJ/m2 Peak Stress (T) 1300 MPa Table 3 Analyses of TPBB specimens: geometrical details and cohesive parameters Specimen T082B T082C T083A T083B T085B T122A T123B T123A T125C T125B T252B T252A T253A T253B
B (mm) 8.18 8.14 8.12 7.90 8.06 12.68 12.58 12.72 12.64 12.48 25.32 25.18 25.24 25.20
BN (mm) 6.98 6.76 6.7 6.46 7.00 10.14 10.12 10.12 10.14 10.10 20.80 20.50 20.52 20.46
a0/W 0.244 0.305 0.347 0.356 0.513 0.275 0.354 0.362 0.430 0.505 0.189 0.200 0.342 0.354
W (mm) 25.10 25.08 25.06 25.06 25.06 25.08 25.08 25.08 25.08 25.02 50.18 50.14 50.20 50.28
Peak stress, Tp (MPa) 1275 1300 1300 1275 1275 1700 1600 1625 1825 1700 1700 1750 1825 1775
Ξ΄p/2 using Eqn. (4) (mm) 0.031738 0.031128 0.031128 0.031738 0.031738 0.023803 0.025291 0.024902 0.022173 0.023803 0.023803 0.023123 0.022173 0.022798
Finite element analyses are conducted on fourteen TPBB specimens using the cohesive zone elements along crack front. Fig. 5 shows computed (a) Load v/s CMOD, (b) Load v/s crack growth (da)and (c) J-Resistance curve in comparison to experimental results for a TPBB specimen T082C for which dimensions are shown in Table 2. Such results have been obtained by varying the value peak stress βTpβ. The results shown in Fig. 5 are for the most optimal value of βTpβ for which best comparison between computed and experimental results have been obtained. Fig. 6, Fig. 7 and Fig. 8 show the best fitted results of TPBB specimens with the experimental results having thickness of 8 mm, 12 mm and 25 mm respectively. The values of best fitted cohesive parameters for all the specimens along with geometric details are tabulated in Table 3. Table 4. Geometrical details & cohesive parameters of a typical TPBB specimen Dimensions Thickness (B) 8.14 mm Thickness after side grooving (BN) 6.76 mm Width (W) 25.08 mm a0/W ratio 0.305 Cohesive parameters determined through present analysis Cohesive Energy (G) 220 KJ/m2 Peak Stress (T) 1300 MPa
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2.4 Determination of βqβ parameter of TPBB specimens by elastic-plastic FEA Standard procedure, cited in the literature, is to first calculate multi-axiality quotient βqβ at the crack tip by carrying out elastic-plastic analysis of the component with initial crack. Based on the calculated value of βqβ, a J-R curve is selected and assumed to be applicable throughout the crack propagation analysis. Similar approach has been adopted in the present work. In place of J-R curve, cohesive parameter βTpβ is used. The value of βTpβ is selected for a particular value of multi-axiality quotient βqβ, which is calculated by carrying out an elastic-plastic analysis of the cracked component. The value of βqβ has been taken to be constant during the crack propagation as is done in case of J-R curve. The stress triaxiality parameters βqβ for TPBB specimens have been determined for initial crack length. For this purpose, elastic-plastic finite element analyses of all fourteen TPBB specimens have been carried out using von Mises yield criterion, Prandtl Reuss flow rule, isotropic hardening law and large strain formulation. True stress-strain data mentioned in section 2.2 is employed yet again. The finite element mesh used in elastic-plastic analysis of TPBB specimen consists of 1656 nodes and 1140 elements. The stress information is taken at the integral points (Gauss points) at the load step when the applied J-integral is equal to the J-initiation of the material. An algorithm to find out βqβ parameter is also shown in Fig. 4. It is observed that in all the cases, the minimum value of βqβ, and hence the maximum constraint, has been obtained at the mid-section of the model across the thickness. The βqβ parameter for all specimens are calculated when the applied J-integral in the specimen reaches a value of 220 kJ/m2. A plot of βqβ v/s distance from the crack tip for TPBB specimen at mid-section is shown in Fig. 9 for a J-integral value close to 220 KJ/m2. The minimum values of βqβ ahead of the crack tip from such plots for all the specimens are shown in Table 5.
Fig. 9 'q' v/s distance from crack tip for T08_2C. Minimum value of 'q' is obtained in Layer 1 is equal to 0.33388 at J ~221.9472 kJ/m2.
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Table 5 The minimum value of 'q' parameter ahead of the crack tip at an applied J~220 kJ/m2 determined for the TPBB specimens Minimum value of βqβ parameter 0.32697 0.33388 0.31896 0.32574 0.30957 0.30772 0.30764 0.30341 0.28148 0.26774 0.27268 0.26977 0.24349 0.23761
Specimen T082B T082C T083A T083B T085B T122A T123B T123A T125C T125B T252B T252A T253A T253B
3. Fracture simulation analysis of Straight pipes with through-wall flaws Using the results of peak stress βTpβ and minimum value of βqβ parameters for TPBB specimens, it is possible to generate correlation between them for βqβ values ranging from 0.237 to 0.333, which is the range shown in Table 5. But the constraint level in real life components is generally lower. Therefore, it is necessary to include the βqβ parameters in the database from large components to increase the range of validity of such correlation. In the present work, experimental data for straight pipes with through wall cracks are used for this purpose. 3.1 Material properties and Experimental data The experimental results of straight pipes used for such purpose are taken from the experiments conducted under the project, named, βPiping component integrity test programβ, carried out by Chattopadhyay et al [16]. Under this project, fracture experiments are carried out on piping components made up of SA333 Grade 6 steel. Table 6 shows geometric details of six pipes selected here for the determination of cohesive parameters. The straight pipes were subjected to four-point bending load. Table 6 Geometrical Details of the Pipes [16] Pipe component
SPBMTWC8-1 SPBMTWC8-2 SPBMTWC8-3 SPBMTWC16-1 SPBMTWC16-2 SPBMTWC16-3
Outer Diameter, OD (mm) 219 219 219 406 406 406
Wall Thickness, t (mm)
Outer span (mm)
Inner span (mm)
15.15 15.10 15.29 32.38 32.15 32.36
4000 4000 4000 5820 5820 5820
1480 1480 1480 1480 1480 1480
Through wall circumferential crack angle, 2ΞΈ (degrees) 65.6 93.9 126.4 96.0 126.3 157.8
3.2 Finite element model Fig. 10 shows schematic diagram of the straight pipe having through-wall circumferential crack subjected to four-point bending load. Due to such arrangement, inner span experiences pure bending moment and hence the crack plane. Elastic-plastic finite element analyses of all pipe components having through wall circumferential cracks have been carried out using von Mises yield criterion, Prandtl Reuss 12 | P a g e
flow rule, isotropic hardening law and large strain formulation A 3D finite element model using 8-node solid brick elements is adopted using quarter symmetricity. The finite element mesh used for cracked pipe consists of 35292 nodes and 28401 elements. The stress information is taken at the integral points (Gauss points) at the load step when the applied J-integral is equal to the J-initiation of the material. A spider web type mesh is employed near the crack tip. It is well known that one needs to have very fine mesh near the crack tip to get correct value of stress triaxiality parameter βqβ. However, such fine mesh is difficult to employ in case of cohesive zone analysis due to exorbitant requirement of computational time. Hence, in the present research work, two different models have been used separately for calculating βqβ and cohesive zone analysis. The mesh used for calculating βqβ, we have used a very small hole (100-micron radius) to model the blunting of crack tip during yielding as the present material is highly ductile (of the order of 40%). Such methodology has been also suggested by many authors in the past such as [17, 19]. The aspect ratio of elements is 1 in the direction of crack propagation. Six layers of elements are taken along the thickness. For cohesive zone analysis, a transition mesh is employed. The finite element mesh used for cracked pipe with cohesive elements consists of 56077 nodes and 46290 elements. Crack growth is simulated by placing cohesive elements along the crack plane. Displacement controlled four-point bending load is applied to simulate the experimental conditions.
Fig. 10 Geometry of straight pipe with through wall crack subjected to four-point bending load.
3.3 Computation of βqβ and cohesive parameters for straight pipe specimens The multi-axiality quotient βqβ for all six straight pipes are determined by carrying out elastic-plastic finite element analyses. Finite element computer code WARP3D [9] available in public domain is again used for this purpose. J-integral along the crack front is also calculated by domain integral method for increasing load. Fig. 11 shows variation of triaxiality parameter βqβ as a function of distance from crack tip for straight pipe SPBMTWC8-1 for all the five layers of elements along thickness. Layer 1 to layer 5 corresponding to outer diameter to inner diameter of the pipe in a sequential manner. These values are plotted for a J-integral value of 220 kJ/m2. The minimum value of βqβ is for layer 4 and is equal to 0.31878. This value is shown in Table 7. This procedure is repeated for all the six straight pipes to compute minimum values of βqβ ahead of the crack front. As followed earlier in case of TPBB specimens, cohesive zone analyses are then carried out for all the six straight pipes for various values of βTpβ with an aim to find out best values of βTpβ which compute load-displacement and load-crack growth curves closest to the measured values. Fig. 12 and Fig. 13 show load-displacement and crack-growth curves for all the pipe specimens. For pipe SPBMTWC8-1, after carrying out analyses for various values of βTpβ, it is seen that a value of Tp = 1200MPa produces computed data which have best comparison with the measured values. Table 7 shows the values of βqβ and corresponding βTpβ parameters for all the six straight pipes.
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Table 7 Stress triaxiality 'q' at an applied J integral of ~220 kJ/m2 and optimized cohesive parameters for straight pipes Piping Component
Minimum value of βqβ parameter
SPBMTWC8-1 SPBMTWC8-2 SPBMTWC8-3 SPBMTWC16-1 SPBMTWC16-2 SPBMTWC16-3
0.31878 0.328761 0.351526 0.332117 0.330341 0.349647
Peak stress Tp (MPa) 1200 1154 900 1166 1133 1077
Ξ΄p/2 using Eqn. (4) (mm) 0.033722 0.035066 0.044963 0.034706 0.035716 0.037574
Fig. 11 Triaxiality parameter 'q' v/s distance from crack tip for pipe specimen SPBMTWC8-1 (The minimum value of 'q' is obtained in Layer 4 and is equal to 0.31878. Layer 1 to Layer 5 is from outer diameter to inner diameter side of the pipe).
Fig. 12 Comparison of Load v/s CMOD with experimental results of pipe components.
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Fig. 13 Comparison of Load v/s crack growth with experimental results of pipe components.
4. Variation of cohesive zone parameter βTpβ with crack tip stress triaxiality 4.1 An Empirical Relation By using the values of peak stress βTpβ and stress triaxiality parameter βqβ (minimum value ahead of crack tip) for all the fourteen TPBB specimens and six piping components, a plot between these two parameters is shown Fig. 14. The equation of best fit sigmoidal curve between these two parameters is as shown below [20] . π(πππ) =
π΄1 β π΄2 1+π
(π β π0)/ππ₯
+ π΄2, where 0.24 < q < 0.35
(9)
Table 8 A1, A2, q0 and dx values of Eq. (9) for different sigmoidal curve fits Upper bound Tl (+3Ο) Best fit ΞΌ Lower bound Tu (-3Ο)
A1 (MPa) 1710.74005 1781.9732 1853.20635
A2 (MPa) 908.59687 979.83002 1051.06317
q0 0.31655 0.31655 0.31655
dx 0.01205 0.01205 0.01205
A significant dependency of peak stress βTpβ on the stress triaxiality is observed in Fig. 14. It is also observed that βTpβ increases with the decrease in the value of βqβ. This means that the value of peak stress βTpβ increases with the increase in constraint value. The value of βTpβ saturates close to a value of 1800 MPa for SA333 Gr. 6 material for a value of βqβ less than 0.26. This is interpreted as material is close to plain strain condition for a value of βqβ less than 0.26. Similarly, the value of βTpβ saturates close to value of 1000 MPa for increase in the value of βqβ value more than 0.36, which may be treated as close to plane stress condition.
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Fig. 14 Plot of parameter 'q' v/s peak stress (Tp) for SA333 Grade 6 steel obtained from the cohesive analysis of TPBB and straight pipe specimens.
4.2 A proposed normal variation of peak stress over upper to lower bounds It is also observed that there is a range of variation of peak stress for a constant value of βqβ. Such variation is attributed to the inhomogeneity in the material microstructure over geometry. This may lead to different microstructures at the crack tip of various TPBB specimens during fabrication with respect to a non-homogenous medium. A range of upper and lower bounds of peak stress with respect to mean value is also shown in Fig. 14. The sigmoidal curve fit parameters for upper and lower bounds are given in Table 8. The maximum probability of having the value of βTpβ for any cracked component is the mean value (Β΅) given by equation (9). Similarly, the minimum probability is at the upper and lower bound values. A normal variation of parameter βTpβ is proposed here over maximum and minimum limits. By considering that the upper and lower bounds lie within Β±3Ο limit, the value of Z score is varied between -3 to +3. Z score for a standard normal distribution is given by Eqn. (10). π=
ππ β π
(10)
ππ π
Where the standard deviation is obtained by using the following equation ππ π =
ππ’ β π
(
3
)
(11)
Here Xu is the upper limit of peak stress (Tp) and ΞΌ is the mean value. The peak stress values of intermediate Z scores can be obtained by using following equation ππ = π + ππ πππ
(12)
As an illustration, the typical values of βTpβ for a constant value of q = 0.31878 for seven values of Z scores are shown in Fig. 15. The values of corresponding probabilities are given in Table 9. The probability density function in terms of Z score is given by
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ππ =
1
π 2π
β π2 2
(13)
Table 9 Values of Z score with corresponding probability of peak stress less than Z score value Z score -3 -1.5 -0.67 0 0.68 1.5 3
π·(π»π β€ π»πππ) 0.135% 6.681% 25.143% 50.000% 75.175% 93.319% 99.869%
Fig. 15 Typical peak stress values at different Z scores for 'q' value of 0.31878.
5. Analysis of straight pipes with through-wall circumferential cracks making use of variation of cohesive parameters 5.1
Determination of cohesive zone parameters.
The cohesive parameters are determined above as a function of stress triaxiality with a normal variation over a bound for SA333 Gr. 6 material. In the present section, the usefulness of these parameters is verified by re-analyzing six straight pipes mentioned above. The 3D finite element models described in Section 3 are used for this purpose. The crack growth is simulated by placing cohesive elements along the crack plane. Displacement controlled four-point bending load is applied to simulate the experimental conditions. Each component is analyzed by using cohesive energy (G) as 220 kJ/m2. The analyses are carried out for seven values of peak stress βTpβ corresponding to probability of occurrence. These values are computed using Equation (12) and are shown in Table 10 for respective values of Z-score. These seven values of βTpβ for each pipe are also shown in Fig. 16 for all six pipes as per their stress triaxiality βqβ values.
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Table 10 Computed values of peak stress (Tp) at different Z scores for all pipes Pipe component
SPBMTWC8-1 SPBMTWC 8-2 SPBMTWC 8-3 SPBMTWC 16-1 SPBMTWC 16-2 SPBMTWC 16-3
Z = -3 (Tl) 1272.6 6 1122.2 3 950.31 6 1081.4 5 1102.3 2 956.93 6
Z = -1.5 1308.27 9 1157.85 6 985.933 3 1117.07 2 1137.94 2 992.552 6
Peak stress, Tp (MPa) Z = -0.67 Z=0 Z =+0.68 (ΞΌ) 1327.987 1343.896 1360.042
1379.513
Z = +3 (Tu) 1415.129
1177.563
1193.472
1209.618
1229.089
1264.705
1005.641
1021.55
1037.696
1057.167
1092.783
1136.779
1152.688
1168.834
1188.305
1223.921
1157.65
1173.559
1189.705
1209.176
1244.792
1012.26
1028.169
1044.315
1063.786
1099.402
Z =+1.5
Fig. 16 Z-score intervals taken for different pipes within the limits.
5.2
Results and discussions
The results are shown in Fig. 17 and Fig. 18 along with the experimental results. It may be seen that the experimental values lie reasonably well within the computed results for all straight pipes except for an 8β pipe having 60Β° crack angle denoted by SPBMTWC8-1.
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Fig. 17 Comparison of Load v/s CMOD for simulation results of different Z scores with experimental result for pipes (a) SPBMTWC8-1, (b) SPBMTWC8-2, (c) SPBMTWC8-3, (d) SPBMTWC16-1, (e) SPBMTWC16-2 and (f) SPBMTWC16-3.
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Fig. 18 Comparison of Load v/s crack growth (da) for simulation results of different Z scores with experimental result for pipes (a) SPBMTWC8-1, (b) SPBMTWC8-2, (c) SPBMTWC8-3, (d) SPBMTWC16-1, (e) SPBMTWC16-2 and (f) SPBMTWC16-3.
6. Conclusions In this work, fracture behavior of a nuclear pressure vessel material SA333 Grade 6 steel was studied using cohesive zone model. A step by step procedure was proposed to predict cohesive zone parameter peak stress βTpβ as a function of stress triaxiality βqβ, in addition to its material dependency. The variation of βTpβ and βqβ as well as normal variation within two bounds have been quantified by numerical analyses employing experimental results of TPBB and pipe specimens. Parameter βTpβ increases with decrease in value of βqβ and reaches a constant value asymptotically which may be represented by a sigmoidal curve. For a constant value of βqβ, there may be statistical variation in the 20 | P a g e
value of peak stress βTpβ due to microstructural changes over the geometry. Such uncertainty may be represented by a normal variation within two bounds. It is shown that experimental load-displacement and load-crack growth data of cracked pipe made up of SA333 Grade 6 steel fall within the values calculated numerically assuming normal variation of βTpβ for a given value of βqβ. It is expected that the methodology presented in this work can be extended to other structural steels to find out cohesive parameter βTpβ. Such material parameters will be useful to the designers to carry out safety analysis of piping components of nuclear and other installations. For this purpose, one needs to carry out elastic-plastic analysis of the flawed component to determine the multiaxiality quotient βqβ at the crack tip. The value of peak stress βTpβ can be determined using the correlation suggested in this work. Peak stress βTpβ then can be used to carry out cohesive zone analysis of the flawed component. References [1] W. Brocks, Plasticity and Fracture, Springer, 2017. [2] A. Needleman, Tvergaard V. and J.W. Hutchinson, "Void growth in Plastic solids.," Topics in Fracture and Fatigue, pp. 145-178, 1992. [3] G. Barenblatt, "The mathematical theory of equilibrium cracks in brittle fracture," Advanced Applied Mechanics, vol. 7, pp. 55-129, 1962. [4] D. Dugdale, "Yielding of steel sheets containing slits," Journal of Mechanics and Physics of solids, vol. 8, pp. 100-104, 1960. [5] A. Cornec, Ingo Schieder and Karl-Heinz Schwalbe, "On the practical application of the cohesive model," Engineering fracture mechanics, vol. 70, pp. 1963-1987, 2002. [6] M. Elices, G.V. Guinea, J. Gomez and J. Planas, "The cohesive zone model: advantages, limitations and challenges," Engineering Fracture Mechanics, vol. 69, pp. 137-163, 2002. [7] K.-H. Schwalbe, Ingo Scheider and Alfred Cornec, "Guidelines for applying cohesive models to the damage behaviorr of engineering materials and structures," Springer, Germany, 2013. [8] Y. Shao, Hong-Ping Zhao, Xi-Qiao Feng and Huajian Gao, "Discontinuous crack-bridging model for fracture toughness analysis of nacre," Journal of the Mechanics and Physics of Solids, vol. 60, pp. 1400-1419, 2012. [9] B. Healy, Arne Gullerud, Kyle Koppenhoefer, Arun Roy, Sushovan Roy Chowdhary, Jason Petti, Matt Walters, Barron Bichon, Kristine Cochhran and et al, "Warp3D-Release 17.7.0 User's Guide," 2016. [10] C. Chen, O. Kolednik, J. Heerens and F.D. Fischer, "Three-dimensional modeling of ductile crack growth: Cohesive zone parameters and crack tip triaxiality," Engineering Fracture Mechanics, vol. 72, pp. 2072-2094, 2005. [11] A. banerjee and R. Manivasagam, "Triaxiality dependent cohesive zone model," Engineering Fracture Mechanics, vol. 76, pp. 1761-1770, 2009. [12] C. R. Chen, O. Kolednik, I. Scheider, T. Seigmund, A. Tatschl and F.D. Fischer, "On the determination of the cohesive zone parameters for the modeling of micro-ductile crack growth in thick specimens," International Journal of Fracture, vol. 120, pp. 517-536, 2003.
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[13] I. Scheider and W. Brocks, "Effect of cohesive law and triaxiality dependence of cohesive parameters in ductile tearing," Fracture of Nano and Engineering Materials and Structures, no. Springer, Dordrecht, 2006. [14] A. E11820--15a, "Standard Test Method for Measurement of Fracture Toughness," ASTM International, 2015. [15] Tarafder,S., Sivaprasad, S., Tarafder, M., Prasad, P., Ranganath, V. R. and Das, S., "Specimen size and constraint effects on J-R curves of SA333 Gr.6 Steel, TPB and C(T) geometries," National Metallurgical Laboratory, Jamshedpur, India, 2000. [16] J. Chattopadhyay, T.V. Pavankumar, B.K. Dutta and H.S. Kushwaha, "Experimental and Analytical study of three point bend specimenn and throughwall circumferentially crack straight pipe," International Journal of Pressure Vessels and Piping, vol. 77, pp. 455-471, 2000. [17] T. V. Pavankumar, J. Chattopadhyay, B.K. Dutta and H.S. Kushwaha, "Transferability of specimen J-R curve to straight pipes with throughwall circumferential flaws," International Journal of Pressure Vessels and Piping, vol. 79, pp. 127-134, 2002. [18] J. Chattopadhyay, "Improved J and COD estimation by GE/EPRI method in elastic to fully plastic transition zone," Engineering Fracture Mechanics, vol. 73, pp. 1959-1979, 2006. [19] Kumar, V., German, M. and Shih, C., "Elastic-Plastic and Fully Plastic Analysis of Crack Initiation, Stable Growth, and Instability in Flawed Cylinders," Elastic-Plastic Fracture: Second Symposium, Vols. 1-Inelastic Crack analysis, no. ASTM International, pp. I-306 to I-353, 1983. [20] J. ResenDiz-MuNoZ, M A Corona-Rivera, J L Fernandez-Munoz, M Zapata-Torres, A MarquezHerrera and V M Ovando-Medina, "Mathematical model of Boltzmann's sigmoidal equation applicabble to the set-up of the RF-magnetron co-sputtering in thin films deposition of BaxSr1xTi03," Bull. Mater. Sci., vol. 40, no. 5, pp. 1043-1047, 2017. [21] M. Mahler and Jarir Aktaa, "Approach for J-R curve determination based on sub-size specimens using a triaxiality dependent cohesive model on a (ferritic martensitic) steel," Engineering Fracture Mechanics, vol. 1144, pp. 222-237, 2015. [22] V. Shanmugam, Ravi Penmetsa, Eric Tuegel and Stephen Clay, "Stochastic modeling of delamination growth in unidirectional composite DCB specimens using cohesive zone models," Composite structures, vol. 102, pp. 38-60, 2013. [23] R. M. Siddiqui, Dattatray, N. Jadhav and Nilesh. R. Raykar, "Prediction of crack initiation load in Throughwall circumferentially cracked pipes using 3D cohesive zone model," International journal of materials science and engineering, vol. 3, no. 3, pp. 183-192, 2015. [24] CSIR, "Characterizing numerical SZW evaluation for determining material fracture toughness (Jszw)," Board of Research on Nuclear Sciences (BRNS), Mumbai, Bhopal, 2014. [25] M. K. Sahu, J. Chattopadhyay and B. K. Dutta, "Fracture studies of straight pipes subjected to internal pressure and bending moment," International Journal of Pressure Vessels and piping, vol. 134, pp. 56-71, 2015. [26] T. V. Pavankumar, M. K. Samal, J. Chattopadhyay, B.K. Dutta, H.S. Kushwaha, E. Roos and M. Seidenfuss, "Transferability of fracture parameters from specimens to component level," International Journal of Pressure Vessels and Piping, vol. 82, no. 5, pp. 386-399, 2005.
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HIGHLIGHTS ο· ο· ο· ο·
Prediction of the load-deflection and load-crack growth variation of cracked pipes employing cohesive zone parameters and stress triaxiality. Peak stress cohesive zone parameter is a strong function of stress triaxiality with a sigmoidal variation. A normal statistical variation in the value of peak stress within two bounds for a constant value of multiaxiality quotient is suggested. Present methodology is useful to predict maximum load carrying capacity of a cracked component for safety analysis.
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