Sandia Fracture Challenge: blind prediction and full calibration to enhance fracture predictability

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1 Sandia Fracture Challenge: blind prediction and full calibration to enhance fracture predictability The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Pack, Keunhwan, Meng Luo, and Tomasz Wierzbicki. Sandia Fracture Challenge: Blind Prediction and Full Calibration to Enhance Fracture Predictability. International Journal of Fracture 186, no. 1 2 (January 10, 2014): pp Springer Netherlands Version Author's final manuscript Accessed Tue Apr 16 22:55:30 EDT 2019 Citable Link Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms

2 Sandia Fracture Challenge Blind Prediction and Full Calibration to Enhance the Fracture Predictability Keunhwan Pack*, Meng Luo, and Tomasz Wierzbicki Impact and Crashworthiness Lab, Massachusetts Institute of Technology, Cambridge, MA 02139, USA address: (K. Pack) (M. Luo) (T. Wierzbicki) Telephone: Fax: Address: 77 Massachusetts Ave., Rm , Cambridge, MA 02139, USA (K.Pack, M. Luo) 77 Massachusetts Ave., Rm , Cambridge, MA 02139, USA (T. Wierzbicki) 1

3 Abstract The Impact and Crashworthiness Lab at Massachusetts Institute of Technology participated in the Sandia Fracture Challenge and predicted the crack initiation and propagation path during a tensile test of a compact tension (CT) specimen with three holes (B, C, and D), using a very limited number of material properties, including uniaxial tensile tests of a dog-bone specimen. The maximum shear stress and Modified Mohr Coulomb (MMC) fracture models were used. The predicted crack path of A-C-E coincided with two out of thirteen experiments performed by Sandia National Laboratories, and the maximum load, as well as the load level at the first and second crack initiation, was accurately captured. However, the crack-tip opening displacements (CODs) corresponding to the initiation of the two cracks were overestimated by 12% and 24%, respectively. After the challenge ended, we received the leftover material from Sandia and did full plasticity and fracture calibration by conducting extra fracture tests, including tensile tests, on a specimen with two symmetric round notches, a specimen with a central hole, and a butterfly specimen with double curvature. In addition, pure shear tests were carried out on a butterfly specimen. Newly identified fracture parameters again predicted the A-C-E crack path, but the force COD response could be reproduced almost perfectly. Detailed calibration procedures and validation are discussed. Furthermore, in order to investigate the influence of the machining quality on the results, a pre-damage value was introduced to the first layer of finite elements around the starter notch, A, and the three holes, B, C, and D. This accelerated shear localization between holes A and D (and between D and C as well) and changed the crack path to A-D-C-E. Parametric study on the pre-damage value showed that there exist two competing crack paths, and the corresponding force COD curve is influenced by the pre-damage value. The effect of mesh size and boundary conditions are also discussed. Keywords Sandia Fracture Challenge, Compact tension, Ductile fracture, Crack propagation, Modified Mohr Coulomb, triaxiality, Lode angle, pre-damage 2

4 1. Introduction Fracture is the physical phenomenon of the separation of a material into two or more parts and the creation of a new surface. On the atomic scale, it means the breakage of bond between neighboring atoms. Predicting fracture is increasingly important as stronger and higher grade materials are developed to enhance performance, crashworthiness, and fuel efficiency in the automotive industries as well as the durability of parts for structural integrity. Increased strength has brought reduced ductility, which leads to premature fracture in the metal forming process and other practical applications. Sandia National Laboratories recently issued an open challenge to the mechanics community to compare the ability to predict fracture in metallic alloys of various methodologies, and identify the most powerful approach. Participants were asked to perform a blind fracture simulation for a tensile test of a compact tension (CT) specimen with three randomly distributed holes (machined from a precipitationhardened stainless steel sheets). Then, they were to make a prediction of crack propagation path, force and crack-tip opening displacement (COD), the point at which a crack first initiates and reinitiates out of the first hole. Also, a representative estimate of the force COD curve was to be submitted for direct comparison with experimental results. It was required that participants use only the few material properties and test results provided by Sandia, and they were not allowed to perform any supplementary mechanical tests for the purpose of calibrating and validating their models. The material data provided by Sandia included the chemical composition, the results from uniaxial tensile tests on a dog-bone specimen in the longitudinal and transverse directions, the results from fracture toughness tests, and a photo of the microstructure. Different models (plasticity or fracture) required different types of experiments for calibration, making the problem very challenging. Microscopically, ductile fracture is induced by the nucleation, growth, and coalescence of voids, which ultimately form a visible macroscopic crack. Fundamental work by McClintock (1968) and Rice and Tracey (1969) found that hydrostatic pressure acting on voids is the main parameter that controls the onset of fracture, as it facilitates void growth. Gurson (1975) integrated the effect of 3

5 mean stress into the plasticity model for the macroscopic description of void growth. Void nucleation and coalescence were later taken into account by Chu and Needleman (1980) and Tvergaard and Needleman (1984). Nahshon and Xue (2009) and Nielson and Tvergaard (2010) modified the original Gurson model to correct the fracture prediction under shear-dominated loading conditions. These attempts lost their micromechanical justification because in the shear dominated loading no voids were observed (Ghahremaninezhad and Ravi-Chandar 2013). Aside from micromechanically inspired models, numerous phenomenological models were suggested on the basis of comprehensive test programs. Bao and Wierzbicki (2005) performed extensive fracture tests on various geometries, evaluated nine fracture models, and concluded that none of them could fit their experimental results. Wierzbicki and Xue (2005) incorporated the third invariant of the deviatoric stress, as a variable in the ductility reference function. More recently, Bai and Wierzbicki (2010) transformed the classical Mohr Coulomb criterion for rock and soil mechanics into the space of the equivalent plastic strain, stress triaxiality, and Lode angle by using a novel plasticity model (Bai and Wierzbkci 2008) for application to ductile metals. Recent studies by Barsoum and Faleskog (2007) and Korkolis and Kyriakides (2008) based on the tests of tubular specimen under biaxial and combined tension/torsion loading, respectively, revealed that the Lode angle parameter as well as the stress triaxiality has a strong influence on fracture initiation. Lou et al. (2012) also introduced the normalized shear stress, a function of the Lode angle, to a damage weighting function to predict the fracture forming limit diagrams. Mohr and Marcadet (2013) suggested the Extended Mohr Coulomb model for the development of a physically consistent damage indicator. The Modified Mohr Coulomb (MMC) model was successfully used by Luo and Wierzbicki (2010) to analyze the shear fracture in the stretch bending process of DP780 sheets. Dunand and Mohr (2011) proved the superior fracture predictive ability of the MMC model over the shear modified Gurson model (Nielson and Tvergaard 2010) using test data for TRIP780 sheets. Beese et al. (2010) applied the plane stress version of the MMC model to aluminum alloy 6061-T6 that showed planar anisotropy. It was reported by Li et al. (2010) that the MMC model is even useful for predicting the shear-induced fracture whose stress state is not available in the conventional forming limit diagram, which is widely used in the 4

6 sheet metal forming community. Recently, Dunand (2013) showed that in the case of low stress triaxiality, fracture is induced by shear instability. In the present research, the MMC fracture model along with the isotropic J2 plasticity model was adopted for the blind fracture simulation, in light of their ability to predict fracture in an un-cracked body. While the MMC model has three parameters, thus requiring at least three types of fracture tests to explicitly identify the parameters, only a single type of fracture test, a conventional uniaxial tensile test on the dog-bone specimen, was available for calibrating the fracture model. Hence, we first employed the maximum shear stress model, which is a special case of the MMC model and has only one material parameter. However, Wierzbicki et al. (2005) reported that the maximum shear stress criterion accurately captures the trend of plane stress fracture in the wide range of the stress triaxiality. To take advantage of its established fracture predicting ability, the full three-parameter MMC model was considered as well, in which case we estimated two parameter values based on our experience in dealing with a large number of similar metallic materials. The force COD curves predicted by both models followed the general trend of the experimental results that showed the same crack path as the simulation. To further improve fracture prediction, we asked Sandia to send the leftover material after the challenge ended and did a full fracture calibration by conducting four types of test. These included tensile tests on a specimen with two symmetric round notches, a specimen with a central hole, and a butterfly specimen with double curvature, as well as the pure shear tests on a butterfly specimen. The newly calibrated MMC model with the optimized stress strain curve was validated by the four tests in terms of the displacement to fracture and then applied to the Sandia problem. The force COD response was almost perfectly reproduced. With the objective of investigating the effect of the machining quality, the concept of pre-damage was introduced. Milled or drilled surfaces are assumed to be damaged during the machining process and thus have a certain extent of predamage related to the final quality. The pre-damage plays an important role in initiating shear localization and can change the crack path and overall load response. Finally, the effect of mesh size and boundary conditions were studied. 5

7 2. Material and plasticity 2.1 Material The material used in the Sandia Fracture Challenge was a 3.15 mm-thick martensitic precipitation-hardened stainless steel sheet of alloy 15-5 PH, supplied by AK Steel. Its excellent mechanical properties together with its high resistance to corrosion up to a temperature of 316ºC enable a wide range of applications including aerospace, chemicals, petrochemicals, food processing, and the paper and general metalworking industries. It was heat-treated by Sandia at around 593 ºC for four hours. The resulting tensile properties lie between the H1075 and H1100 conditions. The chemical composition is summarized in Table Plasticity modeling and validation Plastic deformation of metallic materials can be described with a set of yield function, flow rule, and hardening rule. Figure 1 (Fig.3 of the lead article) shows the dimensions of the dog-bone specimen, and Figure 2 (Fig. 4 of lead article) depicts the results from uniaxial tensile tests on this specimen in the longitudinal (rolling) direction and transverse (90º to the longitudinal) direction, which are provided by Sandia. Instead of force and displacement, engineering stress and strain, measured with a 25.4 mm (1 in) 50% MTS extensometer, were given. We can assume the material to be isotropic since there is negligible change in the engineering stress strain curve with a different material orientation, and thus J2 plasticity in the form of Eq. (1) was adopted. f ( σ, ε ) = σ k( ε ) = 3 J k( ε ) (1) p p p Mises 2 σ denotes the Cauchy stress tensor, p ε is the sum of the work-conjugate incremental plastic strain of the von-mises equivalent stress σ, Mises k is the deformation resistance, which is a function of p ε, and J represents the 2 second invariant of the stress deviator. 6

8 Since the Sandia CT specimen undergoes only a monotonic loading program and does not have a complicated history, such as reverse loading, the associated flow rule with the isotropic hardening was assumed. The normality condition relates the plastic strain increment to the stress components as follows. dε p p f = dε σ (2) The hardening curve is an important component of the plasticity model. Young s modulus was calculated from the initial rectilinear slope of the engineering stress strain curve and the yield strength was estimated using the 0.2% offset method. The Swift power law hardening equation is expressed as swift P ( 0 ) n k = A ε + ε (3) This agrees well with the experimental true stress plastic strain data up to the uniform strain of 0.05 which was obtained from the longitudinal specimen #2 as shown in Fig. 3 (blue). Fracture predictability is strongly influenced by the hardening curve due to its close relationship with necking, especially in the postnecking part. While the stress strain curve up to the uniform strain is uniquely identified in the uniaxial tensile test on the dog-bone specimen, the stress strain curve after necking cannot be determined directly from the experiment without a hydraulic bulge test or special equipment to measure local stress and strain. Usually, fracture is accompanied by a significant amount of necking. With the goal of identifying the hardening curve up to a high level of strain, we utilized a so-called inverse method, where the hardening curve in the post-necking area is adjusted for several strain points (e.g. 0.2, 0.4, 0.6, 0.8, and 1.0) until the force displacement curve (here, engineering stress strain curve) predicted by a simulation correlates well with the one from an experiment. To efficiently adjust the curve with only one parameter, it is assumed that the optimized hardening curve would have a similar shape as the initial Swift fit, and be the maximum true stress before necking ( k max ) plus the multiple of the gap between the initial Swift fit and the flat extension of the maximum true stress (α ). This can be written as Eq. (4). 7

9 opt max ( Swift max ) k = k + α k k (4) To find the optimal multiple, α, finite element (FE) simulations were performed using the commercial FE software, Abaqus/Explicit (2010). An FE model was built up for one-eighth of the dog-bone specimen with reduced-integration eightnode 3D solid elements (C3D8R of the Abaqus element library), exploiting the geometrical symmetry in three orthogonal directions to save computation time. The center of the gauge section was modeled with a fine mesh of 0.15 mm 0.15 mm 0.16 mm (thickness) to generate a reasonable necking shape as well as the converged strain evolution at the critical material point, following the recommendation of Dunand and Mohr (2010). MATLAB code along with a Python script was written so that the multiple α is automatically identified. All the parameters associated with the hardening curve are summarized in Table 2. The optimized stress strain curve is shown as a red solid line in Fig. 3, and is higher than the initial Swift fit. The corresponding engineering stress strain curve showed a perfect agreement with the representative experiment as demonstrated in Fig.4. To strictly validate the plasticity model, not only the global response of the force displacement curve but also the evolution of the local strain quantity needs to be verified experimentally. However, no information beyond the engineering stress strain curve was available. Thus, the plasticity model was assumed to be accurate enough to predict the correct local strain. 3. Fracture modeling 3.1 Characterization of stress states Physically inspired fracture models as well as phenomenological models usually predict fracture in a proportional loading condition. Thus, they are used as a reference weighting function for the calculation of the damage accumulated during a particular plastic strain increment. Considering other researchers successful applications of the MMC model in predicting fracture in various geometries of an un-cracked body (Luo and Wierzbicki 2010; Li et al. 2010; Dunand and Mohr 2011), the MMC model was chosen to be a weighting function 8

10 of our blind fracture simulation. The MMC fracture model can be visualized in the stress space that is parameterized with the stress triaxiality and the normalized Lode angle. It is necessary to define those two stress invariants and explain their physical meaning. The stress triaxiality, the number one parameter for numerous fracture models, is defined as the mean stress normalized by the von-mises equivalent stress, which is expressed as η σ σ m = (5) Mises where σ is one third of the first invariant of a stress tensor, and σ m Mises is related to the second invariant of a stress deviator J 2, which are written as σ m = I / 3 tr 1 = ( σ ) / 3 (6) 3 σ Mises = 3 J2 = σ ': σ ' (7) 2 where σ and σ ' are the Cauchy stress tensor and its deviatoric part. The Lode angle is defined as the angle between the deviatoric principal stress vector and the projection of the maximum principal stress axis on the π (deviatoric) plane, which is a function of the second and third invariants, as shown in Eq. (8) J3 π θ cos = 3/2 0 θ 3 2J 2 3 (8) where J3 is the determinant of a stress deviator, which is written as 1 J 3 = det( σ ') = σ ' σ ': σ ' (9) 3 9

11 The range of the Lode angle, 0 to π / 3, comes from the geometrical constraint of σ1 σ 2 σ3, where σ 1, σ 2, and σ 3 are the maximum, intermediate, and minimum principal stresses, respectively. The Lode angle can be linearly transformed such that it ranges from -1 to 1 with Eq. (10). 6θ θ = 1 ( 1 θ 1) (10) π This parameter is termed the normalized Lode angle. The advantage of representing the stress state using these two stress parameters is that they are independent of the rotation of a coordinate system, and all the stress states, which are in the direction of a principal stress vector, can be described as a combination of the two parameters. Readers interested in the geometrical interpretation of η and θ are referred to Bai and Wierzbicki (2010). 3.2 The MMC fracture model and the maximum shear stress model The MMC fracture model was derived by Bai and Wierzbicki (2010) from the original Mohr-Coulomb criterion, by transforming it from the stress space to the mixed stress-strain space, and thus formulating the equivalent plastic strain to fracture, ˆ ε f, as a function of η and θ. It takes the form of Eq. (11). ( η θ ) = ( c ) ˆ ε, f A 3 θπ c sec 1 c c1 θπ 1 θπ cos + c1 η + sin n (11) The model defines the limit of ductility as an equivalent measure in the proportional loading condition where η and θ are kept constant. The equivalent plastic strain to fracture, ˆ ε f, decays with the stress triaxiality for a fixed θ and is asymmetric to θ for a constant η. Materials under a mechanical 10

12 loading usually experience the change in the stress state during deformation. In such a non-proportional loading condition, a linear incremental damage rule of Eq. (12) is assumed for a monotonic loading, and crack is considered to be initiated when the damage value D reaches unity. The current value of the damage indicator, D, can be interpreted as a percentage of used ductility. D ε f p = (12) 0 ˆ ε ( η, θ ) f dε The MMC model has three parameters, c 1, c 2, and c 3 ( A and n are the strength coefficient and hardening exponent of the Swift law). However, only one type of fracture test, uniaxial tension on the dog-bone specimen performed by Sandia, could be used to calibrate the model. The fracture toughness test provided by Sandia was the tensile test of a fatigue-cracked CT specimen. It contains an extremely sharp pre-crack, so its FE simulation has a strong dependency on the mesh size, and hence it is not appropriate to use the inverse method to identify the model parameters (the inverse method for the fracture calibration is explained in detail in the next section). Moreover, the MMC model was intended for the prediction of crack initiation in an un-cracked body. Hence, toughness test, which characterizes crack propagation, is not a proper choice for the calibration. The limiting case of c 1=0 and c 3 =1 reduces Eq. (11) to the maximum shear stress criterion of Eq. (13), which has only one material parameter and is known to be effective for prediction of plane stress fracture. ˆ ε f ( ) 1 A θπ θ = cos 3 c2 6 1 n (13) Thus, the maximum shear stress criterion was used first. The three-parameter MMC model was also considered in order to exploit its predictive power to the full by estimating c 1 and c 2 based on our experience in testing and calibrating many other similar materials. 3.3 Fracture calibration and validation 11

13 Fracture calibration provides the values for model parameters that make an accurate prediction of fracture. To this end, the hybrid experimental numerical procedure, also called an inverse method, (Dunand and Mohr 2010) was utilized. This method requires an FE simulation of each calibration test. Once the deformation process of each test is perfectly reproduced in terms of the force displacement curve (confirmation of the plasticity model), the most critical element (usually the point of the maximum equivalent plastic strain), which is the potential point of crack initiation, is identified. Then, the history of P ε, η, and θ are extracted up to the experimentally determined displacement to fracture at the identified potential crack site. In general, as material deforms and geometry changes, the stress state keeps changing. To take this history into account, material parameters are optimized such that the damage value calculated for each test based on the history of P ε, η, and θ is as close as possible to unity. This algorithm can be summarized by the following equation. ( ) N f f [ c1, c2, c3 ] = arg min c, c, c 1 D i ( η, θ, ε ) (14) i= 1 For the maximum shear stress model, c 2 was calibrated to be MPa by enforcing the damage value of one for the central element in the minimum cross section (critical material point) when the engineering strain reached the average strain to fracture of two longitudinal dog-bone specimens (all the CT specimens were cut in the longitudinal direction). In the case of the full MMC model, c 1 and c3 were estimated to be and 0.92, respectively, on the basis of our large stock of experience in calibrating similar materials and our deep understanding of η and θ dependency in this type of metallic materials. c 2 was found to be MPa by applying Eq. (14) to the dog-bone specimen. The constructed fracture envelopes for both models are represented in Fig. 5. The black solid line depicts how the stress state varies with the increasing equivalent plastic strain at the critical material point (marked with the white dot in Fig.5a), and the red diamond denotes the final stress state. The purple line on each fracture surface corresponds to the plane stress condition in which η has a unique relationship 12

14 with θ. Notice that the red diamond is not located exactly on the fracture envelopes because it holds only for the proportional loading condition. In order to validate the calibrated fracture models, von-mises plasticity and both fracture models were implemented in Abaqus/Explicit through the user subroutine, VUMAT. The element deletion technique was utilized to describe the crack initiation, and applied to the FE analysis for the dog-bone specimen. As illustrated in Fig. 6, when the engineering strain applied to the specimen gauge section attains the experimentally determined engineering strain to fracture, both models delete the most damaged element (losing its load-carrying capacity), and as a result the engineering stress suddenly starts to drop. Successive element deletion eventually leads to zero stress. The inserted photo in Fig. 6 is the comparison of the cross sectional shape after fracture between the simulation and experiment. The red frame indicates the edge line taken from the simulation, and the photo of fractured cross section was provided by Sandia. The almost perfect match of the external contour is solid evidence for the accuracy of the plasticity and fracture model. 4. Blind fracture simulation The geometry of the specimen for the Sandia Fracture Challenge is based on a conventional CT specimen with a round notch. Sandia added three holes around the round starter notch as shown in Fig. 7 (Fig. 9 of lead article). The starter notch, the three holes, and the backside edge were alphabetically named A, B, C, D, and E as designated in Fig. 8 (Fig. 8 of lead article). Specimens were machined in the longitudinal direction and experiments were done by Sandia after all the participants submitted their blind simulation results. Each specimen was pulled at mm/sec ( in/sec) in a servo-hydraulic MTS 22-kip loading frame at ambient temperature. An Epsilon COD gage measured the amount of opening between two sharp notches. Ten specimens were officially tested for comparison with participants predictions, and another laboratory independently performed three more tests to check repeatability. The results are summarized in section Finite element modeling 13

15 Figure 9 shows the FE model for the Sandia CT specimen meshed with 118,980 reduced-integration eight-node 3D solid elements (C3D8R). Fine mesh of the same size as the one used in the dog-bone specimen was built up around three critical holes without taking advantage of any geometric symmetry, giving consideration to the possibility of asymmetric crack propagation. Twenty elements made up the whole thickness, and the fracture simulation was performed using Abaqus/Explicit with the user material subroutine. Two rigid pins were placed in the two large holes, and surface-to-surface contact with the kinematic contact algorithm was applied. Interaction between the two contact surfaces was assumed to be frictionless. A fixed boundary condition was imposed on the reference point of the lower rigid pin, and the upper rigid pin was pulled at the same velocity as the experiment. Due to the huge number and small size of elements, a fixed mass scaling of 10 8 was introduced. The negligible ratio of kinetic energy to internal energy assured that the analysis could be considered to be quasi-static. 4.2 Crack initiation and propagation The maximum shear stress criterion and the MMC fracture model predicted the same crack path of A-C-E. Figure 10 shows the distribution of the damage indicator around the three critical holes in the mid-plane at different times when the MMC model was utilized. The plastic deformation started on the surface of the starter notch A, the hole C, and the hole D due to geometrically induced stress concentration. Additional tension made the plastic deformation zones of A and D link together, and damage continued to be accumulated on the left and right surfaces of C. Continued loading developed a new plastic deformation zone between A and C, with the plastic zone on both sides of C growing. Almost all the deformation started to concentrate in the region between A and C, followed by diffuse and localized necking. Finally, the ligament lost its resistance to plastic deformation and the first crack initiated inside of the region, closer to C. It propagated both to the right and to the left, and the ligament separated completely. Further displacement accelerated stress concentration on the right surface of C, which formed another crack which propagated to the backside edge E. 14

16 The maximum shear stress model predicted slightly different development of the crack as shown in Fig. 11. The first crack was created at the left surface of C rather than inside the material. The fracture surface of the maximum shear stress criterion has no dependency on η, as represented in Fig. 5a. Thus, the material point that has the stress state of a lower θ leads to fracture first regardless of η value (an inner material point under tensile loading usually suffers from a higher η, which is known to have low ductility). In addition, after the ligament between A and C broke, a second element deletion took place at the right surface of C, but did not propagate as a live crack. Instead, a subsurface crack was observed at a point 1.5 mm away from the right surface of C. The crack progressed to both C and E, ultimately creating a macro crack. Sandia s experiment on specimen 4 exactly duplicated this phenomenon, as seen in Fig Comparison with experiments by Sandia Figure 13a shows the experimental force COD curves provided by Sandia. Eleven out of thirteen experiments showed an A-D-C-E crack path (solid line), and only two specimens featured an A-C-E path (solid w/ symbol), which is consistent with our simulation. For the predominant crack path of A-D-C-E, shear localization between A and D resulted in an early load drop, which is soon followed by breakage between C and D without showing neck development. On the contrary, the A-C-E crack is initiated by uniaxial tensile loading acting on the ligament between A and C, which experiences remarkable necking before a sudden load drop. Afterward, both cases undergo almost constant force until another crack develops, and the force decreases as the crack propagates to E. Figure 13b is a comparison between the force COD curves of the simulations and those of the experiments that showed the same scenario of A-C-E. Compared with the black square data (the two crosses correspond to the first and second crack initiation), the maximum force and the force at the first crack initiation were accurately predicted by both models with less than 1% error. The force level at the second crack initiation was overestimated by only 4.40% for the maximum shear stress model and by only 3.50% for the MMC model. However, corresponding COD values were somewhat overestimated. The COD at the first crack predicted by both models had approximately 12% error, and as for the second crack 15

17 following the load plateau, the error was 14.97% and 23.73% for the maximum shear stress and the MMC model, respectively. The first error is thought to originate from the over-adjusted stress strain curve, which retards the rate of neck formation and the decrease in load after the maximum level (slope is less than two experimental results, Fig. 13b), and thus delays the COD at first crack development. The second error is partially due to the predicted stable crack propagation, which is confirmed by a gentle slope of force COD curves during the first crack propagation. This is caused by inaccurate identification of the stress strain curve, insufficient fracture calibration points, and the mesh size. However, plasticity and fracture modeling based only on simple uniaxial tensile tests of the dog-bone specimen had already made an acceptably accurate prediction of not only crack initiation but also crack propagation. This again proves the predictive power of both the maximum shear stress and MMC model. 4.4 Stress history of critical material points Fracture prediction based on the damage accumulation rule with the reference ductility function is due to the change in the stress state during deformation. In order to better understand the fracture process, it is essential to closely investigate the stress histories of critical material points where fracture initiates. Figure 14 exhibits the variation in η and θ up to the time of fracture (as a function of the equivalent plastic strain) for the first and second crack initiation sites, which are located on the mid-plane, as well as for those of points lying on the surface, which are for comparison. As illustrated in Fig. 14a, points on the mid-plane undergo higher η than points on the surface, thus having a lower ductility and earlier fracture initiation. As for θ, the first crack points do not show a noticeable difference, but the second crack points have a constant difference of 0.2 with the mid-point featuring a smaller θ, which corresponds to a lower ductility. The first (black) and second (blue) crack initiation points depart from almost the same η but go by a different path, with an increasing η for the first and an almost constant η for the second. The stress history of the first initiation site (black) is similar to that of the mid-point in the dog-bone specimen under tension. This is expected, as the ligament between A and C is deformed mainly by uniaxial tensile 16

18 loading. Complicated variation in the stress state reconfirms that the constructed fracture envelope cannot be directly used to predict the onset of fracture and a damage rule should be incorporated. 5. Full calibration After the challenge ended, leftover material was requested with the goal of further enhancing the fracture predictability, and a 270 mm 300 mm sheet was received from Sandia. To improve the plasticity and fracture model, four additional types of tests were performed until fracture. Figure 15 shows the average stress state of the critical material point up to necking for these specimens in the space of η and θ, together with stress states of all the material points in the Sandia CT specimen. It is obvious that the dominant stress states lie in the first quadrant and that the four types of fracture test chosen characterize the stress state to be of utmost importance in the Sandia CT specimen under tensile loading. 5.1 Four calibration tests The new test program was comprised of three tensile tests (on a specimen with two symmetrical round notches, a specimen with a central hole, and a butterfly specimen with double curvature) and pure shear test on the butterfly specimen. Drawings of these three types of specimens are shown in Fig. 16. A symmetrically notched specimen can generate a wide range of stress states in the center of the specimen through a simple change in the radius of the notches. In the present research, a 10 mm radius was chosen. A specimen with a central hole has a large stress concentration at the right and left corners of the hole, but its point-wise stress state at the critical material point is kept almost constant, which is ideal for calibrating the fracture model. A butterfly specimen with double curvature was designed by Bai (2008) so as to induce various stress states at the center of the specimen when loaded in a biaxial loading frame with a combination of horizontal and vertical displacement. Two extreme cases of pure tension and pure shear loading were considered. All the specimens were fabricated in the longitudinal direction except the butterfly specimens for pure shear loading, which were extracted from the transverse direction to reproduce the stress state most similar to 17

19 the one that the ligament between holes A and D (or D and C) underwent. Tests were done twice for each type in the biaxial loading frame of the custom-made hydraulic machine (modified with Instron 8080). The machine could be driven by both displacement (statically) and force (dynamically), so during the pure shear tests, the vertical force was controlled to be zero to satisfy a pure shear condition. For each test, between 200 to 300 pictures of deformation were taken, all the way up to fracture, using a digital camera (QImaging Retiga 1300i with 105 mm Nikon Nikkor lenses), in order to analyze displacement and strain fields on the surface of specimens. These fields were calculated with digital image correlation (DIC) using Vic-2D (Correlated Solutions Inc.) with black and white speckle patterns on the specimen. This technique is strongly recommended in order to exclude the elastic deformation of the vertical or horizontal actuator of the testing machine. Recorded force displacement curves are plotted in circular dots in Fig. 17 with the evolution of local axial logarithmic strain at the specimen center (hollow circle) for the symmetrically notched specimen and butterfly specimen under tension. A detailed explanation is provided in section Optimization of the stress strain curve after necking To inversely identify the fracture parameters, FE simulation was performed on each specimen meshed using reduced-integration eight-node 3D solid elements (C3D8R), with a fine mesh of 0.1 mm built up around the critical area. The stress strain curve optimized with the dog-bone specimen turned out to be overadjusted by the overestimated force displacement response for the four new types of specimen. See, for example, the representative curve for the symmetrically notched specimen illustrated in Fig. 18. This is because the four new types of specimen have a geometrical necking initiator (minimum width or thickness at the center), whereas the dog-bone specimen has a long parallel gauge section. Consequently, the same procedure for identification of the hardening curve done in section 2.2 was carried out again for a specimen with two symmetrical round notches, so as to fit the force displacement curve correctly (red curve in Fig.18). The newly optimized hardening curve was found to lie between the initial Swift fit and the previously optimized hardening curve, and this curve was used for all subsequent FE analysis. 18

20 5.3 Fracture calibration and validation With the confirmation of good agreement of the force displacement curve between the experiment and the simulation for the four types of specimen, three parameters of the MMC model were identified again, following the same procedure described in section 3.3 ( c 1 =0.0312, c 2 = MPa, and c3 =0.9201). The constructed envelope is plotted in Fig. 19 with stress histories of critical material points of each test. Notice that every test but the one on the symmetrically notched specimen goes through a small variation of η and θ during deformation, which is helpful for finding a more reliable fracture surface. The envelope apparently has less dependency on η than the initially obtained fracture surface in Fig. 5b, and its height (ductility) is also much lower (clearly observed for the generalized shear state, θ =0). This new shape of a half tube is attributed to smaller c 1 and c 2 values (readers interested in a detailed parametric study are referred to Bai and Wierzbicki, 2010). The calibrated MMC model was applied to the FE analysis of the four cases, and the force displacement responses were compared with experiments in Fig. 17. For the symmetrically notched specimen and the specimen with a central hole, the evolution of axial logarithmic strain between two points in the center of the surface of each specimen with an initial gap of 2 mm for the notched and 1 mm for the butterfly were measured with a virtual extensometer in Vic-2D. The results were compared with the simulation for stricter validation of the new hardening curve. In the case of the butterfly shear, extreme distortion of speckle patterns in the gauge section prevents accurate measurement of shear strain. For the specimen with a central hole, the maximum axial strain occurs at the intersection of the transverse axis of symmetry and the hole, where the displacement field is not available from Vic-2D (usually DIC is not applicable to the boundary because there are not enough pixels around the point of interest). Even though there is a slight error which comes from the fact that two-dimensional DIC does not take the out-of-plane deformation into account, the evolution of axial strain correlates well with the experiments. The evolution of the equivalent plastic strain at the critical element (a solid red line) is shown as well. Figure 20 summarizes the 19

21 predictability of the displacement to fracture. For the first two tests and the butterfly shear case, it was underestimated by 9.4%, 4.2% and 0.4%, respectively, and in the case of the butterfly tension, it was overestimated by 16.4%. This is caused by the number of specimen types used in the calibration, which is larger than that of the parameters in the MMC model. More calibration points in a wide range of stress states may result in a larger error in predicting the displacement to fracture for each of tests, but the obtained fracture surface is rather said to be more reliable since it can be applied to a more general problem in which stress states are diverse and loading history is complicated. Despite a small error, the calibrated MMC model was proven to be able to cover a range of interest for η and θ. Thus, the model is ready to be used for the fracture simulation of the Sandia CT specimen. 5.4 Re-simulation The fully calibrated MMC model with the newly optimized stress strain curve was used for the FE simulation of the Sandia CT specimen under uniaxial tension. The same finite element model and simulation conditions mentioned in section 4.1 were applied and no changes but the hardening curve and the parameters of the fracture model were made. The crack path was again predicted as A-C-E. There was no difference in the sequence of crack development, compared with the prediction made by the previously calibrated MMC model. However, it is noteworthy that the force COD response was almost perfectly matched with experimental results as shown in Fig. 21. As with previous models, the fully identified MMC model made an accurate prediction of the maximum load and force at the first crack with less than 1% error. The error in the load level at the second crack was only 2.24%. More surprisingly, the predictability of CODs at the first and second cracks was greatly improved (error: 5.93% and 1.94%, respectively). This small error could be reduced even further by using a FE model with finer mesh. This is discussed in section Parametric study 6.1 The effect of pre-damage 20

22 Up to this point, our FE simulation has been performed with the assumption that the machined surfaces of the CT specimens have no roughness or imperfections. According to Fig. 13a, for the tests that showed the A-D-C-E crack path, the displacement at the first load drop has a large spread, as opposed to the test results of the A-C-E crack path. This is thought to be an indication that the machining quality greatly affects the experimental result. The supporting evidence from the previous simulation is that the damage accumulates much more on the surface of A and D than on the inside of the ligament between two holes, as shown in Fig. 22. A recent paper by Ghahremaninezhad and Ravi-Chandar (2013) adds much credibility to the above explanation. Hence, this section is vulnerable to the surface roughness as well as the pre-damage caused by the machining process, and can be a site of the premature crack initiation, leading to shear localization. Moreover, the stress state of the point indicated with a white circle in Fig. 22 is revealed to be close to the generalized shear condition (θ =0), which has the lowest ductility at a fixed η. In contrast, for the A-C-E scenario, the crack is initiated on the inside between A and C, and thus less susceptible to the surface roughness (confirmed by the small variation in the force COD response). To study the influence of the pre-damage on the crack path and force COD curve, various values for the pre-damage, ranging from 0.3 to 0.8, were introduced to the first layer of finite elements around the starter notch and the three holes using the Abaqus option, INITIAL CONDITION (type=solution) as shown in Fig. 23. The force COD curves from the simulations are summarized in Fig. 24 with two representative experimental curves for the A-C-E and A-D-C-E scenarios. The resultant configuration of the CT specimen after fracture for both crack paths is also depicted in Fig. 25. Up to a pre-damage value of 0.5, the crack propagated from A to C to E. However, when the values were larger than 0.5, the crack path was altered to A-D-C-E, and the load response became totally different. Every A- C-E scenario showed the same force COD response as when perfectly machined surfaces were assumed, whereas in the case of A-D-C-E, it was heavily influenced by the pre-damage value. The more damage was introduced, the earlier the load dropped. By contrast, it is notable that the second load drop following the load plateau of 5.3 kn had almost no dependence on the pre-damage value. When the pre-damage value was 0.7, the curve almost perfectly coincided with the 21

23 experiment that exhibited the A-D-C-E path (grey hollow circle). The force and COD at the first and second crack initiation corresponding to the breakage of the ligament between A and D, and D and C, respectively, were within a margin of error of only ± 15% (Two cracks develop consecutively within the first load drop, so the definition of crack initiation is unclear). Also, the time of the third crack initiation was successfully predicted with a margin of error less than 10%. Not only is the shape of the predicted curves compatible with the scatter of experimental results, but also their transition according to the change in the crack path. Although there is no sound physical meaning or basis for the specified damage value, it does turn out to be a reasonable measure of the effects of machining quality in an engineering sense. 6.2 Crack initiation and propagation for the A-D-C-E path The details of the process of crack initiation and propagation for the A-D-C-E scenario were investigated using the case where the pre-damage value of 0.7 was introduced. Figure 26 illustrates the sequence of damage accumulation and crack formation. Just as in the A-C-E scenario, the damage first accumulates between A and D, and D and C. However, before the primary damage zone moves to the ligament between A and C, a surface crack develops on the edges of A and D due to a pre-existing high damage value, which induces a severe local deformation along the damaged zone (shear localization). As more tension is applied, the A-D ligament is broken while another surface crack is being formed between C and D. Finally, the section reaches its limit of ductility and breaks. Continued loading causes another crack to initiate a few millimeters away from C, as reported in the experiment by Sandia (Fig. 12), and it propagates to the backside E. This general sequence of crack initiation and propagation agrees well with the experimental observation. 6.3 The effect of mesh size It is known that fracture simulation inevitably has a mesh size effect. Previous researchers recommend keeping a consistent length scale when calibrating a fracture model and simulating a process of interest, such as the edge length of 22

24 finite elements, their aspect ratio, and the size of subset (facet) in DIC used for the calculation of the displacement field. However, the diverse and complicated geometries of specimens constrain consistency, so there usually exists the problem of a mesh size effect. In this section, with the reference FE model described in section 4.1, the effect of mesh size on the force COD response is studied using two additional FE models: One for coarser mesh (edge length: mm) and the other for finer mesh (0.075 mm). In parallel, two values for the predamage that led to different crack paths for the reference FE model were considered, in order to investigate the influence of the pre-damage at the same time. Fig. 27a summarizes the result when no pre-damage value was introduced. There was no change in the crack path, and all three cases predicted an A-C-E crack path. As mesh size becomes smaller, the force COD curve gets closer to the experimental results. Since the slope of the decrease in load after the first crack initiation does not get steeper, the curve is thought to be almost converged. This result confirms that when the mesh size of the FE model for the interesting geometry (reference: 0.15 mm, fine: mm) is similar to that of the FE model for the calibration tests (0.1 mm), the predicted response is comparable to the experimental results. Fig. 27b depicts the mesh dependency of the force response for the pre-damage value of 0.6, which was the smallest pre-damage that changed the crack path to A-D-C-E. As with the previous case, the identical crack path of A-D-C-E was predicted for all three types of FE models. It is revalidated that the finer mesh shows the better agreement with the experimental results. However, the dependency of the force COD curve on the mesh size is much clearer than when the pre-damage value equals zero. There was still a great increase in the rate of load drop during the first and second crack propagation when the fine mesh was used. The reason for this is that the fracture mechanism for the A-D-C crack is shear localization, rather than necking which is observed in an A-C crack. Wang et al. (2013) reported a strong mesh dependency for a shear dominated loading during the hole-expansion process. 6.4 The effect of boundary condition 23

25 In the fracture simulation of the tensile test on the Sandia CT specimen, frictionless contact between the rigid pins and the specimen was assumed, along with perfect alignment of upper and lower grips, since detailed information on the boundary condition was not provided by Sandia. This implicit assumption might have driven the ligament between holes A and D (or D and C) to better resist shear localization. To investigate this, the effects of different boundary conditions were examined by changing the friction coefficient and intentionally tilting the load line applied to the specimen. For the friction coefficient, three values of 0.1, 0.3, and 0.5 were studied. Friction coefficients are known to be highly dependent on the surface quality, surrounding state, and lubricant. For example, the coefficient in a dry and clean state can be reduced to one fifth in greasy and lubricated state. For the tilt angle, two cases of +5 (clockwise) and -5 (counterclockwise) were considered, as illustrated in Fig. 28. None of conditions altered the crack path, which was predicted to be A-C-E. Tilt angle did not even affect the force COD response, as demonstrated in Fig. 29a. However, the force COD curve was greatly influenced by the friction coefficient, as shown in Fig. 29b. As the coefficient increases, higher force is predicted. As the specimen is pulled upward, rotation between the connecting pins and the specimen is unavoidable, and the required force is clearly proportional to the friction coefficient. In contrast to our initial guess, it was firmly verified that the vagueness of boundary conditions is not responsible for the bifurcation of the crack path. 7. Conclusion In this study, attempts were made to predict not only the location of the initiation of the crack but also the direction of propagation during a tensile test of a modified CT specimen with three holes placed around a round notch. Both plasticity and fracture models were calibrated with only one type of experiment, a simple uniaxial tensile test of a dog-bone specimen. The detailed process of the damage accumulation, crack formation, and crack propagation were investigated using the FE simulation, and the predicted load response was compared with the results of experiments conducted by Sandia. In order to fully exploit the predictive power of the MMC model, a full material characterization was performed by 24

26 testing four new types of specimen, and the newly identified plasticity and fracture models were applied to the fracture simulation of the CT specimen. After observing the spread of experimental results, the influence of the machining quality was investigated by introducing various pre-damage values around the machined notch and holes. A parametric study was performed on the effect of mesh size as well as that of boundary conditions. The conclusions from the present work are as follows. (1) The hardening curve up to a high level of strain and two fracture models (the maximum shear stress model and the MMC model), both of which were calibrated on the dog-bone specimen, together with von-mises yield criterion, the associated flow rule, and isotropic hardening could predict the minor crack path and corresponding force COD curve of the Sandia CT specimen. It is remarkable that a simple uniaxial tensile test, which can be performed in any laboratory environment, can make a satisfactory prediction. (2) The newly optimized stress strain curve and the fully identified MMC fracture model could not only accurately describe the plastic deformation of four new types of specimen, but also predict the displacement to fracture within a small margin of error. (The error is unavoidable when the number of calibration points is larger than that of the parameters) (3) The application of the fully calibrated MMC model to the fracture simulation for the Sandia CT specimens reproduced almost perfectly the shape of the force COD curve for the A-C-E crack path. (4) The study on the effect of machining quality, done by assuming various predamage values around the machined surfaces, reveals that there is a threshold that differentiates two scenarios. Pre-damage less than the threshold results in an A-C- E crack path with almost the same load response. Pre-damage greater than the threshold results in a predominant crack path of A-D-C-E with a force COD curve that is highly sensitive to pre-damage. It is expected that a pre-damage value can be used as a good measure of machining quality in an engineering sense. (5) A parametric study on the mesh size effect reveals that mesh refinement does not change the predicted crack path but does generate force COD predictions that are closer to the experimental results. Also, shear localization has a higher mesh 25

27 dependency than necking instability. The friction coefficient between connecting pins and the specimen turns out to have no effect on the crack path, only on the force level. Neither does tilt angle. Acknowledgements The support of the present work by the MIT/Industry Fracture Consortium is gratefully acknowledged. Thanks are also due to Sandia National Laboratories for providing the material and experimental data. Valuable discussions on the predamage with my colleagues in the lab, Kai Wang and Stephane Marcadet, are deeply appreciated. Reference Abaqus, Reference Manuals v6.10. Dassault Systemes Simulia Corp. Bai Y, Wierzbicki T (2010) Application of extended Mohr Coulomb criterion to ductile fracture. Int J Fract 161 (1):1-20. Bai Y (2008) Effect of loading history on necking and fracture. PhD thesis, Massachusetts Institute of Technology. Bai Y, Wierzbicki T (2008) A new model of metal plasticity and fracture with pressure and Lode dependence. Int J Plast 24 (6): Bao Y, Wierzbicki T (2004) A Comparative Study on Various Ductile Crack Formation Criteria. J Eng Mater Technol 126 (3):11. Barsoum I, Faleskog J (2007) Rupture mechanisms in combined tension and shear Experiments. Int J Solids Struct 44 (6):

28 Beese AM, Luo M, Li Y, Bai Y, Wierzbicki T (2010) Partially coupled anisotropic fracture model for aluminum sheets. Eng Fract Mech 77 (7): Chu CC, Needleman A (1980) Void Nucleation Effects in Biaxially Stretched Sheets. J Eng Mater Technol 102 (3):8. Dunand M, Mohr D (2010) Hybrid experimental numerical analysis of basic ductile fracture experiments for sheet metals. Int J Solids Struct 47 (9): Dunand M, Mohr D (2011) On the predictive capabilities of the shear modified Gurson and the modified Mohr Coulomb fracture models over a wide range of stress triaxialities and Lode angles. J Mech Phys Solids 59 (7): Dunand M (2013) Ductile fracture at intermediate stress triaxialities: experimental investigations and micro-mechanical modeling. PhD thesis, Massachusetts Institute of Technology. Ghahremaninezhad A, Ravi-Chandar K (2013) Crack nucleation from a notch in a ductile material under shear dominant loading. Int J Fract 184: Gurson AL (1975) Continuum theory of ductile rupture by void nucleation and growth. Part I. Yield criteria and flow rules for porous ductile media. Technical Report, Brown University, Providence, RI. Korkolis YP, Kyriakides S (2008) Inflation and burst of aluminum tubes. Part II: An advanced yield function including deformation-induced anisotropy. Int J Plast 24 (9): Li Y, Luo M, Gerlach J, Wierzbicki T (2010) Prediction of shear-induced fracture in sheet metal forming. J Mater Process Technol 210 (14):

29 Lou Y, Huh H, Lim S, Pack K (2012) New ductile fracture criterion for prediction of fracture forming limit diagrams of sheet metals. Int J Solids Struct 49 (25): Luo M, Wierzbicki T (2010) Numerical failure analysis of a stretch-bending test on dual-phase steel sheets using a phenomenological fracture model. Int J Solids Struct 47 (22 23): McClintock F (1968) A criterion of ductile fracture by the growth of holes. J Appl Mech 35 (2):9. Mohr D, Marcadet S (2013) Extended Mohr-Coulomb Model for Predicting the Onset of Ductile Fracture at Low Strss Triaxialities. J Mech Phys Solids, in review Nahshon K, Xue Z (2009) A modified Gurson model and its application to punchout experiments. Eng Fract Mech 76 (8): Nielsen KL, Tvergaard V (2010) Ductile shear failure or plug failure of spot welds modelled by modified Gurson model. Eng Fract Mech 77 (7): Rice JR, Tracey DM (1969) On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 17 (3): Tvergaard V, Needleman A (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 32 (1): Wang K, Greve L (2013) FE simulation on edge fracture considering pre-damage. Int J Fract, in preparation Wierzbicki T, Bao Y, Lee Y-W, Bai Y (2005) Calibration and evaluation of seven fracture models. Int J Mech Sci International Journal of Mechanical Sciences 47 (4 5):

30 Wierzbicki T, Xue L (2005) On the effect of the third invariant of the stress deviator on ductile fracture. Technical report, Massachusetts Institute of Technology, Cambridge, MA. 29

31 Table Table 1. Chemical composition of alloy 15-5 PH in wt% Element Mn P S Si Cr Ni Mo Cu Cb Ta wt% Table 2. Parameters for the hardening curve A ε 0 n k max α 1287 [MPa] [MPa] Figure Fig. 1. Dimension of the dog-bone specimen Eng. stress [MPa] Test Long1 Test Long2 Test Trans1 Test Trans2 Fig. 2. Engineering stress strain curves from uniaxial tensile tests of the dog-bone specimen performed by Sandia Eng. strain [-] 30

32 1600 True stress [MPa] Test data (longitudinal #2) Swift law fit Optimized fit True plastic strain Fig.3. Identification of the optimized true stress strain curve Eng. stress [MPa] Test (Long #2) Swift S-S curve Optimized S-S curve Eng. strain [-] Fig. 4. Engineering stress strain prediction of both hardening curves 31

33 ˆ ε f η θ (a) ˆ f ε η θ (b) Fig. 5. Fracture envelopes: (a) the maximum shear stress model; (b) the MMC fracture model 32

34 Eng. stress [MPa] Test (Long #2) Max. shear stress model The MMC model Eng. strain [-] Fig. 6. Validation of calibrated fracture models on the dog-bone specimen in terms of engineering stress strain curve and cross sectional shape after fracture Fig. 7. Geometry of a modified CT specimen for the Sandia Fracture Challenge B A C D E Fig. 8. Designations for the starter notch, three holes, and backside edge 33

35 y z x x Fig. 9. Finite element model for the Sandia CT specimen Damage Fig. 10. Crack initiation and propagation when the MMC model used (Color coded is the damage indicator) 34

36 Fig. 11. Crack initiation and propagation when the max. shear stress model used E E C E E C E E C Fig. 12. Second crack ahead of hole C: specimen 4 (experiment by Sandia) and simulation with the max. shear stress model 35

37 10 8 Force [kn] Solid line: A-D-C-E Solid line w/ symbol: A-C-E COD [mm] (a) 8 Force [kn] 6 4 Exp. A-C-E 2 Exp. A-C-E Sim (Max. shear stress model) Sim (The MMC model) COD [mm] (b) Fig. 13. (a) Results of tension tests on the Sandia CT specimen (provided by Sandia) (b) Comparison between experiment and simulation (only for A-C-E crack path) 36

38 Equivalent plastic strain st crack (Mid-point) 1st crack (Surface) 2nd crack (Mid-point) 2nd crack (Surface) Triaxiality (a) Equivalent plastic strain st crack (Mid-point) 1st crack (Surface) 2nd crack (Mid-point) 2nd crack (Surface) Lode angle (b) Fig. 14. (a) History of the stress triaxiality during deformation at critical material points (b) History of the normalized Lode angle during deformation at critical material points 37

39 Butterfly shear Hole tension (D8) Notch tension (R10) Butterfly tension Fig. 15. Stress states of the Sandia CT specimen under tensile loading and the average stress state of four additional fracture tests for the full calibration 38

40 (b) (a) (c) Fig. 16. Geometries of three types of specimens: (a) specimen with two symmetrical round notches; (b) specimen with a central hole; (c) butterfly specimen with double curvature 39

41 Force [kn] Notch tension_r10 Exp, Force-displ Sim, Force-displ Displacement [mm] (a) Exp, local axial Log E Sim, local axial Log E PEEQ, ε p Strain [-] Force [kn] Hole tension_d8 10 Exp, force-displ Sim, force-displ 0.4 PEEQ, ε p Displacement [mm] (b) Equivalent plastic strain 30 Butterfly tension Butterfly shear 1.6 Force [kn] Exp, Force-displ Sim, Force-displ Exp, local axial Log E Sim, local axial Log E PEEQ, ε p Strain [-] Force [kn] Exp, force-displ Sim, force-displ PEEQ, ε p Equivalent plastic strain Displacement [mm] Displacement [mm] 0.0 (c) (d) Fig. 17. Comparison of both force displacement and central logarithmic axial strain between experiment and simulation: (a) specimen with two symmetric circular notches; (b) specimen with a central hole; (c) butterfly specimen under pure tension; (d) butterfly specimen under pure shear 40

42 40 30 Force [kn] Experiment Previous S-S curve New S-S curve Displacement [mm] Fig. 18. Prediction of the force displacement curve by previous and new stress strain curves for the symmetrically notched specimen ˆ ε f η θ Fig. 19. Fracture envelope of the MMC model calibrated with four new tests 41

43 2.0 Experiment Simulation Dsipl. to fracture [mm] Notch tension Hole tension Butterly tension Butterly shear Fig. 20. Evaluation of the predicted displacement to fracture 10 8 Force [kn] 6 4 Exp. A-C-E Exp. A-C-E 2 Sim (Max. shear stress) Sim (Previous MMC) Sim (Fully calibrated MMC) COD [mm] Fig. 21. Improvement in the prediction of the force COD curve for the A-C-E crack path 42

44 A B A C D D Fig. 22. Concentration of the damage near upper and lower surfaces in the ligament between A and D Fig. 23. Introduction of the pre-damage around the starter notch (A) and three holes (B, C, and D) 43

45 10 8 Force [kn] Exp. A-C-E Exp. A-D-C-E D=0.0 D=0.3 D=0.4 D=0.5 D=0.6 D=0.7 D= Displacement [mm] Fig. 24. Force COD curves with different pre-damage values B A C D (a) B A C D (b) Fig. 25. Fractured configuration of the Sandia CT specimen: (a) A-C-E path; (b) A-D-C-E path 44

46 Damage Fig. 26. Crack initiation and propagation when the fully calibrated MMC model used with the pre-damage of

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