PSI - Issue 21

Volume 2 1 • 201 9

ISSN 2452-3216

ELSEVIER

1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials, IWPDF 2019, 22-23 August 2019, Ankara, Turkey

Guest Editors: C ihan T eko ğ lu T unca y Y al ç inka y a

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1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials Editorial Cihan Tekog˘ lu a , Tuncay Yalc¸inkaya b a Department of Mechanical Engineering, TOBB University of Economics and Technology, So¨g˜u¨ to¨zu¨ , Ankara 06560, Turkey b Department of Aerospace Engineering, Middle East Technical University, Ankara 06800, Turkey

Keywords: Plasticity; Damage; Fracture Mechanics

The decision of organizing an international workshop in Turkey on plasticity, damage and fracture of engineering materials was taken during ECF 22, 22nd European Conference on Fracture, which was undoubtedly a great success, owing to the encouragements of Prof. Aleksandar Sedmak. Ever increasing demand for materials with usually contra dicting properties such as high strength and high ductility requires a deep understanding of these three phenomena, from micro- to macro-scales. With this perspective, this workshop was co-organized by the Middle East Technical University and TOBB University of Economics and Technology, and held on 22-23 August 2019 in Ankara. The aim of the workshop was to bring together specialists working on both computational and experimental aspects of engi neering materials. The workshop became a great success with 31 oral and 26 poster presentations, and 129 authors from 19 di ff erent countries contributing to the book of abstracts. The poster by Mirac Onur Bozkurt deserves spe cial attention, which was awarded with the best poster prize. The social trip between 24-25 August to Cappadocia, a world heritage site located in Central Anatolia, provided a friendly environment for the participants of the workshop to discuss science while enjoying the beautiful nature of Turkey. The high scientific level of the workshop was set by the brilliant keynote lectures given by Prof. Majid Reza Ay atollahi on mixed mode fracture in brittle polymers, by Prof. Laurence Brassart on the e ff ective rheology of random aggregates in di ff usional creep and sintering, by Prof. Hu¨snu¨ Dal on a multiscale approach for cavity initiation and growth in rubberlike materials, by Prof. Johan Hoefnagels on the use electrons to measure local stresses in polycrys talline materials, by Prof. David Morin on a virtual laboratory for the design of aluminium structures: automotive applications, and by Prof. Cem Tasan on damage resistance through double-transformation: towards new generation TRIP-assisted alloys. We would like to thank all the keynote speakers for their immeasurable contributions to the workshop. This issue presents a selection of research papers presented at the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials, whose process of organizing was made very easy by the kind, attentive and e ffi cient support of the members of the organizing committee: Prof. Hu¨snu¨ Dal, Prof. Mert Efe, and Prof. Ercan Gu¨rses. Finally, let us thank Prof. Francesco Iacoviello, the president of the European Structural Integrity Society (ESIS), for his great support for the workshop, as well as this special issue.

2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers 10.1016/j.prostr.2019.12.079 2210-7843 c ⃝ 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review line: Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers.

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Procedia Structural Integrity 21 (2019) 130–137

© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/ Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers Abstract In this study, dynamic fracture of curved carbon fiber reinforced plastic (CFRP) lam nates under quasi-static loading is inv stigated using xplicit three dimensional (3D) fin te ele en method in conjuncti n with Cohesive Zone Modelling (CZM). The s mulations are based the experimental studies conducted by Tasd mir (2018). Three dimension l finite element models of two different p rchitectures (unidirecti nal and fabric l mi ate) are gener ted corresponding to the experiment l co f gurations. The computational resul s show good correlation with the experimental results in which a major delamination is observed approxim ely at 35% of the thickness for both un dir ctional and fabric curved lamin t s. It is also obs rved tha d laminatio initiates at the half width of the laminate for both specimen configurations. For the fabric a i ate, it i int res ing to b erve hat th dela inatio initiates at the c nter of the width i stead of the free-edges where a material mismatch exists between different layer orientations (Cao et al., 2019; Lagunegrand et al., 2006; Solis et al., 2018). Finite element analysis results are consistent with experiments in terms of main delamination location in thickness direction. © 2019 The Autho s. Publ shed by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/ Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials 3D Simulation of Dynamic Delamination in Curved Composite Laminates Tamer Tahir Ata a,b , Demirkan Coker a, * a Middle East Technical University, Department of Aerospace Engineering, Ankara 06800, Turkey b Turkish Aerospace Industries (TAI), Ankara 06980, Turkey Abstract In this study, dynamic fracture of curved carbon fiber reinforced plastic (CFRP) laminates under quasi-static loading is investigated using explicit three dimensional (3D) finite element method in conjunction with Cohesive Zone Modelling (CZM). The simulations are based on the experimental studies conducted by Tasdemir (2018). Three dimensional finite element models of two different ply architectures (unidirectional and fabric laminate) are generated corresponding to the experimental configurations. The computational results show good correlation with the experimental results in which a major delamination is observed approximately at 35% of the thickness for both unidirectional and fabric curved laminates. It is also observed that delamination initiates at the half width of the laminate for both specimen configurations. For the fabric laminate, it is interesting to observe that the delamination initiates at the center of the width instead of the free-edges where a material mismatch exists between different layer orientations (Cao et al., 2019; Lagunegrand et al., 2006; Solis et al., 2018). Finite element analysis results are consistent with experiments in terms of main delamination location in thickness direction. 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials 3D Simulation of Dynamic Delamination in Curved Composite Laminates Tamer Tahir Ata a,b , Demirkan Coker a, * a Middle East Technical University, Department of Aerospace Engineering, Ankara 06800, Turkey b Turkish Aerospace Industries (TAI), Ankara 06980, Turkey Keywords: Delamination; Cohesive Zone Modelling; Dynamic Fracture

Keywords: Delamination; Cohesive Zone Modelling; Dynamic Fracture

* Corresponding author. Tel.: +9-0312-210-4253; fax: +9-0312-210-4250. E-mail address: coker@metu.edu.tr

2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/ Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers 2452 3216 © 2019 Th Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/ Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers * Corresponding author. Tel.: +9-0312-210-4253; fax: +9-0312-210-4250. E mail address: coker@metu.edu.tr

2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers 10.1016/j.prostr.2019.12.094

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1. Introduction Most of the primary and secondary structures include curved shape parts in the aircraft and wind turbine structures. Due to the presence of curved regions at which significant interlaminar stresses are developed, delamination followed by a reduced load carrying capability of the component may cause catastrophic failure of the whole structure. Therefore, it is important to understand the mechanisms of delamination in curved laminated composites. One of the pioneering studies on delamination in curved laminated composites was performed by Chang and Springer (1986) in which the in-plane failure is predicted using the Tsai-Hill criterion whilst out-of-plane failure is predicted by a quadratic stress criterion proposed by the authors. Sun and Kelly (1988) investigated two possible failure modes (matrix cracking and delamination) of composite angle structures through experimentation and analysis. They concluded that the initial delamination crack growth is unstable. In the early 1990’s, Martin (1992) worked on unstable delamination in unidirectional curved composite laminates under quasi-static loading both experimentally and numerically. In 1996, Wisnom (1996) studied the anticlastic curvature in pure bending both in 3D and 2D models assuming generalized plane strain. He addressed the significant variation of stresses across the width of the specimen as a result of the comparison between 3D and 2D models. At the beginning of 2000s, emergence and growth of delamination in L-shaped composite laminates are investigated by Pettermann et al. (2009). They concluded that after a certain delamination size, the delamination propagates in a stable manner. The majority of these investigations on delamination in curved composited were conducted considering the failure process as static. In a series of recent studies, Gozluklu and Coker (2012, 2016) have demonstrated that delamination of L-shaped composite materials is highly dynamic. They performed 2D explicit FEA in conjunction with cohesive zone elements. The dynamic nature of failure of curved beams under quasi-static loading was also shown with experimental studies by Coker and coworkers (Gozluklu et al., 2015; Uyar et al., 2015; Yavas et al., 2014). In this study, finite element models with cohesive elements at the layer interfaces are generated by using 3D elements in ABAQUS software. The analyses are performed in explicit solver since previous studied revealed that the delamination of L-shaped composite materials is highly dynamic. As a result of the 3D simulations, initial failure location and propagation path of the delamination inside the part are clearly observed. 2. Computational Method 2.1. Material The material used for unidirectional laminate is AS4/8552 unidirectional prepreg with cured ply thickness of 0.188 mm and density of 1580 kg/m 3 . The mechanical and interface properties of this material are provided in Table 1. All other values except interface strengths are directly taken from Camanho et al. (2009). Interface strengths are obtained from experiments (Ata, 2019) conducted according to ASTM Standard D6415 (2006) and ASTM Standard D2344 (2006). The curve fit factor, η , for B-K criterion is taken as 1.45.

Table 1. Mechanical and interface properties of Hexply AS4/8552 UD prepreg and AS4/8552 5HS fabric.

AS4/8552 UD Prepreg

AS4/8552 5HS Fabric

Elastic Properties

E 11 =135 GPa; E 22 = E 33 =9.6 GPa;

E 11 = E 22 =64 GPa; E 33 =8.5 GPa;

ѵ 12 = ѵ 13 =0.32; ѵ 23 =0.487

ѵ 12 = 0.046; ѵ 13 =ѵ 23 =0.30

G 12 = G 13 =5.3 GPa; G 23 =3.4 GPa; 0 = 79.07; 0 = 0 = 106.4 G I,C =0.28; G II,C = G III,C = 0.79

G 12 = 4.9 GPa; G 13 =G 23 =3.7 GPa; 0 = 53; 0 = 0 = 79 G I,C =0.375;G II,C = G III,C = 1.467

Interface Strength (MPa)

Fracture Toughness (N/mm)

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The material used for fabric laminate is AS4/8552 5HS fabric with cured ply thickness of 0.280 mm and density of 1570 kg/m 3 . The mechanical and interface properties are provided in Table 1 as done for UD laminate. Mechanical properties are taken from the material specification datasheet (Marlett, 2011). The average of compressive and tensile measured mean values is taken in order to obtain modulus values in warp and weft directions. Interlaminar normal and shear strengths are taken as 53 and 79 MPa, respectively (Hexcel, 2016). Fracture toughness values for each mode are taken from Gozluklu (2014). The interface stiffnesses of both materials are calculated by using the below given closed-form expression derived by Turon et al. (2007):

t K E 3  

(1)

Wave speeds in both AS4/8552 UD Prepreg and AS4/8552 5HS Fabric are calculated using the formulas from Coker et al. (2001) and provided in Table 2.

Table 2. Material wave speeds for AS4/8552 UD prepreg and AS4/8552 5HS fabric.

( / ) c m s l 

( / ) c m s s

( / ) V m s c

( / ) c m s R

( / ) c m s l 

UD Prepreg

9377

2852

1831

1816

8045

5HS Fabric

6434

6434

1767

1761

6243

2.2. Geometry and Boundary Conditions

The geometry of the curved CFRP laminate is shown schematically in Fig. 1 for both unidirectional and fabric specimens. The upper and lower arm lengths (l) are 66.36 mm. Inner radius (r i ) and width (w) of the considered specimens are 8.0 mm and 25 mm, respectively. The unidirectional laminate, [ 0 ] 30 , is composed of 30 unidirectional plies of carbon fiber reinforced plastic with a ply thickness of 0.188 mm which corresponds to 5.64 mm total thickness. The fabric laminate, [(45/0) 7 ,45/45/0/45], is composed of 18 5HS fabric plies of carbon fiber reinforced plastic with a ply thickness of 0.28 mm which corresponds to a total thickness of 5.04 mm. Schematic of the experimental configuration (Tasdemir and Coker, 2019) and finite element idealization of load and boundary conditions are shown in Fig. 2. The freely rotating pin and bolts are not considered in the finite element model. This connection and boundary condition case are simulated with kinematic couplings which transfers applied boundary conditions to the specimen from upper and lower load introduction points. The remaining parts of the specimen (from bolt attachment region to free edge) are not modelled since they have no contribution to the stiffness and are far away from the considered curve region. The finite element model of the specimen is allowed to move in the y-direction at the upper load introduction point and rotation around the z-axis is allowed at both upper and lower load introduction points. Allowing rotation around the z-axis accommodates a freely rotating pin clearly. All other degrees of freedom (displacement and rotational degrees of freedom are referred as U and R, respectively) are fixed at both load introduction points. The maximum applied displacement is set to 7 mm and applied at the upper load introduction point as shown below in figure. Load is applied to the specimen with a smooth-step amplitude in order to simulate quasi-static loading. 2.3. Finite Element Model In the three-dimensional finite element model, the bulk region was modelled by reduced integration continuum solid elements (C3D8R) and interfaces between adjacent layers was modelled by 3D cohesive elements (COH3D8). The three-dimensional finite element model of the unidirectional laminate consists of only one layer of cohesive

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elements since only the critical interface predicted by analytical approach is considered. This finite element model includes 936320 linear hexahedral elements of type C3D8R and 46816 linear hexahedral elements of type COH3D8. The total number of elements is 983136 and the total number of nodes is 1043616 with a total of 3130854 degrees of freedom. The three-dimensional finite element model of the fabric laminate includes 17 interfaces and all of them are modelled with cohesive elements and each solid layer is modelled with only one element. This finite element model includes 874800 linear hexahedral elements of type C3D8R and 826200 linear hexahedral elements of type COH3D8. The total number of elements is 1701000 and the total number of nodes is 1772318 with a total of 5316960 degrees of freedom.

w

l

Upper Arm

r i

Curved Region

Lower Arm

l

t

Fig. 1. Specimen geometry.

Kinematic Coupling

U

y (t)

U x =U z =0 R x =R y =0

U x =U y =U z =0 R x =R y =0

Kinematic Coupling

Fig. 2. Schematic of the experimental configuration (Tasdemir and Coker, 2019) and finite element idealization of load and boundary conditions.

5

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2.4. Cohesive Zone Modelling Local stress goes to infinity in the vicinity of the crack tip according to the theory of elasticity that is why fracture mechanics approach is preferred generally in crack initiation and propagation studies. One of the most common fracture mechanics application is cohesive zone modelling (CZM). In this study, bilinear CZM with quadratic damage initiation criterion is employed. The quadratic nominal stress criterion for mixed-mode loading can be expressed as;

2

2

2

t

  

  

  

  

  

  

t t

t t

I

f







0 II II

0 III III

(2)

0

t

I

in which t I , t II , t III are the tractions in each fracture mode as normal, shear, and tearing, respectively. Damage initiates when this equation equals to one. The superscript “0” in denominator of each term is used to express the interfacial strength of that fracture mode. The symbol (< >) used in the normal stress component refers to the Macaulay bracket and it is defined as follow:   2 I I I t t t   (3) As can be understood from the Eqn. 3, the Macaulay bracket in the first term implies that compressive stresses do not cause damage. The interaction between different fracture modes is taken into account both for the onset and propagation of delamination through quadratic nominal stress criterion and Benzeggagh and Kenane (1996) criterion, respectively. The mixed-mode damage propagation criterion is given as; where G IC and G IIC are the fracture toughness values in Mode-I and Mode-II, respectively. G I , G II , and G III are the strain energy release rates for each related fracture mode. The parameter ( η ) is the curve fitting factor obtained from mixed-mode bending (MMB) experiments. 3. Results and Discussion Fig. 3 represents the evolution of damage in the 12 th interface with time after the peak load is attained at which delamination is nucleated in the center of the width in the UD specimen. The shaded areas represent the delamination region inside the specimen. Delamination onset is observed to occur at the center of the width. The delamination then grows in both the transverse (through the width) direction and the longitudinal (along the beam length) direction. When the transverse crack reaches the edge of the specimen at ∆ t= 11 µs, it nucleates an edge crack that propagates in the longitudinal direction along the beam length. Afterwards, a single crack front which consists of the center and edge cracks propagates along the beam length. At 14 µs from delamination initiation, edge crack reaches the specimen arms where it travels faster than center crack for 12 µs. At ∆ t= 53 µs, the edge crack begins to slow down and the center crack catches the edge crack after which the crack front moves at a small speed to the end of the specimen arms.             II    G G G G G G G G G , III IC IIC IC equiv C III II I (4)

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Fig. 3. Delamination propagation path inside the curved unidirectional laminate.

Fig. 4 represents the evolution of damage in the 6 th interface with time after the peak load is attained at which delamination is nucleated in the center of the width in the fabric specimen. As before, the shaded areas represent the delamination region inside the specimen. Since elastic property mismatch between plies with different orientations induces stress concentrations near free edges, it is expected that the delamination initiates at the free edge. Contrary to this expectation, delamination initiated at the half-width of the specimen in the same manner as in the unidirectional laminate. Transverse crack reaches the edge at ∆t= 1 9 µs. After 24 µs from delamination initiation, edge crack reaches the end of the curved region and continues to propagate in the specimen arms. The edge crack reaches intersonic speed at the arm region as shown in the next section , it passes the center crack between ∆t= 35 µs and ∆t= 69 µs. Afterwards, the edge crack slows down, and the center crack catches up with the edge crack moving as a single uniform crack front.

Fig. 4. Delamination propagation path inside the curved fabric laminate.

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Variation of edge crack tip speed with respect to time for UD (red circle marker) and fabric (blue square marker) laminates are shown in Fig. 5. The initial time is taken to be the time at the peak load which also corresponds to the initiation of delamination at the center of the width of the specimen. Shear and critical wave speeds are also shown for both UD and fabric laminates. The edge crack in the UD specimen initiates at 11 µs after peak load and reaches a maximum speed of 5500 m/s in 4 µs at the end of curved region (EOCR UD). The crack tip travels at intersonic speeds for 5 µs as it gradually slows down to sub-Rayleigh wave speeds and finally arrests at 30 µs. In the fabric specimen, the edge crack initiates at 16 µs after the peak load. The edge crack attains an intersonic speed of 2400 m/s in 2 µs. Then it decreases to sub-Rayleigh wave speed of approximately 1700 m/s for 2 µs. However, it immediately increases to a maximum crack tip speed of 2700 m/s at the end of the curved region (EOCR Fabric). Afterwards, at the arm region the crack gradually slows down zero crack tip speed in 13 µs. In both cases, the crack initiates under pure Mode-I loading and attain intersonic speeds at the end of the curved region where shear dominated mixed mode loading is observed. This is consistent with the literature where intersonic speeds are observed in composites only under shear dominated loading (Coker and Rosakis, 2001).

9000

V c_UD

8000

EOCR UD

EOCR Fabric

7000

V c_Fabric

6000

UD Laminate Edge Crack Fabric Edge Crack

5000

4000

3000

Crack Tip Speed (m/s)

C S_UD

2000

C S_Fabric

1000

0

0

5

10

15

20

25

30

35

40

t (µs)

Fig. 5. Edge crack tip speed as a function of time for UD and fabric laminate.

4. Conclusion In this study, 3D explicit finite element analyses of dynamic delamination in UD ([0] 30 ) and fabric ([(45/0) 7 ,45/45/0/45]) curved CFRP laminates are performed under quasi - static loading. The conclusions of this computational study are as follows:  Delamination nucleates at the center of the width for both specimens and at the interface where the radial stress reaches the maximum value. For the fabric laminate, it is interesting to observe that the delamination initiates at the center of the width instead of the free-edges where a material mismatch exists between different layer orientations (Cao et al., 2019; Lagunegrand et al., 2006; Solis et al., 2018).  Delamination initiates at sub-Rayleigh wave speeds under Mode-I condition for both UD and Fabric laminates and reaches intersonic speeds as the delamination grows towards the arms.  For UD laminate, the delamination is initiated at the 12 th interface. This location is determined from 2D

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analysis results and is in good correlation with the experimental results (Tasdemir, 2018) in which the delamination occurs to initiate approximately at 35% of the thickness from inner radius.  For fabric laminate, the delamination is initiated at the 6 th interface which corresponds to 33% of the thickness from inner radius. This results shows good correlation with the experimental results (Tasdemir, 2018) in which the main delamination is also observed at the 33% of the thickness from inner radius.  The propagation of delamination in the fabric laminate is slower than the UD laminate. Acknowledgements The authors would like to acknowledge to RÜZGEM, METU Center for Wind Energy and Turkish Aerospace Industries (TAI) for their continuous support. References ASTM International. Standard D2344 - standard test method for short-beam strength of polymer matrix composite materials. West Conshohocken (PA); 2006. http://dx.doi.org/10.1520/D2344. ASTM International. Standard D6415 - standard test method for measuring the curved beam strength of a fiber-reinforced. West Conshohocken (PA); 2006. http://dx.doi.org/10.1520/D6415. Ata, T. T., 2019. 2D and 3D Finite Element Analyses of Dynamic Delamination in Curved CFRP Laminates. MSc thesis, Middle East Tech Univ. Benzeggagh M.L., Kenane M., 1996. Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos. Sci. Technol. 56, pp. 439–449. Cao D., Hu H., Duan Q., Song P., Li S., 2019. Experimental and three-dimensional numerical investigation of matrix cracking and delamination interaction with edge effect of curved composite laminates. Composite Structures 225. Chang F., Springer G.S., 1986. The strength of fiber reinforced composite bends. Journal of composite materials 20, pp. 30-45. Coker D., Rosakis A.J., 2001. Experimental observations of intersonic crack growth in asymmetrically loaded unidirectional composite plates. Philosophical Magazine A 81(3), pp. 571-595. Gozluklu B., Coker D., 2012. Modeling of the dynamic delamination of L-shaped unidirectional laminated composites. Composite Structures 94, pp. 1430-1442. Gozluklu B., 2014. Modelling of Intersonic Delamination in Curved-Thick Composite Laminates under Quasi-Static Loading, Middle East Technical University. Gozluklu B., Uyar I., Coker D., 2015. Intersonic delamination in curved thick composite laminates under quasi-static loading. Mechanics of Materials 80, pp. 163-182. Gozluklu B., Coker D., 2016. Modeling of dynamic crack propagation using rate dependent interface model. Theoretical and Applied Fracture Mechanics 85, pp. 191-206. Hexcel Corporation, 2016. HexPly 8552 Prod. Data Sheet, pp. 1–6. Lagunegrand L., Lorriot T., Harry R., Wargnier H., Quenisset J.M., 2006. Initiation of free-edge delamination in composite laminates. Composites Science and Technology 66 (10), pp. 1315-1327. Lopes C. S., Camanho P. P., Gürdal Z., Maimí P., González E. V., 2009. Low Velocity Impact Damage on Dispersed Stacking Sequence Laminates. Part II: Numerical simulations. Composites Science and Technology 69 (7–8), pp. 937–947. Marlett K., 2011. Hexcel 8552SAS4 Plain Weave Fabric Prepreg 193 gsm& 38% RC Qualification Material Property Data report. National Institute for Aviation Research, Wichita State University, CAM-RP-20110-006. Martin R.H., 1992. Delamination failure in a unidirectional curved composite laminate. Composite Materials Testing 10, pp. 365–83. Solis A., Sánchez-Sáez S., Barbero E., 2018. Influence of ply orientation on free-edge effects in laminates subjected to in-plane loads. Composites Part B: Engineering 153, pp. 149-158. Sun C.T., Kelly S.R., 1988. Failure in Composite Angle Structures Part I: Initial Failure. Journal of Reinforced Plastics and Composites 7, pp 220 232. Tasdemir B., Coker D., 2019. Comparison of damage mechanisms in curved composite laminates under static and fatigue loading. Composite Structures 213, pp. 190-203. Tasdemir, B., 2018. Fatigue and static behavior of curved composite laminates. MSc thesis, Middle East Techn Univ. Turon A., Davila C.G., Camanho P.P., Costa J., 2007. An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Engineering Fracture Mechanics 74, pp. 1665-1682. Uyar I ., Arca M.A., Gozluklu B., Coker D., 2015. Experimental observations of dynamic delamination in curved [0] and [0/90] composite laminates. Fracture, Fatigue, Failure, and Damage Evolution 5, pp. 189-196. Wimmer G., Kitzmüller W., Pinter G., Wettemann T., Pettermann H.E., 2009. Computational and Experimental Investigation of Delamination in L-shaped Laminated Composite Components. Engineering Fracture Mechanics 76, pp. 2810-2820. Wisnom M., 1996. 3D finite element analysis of curved beams in bending. J Comp Mater 30, pp. 1178–1190. Yavas D., Gozluklu B., Coker D., 2014. Investigation of Crack Growth Along Curved Interfaces in L-shaped Composite and Polymers. Fracture and Fatigue 7, pp. 45-50.

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2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers 10.1016/j.prostr.2019.12.087 ∗ Corresponding author. Tel.: + 90-312-210-4258 ; fax: + 90-312-210-4250. E-mail address: yalcinka@metu.edu.tr 2210-7843 c ⃝ 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review line: Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers. DP steels belong to a group of advanced high strength steels which are mainly developed for the needs of auto motive industry in 1970’s, when low carbon, low alloy steels were in demand. Low alloy content of dual-phase steels have provided high elongation and strength with improved formability along with fatigue and crash resistance with an extra advantage of being light and a ff ordable (see e.g. Tasan et al. (2015) for an overview on the subject). DP steels are composed of brittle martensite islands distributed in a ductile ferrite matrix. The macroscopic mechanical properties ofDP steels are strongly related to their complex microstructure, which on the other hand comes with interesting local ization and failure mechanisms at the microscopic scale. Mechanical response of dual-phase steels can be accurately ∗ Corresponding author. Tel.: + 90-312-210-4258 ; fax: + 90-312-210-4250. E-mail address: yalcinka@met .edu.tr 2210-7843 c ⃝ 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review line: Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers Abstract The aim of this paper is to investigate the e ff ects of microstructural parameters such as the volume fraction, morphology and spatial distribution of the martensite phase and the grain size of the ferrite phase on the plasticity and localized deformation of dual-phase (DP) steels. For this purpose, Voronoi based representative volume elements (RVEs) are subjected to proportional loading with constant stress triaxility. Two alternative approaches are employed in a comparative way to model the plastic response of the ferrite phase, namely, micromechanically motivated crystal plasticity and phenomenological J2 flow theory. The plastic response of the martensite phase, however, is modeled by the J2 flow theory in all the calculations. The predictions of both approaches closely agree with each other in terms of the overall macroscopic response of the DP steels, while clear di ff erences are observed in the localized deformation patterns. The results of the present study are also compared with experimental and computational findings from the literature. c ⃝ 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review line: Peer-review under responsibility of the 1st Internati nal Workshop on Plasticity, Damage and Fracture of Engineering Materials organizers. Keywords: Dual-phase Steel; Representative Volume Element; Crystal Plasticity; Triaxiality; Localization 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials A Micromechanics Based Num rical Investigation of Dual Phase Steels Tuncay Yalc¸inkaya a, ∗ , Go¨nu¨ l O¨ yku¨ Gu¨ngo¨r a , Serhat Onur C¸ akmak a , Cihan Tekog˘ lu b a Middle East Technical University, Department of Aerospace Engineering, Ankara 06800, Turkey b Department of Mechanical Engineering, TOBB University of Economics and Technology, So¨g˜u¨ to¨zu¨ , Ankara 06560, Turkey Abstract The aim of this paper is to investigate the e ff ects of microstructural parameters such as the volume fraction, morphology and spatial distribution of the martensite phase and the grain size of the ferrite phase on the plasticity and localized deformation of dual-phase (DP) steels. For this purpose, Voronoi based representative volume elements (RVEs) are subjected to proportional loading with constant stress triaxility. Two alternative approaches are employed in a comparative way to model the plastic response of the ferrite phase, namely, micromechanically motivated crystal plasticity and phenomenological J2 flow theory. The plastic response of the martensite phase, however, is modeled by the J2 flow theory in all the calculations. The predictions of both approaches closely agree with each other in terms of the overall macroscopic response of the DP steels, while clear di ff erences are observed in the localized deformation patterns. The results of the present study are also compared with experimental and computational findings from the literature. c ⃝ 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review line: Peer-review under responsibility of the 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materi ls organize s. Keywords: Dual-phase Steel; Representative Volume Element; Crystal Plasticity; Triaxiality; Localization 1st International Workshop on Plasticity, Damage and Fracture of Engineering Materials A Micromechanics Based Numerical Investigation of Dual Phase Steels Tuncay Yalc¸inkaya a, ∗ , Go¨nu¨ l O¨ yku¨ Gu¨ngo¨r a , Serhat Onur C¸ akmak a , Cihan Tekog˘ lu b a Middle East Technical University, Department of Aerospace Engineering, Ankara 06800, Turkey b Department of Mechanical Engineering, TOBB University of Economics and Technology, So¨g˜u¨ to¨zu¨ , Ankara 06560, Turkey 1. Introduction 1. Introduction DP steels belong to a group of advanced high strength steels which are mainly developed for the needs of auto motive industry in 1970’s, when low carbon, low alloy steels were in demand. Low alloy content of dual-phase steels have provided high elongation and strength with improved formability along with fatigue and crash resistance with an extra advantage of being light and a ff ordable (see e.g. Tasan et al. (2015) for an overview on the subject). DP steels are composed of brittle martensite islands distributed in a ductile ferrite matrix. The macroscopic mechanical properties ofDP steels are strongly related to their complex microstructure, which on the other hand comes with interesting local ization and failure mechanisms at the microscopic scale. Mechanical response of dual-phase steels can be accurately

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represented only if both the ferrite and martensite phases are modeled realistically. An accurate model should take into account microstructural features of DP steels such as the volume fraction, morphology, carbon content and spatial distribution of the martensite, and the grain size of ferrite (see e.g. Bag et al. (1999); Kang et al. (2007); Avramovic Cingara et al. (2009); Kadkhodapour et al. (2011c)). Therefore, micromechanical modeling of dual-phase steels is crucial to understand and capture their bulk and local constitutive response. In this context, the crystal plasticity finite element approach is a good candidate to take into account various e ff ects at the grain scale. There have been several studies addressing these materials through both experiments and crystal plasticity modelling using representative vol ume elements (see e.g. Kim et al. (2012); Choi et al. (2013); Al-Rub et al. (2015); Jafari et al. (2016); Bong et al. (2017)). In general, the modelling and comparison with experiments have been conducted at regions where uniaxial loading conditions are assumed to occur, yet the e ff ect of stress triaxality has not been discussed before. In the present study, four di ff erent representative volume elements (RVEs) are generated with di ff erent martensite volume fractions and spatial distributions to simulate the overall macroscopic as well as the microscopic behavior of DP steels under constant stress triaxiality loading conditions. The focus of the study has been directed on the similar ities and di ff erences between the two modelling approaches, i.e. crystal and phenomenological plasticity models. All the RVEs used in this study are three-dimensional (3D) and they are produced by polycrystal generation and meshing software Neper; see Quey et al. (2011). Before proceeding with the simulations for the DP steels, crystal plas ticity parameters for the ferrite phase are identified by comparing the overall mechanical response of a 200-grain RVE (containing only randomly oriented ferrite grains) with the experimental tensile data presented in Lai et al. (2016). Once the crystal plasticity parameters are identified, four (approximately) 400-grain RVEs are generated, referred to as DP1, DP2, DP3 and DP4 in the following, each representing a di ff erent DP steel with di ff erent microstructural features (see Fig. 1, where the green and white zones respectively correspond to the ferritic and martensitic phases). The microstructural features for these four RVEs are given in Table 1. All the finite element (FE) calculations in this study are performed by using the commercial software ABAQUS, and all the RVEs are meshed by ten node tetrahedral elements, referred to as C3D10 in ABAQUS terminology. Table 1: Microstructural characteristics of investigated dual-phase steels. Listed data are taken from Lai et al. (2016). 2. Micromechanical model 2.1. Representative volume element generation

Steel

V m ( % )

d f ( µ m )

d m ( µ m )

DP1 DP2 DP3 DP4

15 19 28 37

6.5 5.9 5.5 4.2

1.2 1.5 2.1 2.4

2.2. Constitutive behaviour of di ff erent phases

For the first numerical approach, rate-independent von Mises elastoplastic theory with isotropic hardening is as signed for both the ferrite and martensite grains. The following phenomenological flow equations are used:

θ f β (1 − exp ( − βε P )) f or σ y , f < σ tr y

(1)

σ y , f = σ y 0 , f +

tr y + θ IV ( ε P − ε tr

P ) f or σ y , f > σ tr y

(2)

σ y , f = σ

θ f − θ IV β

σ tr

y = σ y 0 , f +

(3)

ln (

θ f θ IV )

1 β

ε tr

P =

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(a)

(b)

(c)

(d)

Fig. 1: Artificially generated dual-phase steel microstructures that belong to DP1 (a), DP2 (b), DP3 (c), and DP4 (d).

The parameters for the ferrite phase are taken from Lai et al. (2016) and presented below in Table 2, where σ y , f is the current yield stress, σ y 0 , f is the initial yield stress, α f and β are parameters that are related to the average ferrite grain size. θ f is the initial, θ IV is the stage-IV hardening rate and it is taken to be 100 MPa for all the steels investigated in this study. Finally, σ tr y and ε tr P respectively represent the flow stress and the plastic strain. Table 2: Parameter set used for ferrite flow curves(Lai et al. (2016)).

Steel

σ y 0 , f ( MPa )

α f ( GPa )

β ( GPa )

θ f ( MPa )

DP1 DP2 DP3 DP4

250 279 300 307

4.9

11 13 17 20

4895 5980 8925

6

8.9

10.3

10260

The flow behavior of the martensitic phase is governed by the phenomenological equations and parameter sets given by Pierman et al. (2014): σ y , m = σ y 0 , m + k m (1 − exp ( − ε P n m )) (4) where σ y , m is the current yield stress, ε P is the accumulated plastic strain, and σ y 0 , m , k m , n m are material parameters. C m is the martensite carbon content in wt%, whose e ff ect on strain hardening is given below σ y 0 , m = 300 + 1000 C 1 / 3 m . (5) The hardening modulus k m reads

1 n m   a +

q  

bC m 1 + ( C m

k m =

(6)

C 0 )

with a = 33 GPa, b = 36 GPa, C 0 = 0.7, q = 1.45, n m = 120, C m = 0.3 wt%. For the second numerical approach, the crystal plasticity constitutive model is assigned to ferrite phase (see Huang (1991)) while martensite is still governed by the J2 plasticity with isotropic hardening. The plastic slip rate in each slip system, ˙ γ ( α ) , is obtained through a classical power law relation,

0 � � � � � � τ ( α )

g ( α ) � � � � � � 1 / m h αβ � � � ˙ γ β � � �

˙ γ ( α ) = ˙ γ

sign( τ ( α ) )

(7)

where, τ ( α ) is the resolved Schmid stress on the slip systems, ˙ γ

( α ) is the slip resistance on

0 is the initial slip rate, and g

each slip system, which governs the hardening of the material and evolves according to

n ∑ β = 1

˙ g ( α ) =

(8)

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where h αβ is the latent hardening matrix. This matrix measures the strain hardening due to shearing of slip system β on slip system α and it is defined as h αβ = q αβ h ( β ) (9) where q αβ is the latent hardening matrix and h ( β ) represents the self-hardening rate, for which a simple form is used (see e.g. (Peirce et al., 1982))

2 � � � � � h 0 γ

g s − g 0 � � � � �

h αα = h

0 sech

(10)

with h 0 , g 0 , g s are initial hardening rate, the initial slip resistance and saturation value of the slip resistance, respec tively. The exponent a is considered as a constant material parameter. The above presented relations summarizes the main equations for the calculation of plastic slip in each slip system in single crystal plasticity framework. After ob taining the plastic slip value in each slip system, the plastic strain and the stress should be calculated. For more details on the plastic strain decomposition, the incremental calculation of plastic strain and stress the readers are referred to the literature (see e.g. Huang (1991), Yalcinkaya et al. (2008)). The model is employed here for the simulation of plastic behavior polycrystal aggregates, where the grain structure is obtained through Voronoi tesselation using Neper software. In each grain with di ff erent orientation the crystal plasticity model runs resulting in a heterogeneous stress and strain distribution.

2.3. Boundary conditions

It is well established that the stress triaxiality ( T ), which is defined as T = Σ h Σ eq

Σ 11 +Σ 22 +Σ 33 3

(11)

Σ h =

2 √ (

2 + ( Σ

2 + ( Σ

2

1 √

Σ 11 − Σ 22 )

11 − Σ 33 )

33 − Σ 22 )

Σ eq =

with Σ h and Σ eq being respectively the hydrostatic and equivalent von Mises stresses, has a pronounced e ff ect on damage, localization and fracture. In the FE calculations performed here, axisymmetric tension is imposed on the RVEs, while keeping the stress triaxiality constant throughout the entire loading. T = 1 / 3 corresponds to uniaxial tensile loading. For T > 1 / 3, the RVE represents a material point in the center of the minimum cross-section of a notched tensile sample, where the stress triaxiality remains more or less constant during deformation (see e.g Tekog˘ lu and Pardoen (2010) and references therein). In order to enforce periodicity, all the faces of an RVE are kept straight during the entire loading. For this purpose, first, three arbitrary nodes, M 1 , M 2 , and M 3 are selected respectively on the right, top, and front surfaces of the RVE; see Fig. 2. Then u i ( i ∈ { 1 , 2 , 3 } ) displacements of all the other nodes on the surface which contains node M i are coupled to the u i displacement of node M i . Similarly, u i displacements of all the nodes on the surface opposite to the one which contains node M i are coupled to the negative value of the u i displacement of node M i . These couplings are achieved by the following linear equations u 1 ( L 1 , x 2 , x 3 ) − u M 1 = 0 , u 1 (0 , x 2 , x 3 ) + u M 1 = 0 , u 3 ( x 1 , x 2 , L 3 ) − u M 3 = 0 , u 3 ( x 1 , x 2 , 0) + u M 3 = 0 , u 2 ( x 1 , L 2 , x 3 ) − u M 2 = 0 , u 2 ( x 1 , 0 , x 3 ) = 0 . (12)

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5

 

 

 

 

 

 

 

 

 

 



 





 

Fig. 2: A unit cell showing nodes M 1 , M 2 , M 3 and surface names. Node M 1 is on right, M 2 is on top, M 3 is on front surface. Displacement u 1 of node M 1 , u 2 of node M 2 and u 3 of node M 3 are coupled to selected surfaces.

The stress ratios that need to be imposed on the RVE to keep T constant reads

3 T − 1 3 T + 2

Σ 11 = Σ 33 = (13) where the predominant loading is taken to be applied in the x 2 direction. For the nodes on the bottom surface of the RVE, u 2 displacements are fixed while imposing zero tractions in the x 1 and x 3 directions. On the reaming surfaces of the RVE, uniformly distributed loads acting in the surface normal directions are imposed, again letting the tractions in the directions perpendicular to the surface normals to be zero. As a result, the top surface of the RVE is subjected to Σ 22 , left and right surfaces to Σ 11 , and front and back surfaces to Σ 33 ; see Fig. 2. The stress ratios are kept constant and equal those given in Eq. (13) by using the Riks algorithm provided by ABAQUS (see Simulia (2010)). The method to keep the stress triaxality constant described above works perfectly fine for the calculations performed in this paper, where there is no softening. For a more general method to perform RVE calculations under constant stress triaxiality, the reader is referred to Tekog˘ lu (2014). Σ 22 ,

2.4. Overall response of the RVEs

In order to determine the overall response of the RVEs, the fundamental theorem of homogenization

1 V ∫ V

σ i j dV

(14)

Σ i j =

is employed, which relates mesoscopic stress tensor components Σ i j ( i , j ∈ { 1 , 2 , 3 } ) for an RVE with a volume V , to the local Cauchy stress components σ i j in the RVE. Accordingly, Σ i j for an RVE reads Σ i j = ∑ N m = 1 ( ∑ p q = 1 σ { q } i j v { q } ) { m } V (15) where N is the number of elements, p is the total number of integration points ( p = 4 for C3D10 elements), and v is the local volume value at the corresponding integration point. The total volume V of the RVE, which remains as a rectangular prism in the entire course of the deformation, is calculated by simply multiplying the current edge lengths of the RVE: V = L 1 × L 2 × L 3 . The mesoscopic principal strain components for the RVE, E ii , are given by E ii = ln ( L i L i 0 ) , (16)