Ductile failure prediction of U-notched bainitic functionally graded steel specimens using the equivalent material concept combined with the averaged strain energy density criterion Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

УДК 539.422.23

Ductile failure prediction of U-notched bainitic functionally graded steel specimens using the equivalent material concept combined with the averaged strain energy density criterion

H. Salavati1 and H. Mohammadi2

1 Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, 76169-14111, Iran 2 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, 15875-4413, Iran

In this paper, the ductile fracture of bainitic functionally graded steel has been studied. Fracture tests was performed on U-notched specimens made of bainitic functionally graded steel under mode I. The averaged strain energy density criterion combined with equivalent material concept was employed to predict the ductile fracture of bainitic functionally graded steel. For this purpose, first, based on equivalent material concept, the mechanical properties of virtual brittle functionally graded steel were obtained. Then the averaged value of strain energy density over a well-defined control volume was calculated by finite element analysis for U-notched virtual brittle functionally graded steel. After that, the fracture loads was obtained based on the averaged strain energy density criterion. The agreement between experimental fracture loads and theoretical predictions was good.

Keywords: functionally graded steel, strain energy density, equivalent material concept, U-notch, mode I

DOI 10.24411/1683-805X-2018-14008

Оценка вязкого разрушения образцов с U-образным надрезом из функционально-градиентной бейнитной стали на основе концепции эквивалентного материала и критерия усредненной плотности энергии деформации

H. Salavati1 and H. Mohammadi2

1 Керманский университет им. Шахида Бахонара, Керман, 76169-14111, Иран 2 Технологический университет им. Амира Кабира, Тегеран, 15875-4413, Иран

В статье изучено вязкое разрушение функционально-градиентной бейнитной стали. Испытания на разрушение проводили с использованием образцов с U-образным надрезом из функционально-градиентной бейнитной стали при нагружении типа I. Оценку вязкого разрушения образцов проводили с использованием критерия усредненной плотности энергии деформации и концепции эквивалентного материала. Для этого сначала на основе концепции эквивалентного материала были определены механические свойства виртуальной хрупкой функционально-градиентной стали. Затем с помощью конечно-элементного анализа рассчитывали усредненное значение плотности энергии деформации в некотором контрольном объеме образцов с U-образным надрезом из хрупкой функционально-градиентной стали. Значения разрушающей нагрузки определяли на основе критерия усредненной плотности энергии деформации. Показано хорошее соответствие между экспериментальными и теоретическими значениями разрушающей нагрузки.

Ключевые слова: функционально-градиентная сталь, плотность энергии деформации, концепция эквивалентного материала, U-образный надрез, отрыв

1. Introduction

Prediction of fracture in presence of stress concentrator parts such as notch is a significant issue in the field of structural integrity. The stress field in the vicinity of crack [1, 2]

and notch tip [3] has been studied in various contributions. In addition, three dimensional effects on stress field have been investigated [4]. Different researchers proposed some approaches to predict the fracture of specimens with notches

© Salavati H., Mohammadi H., 2018

Chemical composition of original ferritic and austenitic steels [23]

Table 1

C, % Ni, % Cr, % Mo, % Cu, % Si, % Mn, % S, % P, %

AISI1020 (y) 0.01 9.58 16.69 1.89 0.43 0.53 1.5 0.04 0.04

316L (a ) 0.11 0.07 0.12 0.02 0.29 0.19 0.63 0.08 0.01

[5-13]. A recent approach [14] based on the averaged strain energy density (SED) in a structural control volume has attracted serious attention. This criterion was originally proposed for prediction of fracture of brittle and quasi-brittle materials and it was applied to different types of notch and different fracture modes [15, 16]. Recently, by using the equivalent material concept (EMC) [17], application of the averaged strain energy density criterion has been extended for prediction of ductile fracture [18]. As the equivalent material concept suggests, ductile material having valid fracture toughness is equated with a virtual brittle material having the same elastic modulus and the same fracture toughness, but a different tensile strength. The tensile strength of the equivalent brittle material can be determined by assuming the same values of the tensile strain energy density required for both ductile and virtual brittle materials for crack initiation [17, 18].

Recently, the averaged strain energy density criterion was employed to predict the fracture of notched specimens made of different types of materials such as polyurethane [19], titanium alloys [20], tungsten-copper functionally graded material (FGM) [21, 22] and functionally graded steels [23, 24].

To predict the fracture of notched specimens made of functionally graded materials by the averaged strain energy density criterion, unlike homogeneous materials for which the outer boundary of the control volume is a circular arc, the strain energy density is averaged over a well-

defined control volume which its outer boundary is an oval arc [21-24]. In these works, the notch depth was in brittle or semi-brittle region. Therefore, the strain energy density approach could predict the critical fracture load precisely and it was not necessary to use the equivalent material concept criterion.

In this paper, ductile fracture of bainitic functionally graded steel (FGS) is studied. In the experimental program, bainitic functionally graded steel was produced by electro-slag remelting (ESR). The experiments was carried out on U-notched specimens under mode I loading condition. In the theoretical section, equivalent material concept combined with the averaged strain energy density criterion in functionally graded materials was employed to predict the ductile fracture of bainitic functionally graded steel. The accuracy of the predictions was acceptable.

2. Experimental procedure

Fracture experiments were carried out on bainitic functionally graded steel. The fabrication procedure of this material has been explained in elsewhere [23]. The chemical composition ofthe two materials is summarised in Table 1. Figure 1 shows the fabrication scheme of the functionally graded steel.

Figure 2 shows the Vickers microhardness profile of the aPy composite.

The aPy specimens drawn from the ingots were 90 mm in length, 18 mm in width and 9 mm in thickness. The

Fig. 1. The fabrication scheme of the U-notched FGS specimens

Fig. 2. Vickers microhardness profile versus width in aPy functionally graded steel

geometry is in agreement with ASTM E1820 for the crack arrester configuration. A U-notch was drawn from ferritic steel side of each specimen with a notch root radius of 1 mm. Three different notch depths of 9, 11 and 14 mm were considered in the experiments.

To measure the critical fracture load, three-point bending load was used. The span length between two supports was set to be equal to 72 mm. The load was applied normally to the interface layers at the notch bisector line in order to obtain mode I loading condition (Fig. 3). The tests were performed by a ZWICK 1494 testing machine under load displacement control with constant displacement-rate of 1 mm/min. The load-displacement curves were recorded and used to obtain the critical fracture load.

3. Mechanical properties

The mechanical properties of single phase steels present in the considered functionally graded steels are summarized in Table 2 [24, 25]. As is stated in [23], the thickness of the bainitic layer is approximately equal to 2 mm. In addition, the thickness of a and y graded regions are 2.5 and 3.5 mm, respectively. Moreover, the thickness of the original ferritic a and the original austenitic layers & are 4.5 and 5.5 mm, respectively. The mechanical properties (ultimate tensile strength and plain strain fracture toughness) of a and y graded regions can be described by using exponential function as follow:

X-4.5, (o ut)p

Out (X)( a) = (Ou )

2.5 (a, J„

ut )aA

(1)

aut(x)( Y) = (aut )p e

In—^ 3-5 (Out)p

KIc ( x)( a) = ( KIc )a 0 e

x-4.5 ln (Kic)p

2-5 (Kic)a0

X-9,„( KIc )Y 0

Kc(x)(Y) = (Kic)p e3'5 (Klc)p

(2)

(3)

(4)

where (aut)a0,(aut)p and (aut)Yo are the ultimate tensile strength corresponding to a0, P and y0, respectively, and (KIc)a0, (KIc)p and (KIc)Yo are the plane strain fracture toughness corresponding to a0, P and y0, respectively, as shown in Table 2.

4. Fracture criterion based on averaged strain energy density (ASED) in FGMs and equivalent material concept

According to the averaged strain energy density criterion, brittle fracture occurs when the averaged value of the strain energy density over a well-defined control volume

reaches a critical value Wc [14]: a 2

Wc = —uL. c 2E

(5)

The critical length Rc can be evaluated as follow under plane strain conditions [26]:

Rc =

(1 + v)(5 -8 v)

4n

K

(6)

where KIc is the fracture toughness, aut is the ultimate tensile stress and v is the Poisson's ratio.

For an embedded crack or notch in a FGM specimen, it is assumed that the properties of the material in which the crack or notch tip is placed plays a key role in initiation of fracture of the specimen. Accordingly, the criterion states that fracture initiates when the averaged value of strain energy density over a well-defined control volume reaches the corresponding to the notch tip value of Wc. In a non-homogeneous medium with a smooth unidirectional variation of mechanical properties in the direction x (along the notch depth), the value of Wc can be determined as follow [23]:

F

P ) = = 1 T a i Î W = 18

H_5 = 72_M D O

Fig. 3. Notched beam specimen geometries (dimensions are in mm)

Table 2

Mechanical properties of single phase steels present in the considered functionally graded steels [25]

Yield stress ay, MPa Ultimate tensile strength aut, MPa Fracture toughness KIc, MPa-m05 Poisson's ratio v Elasticity modulus E, GPa Area under stress-strain curve, MPa autEMC> MPa

Ferrite 245 425 45.72 0.33 207 71.4 5437

Bainite 1025 1125 82.08 0.33 207 120.2 7054

Austenite 200 480 107.77 0.33 207 155.6 8026

Wc =

aUt(a)

(7)

2 E (a)

where a is the notch depth.

Unlike homogeneous materials, in functionally graded materials due to a gradual change in material properties, the outer boundary of control volume is no longer a circular arc, but rather an oval arc is assumed. This notion has been used in a number of previous studies on fracture of notched specimens made of functionally graded materials [21, 22, 27]. For a U-notched functionally graded material specimen with a material variation in the direction x under mode I, the outer boundary can be determined by a numerical approach using the following equations. For more details please see Ref. [23]:

x = a -p/2 + (R.(x) + p/2)cos6, y = (Rc( x) + p/ 2)sin 6,

(8)

Rc( x) =

(1 + v)(5 -8 v)

4n

Kfc( x)

aut( x)

Y

where x and y are the coordinates of a point on the outer boundary, 6 is the corresponding angle to that point, a is notch depth, and Rc (x) is the critical length as a function of coordinate x.

The mentioned criterion has been used in different investigations to predict the fracture of notched specimens

made of functionally graded materials [21-23]. However, in this study, as the notch root has been located in a very ductile region, the mentioned criterion cannot give reasonably accurate results. So, the mentioned criterion is modified by using equivalent material concept [17, 28]. As is proposed by Torabi [17], a ductile material having valid fracture toughness is equated with a virtual brittle material having the same elastic modulus and the same fracture toughness, but different tensile strength. The tensile strength of the equivalent brittle material can be determined by assuming the same values of the tensile strain energy density required for both ductile and virtual brittle materials for crack initiation:

ut EMC

2 E

= (SED),

ductile •

(9)

In Eq. (9), a,

ut EMC

is the ultimate tensile strength of the virtual brittle material, E is the elasticity modulus, and (SED)

ductile

is the area under stress-strain curve of the ductile material until necking.

In this paper, the bainitic functionally graded steel has been modelled as a virtual brittle one. To this end, the ultimate tensile strength of single phase steels in the virtual brittle functionally graded steel was calculated by using the values of the area under stress-strain curves provided in Table 2. The values of the ultimate tensile strength of

1.177

1.158

1.140

1.122

1.103

1.085

1.066

2.921 ■ 10-6

2.833 ■ 10-6

2.745 ■ 10-6

2.657 ■ 10-6

2.569 ■ 10-6

2.481 ■ 10-6

2.393 ■ 10-6

b

Fig. 4. Contour lines

of maximum principal stress (a) and strain energy density for notch depth a

= 14 mm (b)

Table 3

Theoretical values of SED and fracture load Fth together with experimental results Fexp (the SED in this table has been evaluated by applying F = 1 N in finite element models)

a, mm Gut EMC' MPa E, GPa Wcemc, MJ/m3 SED, J/m3 Fth> KN Fexp, KN Fexp / Fth

8 7054 120.2 0.172 26.4

9 7054 120.2 0.238 22.5 24.9 1.11

10 7319 129.4 0.343 19.4

11 7594 207 139.3 0.505 16.6 15.3 0.92

12 7879 150.0 0.819 13.5

13 8026 155.6 1.391 10.6

14 8026 155.6 2.523 7.9 7.08 0.90

15 8026 155.6 5.697 5.2

single phase steels in the virtual brittle functionally graded steel GutEMC have been summarized in Table 2. In the virtual brittle functionally graded steel, the ultimate tensile strength of a and y graded regions can be described by Eqs. (1) and (2), respectively, by replacing aut with autemc- After describing the virtual ultimate tensile strengths of each point, the control volume in the virtual brittle functionally graded steel can be determined by Eq. (8)

by replacing G^ with emc •

5. Application of EMC-ASED to predict the fracture loads of U-notched functionally graded steel specimens and results

In order to obtain the averaged value of strain energy density over the control volume, some finite element analyses were carried out by using ABAQUS software version 6.11. It should be noted that the Young's modulus (E = = 207 GPa) and the Poisson's ratio (v = 0.33) have been assumed to be constant along the specimen width. All the finite element analyses were carried out under plane strain conditions and linear elastic hypothesis. Eight-node elements were used in analyses.

By applying an arbitrary load in ABAQUS the fracture load can be obtained as follow:

ap

xT20

ctf O

« 15

% 10

a

ph

Experimental

results

— Numerical

predictions

10 12 Notch depth, mm

14

SED

(10)

Fig. 5. Comparison between experimental results and numerical predictions

where Fap is the applied load in the numerical model, Fth is the theoretical fracture load, SED is the averaged value of strain energy density over the control volume and Wc is the critical value of strain energy density corresponding to the notch tip.

The notch depths from 8 to 15 mm have been considered in numerical analyses in order to assess the effect of notch depth on fracture load. Figure 4 shows the contour lines of maximum principal stress and strain energy density obtained from finite element analyses.

Table 3 summarizes the theoretical predictions and experimental results. As can be seen from the table, the agreement is satisfactory. The experimental and theoretical results have been depicted in Fig. 5. As can be seen from Fig. 5, the experimental and theoretical results follow the same trend.

6. Conclusion

In this paper, equivalent material concept in conjunction with the averaged strain energy density criterion in functionally graded materials was employed to predict the ductile fracture of bainitic functionally graded material. The accuracy of the predictions was reasonable.

References

1. Rice J.R. Some remarks on elastic crack-tip stress fields // Int. J. Solids

Struct. - 1972. - V. 8. - P. 751-758. - doi 10.1016/0020-7683(72) 90040-6.

2. Rousseau C.E., Tippur H. V. Evaluation of crack tip fields and stress intensity factors in functionally graded elastic materials: Cracks parallel to elastic gradient // Int. J. Fract. - 2002. - V. 114. - P. 87-112. -doi 10.1023/a:1014889628080.

3. Filippi S., Lazzarin P., Tovo R. Developments of some explicit formulas

useful to describe elastic stress fields ahead of notches in plates // Int. J. Solids Struct. - 2002. - V. 39. - P. 4543-4565. - doi 10.1016/ S0020-7683(02)00342-6.

4. Pook L.P., Berto F., Campagnolo A. State of the art of corner point singularities under in-plane and out-of-plane loading // Eng. Fract. Mech. - 2017. - V. 174. - P. 2-9. - doi 10.1016/j.engfracmech. 2016.10.001.

5. Lin T., Evans A.G., Ritchie R.O. Statistical analysis of cleavage fracture

ahead of sharp cracks and rounded notches // Acta Metall. - 1986. -V. 34. - P. 2205-2216. - doi 10.1016/0001-6160(86)90166-5.

6. Novozhilov V.V. On a necessary and sufficient criterion for brittle strength // J. Appl. Math. Mech. - 1969. - V. 33. - P. 201-210. - doi 10.1016/0021-8928(69)90025-2.

7. Seweryn A. Brittle fracture criterion for structures with sharp notches // Eng. Fract. Mech. - 1994. - V. 47. - P. 673-681. - doi 10.1016/ 0013-7944(94)90158-9.

8. Taylor D. Predicting the fracture strength of ceramic materials using the theory of critical distances // Eng. Fract. Mech. - 2004. - V. 71. -P. 2407-2416. - doi 10.1016/j.engfracmech.2004.01.002.

9. Leguillon D. A criterion for crack nucleation at a notch in homogeneous

materials // Comptes Rendus l'Academie Des Sci. IIB Mech. - 2001. -V. 329. - P. 97-102. - doi 10.1016/S1620-7742(01)01302-2.

10. Gomez F.J., Elices M., Planas J. The cohesive crack concept: application to PMMA at -60°C // Eng. Fract. Mech. - 2005. - V. 72. -P. 1268-1285. - doi 10.1016/j.engfracmech.2004.09.005.

11. Elices M., Guinea G.V., Gomez J., Planas J. The cohesive zone model: advantages, limitations and challenges // Eng. Fract. Mech. - 2002. -P. 137-163. - doi 10.1016/S0013-7944(01)00083-2.

12. Ayatollahi M.R., Torabi A.R. Investigation of mixed mode brittle fracture in rounded-tip V-notched components // Eng. Fract. Mech. -2010. - V. 77. - P. 3087-3104. - doi 10.1016/j.engfracmech. 2010.07.019.

13. Oh C.-K., Kim Y.-J., Baek J.-H., Kim Y.-P., Kim W.-S. Ductile failure analysis of API X65 pipes with notch-type defects using a local fracture criterion // Int. J. Press. Vessel. Pip. - 2007. - V. 84. - P. 512-525. -doi 10.1016/j.ijpvp.2007.03.002.

14. Berto F., Lazzarin P. Recent developments in brittle and quasi-brittle failure assessment of engineering materials by means of local approaches // Mater. Sci. Eng. R Rep. - 2014. - V. 75. - P. 1-48. - doi 10.1016/j.mser.2013.11.001.

15. Torabi A.R., Campagnolo A., Berto F. Local strain energy density to predict mode II brittle fracture in Brazilian disk specimens weakened by V-notches with end holes // Mater. Des. - 2015. - V. 69. - P. 22-29.

16. Lazzarin P., Berto F., Ayatollahi M.R. Brittle failure of inclined keyhole notches in isostatic graphite under in-plane mixed mode loading // Fatig. Fract. Eng. Mater. Struct. - 2013. - V. 36. - P. 942-955. - doi 10.1111/ffe.12057.

17. Torabi R. Estimation of tensile load-bearing capacity of ductile metallic materials weakened by a V-notch: The equivalent material

concept // Mater. Sci. Eng. A. - 2012. - V. 536. - P. 249-255. -doi 10.1016/j.msea.2012.01.007.

18. Torabi R., Berto F., Campagnolo A., Akbardoost J. Averaged strain energy density criterion to predict ductile failure of U-notched Al 6061-T6 plates under mixed mode loading // Theor. Appl. Fract. Mech. -2017. - V. 91. - P. 86-93. - doi 10.1016/j.tafmec.2017.04.010.

19. Marsavina L., Berto F., Negru R., Serban D.A., Linul E. An engineering approach to predict mixed mode fracture of PUR foams based on ASED and micromechanical modelling // Theor. Appl. Fract. Mech. -2017. - V. 91. - P. 148-154. - doi 10.1016/j.tafmec.2017.06.008.

20. Berto F., Campagnolo A., Welo T. Local strain energy density to assess the multiaxial fatigue strength of titanium alloys // Fract. Struct. Integr. - 2016. - P. 69-79.

21. Mohammadi H., Salavati H., Alizadeh Y., Abdullah A., Berto F. Fracture investigation of U-notch made of tungsten-copper functionally graded materials by means of strain energy density // Fatig. Fract. Eng. Mater. Struct. - 2017. - doi: 10.1111/ffe.12616.

22. Salavati H., Mohammadi H., Yusefi A., Berto F. Fracture assessment of V-notched specimens with end holes made of tungsten-copper functionally graded material under mode I loading // Theor. Appl. Fract. Mech. - 2017. - doi 10.1016/j.tafmec.2017.06.013.

23. Salavati H., Alizadeh Y., Berto F. Fracture assessment of notched bainitic functionally graded steels under mixed mode (I + II) loading // Phys. Mesomech. - 2015. - V. 18. - No. 4. - P. 307-325.

24. Mehran S., Rouhi S., Ramzani B., Barati E. Fracture analysis of functionally graded materials with U- and V-notches under mode I loading using the averaged strain-energy density criterion // Fatig. Fract. Eng. Mater. Struct. - 2012. - V. 35. - P. 614-627. - doi 10.1111/ j.1460-2695.2011.01655.x.

25. Nazari A., Mohandesi J.A., Riahi S. Modified modeling fracture toughness of functionally graded steels in crack divider configuration // Int. J. Damage Mech. - 2011. - V. 20. - P. 811-830. - doi 10.1177/ 1056789510382851.

26. Yosibash Z., Bussiba A., Gilad I. Failure criteria for brittle elastic materials // Int. J. Fract. - 2014. - V. 125. - P. 307-333. - doi 10.1023/ B:FRAC.0000022244.31825.3b.

27. Barati E., Aghazadeh Mohandesi J., Alizadeh Y The effect of notch depth on J-integral and critical fracture load in plates made of functionally graded aluminum-silicone carbide composite with U-notches under bending // Mater. Des. - 2010. - V. 31. - P. 4686-4692. - doi 10.1016/j.matdes.2010.05.025.

28. Torabi R., Campagnolo A., Berto F. Mixed mode I/II crack initiation from U-notches in Al 7075-T6 thin plates by large-scale yielding regime // Theor. Appl. Fract. Mech. - 2016. - doi 10.1016/j.tafmec.2016. 08.002.

Поступила в редакцию 13.02.2018 г.

Сведения об авторах

Hadi Salavati, Assist. Prof., Shahid Bahonar University of Kerman, Iran, [email protected]

Hosein Mohammadi, MSc student, Amirkabir University of Technology, Iran, [email protected]