CC BY
33
УДК 639.42
Fracture investigation of V-notch made of tungsten-copper functionally graded materials
H. Mohammadi1, H. Salavati2, Y. Alizadeh1, F. Berto3, and S.V. Panin4
1 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, 15875-4413, Iran
2 Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, 76169-14111, Iran
3 Department of Engineering Design and Materials, Norwegian University of Science and Technology,
Trondheim, 7491, Norway 4 Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634055, Russia
The fracture of V-notches with end holes made of tungsten-copper functionally graded material under mode I has been studied in this paper. The averaged strain energy density over a well-defined control volume was employed to predict the fracture loads. A numerical approach was used to determine the outer boundary of the control volume. Mechanical properties such as elasticity modulus, Poisson's ratio, fracture toughness KIc, and ultimate tensile stress have been considered to obey the power law function through the specimen width.
Keywords: functionally graded materials, tungsten, copper, V-notches with end hole, mode I loading, strain energy density
Исследование разрушения вольфрамо-медных функциональных градиентных материалов с V-образным надрезом
H. Mohammadi1, H. Salavati2, Y. Alizadeh1, F. Berto3, and S.V. Panin4
1 Технологический университет им. Амира Кабира, Тегеран, 15875-4413, Иран
2 Керманский университет им. Шахида Бахонара, Керман, 76169-14111, Иран
3 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия 4 Институт физики прочности и материаловедения СО РАН, Томск, 634055, Россия
Изучено разрушение вольфрамо-медных функциональных градиентныж материалов с V-образным надрезом с отверстием в вершине при нагружении типа I. Проведена оценка разрушающей нагрузки с использованием усредненной плотности энергии деформации в заданном контрольном объеме. В рамках численного подхода определена внешняя граница контрольного объема. Изменение механических свойств (модуль упругости, коэффициент Пуассона, вязкость разрушения KIc, предел прочности при растяжении) по ширине образца представлено в виде степенной функции.
Ключевые слова: функциональные градиентные материалы, вольфрам, медь, V-образный надрез с отверстием в вершине, нормальный отрыв, плотность энергии деформации
1. Introduction
Prediction of fracture in key-hole notches and V- notches with end holes (VO-notches) which are appeared in structures due to applying a conventional repairing method on
cracked U- and V-notches, respectively, have been taken
into consideration in recent years [1]. Zappalorto and Laz-zarin [2] presented some closed form expressions of stress distribution for VO-notches. Based on fictitious notch rounding (FNR) concept, several investigations have been done on VO-notches [3-8]. The point-stress [9, 10], the mean-
stress [1, 9-11], the maximum tangential stress [1, 11], and the averaged strain energy density [12-14] criteria have been used to assess the fracture of VO-notches under different fracture modes. Compressive [12] and tensile [14] brittle fracture of VO-notches have been studied by means of averaged strain energy density criterion. The compressive mode was assessed using specimens made of isostatic polycrystalline graphite and the tensile mode was investigated using polymethyl methacrylate (PMMA) specimens. It was demonstrated that the averaged strain energy density
© Mohammadi H., Salavati H., Alizadeh Y., Berto F., Panin S.V., 2017
Table 1
Chemical composition and theoretical density of layers
Layer no. WBA, vol % Cu, vol % Chemical composition of layers, wt % Theoretical density of layers, g/cm3
W Ni Mn Cu
1 100 0 90.00 4.00 3.33 2.67 17.176
2 80 20 71.61 3.18 2.65 22.56 15.678
3 60 40 53.41 2.37 1.98 42.23 14.1077
4 40 60 35.42 1.57 1.31 61.70 12.472
5 20 80 17.61 0.78 0.65 80.95 10.758
6 0 100 0.00 0.00 0.00 100.00 8.960
is applicable for VO-notches under both tensile and compressive mode. Recently, Torabi et al. [13] investigated pure mode II brittle fracture of VO-notched specimens made of PMMA. The averaged strain energy density was employed to predict the fracture loads and a good agreement was observed between experimental and theoretical results. Although various works have been done on VO-notches, no assessment have been done on fracture of functionally graded materials (FGM).
Regarding brittle or quasi-brittle static fracture of FGMs, some papers have been published on U- [15-17] and V-notches [18-21]. The static fracture of martensitic functionally graded steels was investigated by Barati et al. and the bainitic one was studied by Salavati et al. In both works, the elasticity modulus and the Poisson's ratio was assumed to be constant, while the ultimate tensile stress and the fracture toughness KIc was considered to vary exponentially through the specimen width. In another work, Barati et al. [15] assessed the effect of notch depth on J-integral and fracture load of U-notched Al-SiC specimens under mode I loading condition. The fracture loads were predicted using three criteria, namely the point-stress, the mean-stress, and the averaged strain energy density. A good agreement was observed between theoretical results which were obtained using averaged SED and MS criteria and experimental ones.
In this paper, the fracture of W-Cu FGM specimens has been studied under mode I loading. Mechanical properties (elasticity modulus, Poisson's ratio, fracture toughness KIc and ultimate tensile stress) have been considered to obey the power law function through the specimen width. The averaged strain energy density criterion has been used to predict the critical fracture loads.
2. Experimental section
2.1. Material and experimental tests
In this paper, W-Cu FGM specimens having six layers have been fabricated by powder metallurgy method. There is a gradual change from a tungsten-based alloy (WBA) with the chemical composition of W-Ni-Mn-Cu respectively as 90-4-3.33-2.67 wt % to pure copper. High pu-
rity (>99%) fine-grained (particle size of 7.5 to 30 ^m) elemental powders were used as received without any modification. The six-layered W-Cu FGM specimens were produced with the content of WBA as 100, 80, 60, 40, 20, and 0 (vol %) in each layer, respectively. Table 1 shows the chemical composition and theoretical density of each layer.
The composition of WBA (layer 1) was made mixing tungsten, copper, nickel, and manganese powders in a high-energy ball milling system. In order to obtain the composition of layers 2 to 5, WBA and copper powders were mixed in a ball milling system with respect to the content of each item (Table 1). The powders of layers 1 to 6 were stacked layer-by-layer into a D2 steel die with the diameter of 25 mm
)
Fig. 1. Fabrication procedure of W-Cu FGM specimens
Fig. 2. SENB specimen geometries (dimension are in mm)
and height of 50 mm. Thicknesses of the layers were considered to be 1.0, 0.5, 0.5, 0.5, 0.5, and 1.0 mm for each layer, respectively. So the amount of powder of each layer was determined using the theoretical densities. The stacked powders were pressed at room temperature under a pressure of 500 MPa. The green compact was ejected from the die. The sintering process was performed in a vacuum furnace at a heating rate of 100C/min. Sintering temperature and pressure were set to be 10500C and 10-5 mbar, respectively. Specimens were sintered at that temperature for 3 hours and were cooled in the furnace. The fabrication procedure of W-Cu FGM specimens is depicted in Fig. 1. Both sides of the sintered specimens were ground and the final thickness of 4 mm was reached by grinding. Then notched specimens were drawn by a high precision electro-discharge wire cut machine.
Three-point bending tests were performed on single edge VO-notched (SENB) specimens. Thickness, width, and length of the specimens were set to be equal to 2, 4, and 18 mm, respectively. Dimensions were in agreement with ASTM E1820. The span length between two supports was set to be 16 mm. The load was applied on the notch bisector line in order to obtain mode I loading condition. Figure 2 illustrates loading condition and geometry of the specimens. VO-notches were drawn from WBA side. Three notch tip radii p of 0.3, 0.4, 0.6 mm and three notch opening angles 2a of 600, 900, and 1200 were considered in experimental tests. In all specimens, the depth of the center of curvature b was equal to 1 mm. In total, nine different geometrical configurations were prepared.
For each geometrical configuration, three specimens were tested providing 27 new experimental data for mode I loading in W-Cu FGM. The experiments were performed by a ZWICK 1494 testing device under displacement control with a constant displacement rate of 0.05 mm/min. A VO-notched specimen before and after three-point bending test is shown in Fig. 3. The fracture loads of each test and also the mean value Fexp are summarized in Table 2.
2.2. Mechanical properties
In order to present some functions for mechanical properties, the specimens were assumed to consist 3 regions, WBA region, copper region, and the graded region which
Fig. 3. A VO-notched W-Cu FGM specimen before (a) and after three-point bending test (b)
connects the WBA region to copper region (Fig. 2). In other words, 4 innermost layers (layers 2 to 5) were considered as the graded region. The thicknesses of regions were determined using metallography techniques. The averaged thickness of WBA region, the graded region, and copper region were obtained to be 0.95, 2.30, and 0.75 mm, respectively. The power law function was employed to describe mechanical properties of the graded region. Using the properties of boundary conditions, mechanical properties of the graded region such as elasticity modulus, Pois-son's ratio, ultimate tensile stress, and fracture toughness KIc can be obtained using the following functions:
E(Z) - [ECu - E WBA ]
2Z + h 2h
+ E
'WBA'
v ( Z ) - [v
Cu '
' WBA
2Z +
/ \n
+ v
Gui Z) - fa. -aiÉ 1
2h
f2Z + h^1
WBA'
V
KIc (Z ) - [ KIc. - KIc™. 1
2h
2Z+ 2h
(1)
+ G„
7
\n2
+ K
Table 2
Experimental values of fracture load FJ, F2, and F3 and mean value F
p, mm
0.3
0.4
0.6
0.3
0.4
0.6
0.3
0.4
0.6
b, mm
2a
60°
60°
60°
90°
90°
90°
120°
120°
120°
F1, N
103.5
119.9
118.7
115.1
105.4
120.1
87.6
105.4
104.1
99.2
96.8
120.6
95.3
115.4
131.0
82.0
113.2
108.9
85.5
98.5
124.9
90.3
101.1
106.4
95.6
103.3
135.7
F , N
A exp ' Ly
96.1
105.1
121.4
100.2
107.3
119.2
8.4
107.3
116.2
Mechanical properties of boundary layers
Table 3
Boundary regions Elasticity modulus E, GPa Poisson's ratio v Ultimate tensile stress aut, MPa Fracture toughness K1c, MPa-m05 n ni n2
Cu 112.7 0.34 188.9 88.5 0.9646 1.312 2.78
WBA 289.5 0.29 447.3 4.52
where E is the elasticity modulus, v is the Poisson's ratio, Gut is the ultimate tensile stress, KIc is the fracture toughness, n, nx, n2 are the power law exponents, Z is the thickness coordinate of the graded region - h/2 < Z < h/2, and h is the thickness of the graded region (h = 2.3 mm). Indexes Cu and WBA shows the properties of copper and WBA. The boundary condition properties and power law exponents of the Eq. (1) are presented in Table 3.
3. Averaged SED criterion over a well-defined control volume
The averaged SED criterion states that brittle fracture occurs when the averaged value of the SED over a well-defined control volume is equal to a critical value Wc [22]. Wc is a material-dependent value which is independent of notch geometry. For brittle materials, Wc can be evaluated as follow [22]:
Wc . c 2E
(2)
For a VO-notch under mode I loading, the control volume assumes the crescent shape which is centered in relation to the notch bisector line as shown in Fig. 4. Rc is the critical length which is measured along the notch bisector line.
The critical length Rc can be evaluated as follow under plane strain conditions [23, 24]:
Rc =
(1 + v)(5 -8 v)
4n
K,
where KIc is the fracture toughness,
(3)
Gut is the ultimate
tensile stress and v is the Poisson's ratio. The outer radius of the control volume is equal to Rc + r0 as shown in Fig. 4. The value of r0 can be determined as follow [25]: q -1 2n- 2a
ft =-
-p, q -
(4)
q n
In homogeneous materials, Rc is constant all over the specimen. In comparison, in nonhomogeneous materials, Rc varies point by point due to a gradual change in material properties. In a nonhomogeneous medium with a smooth unidirectional variation in mechanical properties in the X-direction (along the notch depth), the outer boundary of the control volume is assumed oval shape as shown schematically in Fig. 5.
For a VO-notched FGM specimen with a material variation in the X-direction under mode I, the outer boundary can be determined by a numerical approach using the following equations. For more details please see Ref. [19]:
x = a - r0 + (Rc(x) + r0)cos6,
j = ( Rc( x ) + ft)) sin 0,
(1 + v( x ))(5 -8v( x ))
(5)
Rc( x):
4n
Kic( x)
a ut( x)
\2
Fig. 4. Control volume for a VO-notched homogeneous specimen under mode I
Fig. 5. Control volume for a VO-notched FGM specimen under mode I
Fig. 6. Control volume boundary in the case p = 0.1
where x and y are the coordinates of a point on the outer boundary, 0 is the corresponding angle to that point (see Fig. 5), a is notch depth, Rc(x) is the critical length as a function of X-coordinate, and r0 can be evaluated by Eqs. (4).
4. The SED criterion in fracture assessment of VO-notched W-Cu FGM specimens
As stated before, according to SED criterion, fracture occurs when the averaged strain energy density over a well-defined control volume reaches its critical value Wc. In functionally graded materials, Wc is not constant all over the specimen. The initiation of fracture depends on the value of Wc corresponding to the notch tip. As mentioned earlier, Rc changes point by point. Figure 6 shows two cases of control volume for a constant notch tip radius and notch opening angle and different notch depths equal to 1.10 and 1.35 mm. It can be observed from these figures that by increasing the notch depth, the control volume becomes larger which is due to the fact that the fracture toughness of the material increases and the ultimate tensile stress decreases.
In order to obtain the averaged value of SED over the control volume, some finite element analyses were carried out by using ABAQUS software version 6.11. Various geometries have been considered in numerical analyses keeping constant the notch center of curvature depth (b= 1 mm). For each geometry, a model was created and the boundary of control volume which was obtained by a numerical approach was accurately defined. All the finite element analyses were carried out under plane strain conditions and linear elastic hypothesis. Eight-node elements were used in
analyses. The contour lines of the maximum principal stress and strain energy density are shown in Fig. 7. Table 4 present the values of Wc, the averaged strain energy density SED, and theoretical fracture load Fth for different values of notch tip radius, notch depths, and notch opening angles. In Table 4, the values of mechanical properties such as elasticity modulus, Poisson's ratio, ultimate tensile stress, and fracture toughness Klc for the point which the notch tip is located there are presented.
A comparison between average values of experimental fracture loads Fexp and theoretical ones Fth calculated by SED criterion has been made in Table 5. A good agreement can be clearly seen from this table.
In addition in Fig. 8, a comparison between experimental fracture loads and theoretical predictions was made for notch opening angles 60o, 90o, and 120°. The plots are given for a constant center of curvature depth (b = 1 mm). In other words, in these plots, the notch depth a varies by changing the notch tip radius. The agreement between experimental and theoretical fracture loads is quite satisfactory. The trend of experimental and theoretical fracture loads are in a good agreement.
A synthesis in terms of the square root value of the SED averaged over the control volume, normalized with respect to the critical energy of the material, as a function of the notch tip radius is shown in Fig. 9. The normalized scatter band is plotted based on new data for SENB VO-notched W-Cu FGM specimens under mode I loading. At first sight, it can be noted that the scatter is almost independent of the notch tip radius. Almost all values fall inside a band ranges from 0.8 to 1.2. The synthesis confirms that SED is applicable to predict the fracture load of VO-notched specimens made of functionally graded materials.
Max. principal stress r- 9.715 - 9.128 8.541 7.954 ^ 7.368
a
SED (Avg:75%) a 1.718 • 10-4 =: 1.512 • 10 1.306 . 10
_
1.101 . 10 8.949 . 10
b
Fig. 7. Contour lines of the maximum principal stress (a) and the SED (b) in the case p = 0.4 mm and 2a = 60° with a = 1.4 mm
Table 4
Theoretical values of SED and fracture load Fth (the SED in this table has been evaluated applying F = 1 N
in finite element models)
Model no. p, mm a, mm 2a E, GPa V out, MPa KIc, MPa-m05 Wc, MJ/m3 SED, J/m3 Fth, N
1 0.05 1.05 60° 280.911 0.292 443.076 4.534 0.349 61.140 75.599
2 0.10 1.10 60° 276.800 0.294 440.110 4.562 0.350 53.596 80.797
3 0.15 1.15 60° 272.738 0.295 436.813 4.615 0.350 46.355 86.868
4 0.20 1.20 60° 268.712 0.296 433.246 4.696 0.349 41.104 92.180
5 0.25 1.25 60° 264.715 0.297 429.448 4.812 0.348 37.864 95.916
6 0.30 1.30 60° 260.741 0.298 425.446 4.968 0.347 35.638 98.689
7 0.35 1.35 60° 256.788 0.299 421.262 5.169 0.346 33.875 100.998
8 0.40 1.40 60° 252.852 0.300 416.911 5.421 0.344 32.268 103.206
9 0.45 1.45 60° 248.932 0.301 412.406 5.727 0.342 31.025 104.933
10 0.50 1.50 60° 245.025 0.303 407.758 6.093 0.339 29.689 106.902
11 0.55 1.55 60° 241.131 0.304 402.976 6.524 0.337 28.185 109.302
12 0.60 1.60 60° 237.249 0.305 398.068 7.023 0.334 25.832 113.700
13 0.05 1.05 90° 280.911 0.292 443.076 4.534 0.349 61.361 75.443
14 0.10 1.10 90° 276.800 0.294 440.110 4.562 0.350 54.126 80.401
15 0.15 1.15 90° 272.738 0.295 436.813 4.615 0.350 46.944 86.321
16 0.20 1.20 90° 268.712 0.296 433.246 4.696 0.349 42.296 90.871
17 0.25 1.25 90° 264.715 0.297 429.448 4.812 0.348 38.885 94.649
18 0.30 1.30 90° 260.741 0.298 425.446 4.968 0.347 36.195 97.927
19 0.35 1.35 90° 256.788 0.299 421.262 5.169 0.346 34.204 100.511
20 0.40 1.40 90° 252.852 0.300 416.911 5.421 0.344 32.600 102.681
21 0.45 1.45 90° 248.932 0.301 412.406 5.727 0.342 31.319 104.440
22 0.50 1.50 90° 245.025 0.303 407.758 6.093 0.339 30.012 106.324
23 0.55 1.55 90° 241.131 0.304 402.976 6.524 0.337 28.432 108.827
24 0.60 1.60 90° 237.249 0.305 398.068 7.023 0.334 26.037 113.251
25 0.05 1.05 120° 280.911 0.292 443.076 4.534 0.349 55.140 79.606
26 0.10 1.10 120° 276.800 0.294 440.110 4.562 0.350 51.455 82.461
27 0.15 1.15 120° 272.738 0.295 436.813 4.615 0.350 46.218 86.996
28 0.20 1.20 120° 268.712 0.296 433.246 4.696 0.349 41.562 91.670
29 0.25 1.25 120° 264.715 0.297 429.448 4.812 0.348 38.963 94.554
30 0.30 1.30 120° 260.741 0.298 425.446 4.968 0.347 36.426 97.616
31 0.35 1.35 120° 256.788 0.299 421.262 5.169 0.346 34.550 100.006
32 0.40 1.40 120° 252.852 0.300 416.911 5.421 0.344 32.942 102.146
33 0.45 1.45 120° 248.932 0.301 412.406 5.727 0.342 31.751 103.726
34 0.50 1.50 120° 245.025 0.303 407.758 6.093 0.339 30.309 105.802
35 0.55 1.55 120° 241.131 0.304 402.976 6.524 0.337 28.711 108.296
36 0.60 1.60 120° 237.249 0.305 398.068 7.023 0.334 26.206 112.886
5. Conclusion VO-notched specimens made of W-Cu FGM under mode I
In the present research, the averaged strain energy den- loading. The main findings of this research have been sum-sity criterion was employed to predict the critical load of marized as follows.
Table 5
Comparison between experimental Fexp and theoretical Ft
fracture loads
p, mm b, mm 2a F N 1 exp' N F*, N Fexp / Fth
0.3 1 60o 96.1 98.689 0.973
0.4 1 60o 105.1 103.206 1.018
0.6 1 60o 121.4 113.700 1.068
0.3 1 90o 100.2 97.927 1.024
0.4 1 90o 107.3 102.681 1.045
0.6 1 90o 119.2 113.251 1.052
0.3 1 120o 88.4 97.616 0.906
0.4 1 120o 107.3 102.146 1.050
0.6 1 120o 116.2 112.886 1.029
180 120H 60
Ph
2a = 60° b = 1 mm
o Experimental results -Theoretical solution
"ra
0.0 0.2 0.4 0.6
Notch tip radius, mm
12060)
2a = 90° b = 1 mm
TU
0.0 0.2 0.4 0.6
Notch tip radius, mm
¡z; 180
CÖ O
<D
3
120-
60
2a = 120° » = 1 mm
m
0.0 0.2 0.4 0.6
Notch tip radius, mm
Fig. 8. Comparison between experimental and theoretical fracture loads for b = 1 mm and 2a = 60° (a), 90° (b), and 120° (c)
1.4
^1.0
0.6
o 2a = 60°
o 2a = 90°
□ 2a =120°
------^— s
s s s
0.2
0.4
0.6 p, mm
Fig. 9. Scatter band summarizing new data for VO-notched W-Cu FGM under mode I
The approach based on the SED was successfully extended to the FGMs with variant elasticity modulus and Poisson's ratio. The outer boundary of the control volume was determined numerically.
The average discrepancy between experimental and theoretical fracture loads was found to be 4.5%.
The shape and size of control volume were appropriate for W-Cu functionally graded materials.
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Поступила в редакцию 02.09.2016 г.
Сведения об авторах
Hosein Mohammadi, MSc student, Amirkabir University of Technology, Iran, [email protected] Hadi Salavati, Assist. Prof., Shahid Bahonar University of Kerman, Iran, [email protected] Yoness Alizadeh, Assoc. Prof., Amirkabir University of Technology, Iran, [email protected] Filippo Berto, Prof., Norwegian University of Science and Technology, Norway, [email protected] Sergey V Panin, Prof., ISPMS SB RAS, [email protected]
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