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Estimation of energy-dissipative process development during crack propagation in an intermetallic alloy
Materials Science and Engineering, A169 (1993) L9-L 11
L9
Letter
Estimation of energy-dissipative process development during crack propagation in an intermetallic alloy
possibilities of the energy approach for investigations of the development of energy dissipation processes in stable fracture processes.
S. M. Barinov
2. Experimental details
High-Tech Ceramics Research Centre, Russian Academy of Sciences, Ozernaya 48, 119361 Moscow (Russian Federation) (Received October 29, 1992; in revised form March 29, 1993)
Abstract A technique based on the non-linear energy principles of fracture for evaluation of the elastic and irreversible constituents of the crack growth resistance is described. These contributions were determined for two Ni3Al-based cast alloys. It was shown that the irreversible part of the crack propagation resistance goes through a maximum with the crack increment, while the linear elastic constituent is generally constant. An alloy containing Cu develops an energy dissipation zone to a greater extent than does an Mo-doped alloy.
1. Introduction The increasing use of brittle materials, e.g. intermetallic alloys, in a wide range of engineering fields demands the detailed characterization of their mechanical behaviour. A number of energy-dissipative processes can operate during crack propagation in these materials, causing marked non-linearity in the load-displacement relationship and rising crack propagation resistance behaviour. It is known that the increasing crack extension resistance is governed by the evolution of the energy dissipation zone (process zone) at the crack tip [1]. To characterize the dissipative zone, its linear dimension is generally used [2-4]. However, increasing energy dissipation density in the process zone may also contribute to the increase in the dissipative energy. Therefore an energy approach based on the global energy balance concept can be useful to describe the total energy dissipation development (zone size and dissipation density) during stable crack growth. Besides, the energy approach is not restricted to the limitations of linear elastic fracture mechanics. The aim of this work was to show, using some model intermetallic alloys, the 0921-5l)93/93/$6.0()
Experiments with specimens of the intermetallic alloys NiaAI-5wt.%Cu and NiaA1-2wt.%Mo were performed. The alloys were fabricated by arc melting in a vacuum furnace and casting of the melts into water-cooled copper moulds. After cutting from ingots, the specimens were finely ground, polished and annealed to avoid residual stresses. The geometry of specimens for mechanical tests was defined by the thickness B - - 4 mm, the width W= 6 mm and the length l--45 mm. The side edge notch was machined with a dimond wheel saw; the notch depth was 0.5 W and the notch tip curvature radius was about 50/~m. The specimens were deformed with a stiff threepoint-loading adjustment that allowed subcritical stable crack growth from the notch tip, using an Instron-type machine, at a cross-head speed of 8 x 10- 7 m s- 1. The span was 24 mm. Using loading-unloading procedures during the stable fracture process, the empirical values of the non-linear fracture resistance parameters were estimated as follows according to ref. 5. It can be assumed that the external force input U is the sum of the elastic stored energy U~, the irreversible energy dissipation Ud and the elastic fracture energy Uef, i.e. U = Ue+ Ua+ Uef
(1)
The specific elastic fracture energy can be defined as [5]
G -dU~-d(u-U~-Ua) dS
(2)
dS
and the crack growth resistance R is
R-
d ( U - U~) dS
(3)
where dS denotes the increment of the fractured surface area. According to ref. 5, the values of Uef and © 1993 - Elsevier Sequoia. All rights reserved
L 10
Letter
U~ may be estimated by the following procedure, which is illustrated in Fig. 1 for the specimen of Ni3A1-5wt.%Cu alloy. Consider an arbitrary point Y of the load ( P ) displacement (u) diagram. If the unloading operation is performed from this point, the extension of the unloading line to the crossing with the u axis indicates the residual irreversible displacement u d at the given point of the fracture process. By shifting such unloading lines to the left by the appropriate increments of residual deformation, the area of the total load-displacement diagram can be separated into distinct parts. One is the area enclosed by the figure OAX'Y'Z', which is the "distinguished elastic diagram". The difference between the real diagram enclosed by OAXYZ and the elastic diagram OAX'Y'Z' corresponds to the irreversible energy dissipation Ud. Under the assumption that AS/S < 1, where S is a momentary value of the fracture area, the fracture parameters G and R can be estimated using eqns. (2) and (3). Consider again an arbitrary point Y of the load-displacement diagram. Suppose that the crack is extended quasi-statically with the loading path from point A to point Y. According to ref. 5, the area enclosed within the region OAXYB is the additional irrecoverable energy consumed in the loading process from A to Y. Assuming that the residual elastic energy can be negligibly small at point B, the additional irrecoverable energy divided by the crack surface increment is equivalent to the crack growth resistance R. The triangular area enclosed by OX'Y'O is related to the specific elastic fracture energy according to eqn. (2). To estimate AS values, the following procedure was used. According to ref. 6, an increment of compliance,
200 x"
-20, of the side edge notch beam specimen due to the crack increment a - a0 can be expressed as y2L2(1 - r 2) {
(4)
where Y is the shape-correcting coefficient for the stress intensity factor [6], L is the span, v is the Poisson ratio and E is the elasticity modulus. For specimens with an a/W ratio in the range 0.5-0.8 the value of the coefficient Y does not depend on a/W and is equal to about 4. A more detailed estimation gives Y= 3.78 [7]. Utilizing loading-unloading procedures, the values of compliance and hence the crack length can be determined.
3.
Results and discussion
Figures 2 and 3 show the dependence of the specific elastic fracture energy G and the crack growth resistance R on the crack length for Ni3A1-5wt.%Cu and Ni3A1-2wt.%Mo alloys respectively. The crack growth resistance increases substantially with increasing crack length, while the linear fracture toughness is approximately constant in the Cu-doped alloy but decreases somewhat with crack extension in the Mo-doped alloy. However, the R curve behaviour has a failing tendency in the final stages of the fracture process. The Mo-doped alloy shows reduced R values compared with the Cu-doped alloy. The ratio R/G indicates the material condition [5]. The states R/G< 1, R/G = 1 and R/G >>1 correspond to elastic (or brittle), elastic-plastic and plastic bodies respectively. The
E "5
x
1 ~iW-a) 2 (W-ao) 2
15-EB
2-20
1
4.0 R
Z
-~
q2
150
A
'-
D O ._J
•
3.0
Y
100
~D ¢.) cO "~
. . . . . . .
0
0.0
(,b
. . . . . . .
2.0 Deflection,
"6
~ . . . . . . . . .
4.0
6.0
1 0 -4 m
Fig.; 1. Load-displacement diagram for side edge notch beam specimen of Ni3A1-5wt.%Cu alloy.
8
U_
8.0
1.0
0.0
I I I II
2.00
I III
2.50
I II
I I I I 11
I I I I I I I II
3.00 Crock
[ I I I I I [ II
3.50 length,
I
4.00
1 0 -5 m
Fig. 2. Specific elastic fracture energy G and crack growth resistance R vs. crack lengthfor Ni3AI-5wt.%Cualloy.
Letter
E
3.0 R
*o ,e--
n"
2,0
c'121
-'-' 1.0
0.0 kl_
IIIll1111[lllllllli[lllllllll]lll[lllJlllllllllll~Liilrl[i
I
2.00 2,50 3.00 3.50 4.00 4.50 5.00 Crock
length,
1 0 -3 m
L 11
The difference between R and G depends on the contributions of energy consumption due to irreversible processes. Figure 4 shows the dependence of R - G on the crack increment for both alloys. The general form of the results indicates the energy dissipation development during the crack propagation process. It increases in the initial stages of the fracture process and then has a tendency to fail. The maximal values of R - G for both alloys investigated correspond to the load maximum on the load-displacement diagrams. Apparently, it can be assumed that the value of R - G characterizes the energy-dissipative zone near the crack tip. In this case the parameters of the dissipative zone, e.g. linear size and dissipation density, reach a critical state at the instant of transition from rising to failing line-displacement behaviour.
Fig. 3. Specific elastic fracture energy G and crack growth resistance R vs. crack length for Ni~Al-2wt.%Mo alloy. 4.
A technique was proposed on the basis of the nonlinear energy principle of fracture considered by Sakai et al. [5] to estimate the development of energy dissipation processes during crack propagation in two intermetallic alloys. It was shown that the crack growth resistance can be divided into elastic and irreversible contributions using load-unloading operations. On the whole, the elastic part does not depend on the crack increment, while the irreversible component of the crack growth resistance goes through a maximum at the instant of transition from rising to failing loaddisplacement behaviour. In the Cu-containing alloy the energy dissipation process development is more pronounced than in the Mo-doped alloy.
4.0
? E 1
3.0
I
c~
~
.~
2.0(~
~
v
1.0
0.0 0.00
.
.
Crock
increment,
Conclusions
.
2.00
1 0 -3 m
Fig. 4. Dependence of non-linear part of crack growth resistance, R - G, on crack increment in alloys containing Cu (curve 1) and Mo (curve 2).
ratio R I G for an alloy containing Cu is greater than that for an Mo-doped alloy. The "saturated" values of R / G are 7.15 and 2.68 for these alloys respectively. Therefore the Cu-doped alloy is more plastic (ductile) than the Mo-doped one.
References
1 2 3 4
A.G. Evans, J. Am. Ceram. Soc., 73(1990) 187-201. G.R. Irwin, Appl. Mech. Res., 3 (1964) 65-72. A. Kelly, Strong Solids, Mir, Moscow, 1976, pp. 192-194. J. F. Knott, Fundamentals' of Fracture Mechanics, Metallurgia, Moscow, 1978, pp. 64-76. 5 M. Sakai, K. Urashima and M. Inagaki, J. Am. Ceram. Soc., 66 (1983) 868-874. 6 W. K. Wilson, Eng. Fract. Mech., 2(1970) 169-172. 7 S.M. Barinov and Yu. L. Krasulin, Ind. Lab., 4 (1985) 56-58.
L9
Letter
Estimation of energy-dissipative process development during crack propagation in an intermetallic alloy
possibilities of the energy approach for investigations of the development of energy dissipation processes in stable fracture processes.
S. M. Barinov
2. Experimental details
High-Tech Ceramics Research Centre, Russian Academy of Sciences, Ozernaya 48, 119361 Moscow (Russian Federation) (Received October 29, 1992; in revised form March 29, 1993)
Abstract A technique based on the non-linear energy principles of fracture for evaluation of the elastic and irreversible constituents of the crack growth resistance is described. These contributions were determined for two Ni3Al-based cast alloys. It was shown that the irreversible part of the crack propagation resistance goes through a maximum with the crack increment, while the linear elastic constituent is generally constant. An alloy containing Cu develops an energy dissipation zone to a greater extent than does an Mo-doped alloy.
1. Introduction The increasing use of brittle materials, e.g. intermetallic alloys, in a wide range of engineering fields demands the detailed characterization of their mechanical behaviour. A number of energy-dissipative processes can operate during crack propagation in these materials, causing marked non-linearity in the load-displacement relationship and rising crack propagation resistance behaviour. It is known that the increasing crack extension resistance is governed by the evolution of the energy dissipation zone (process zone) at the crack tip [1]. To characterize the dissipative zone, its linear dimension is generally used [2-4]. However, increasing energy dissipation density in the process zone may also contribute to the increase in the dissipative energy. Therefore an energy approach based on the global energy balance concept can be useful to describe the total energy dissipation development (zone size and dissipation density) during stable crack growth. Besides, the energy approach is not restricted to the limitations of linear elastic fracture mechanics. The aim of this work was to show, using some model intermetallic alloys, the 0921-5l)93/93/$6.0()
Experiments with specimens of the intermetallic alloys NiaAI-5wt.%Cu and NiaA1-2wt.%Mo were performed. The alloys were fabricated by arc melting in a vacuum furnace and casting of the melts into water-cooled copper moulds. After cutting from ingots, the specimens were finely ground, polished and annealed to avoid residual stresses. The geometry of specimens for mechanical tests was defined by the thickness B - - 4 mm, the width W= 6 mm and the length l--45 mm. The side edge notch was machined with a dimond wheel saw; the notch depth was 0.5 W and the notch tip curvature radius was about 50/~m. The specimens were deformed with a stiff threepoint-loading adjustment that allowed subcritical stable crack growth from the notch tip, using an Instron-type machine, at a cross-head speed of 8 x 10- 7 m s- 1. The span was 24 mm. Using loading-unloading procedures during the stable fracture process, the empirical values of the non-linear fracture resistance parameters were estimated as follows according to ref. 5. It can be assumed that the external force input U is the sum of the elastic stored energy U~, the irreversible energy dissipation Ud and the elastic fracture energy Uef, i.e. U = Ue+ Ua+ Uef
(1)
The specific elastic fracture energy can be defined as [5]
G -dU~-d(u-U~-Ua) dS
(2)
dS
and the crack growth resistance R is
R-
d ( U - U~) dS
(3)
where dS denotes the increment of the fractured surface area. According to ref. 5, the values of Uef and © 1993 - Elsevier Sequoia. All rights reserved
L 10
Letter
U~ may be estimated by the following procedure, which is illustrated in Fig. 1 for the specimen of Ni3A1-5wt.%Cu alloy. Consider an arbitrary point Y of the load ( P ) displacement (u) diagram. If the unloading operation is performed from this point, the extension of the unloading line to the crossing with the u axis indicates the residual irreversible displacement u d at the given point of the fracture process. By shifting such unloading lines to the left by the appropriate increments of residual deformation, the area of the total load-displacement diagram can be separated into distinct parts. One is the area enclosed by the figure OAX'Y'Z', which is the "distinguished elastic diagram". The difference between the real diagram enclosed by OAXYZ and the elastic diagram OAX'Y'Z' corresponds to the irreversible energy dissipation Ud. Under the assumption that AS/S < 1, where S is a momentary value of the fracture area, the fracture parameters G and R can be estimated using eqns. (2) and (3). Consider again an arbitrary point Y of the load-displacement diagram. Suppose that the crack is extended quasi-statically with the loading path from point A to point Y. According to ref. 5, the area enclosed within the region OAXYB is the additional irrecoverable energy consumed in the loading process from A to Y. Assuming that the residual elastic energy can be negligibly small at point B, the additional irrecoverable energy divided by the crack surface increment is equivalent to the crack growth resistance R. The triangular area enclosed by OX'Y'O is related to the specific elastic fracture energy according to eqn. (2). To estimate AS values, the following procedure was used. According to ref. 6, an increment of compliance,
200 x"
-20, of the side edge notch beam specimen due to the crack increment a - a0 can be expressed as y2L2(1 - r 2) {
(4)
where Y is the shape-correcting coefficient for the stress intensity factor [6], L is the span, v is the Poisson ratio and E is the elasticity modulus. For specimens with an a/W ratio in the range 0.5-0.8 the value of the coefficient Y does not depend on a/W and is equal to about 4. A more detailed estimation gives Y= 3.78 [7]. Utilizing loading-unloading procedures, the values of compliance and hence the crack length can be determined.
3.
Results and discussion
Figures 2 and 3 show the dependence of the specific elastic fracture energy G and the crack growth resistance R on the crack length for Ni3A1-5wt.%Cu and Ni3A1-2wt.%Mo alloys respectively. The crack growth resistance increases substantially with increasing crack length, while the linear fracture toughness is approximately constant in the Cu-doped alloy but decreases somewhat with crack extension in the Mo-doped alloy. However, the R curve behaviour has a failing tendency in the final stages of the fracture process. The Mo-doped alloy shows reduced R values compared with the Cu-doped alloy. The ratio R/G indicates the material condition [5]. The states R/G< 1, R/G = 1 and R/G >>1 correspond to elastic (or brittle), elastic-plastic and plastic bodies respectively. The
E "5
x
1 ~iW-a) 2 (W-ao) 2
15-EB
2-20
1
4.0 R
Z
-~
q2
150
A
'-
D O ._J
•
3.0
Y
100
~D ¢.) cO "~
. . . . . . .
0
0.0
(,b
. . . . . . .
2.0 Deflection,
"6
~ . . . . . . . . .
4.0
6.0
1 0 -4 m
Fig.; 1. Load-displacement diagram for side edge notch beam specimen of Ni3A1-5wt.%Cu alloy.
8
U_
8.0
1.0
0.0
I I I II
2.00
I III
2.50
I II
I I I I 11
I I I I I I I II
3.00 Crock
[ I I I I I [ II
3.50 length,
I
4.00
1 0 -5 m
Fig. 2. Specific elastic fracture energy G and crack growth resistance R vs. crack lengthfor Ni3AI-5wt.%Cualloy.
Letter
E
3.0 R
*o ,e--
n"
2,0
c'121
-'-' 1.0
0.0 kl_
IIIll1111[lllllllli[lllllllll]lll[lllJlllllllllll~Liilrl[i
I
2.00 2,50 3.00 3.50 4.00 4.50 5.00 Crock
length,
1 0 -3 m
L 11
The difference between R and G depends on the contributions of energy consumption due to irreversible processes. Figure 4 shows the dependence of R - G on the crack increment for both alloys. The general form of the results indicates the energy dissipation development during the crack propagation process. It increases in the initial stages of the fracture process and then has a tendency to fail. The maximal values of R - G for both alloys investigated correspond to the load maximum on the load-displacement diagrams. Apparently, it can be assumed that the value of R - G characterizes the energy-dissipative zone near the crack tip. In this case the parameters of the dissipative zone, e.g. linear size and dissipation density, reach a critical state at the instant of transition from rising to failing line-displacement behaviour.
Fig. 3. Specific elastic fracture energy G and crack growth resistance R vs. crack length for Ni~Al-2wt.%Mo alloy. 4.
A technique was proposed on the basis of the nonlinear energy principle of fracture considered by Sakai et al. [5] to estimate the development of energy dissipation processes during crack propagation in two intermetallic alloys. It was shown that the crack growth resistance can be divided into elastic and irreversible contributions using load-unloading operations. On the whole, the elastic part does not depend on the crack increment, while the irreversible component of the crack growth resistance goes through a maximum at the instant of transition from rising to failing loaddisplacement behaviour. In the Cu-containing alloy the energy dissipation process development is more pronounced than in the Mo-doped alloy.
4.0
? E 1
3.0
I
c~
~
.~
2.0(~
~
v
1.0
0.0 0.00
.
.
Crock
increment,
Conclusions
.
2.00
1 0 -3 m
Fig. 4. Dependence of non-linear part of crack growth resistance, R - G, on crack increment in alloys containing Cu (curve 1) and Mo (curve 2).
ratio R I G for an alloy containing Cu is greater than that for an Mo-doped alloy. The "saturated" values of R / G are 7.15 and 2.68 for these alloys respectively. Therefore the Cu-doped alloy is more plastic (ductile) than the Mo-doped one.
References
1 2 3 4
A.G. Evans, J. Am. Ceram. Soc., 73(1990) 187-201. G.R. Irwin, Appl. Mech. Res., 3 (1964) 65-72. A. Kelly, Strong Solids, Mir, Moscow, 1976, pp. 192-194. J. F. Knott, Fundamentals' of Fracture Mechanics, Metallurgia, Moscow, 1978, pp. 64-76. 5 M. Sakai, K. Urashima and M. Inagaki, J. Am. Ceram. Soc., 66 (1983) 868-874. 6 W. K. Wilson, Eng. Fract. Mech., 2(1970) 169-172. 7 S.M. Barinov and Yu. L. Krasulin, Ind. Lab., 4 (1985) 56-58.