- Email: [email protected]
Numerical simulation of the essential fracture work method
175
NUMERICAL SIMULATION OF THE ESSENTIAL FRACTURE WORK METHOD Xiao-hong Chen 1, Yiu-Wing Mai l, Pin Tong2 and Liang-chi Zhang I
1Centrefor Advanced Materials Technology (CAMT) Department of Mechanical & Mechatronic Engineering J07 The University of Sydney Sydney, NSW 2006, Australia :Department of Mechanical Engineering Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong
ABSTRACT The Essential Fracture Work Method was simulated by elastic-plastic finite element analysis by using the crack-tip opening angle fracture criterion (CTOA) and the constitutive relation of the material under consideration. The load-displacement curves and JR-curves for samples with different ligament length could be calculated for any given specimen geometry. Finite element analyses was carried out for deep double-edge-notched tension (DENT), deep centre-notched tension (DCNT), single-edge-notched tension (SENT) and centre-lined ligament loading (CLLL) samples. For a grade of high-density polyethylene (HDPE) thin sheets very good agreement between numerical predicted results and experimental data was obtained for DENT and DCNT specimens. Linear relationships between the specific total fracture work and ligament length as well as the final elongation and ligament length were found for all DENT, DCNT, SENT and CLLL samples. The equivalence between we and Jc was also confirmed.
KEYWORDS Essential work of fracture method, numerical simulations, J-integral, specific essential fracture work.
INTRODUCTION
Fracture mechanics provides a reproducible and theoretically sound approach to determine the fracture toughness, which is a measure of the resistance to crack propagation of a material. Fracture characterisation of new ductile materials, such as polymeric thin films and toughened polymers and polymer blends, has greatly stimulated the development of fracture mechanics, which is, in turn, essential for material design and modification to obtain optimal combination of stiffness, strength and toughness. The Griffith energy-balance theory has laid the foundation for fracture mechanics. The energy release rate (G) and stress-intensity factor (K) in linear elastic fracture mechanics (LEFM) are widely used to characterise fracture toughness of glassy polymeric materials subjected to brittle
176
X.H. CHEN, Y.- W. MAL P. TONG, L.C. ZHANG
fracture [1]. If plastic tlow occurs, the energetic approach becomes more complicated. The formation of a large plastic zone prior to crack initiation violates the limit of small-scale yielding condition. The J-integral concept by Rice [2] based upon the deformation theory of plasticity (i.e. non-linear elasticity) is used as an alternative since deformation plasticity and incremental plasticity are equivalent under the conditions of proportional loading or total loading [3, 4]. Nevertheless, crack advance in an elastic-plastic material involves elastic unloading and non-proportional loading around the crack-tip, neither of which can be adequately accommodated by deformation plasticity. Hence, the J-integral theory might break down for a combination of significant plasticity and crack growth [5, 6]. Moreover, it is recognised that it may not be proper to use a "blunting line" to determine the critical value of J-integral from the JR-curve for some ductile polymers and toughened blends [7, 8]. It is also difficult and expensive to be adopted as a standard method to evaluate the impact fracture toughness. Additionally, the J-integral method is almost impossible for fracture toughness characterisation of polymeric thin films. Accordingly, Cotterell, Mai and their coworkers [9-12], using the original ideas of Broberg [13-15], have developed a simple and elegant method, the Essential Work of Fracture (EWF) method, to overcome these aforementioned deficiencies of the J-integral technique. The EWF method has now been taken up by many research groups for experimental measurement of fracture toughness for thin metal sheets, news prints and papers, polymeric thin films, toughened plastics and blends [ 16-25]. Although much work has been done on experimental investigation of the essential and nonessential fracture works, relative few studies are concerned with numerical simulation. Marchal and Delannay [26] did carry out an experimental and numerical investigation of fracture in double-edge-notched tension steel plates. Good agreement was found between the experimental and numerical load-displacement curves up to the maximum load, thickness reduction, CTOD and J-integral. However, the total fracture work, the essential and non-essential works of fracture were not given numerically since the load-displacement curves were only calculated up to the maximum load. In this paper, we will numerically simulate the essential fracture work method. The loaddeflection curves up to total fracture and the J-integral estimates for samples of various material properties can be calculated for any given specimen geometry and size with a suitable crack growth criterion.
FINITE ELEMENT MODELLING
Since crack growth is a local phenomenon, there exists an end region near the crack tip that is crucial for the fracture process. As crack growth is accompanied by permanent deformation of the surrounding material, plastic dissipation in the outer region is not directly associated with the fracture process. An energy criterion can be applied by the assumption that the end-region properties are conserved, that is, they are material properties independent of crack length and extemal load. The total work of fracture Wf can be separated into the essential fracture work imported into the fracture process zone and non-essential fracture work absorbed by the outer region, that is, Wf = We + Wp
(1)
177
Numerical Simulation o f the Essential Fracture Work Method
The specific essential fracture work can be conveniently determined using the deeply cracked specimens, where the height of the outer plastic region may be proportional to the ligament length [9-12, 19-21]. The plastic strain energy is, hence, proportional to the plastic zone volume /TBl 2, where B is initial specimen thickness, l ligament length and/6' plastic zone shape factor. On the other hand, the generalised surface energy is proportional to the new crack surface area BI at the stage of final failure. Hence we have the following general expression for the specific total fracture work wf (=Wy/BI) wf = we +/~pl
(2)
where w e and flWp are the specific essential and non-essential works of fracture, respectively. If w e is a material property and Wp and fl are both independent of l in all tested specimens, there should be a linear relationship when w f is plotted against l in accordance with Eq. 2. By extrapolation of this line to zero ligament length, the interception at the Y-axis and the slope of the line give w e and f l W p , respectively. The load-deflection curves up to the total fracture as well as the J-integrals can be calculated for any given specimen geometry and size with a crack growth criterion. We use the J2 isotropic hardening yield criterion with the true stress-strain relation provided by Mai and Powell [19] for the high-density polyethylene: ~ = 1 2 0 ~ ~ (MPa) with a 2% offset yield strength of 27 MPa. The elastic-plastic crack growth analysis is carried out by the ABAQUS program on an ALPHA STATION at the CAMT with the "DEBOND" and "FRACTURE CRITERION" options [27] for four types of samples (DENT, DCNT, SENT and CLLL), as shown in Fig. 1. Since the users are required to define two distinct initially bonded contact surfaces between which the crack will propagate, potential crack surfaces are modelled as slave and master contact surfaces. The "DEBOND" option is used within the step definition to specify that crack growth may occur between the two surfaces that are initially partially bonded. The crack opening displacement (COD) criterion can be used in ABAQUS to assess crack growth, which can be taken as either the crack mouth opening displacement (CMOD) or the crack-tip opening angle (CTOA) [27]. The CTOA can be easily reinterpreted as the crack opening at a distance behind the current crack tip. We adopt the CTOA criterion in our analysis. Before the CTOA reaches a given critical value, the nodal pairs between the two contact surfaces are tied together and the crack tip becomes blunted with increasing load. When the CTOA criterion is satisfied, we replace the nodal constraints with surface forces at the slave node and the corresponding point on the master surface. Thereafter, upper and lower crack surfaces start to separate with decreasing surface traction. How this surface traction is reduced to zero with numerical incremental step length is controlled by a debonding amplitude curve in the "DEBOND" option. According to the ABAQUS User's Manual [27], the best choice of the amplitude curve depends on the material properties, specified loading and the crack growth criterion. It is suggested that a linear amplitude curve should be normally adequate for rate-independent materials whereas the stresses should be ramped off more slowly at the beginning of debonding for rate-dependent materials. Although HDPE is rate-sensitive, for convenience, we chose a linear function for the amplitude curve in our numerical simulation as an approximation. We found that the results
178
X.H. CHEN, Y.-W. MAI, P. TONG, L.C. ZHANG
were not affected by the unloading procedure so long as the numerical incremental step length does not exceed 0.005. The total numerical step length for the entire simulation is around 0.5.
Fig. 1. Specimen geometry for DENT, DCNT, SENT and CLLL samples. (a) DENT, (b) DCNT, (c) SENT, (d) CLLL. Two input parameters are needed for the CTOA criterion, i.e. the normal displacement between the two crack surfaces and the distance behind the current crack tip. The distance parameter specifies the position behind the current crack tip where the normal displacement is computed by ABAQUS through interpolating the values at the adjacent nodes. We use a constant critical CTOA of 28 ~ for DENT, DCNT, SENT and CLLL samples with input parameters of 1.065 mm for the distance and 1 mm for the normal displacement. There is some sensitivity to the numerical simulations as may be affected by the choice of these two length parameters. A critical CTOA of 28 ~ was based on experimental measurements from the DENT geometry.
N U M E R I C A L RESULTS AND COMPARISON WITH EXPERIMENTAL DATA
The eight-node quadrilateral plane-stress elements are adopted for automatic mesh generation in one-quarter of DENT and DCNT samples and one-half of SENT and CLLL samples due to symmetry by the PATRAN program. The real specimen thickness used in the experiments is 3 mm [19]. The deformed section area and the section force can be output directly by the ABAQUS program.
Numerical Simulation of the Essential Fracture Work Method
179
Fig. 3. (a)-(d) Plots for DCNT samples with ligament length 12 mm. (a) Undeformed and deformed meshes at crack initiation, (b) Von Mises stress at crack initiation, (c) equivalent plastic strain at crack initiation, and (d) equivalent plastic strain at final fracture.
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X..H. CHEN, Y.- W. MAL P. TONG, L. C. ZHANG
Fig. 5. (a)-(d) Plots for CLLL samples with ligament length 6 mm. (a) Undeformed and deformed meshes at crack initiation, (b) Von Mises stress at crack initiation, (c) equivalent plastic strain at crack initiation, and (d) equivalent plastic strain at final fracture.
Numerical Simulation of the Essential Fracture Work Method
181
Figs. 2-5 show the undeformed and deformed meshes at crack initiation, Von Mises stress at crack initiation, equivalent plastic strain at crack initiation and at final failure in the central region around the crack-tip for DENT, DCNT, SENT and CLLL samples, respectively. For the DENT and DCNT geometry the ligament length was 12 mm; and for the SENT and CLLL geometry the ligament length was 6 mm. It is demonstrated that the ligaments are fully yielded in the DENT and DCNT samples and nearly fully yielded in the SENT and CLLL samples at incipient crack initiation. The outer plastic zones at both crack initiation and final failure are indeed elliptic. The latter has a larger size than the former. We can find a thin layer of highly deformed fracture process zone embedded in the plastic region at the final stage of failure. The shapes of the plastic zone are different for DENT, DCNT, SENT and CLLL samples, which indicates that the shape factor flshould be different for the four geometry. The load-displacement curves (P-A ) are obtained for a selection of DENT, DCNT, SENT and CLLL samples so that the areas under these curves can be calculated by integration. Comparisons of the numerical load-displacement curves with the experimental data [19] for the DENT samples are shown in Fig. 6 and the agreement is generally good. It is noted that all these curves have a similar shape. It is also found that there exists a linear relation between the specific total fracture work and ligament length for all the four types of samples. Hence, the specific essential fracture work we can be determined by extrapolating the straight line to zero ligament length.
1=6.32 m m
~,3
-
............ 1=9 m m 1=12 m m I=15.23 m m 1=20 m m
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I.!_
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0
2
I''''
4
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6
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8
10
I''''
12
14
D i s p l a c e m e n t (mm) Fig. 6. Numerical load-displacement curves (P-A) for a selection of DENT samples with Z = 150 mm in comparison with experimental data [19]. Thick lines: experimental result. Thin lines: numerical simulation. The specific total fracture work wf is plotted against the initial ligament length l by linear regression for DENT and DCNT samples in comparison with experimental data in Figs.7-8 and for SENT and CLLL samples in Fig.9, respectively. The J-integral is calculated on four contours surrounding the crack-tip. The smaller the contour label, the closer is the contour to the crack-tip. As an example, the numerical J-integral is plotted against the accumulated incremental crack length up to 3 mm for DCNT samples in Fig. 10, in comparison with the experimental data [ 19]. The agreement is good particularly after-1 mm crack growth. Similar plots for DENT, SENT and CLLL samples have been obtained. Jc and dJR/da are given by the
X.H. CHEN, Y.-W. MA1, P. TONG, L.C. ZHANG
182
interception of the outermost contour J-19 at the Y-axis and the slope of the linear regression of the outermost contour integral Jr-19 up to 3 mm.
500
500 o
E
400
400
300
300
200
200 . o
100
100
E
v
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I
I
I
I
I
I
5
10
15
20
25
30
35
l(mm) Fig. 7. Specific total fracture work wI versus initial ligament length l for DENT samples in comparison with experimental data [ 19]. The dashed line represents the numerical Jc initiation results.
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10
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I (mm) Fig. 8. Specific total fracture work Wf v e r s u s initial ligament length l for DCNT samples in comparison with experimental data [ 19]. Dashed line represents Jc initiation results.
Numerical Simulation of the Essential Fracture Work Method
./
500
183
500
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400
400 r
E
300
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E
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2oo
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5
10
15
20
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30
100
35
/(mm) Fig. 9. Specific total fracture work wi versus initial ligament length l for SENT and CLLL samples. Chained line represents Jc initiation results.
Fig. 10. Numerical JR-curves for four contours versus accumulated incremental crack growth to 3 mm in DCNT samples. Experimental data (A, ~o) taken from reference [19] is also shown. The outermost contour integral J-19 versus accumulated incremental crack length till final failure is shown in Fig. 11 for all four types of samples. We can see that there is a transition at Aa = 1.065 mm, which happens to be the length parameter chosen to define the CTOA. (This transition is sensitive to the two length parameters used in the calculations and further work is needed to clarify this effect). The outmost contour integral (J-19) increases with crack extension almost linearly after the transition for all four geometry. Thus the assumption that d J J d a is constant during crack propagation is valid approximately and so we have the equality
184
X.H. CHEN, Y.- W. MAI, P. TONG, L.C. ZHANG
d J R / d a = 2flWp for CLLL and SENT samples and the equality d J n I da = 4flWp for DENT
and DCNT samples, which were given in [12, 19].
Fig. 11. Outermost integral J-19 versus accumulated incremental crack growth to final failure in all four geometry. TABLE 1. Comparison of W e (kJ/m 2) and flWp (MJ/m 3) with Jc (kJ/m2) and dJRIda (MJ/m 3)
J~
W e
DENT DCNT
Exp. 35.4 36.5
Num. 36.6 36.7
Exp. 34.0 30.0
SENT CLLL
-
-
Num. 40.5 40.3
ddRIda Exp. 56.5 41.6
J~
W e
Exp.
Num. 31.1 32.8
Exp. Num. 34.4 30.9 -
Num. 58.9 43.6
Exp. 43.0 40.0
dJ,,/aa
2/~w. Exp. -
-
Num. 42.0 52.2
Num. 42.6 37.7
Exp. -
Num. 35.6 38.9
The numerical specific essential work of fracture (We) , non-essential work of fracture (flWp), critical J-integral at crack initiation (Jc), and dJR/da for crack growth are listed in Table 1 for the four types of specimens in comparison with the experimental data. The numerical specific essential work of fracture (w e ) has almost the same value of 37 kJ/m 2 for DENT and DCNT samples, which agrees well with the experimental result of 36 kJ/m 2. The numerical specific essential work of fracture ( w e ) also has almost the same value of 40 kJ/m 2 for SENT and CLLL samples. The numerical and experimental J-integrals at crack initiation (Jc) are within the range of 32.5 + 2 kJ/m 2 for all the four types of samples. Hence, w e and Jc can be taken as a material property independent of specimen geometry with the former slightly larger than the latter. However, there is a bigger difference in the specific non-essential works of fracture for the four types of specimen geometry. The numerical and experimental 4~Wp are 57.5 + 2 MJ/m 3 for DENT samples and 42.6 + 1 MJ/m 3 for DCNT samples, which correspond to the experimental and numerical dJR/da around 43 and 39 MJ/m 3. The numerical 2flWp is 42 MJ/m 3
Numerical Simulation of the Essential Fracture Work Method
185
for SENT samples and 52 MJ/m 3 for CLLL samples, which correspond to numerical dJR/da around 36 and 39 MJ/m 3. It turns out that DENT has the most plastic dissipation, followed by CLLL, DCNT and SENT, sequentially. Fig. 12 also shows that a straight-line relationship also exists when the final elongation at break 8f from the load-displacement curves is plotted against the ligament length l for all the four types of samples. The estimated averaged critical crack-tip opening displacement (CTOD) by extrapolating the straight lines to zero ligament length is 0.98 mm for DENT and 0.96 mm for DCNT, which agree very well with the measured value of 1 mm. But somewhat higher values are obtained for SENT and CLLL samples. 20
9
/
..'
//
..: ... ..
SENT: 6f=1.30+0.76/,"'
/ /
/
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\ DeNT (Experimental data \DENT: 6f=0.96+0.37/
' ~ "XDENT(Experimental data) DENT: 6f=0.98+0.46/ 5
10
15
20
25
30
35
/(mm)
Fig. 12. Final elongation ~versus ligament length I for all four types of samples in comparison with experimental data for DENT (A) and DCNT(E) samples [19].
CONCLUSIONS
The validity of numerical simulation has been demonstrated by comparing the predicted loaddeflection curves and JR-curves for a variety of specimen geometry with available experimental results. Good agreement is obtained. It is demonstrated that there exists a thin and highly deformed layer embedded in the plastic region that can be taken as the inner fracture process zone. Straight-line relationships between the specific total fracture work (wf) and ligament length (/) as well as the final elongation ( d f ) and the ligament length (/) are found for all four types of specimen geometry adopted in the numerical simulation. It essential fracture work ( w e ) and the J-integral at crack initiation (Jc) property independent of specimen geometry and external load with than the latter. There also exists a fairly good correlation between
is shown that the specific can be taken as a material the former slightly larger the specific non-essential
fracture work (flWp) and the tearing modulus ( T R = ( E / c r Z ) ( d J R / d a ) ) [1, 6]. Thus, the essential fracture work method has been successfully simulated numerically in good comparison with experimental results.
186
X.H. CHEN, Y.- W. MAI, P. TONG, L.C. ZHANG
ACKNOWLEDGMENTS
Y-WM wishes to thank the Australian Research Council (ARC) for the financial support of this project. XHC is Australian Postdoctoral Research Fellow funded by the ARC.
REFERENCES
1. Atkins, A.G. and Mai, Y.-W. (1985) Elastic and Plastic Fracture, Ellis Horwood Limited, Chichester, UK. 2. Rice, J. R. (1968) J. AppL Mech. 35, 379-386. 3. Budiansky, B. (1959) Tran. ASME. J. Appl. Mech. 26, 259-264. 4. Yu, T.X. and Zhang, L.C. (1996) Plastic Bending, World Scientific, Singapore, Chapter 6 and Appendix. 5. Kanninen, M.F. and Popelar, C.H. (1985) Advanced Fracture Mechanics, Oxford University Press, p 68, Oxford, UK. 6. Anderson, T. L. (1991) Fracture Mechanics: Fundamentals and Applications, CRC Press, p 174, Boca Raton, Florida, USA. 7. Hashime, S. and Williams, J. G. (1986) Polymer 27, 384. 8. Narisawa, I. and Takemori, M.T. (1990) Polym. Eng. Sci. 30,1341. 9. Cotterell, B. and Reddel, J.K. (1977) Int. J. Fract. 13, 267-277. 10. Mai, Y.-W. and Pilko, K.M.(1979) J. Mater. Sci. 14, 386. 11. Mai, Y.-W. and Cotterell, B. (1980) J. Mater. Sci. 15, 2296-2306. 12. Mai, Y.-W. and Cotterell, B. (1986) Int. J. Fract. 32, 105-125. 13. Broberg, K.B. (1968) Int. J. Fracture Mech. 4,11. 14. Broberg, K.B. (1971) J. Mech. Phys. Solids 19, 407-418. 15. Broberg, K.B. (1975) J. Mech. Phys. Solids 23, 215-237. 16. Broberg, K.B. (1982) Engineering Fracture Mechanics 16, 497-515. 17. Wnuk, M.P. and Read, D.T. (1986) Int. J. Fract. 31,161-171. 18. Saleemi, A.S. and Nairn, J.A. (1990) Polym. Eng. Sci. 30, 211-218. 19. Mai, Y.-W. and Powell, P. (1991) Journal of Polymer Science, Part B: Polymer Physics 29, 785-793. 20. Mai, Y.-W. (1993) Int. J. Mech. Sci. 35, 995-1005. 21. Wu, J.S. and Mai, Y.-W. (1996) Polymer Engineering and Science 36, 2275-2288. 22. Levita, G., Parisi, L. and Marchetti, A. (1994) J. Mater. Sci. 29, 4545-4553. 23. Levita, G. (1997) Polymer Engineering and Science 36, 2534-2541. 24. Karger-Kocsis, J., Czigany, T. and Moskala, E.J. (1997) Polymer 38, 4587-4593. 25. Hashemi, S. (1997) Polymer Engineering and Science 37, 912-921. 26. Knockaert, R., Doghri, I., Marchal, Y., Pardoen, T. and Delannay, F. (1996) International Journal of Fracture 81,383-399. 27. ABAQUS/Standard User's Manual, Version 5.7, Hibbitt, Karlsson & Sorensen, INC. (1997), Section 7.8.
NUMERICAL SIMULATION OF THE ESSENTIAL FRACTURE WORK METHOD Xiao-hong Chen 1, Yiu-Wing Mai l, Pin Tong2 and Liang-chi Zhang I
1Centrefor Advanced Materials Technology (CAMT) Department of Mechanical & Mechatronic Engineering J07 The University of Sydney Sydney, NSW 2006, Australia :Department of Mechanical Engineering Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong
ABSTRACT The Essential Fracture Work Method was simulated by elastic-plastic finite element analysis by using the crack-tip opening angle fracture criterion (CTOA) and the constitutive relation of the material under consideration. The load-displacement curves and JR-curves for samples with different ligament length could be calculated for any given specimen geometry. Finite element analyses was carried out for deep double-edge-notched tension (DENT), deep centre-notched tension (DCNT), single-edge-notched tension (SENT) and centre-lined ligament loading (CLLL) samples. For a grade of high-density polyethylene (HDPE) thin sheets very good agreement between numerical predicted results and experimental data was obtained for DENT and DCNT specimens. Linear relationships between the specific total fracture work and ligament length as well as the final elongation and ligament length were found for all DENT, DCNT, SENT and CLLL samples. The equivalence between we and Jc was also confirmed.
KEYWORDS Essential work of fracture method, numerical simulations, J-integral, specific essential fracture work.
INTRODUCTION
Fracture mechanics provides a reproducible and theoretically sound approach to determine the fracture toughness, which is a measure of the resistance to crack propagation of a material. Fracture characterisation of new ductile materials, such as polymeric thin films and toughened polymers and polymer blends, has greatly stimulated the development of fracture mechanics, which is, in turn, essential for material design and modification to obtain optimal combination of stiffness, strength and toughness. The Griffith energy-balance theory has laid the foundation for fracture mechanics. The energy release rate (G) and stress-intensity factor (K) in linear elastic fracture mechanics (LEFM) are widely used to characterise fracture toughness of glassy polymeric materials subjected to brittle
176
X.H. CHEN, Y.- W. MAL P. TONG, L.C. ZHANG
fracture [1]. If plastic tlow occurs, the energetic approach becomes more complicated. The formation of a large plastic zone prior to crack initiation violates the limit of small-scale yielding condition. The J-integral concept by Rice [2] based upon the deformation theory of plasticity (i.e. non-linear elasticity) is used as an alternative since deformation plasticity and incremental plasticity are equivalent under the conditions of proportional loading or total loading [3, 4]. Nevertheless, crack advance in an elastic-plastic material involves elastic unloading and non-proportional loading around the crack-tip, neither of which can be adequately accommodated by deformation plasticity. Hence, the J-integral theory might break down for a combination of significant plasticity and crack growth [5, 6]. Moreover, it is recognised that it may not be proper to use a "blunting line" to determine the critical value of J-integral from the JR-curve for some ductile polymers and toughened blends [7, 8]. It is also difficult and expensive to be adopted as a standard method to evaluate the impact fracture toughness. Additionally, the J-integral method is almost impossible for fracture toughness characterisation of polymeric thin films. Accordingly, Cotterell, Mai and their coworkers [9-12], using the original ideas of Broberg [13-15], have developed a simple and elegant method, the Essential Work of Fracture (EWF) method, to overcome these aforementioned deficiencies of the J-integral technique. The EWF method has now been taken up by many research groups for experimental measurement of fracture toughness for thin metal sheets, news prints and papers, polymeric thin films, toughened plastics and blends [ 16-25]. Although much work has been done on experimental investigation of the essential and nonessential fracture works, relative few studies are concerned with numerical simulation. Marchal and Delannay [26] did carry out an experimental and numerical investigation of fracture in double-edge-notched tension steel plates. Good agreement was found between the experimental and numerical load-displacement curves up to the maximum load, thickness reduction, CTOD and J-integral. However, the total fracture work, the essential and non-essential works of fracture were not given numerically since the load-displacement curves were only calculated up to the maximum load. In this paper, we will numerically simulate the essential fracture work method. The loaddeflection curves up to total fracture and the J-integral estimates for samples of various material properties can be calculated for any given specimen geometry and size with a suitable crack growth criterion.
FINITE ELEMENT MODELLING
Since crack growth is a local phenomenon, there exists an end region near the crack tip that is crucial for the fracture process. As crack growth is accompanied by permanent deformation of the surrounding material, plastic dissipation in the outer region is not directly associated with the fracture process. An energy criterion can be applied by the assumption that the end-region properties are conserved, that is, they are material properties independent of crack length and extemal load. The total work of fracture Wf can be separated into the essential fracture work imported into the fracture process zone and non-essential fracture work absorbed by the outer region, that is, Wf = We + Wp
(1)
177
Numerical Simulation o f the Essential Fracture Work Method
The specific essential fracture work can be conveniently determined using the deeply cracked specimens, where the height of the outer plastic region may be proportional to the ligament length [9-12, 19-21]. The plastic strain energy is, hence, proportional to the plastic zone volume /TBl 2, where B is initial specimen thickness, l ligament length and/6' plastic zone shape factor. On the other hand, the generalised surface energy is proportional to the new crack surface area BI at the stage of final failure. Hence we have the following general expression for the specific total fracture work wf (=Wy/BI) wf = we +/~pl
(2)
where w e and flWp are the specific essential and non-essential works of fracture, respectively. If w e is a material property and Wp and fl are both independent of l in all tested specimens, there should be a linear relationship when w f is plotted against l in accordance with Eq. 2. By extrapolation of this line to zero ligament length, the interception at the Y-axis and the slope of the line give w e and f l W p , respectively. The load-deflection curves up to the total fracture as well as the J-integrals can be calculated for any given specimen geometry and size with a crack growth criterion. We use the J2 isotropic hardening yield criterion with the true stress-strain relation provided by Mai and Powell [19] for the high-density polyethylene: ~ = 1 2 0 ~ ~ (MPa) with a 2% offset yield strength of 27 MPa. The elastic-plastic crack growth analysis is carried out by the ABAQUS program on an ALPHA STATION at the CAMT with the "DEBOND" and "FRACTURE CRITERION" options [27] for four types of samples (DENT, DCNT, SENT and CLLL), as shown in Fig. 1. Since the users are required to define two distinct initially bonded contact surfaces between which the crack will propagate, potential crack surfaces are modelled as slave and master contact surfaces. The "DEBOND" option is used within the step definition to specify that crack growth may occur between the two surfaces that are initially partially bonded. The crack opening displacement (COD) criterion can be used in ABAQUS to assess crack growth, which can be taken as either the crack mouth opening displacement (CMOD) or the crack-tip opening angle (CTOA) [27]. The CTOA can be easily reinterpreted as the crack opening at a distance behind the current crack tip. We adopt the CTOA criterion in our analysis. Before the CTOA reaches a given critical value, the nodal pairs between the two contact surfaces are tied together and the crack tip becomes blunted with increasing load. When the CTOA criterion is satisfied, we replace the nodal constraints with surface forces at the slave node and the corresponding point on the master surface. Thereafter, upper and lower crack surfaces start to separate with decreasing surface traction. How this surface traction is reduced to zero with numerical incremental step length is controlled by a debonding amplitude curve in the "DEBOND" option. According to the ABAQUS User's Manual [27], the best choice of the amplitude curve depends on the material properties, specified loading and the crack growth criterion. It is suggested that a linear amplitude curve should be normally adequate for rate-independent materials whereas the stresses should be ramped off more slowly at the beginning of debonding for rate-dependent materials. Although HDPE is rate-sensitive, for convenience, we chose a linear function for the amplitude curve in our numerical simulation as an approximation. We found that the results
178
X.H. CHEN, Y.-W. MAI, P. TONG, L.C. ZHANG
were not affected by the unloading procedure so long as the numerical incremental step length does not exceed 0.005. The total numerical step length for the entire simulation is around 0.5.
Fig. 1. Specimen geometry for DENT, DCNT, SENT and CLLL samples. (a) DENT, (b) DCNT, (c) SENT, (d) CLLL. Two input parameters are needed for the CTOA criterion, i.e. the normal displacement between the two crack surfaces and the distance behind the current crack tip. The distance parameter specifies the position behind the current crack tip where the normal displacement is computed by ABAQUS through interpolating the values at the adjacent nodes. We use a constant critical CTOA of 28 ~ for DENT, DCNT, SENT and CLLL samples with input parameters of 1.065 mm for the distance and 1 mm for the normal displacement. There is some sensitivity to the numerical simulations as may be affected by the choice of these two length parameters. A critical CTOA of 28 ~ was based on experimental measurements from the DENT geometry.
N U M E R I C A L RESULTS AND COMPARISON WITH EXPERIMENTAL DATA
The eight-node quadrilateral plane-stress elements are adopted for automatic mesh generation in one-quarter of DENT and DCNT samples and one-half of SENT and CLLL samples due to symmetry by the PATRAN program. The real specimen thickness used in the experiments is 3 mm [19]. The deformed section area and the section force can be output directly by the ABAQUS program.
Numerical Simulation of the Essential Fracture Work Method
179
Fig. 3. (a)-(d) Plots for DCNT samples with ligament length 12 mm. (a) Undeformed and deformed meshes at crack initiation, (b) Von Mises stress at crack initiation, (c) equivalent plastic strain at crack initiation, and (d) equivalent plastic strain at final fracture.
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X..H. CHEN, Y.- W. MAL P. TONG, L. C. ZHANG
Fig. 5. (a)-(d) Plots for CLLL samples with ligament length 6 mm. (a) Undeformed and deformed meshes at crack initiation, (b) Von Mises stress at crack initiation, (c) equivalent plastic strain at crack initiation, and (d) equivalent plastic strain at final fracture.
Numerical Simulation of the Essential Fracture Work Method
181
Figs. 2-5 show the undeformed and deformed meshes at crack initiation, Von Mises stress at crack initiation, equivalent plastic strain at crack initiation and at final failure in the central region around the crack-tip for DENT, DCNT, SENT and CLLL samples, respectively. For the DENT and DCNT geometry the ligament length was 12 mm; and for the SENT and CLLL geometry the ligament length was 6 mm. It is demonstrated that the ligaments are fully yielded in the DENT and DCNT samples and nearly fully yielded in the SENT and CLLL samples at incipient crack initiation. The outer plastic zones at both crack initiation and final failure are indeed elliptic. The latter has a larger size than the former. We can find a thin layer of highly deformed fracture process zone embedded in the plastic region at the final stage of failure. The shapes of the plastic zone are different for DENT, DCNT, SENT and CLLL samples, which indicates that the shape factor flshould be different for the four geometry. The load-displacement curves (P-A ) are obtained for a selection of DENT, DCNT, SENT and CLLL samples so that the areas under these curves can be calculated by integration. Comparisons of the numerical load-displacement curves with the experimental data [19] for the DENT samples are shown in Fig. 6 and the agreement is generally good. It is noted that all these curves have a similar shape. It is also found that there exists a linear relation between the specific total fracture work and ligament length for all the four types of samples. Hence, the specific essential fracture work we can be determined by extrapolating the straight line to zero ligament length.
1=6.32 m m
~,3
-
............ 1=9 m m 1=12 m m I=15.23 m m 1=20 m m
"~
I.!_
' ' ' ' 1 ' ' ' '
0
2
I''''
4
I''''-I''''
6
I''''
8
10
I''''
12
14
D i s p l a c e m e n t (mm) Fig. 6. Numerical load-displacement curves (P-A) for a selection of DENT samples with Z = 150 mm in comparison with experimental data [19]. Thick lines: experimental result. Thin lines: numerical simulation. The specific total fracture work wf is plotted against the initial ligament length l by linear regression for DENT and DCNT samples in comparison with experimental data in Figs.7-8 and for SENT and CLLL samples in Fig.9, respectively. The J-integral is calculated on four contours surrounding the crack-tip. The smaller the contour label, the closer is the contour to the crack-tip. As an example, the numerical J-integral is plotted against the accumulated incremental crack length up to 3 mm for DCNT samples in Fig. 10, in comparison with the experimental data [ 19]. The agreement is good particularly after-1 mm crack growth. Similar plots for DENT, SENT and CLLL samples have been obtained. Jc and dJR/da are given by the
X.H. CHEN, Y.-W. MA1, P. TONG, L.C. ZHANG
182
interception of the outermost contour J-19 at the Y-axis and the slope of the linear regression of the outermost contour integral Jr-19 up to 3 mm.
500
500 o
E
400
400
300
300
200
200 . o
100
100
E
v
0
I
I
I
I
I
I
5
10
15
20
25
30
35
l(mm) Fig. 7. Specific total fracture work wI versus initial ligament length l for DENT samples in comparison with experimental data [ 19]. The dashed line represents the numerical Jc initiation results.
500 -
t 500
400
400
E 300
300 E
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DCNT:
"~ u
~:~
200
200 ~
.
0
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.
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.
.
.
I
v
I
i
i
i
5
10
15
20
25
30
.
.
0
35
I (mm) Fig. 8. Specific total fracture work Wf v e r s u s initial ligament length l for DCNT samples in comparison with experimental data [ 19]. Dashed line represents Jc initiation results.
Numerical Simulation of the Essential Fracture Work Method
./
500
183
500
/
400
400 r
E
300
300
E
40.49+20.991
v
g
200 . o
CLLL: wf=40.26+26.12/
2oo
CLLL: J0=32.71 SENT:J0=32.56
100 .
.
.
.
.
I
.
I
I
I
I
I
I
5
10
15
20
25
30
100
35
/(mm) Fig. 9. Specific total fracture work wi versus initial ligament length l for SENT and CLLL samples. Chained line represents Jc initiation results.
Fig. 10. Numerical JR-curves for four contours versus accumulated incremental crack growth to 3 mm in DCNT samples. Experimental data (A, ~o) taken from reference [19] is also shown. The outermost contour integral J-19 versus accumulated incremental crack length till final failure is shown in Fig. 11 for all four types of samples. We can see that there is a transition at Aa = 1.065 mm, which happens to be the length parameter chosen to define the CTOA. (This transition is sensitive to the two length parameters used in the calculations and further work is needed to clarify this effect). The outmost contour integral (J-19) increases with crack extension almost linearly after the transition for all four geometry. Thus the assumption that d J J d a is constant during crack propagation is valid approximately and so we have the equality
184
X.H. CHEN, Y.- W. MAI, P. TONG, L.C. ZHANG
d J R / d a = 2flWp for CLLL and SENT samples and the equality d J n I da = 4flWp for DENT
and DCNT samples, which were given in [12, 19].
Fig. 11. Outermost integral J-19 versus accumulated incremental crack growth to final failure in all four geometry. TABLE 1. Comparison of W e (kJ/m 2) and flWp (MJ/m 3) with Jc (kJ/m2) and dJRIda (MJ/m 3)
J~
W e
DENT DCNT
Exp. 35.4 36.5
Num. 36.6 36.7
Exp. 34.0 30.0
SENT CLLL
-
-
Num. 40.5 40.3
ddRIda Exp. 56.5 41.6
J~
W e
Exp.
Num. 31.1 32.8
Exp. Num. 34.4 30.9 -
Num. 58.9 43.6
Exp. 43.0 40.0
dJ,,/aa
2/~w. Exp. -
-
Num. 42.0 52.2
Num. 42.6 37.7
Exp. -
Num. 35.6 38.9
The numerical specific essential work of fracture (We) , non-essential work of fracture (flWp), critical J-integral at crack initiation (Jc), and dJR/da for crack growth are listed in Table 1 for the four types of specimens in comparison with the experimental data. The numerical specific essential work of fracture (w e ) has almost the same value of 37 kJ/m 2 for DENT and DCNT samples, which agrees well with the experimental result of 36 kJ/m 2. The numerical specific essential work of fracture ( w e ) also has almost the same value of 40 kJ/m 2 for SENT and CLLL samples. The numerical and experimental J-integrals at crack initiation (Jc) are within the range of 32.5 + 2 kJ/m 2 for all the four types of samples. Hence, w e and Jc can be taken as a material property independent of specimen geometry with the former slightly larger than the latter. However, there is a bigger difference in the specific non-essential works of fracture for the four types of specimen geometry. The numerical and experimental 4~Wp are 57.5 + 2 MJ/m 3 for DENT samples and 42.6 + 1 MJ/m 3 for DCNT samples, which correspond to the experimental and numerical dJR/da around 43 and 39 MJ/m 3. The numerical 2flWp is 42 MJ/m 3
Numerical Simulation of the Essential Fracture Work Method
185
for SENT samples and 52 MJ/m 3 for CLLL samples, which correspond to numerical dJR/da around 36 and 39 MJ/m 3. It turns out that DENT has the most plastic dissipation, followed by CLLL, DCNT and SENT, sequentially. Fig. 12 also shows that a straight-line relationship also exists when the final elongation at break 8f from the load-displacement curves is plotted against the ligament length l for all the four types of samples. The estimated averaged critical crack-tip opening displacement (CTOD) by extrapolating the straight lines to zero ligament length is 0.98 mm for DENT and 0.96 mm for DCNT, which agree very well with the measured value of 1 mm. But somewhat higher values are obtained for SENT and CLLL samples. 20
9
/
..'
//
..: ... ..
SENT: 6f=1.30+0.76/,"'
/ /
/
/// zx~
'""~////
15
Ckkk:6i,\=1.24+0, ....../ E
E
%...... /
lO
oO
..:'
Z
/
/
. / nN/." ....)/ ~ / . / " ~ 9' S / / ~ ~"~ 0
~/
\ DeNT (Experimental data \DENT: 6f=0.96+0.37/
' ~ "XDENT(Experimental data) DENT: 6f=0.98+0.46/ 5
10
15
20
25
30
35
/(mm)
Fig. 12. Final elongation ~versus ligament length I for all four types of samples in comparison with experimental data for DENT (A) and DCNT(E) samples [19].
CONCLUSIONS
The validity of numerical simulation has been demonstrated by comparing the predicted loaddeflection curves and JR-curves for a variety of specimen geometry with available experimental results. Good agreement is obtained. It is demonstrated that there exists a thin and highly deformed layer embedded in the plastic region that can be taken as the inner fracture process zone. Straight-line relationships between the specific total fracture work (wf) and ligament length (/) as well as the final elongation ( d f ) and the ligament length (/) are found for all four types of specimen geometry adopted in the numerical simulation. It essential fracture work ( w e ) and the J-integral at crack initiation (Jc) property independent of specimen geometry and external load with than the latter. There also exists a fairly good correlation between
is shown that the specific can be taken as a material the former slightly larger the specific non-essential
fracture work (flWp) and the tearing modulus ( T R = ( E / c r Z ) ( d J R / d a ) ) [1, 6]. Thus, the essential fracture work method has been successfully simulated numerically in good comparison with experimental results.
186
X.H. CHEN, Y.- W. MAI, P. TONG, L.C. ZHANG
ACKNOWLEDGMENTS
Y-WM wishes to thank the Australian Research Council (ARC) for the financial support of this project. XHC is Australian Postdoctoral Research Fellow funded by the ARC.
REFERENCES
1. Atkins, A.G. and Mai, Y.-W. (1985) Elastic and Plastic Fracture, Ellis Horwood Limited, Chichester, UK. 2. Rice, J. R. (1968) J. AppL Mech. 35, 379-386. 3. Budiansky, B. (1959) Tran. ASME. J. Appl. Mech. 26, 259-264. 4. Yu, T.X. and Zhang, L.C. (1996) Plastic Bending, World Scientific, Singapore, Chapter 6 and Appendix. 5. Kanninen, M.F. and Popelar, C.H. (1985) Advanced Fracture Mechanics, Oxford University Press, p 68, Oxford, UK. 6. Anderson, T. L. (1991) Fracture Mechanics: Fundamentals and Applications, CRC Press, p 174, Boca Raton, Florida, USA. 7. Hashime, S. and Williams, J. G. (1986) Polymer 27, 384. 8. Narisawa, I. and Takemori, M.T. (1990) Polym. Eng. Sci. 30,1341. 9. Cotterell, B. and Reddel, J.K. (1977) Int. J. Fract. 13, 267-277. 10. Mai, Y.-W. and Pilko, K.M.(1979) J. Mater. Sci. 14, 386. 11. Mai, Y.-W. and Cotterell, B. (1980) J. Mater. Sci. 15, 2296-2306. 12. Mai, Y.-W. and Cotterell, B. (1986) Int. J. Fract. 32, 105-125. 13. Broberg, K.B. (1968) Int. J. Fracture Mech. 4,11. 14. Broberg, K.B. (1971) J. Mech. Phys. Solids 19, 407-418. 15. Broberg, K.B. (1975) J. Mech. Phys. Solids 23, 215-237. 16. Broberg, K.B. (1982) Engineering Fracture Mechanics 16, 497-515. 17. Wnuk, M.P. and Read, D.T. (1986) Int. J. Fract. 31,161-171. 18. Saleemi, A.S. and Nairn, J.A. (1990) Polym. Eng. Sci. 30, 211-218. 19. Mai, Y.-W. and Powell, P. (1991) Journal of Polymer Science, Part B: Polymer Physics 29, 785-793. 20. Mai, Y.-W. (1993) Int. J. Mech. Sci. 35, 995-1005. 21. Wu, J.S. and Mai, Y.-W. (1996) Polymer Engineering and Science 36, 2275-2288. 22. Levita, G., Parisi, L. and Marchetti, A. (1994) J. Mater. Sci. 29, 4545-4553. 23. Levita, G. (1997) Polymer Engineering and Science 36, 2534-2541. 24. Karger-Kocsis, J., Czigany, T. and Moskala, E.J. (1997) Polymer 38, 4587-4593. 25. Hashemi, S. (1997) Polymer Engineering and Science 37, 912-921. 26. Knockaert, R., Doghri, I., Marchal, Y., Pardoen, T. and Delannay, F. (1996) International Journal of Fracture 81,383-399. 27. ABAQUS/Standard User's Manual, Version 5.7, Hibbitt, Karlsson & Sorensen, INC. (1997), Section 7.8.