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Prediction of fracture instability in thin sheets by finite element method
0013-7944484 $3.00 + .wl Pe~onprcSSLtd.
PREDICTION OF FRACTURE INSTABILITY IN THIN SHEETS BY FINITE ELEMENT METHOD S. C. MISHRAt Department of Applied Mechanics, R.E.C., Rourkela, India and B. K. PARIDA Department of Aeronautical Engineering, Indian Institute of Technology, Kharagpur, India Abstract-A fracture instability study has been made for narrow thin panels with central cracks. The quasi-static energy approach in conjunction with finite element method has been employed for the prediction of critical crack length corresponding to a given applied stress level. The corresponding plane-stress fracture toughness, &< has been shown to differ considerably with regard to that obtained from tests following the standard practice. However, the plane stress fracture toughness value obtained experimentally based on the predicted critical crack length is found to be in close agreement with the theoretically obtained value.
RECENT advances in linear elastic fracture mechanics (LEFM) as a quantitative aid to fail safe design of structures has enabled the designers to focus their attention on crack initiation characteristics as a means of fracture control. This is because of the increasing awareness amongst the design engineers for the best utilization of the existing high strength materials to their full potential. Consequently the prediction of fracture instability has become important for predicting the occurence of catastrophic inservice failures of structural components. The fracture ins~bility problem was first studied by Irwin[l] through the concept of crack extension force. Subsequently Krafft et al.f2], Srawley et al. [3], Clausing[4], Glucklich[5], Gurney and Hunt[6], Parida[7] etc. have studied different aspects of crack instability problem. Irwin and Gurney’s findings of fracture instability involved the balance between the fracture surface energy of the material and strain energy stored in the system. Gurney’s analysis was based on quasi-static energy approach, which was quite valuable in solving linear and non-linear problems. Parida’s finding for the crack instability problem was that before the commencement of unstable crack growth leading to ultimate fracture, the crack-edges in thin sheets undergo local buckling, which can be construed as a safe limit for stable crack growth under a given loading condition, Wheeler et al.[8] have made extensive study of the fracture instability problem through experimental techniques for obtaining the R-Curve, which is a measure of fracture toughness used for selection and quality assurance of thin section materials. For solving the crack instability problem in any engineering structure with arbitrary crack configurations and loading one has to naturally rely upon one of the numerical method, amongst which FEM is well suited. The application of FEM to predict fracture instability was attempted by Anderson[9], who used the R-curve concept. Subsequently Parks[lO] and Chow er al.[ll] etc. used FEM in predicting fracture instability with a somewhat different approach to the problem. Among the stability criteria proposed by various investigators, the quasi-static energy analysis advocated by Gumey[6] is considered to be quite versatile in eharacterising the stability of crack propagation in structures. The stability criteria based on this analysis are expressed as
For dP/P > 0 under load-controlled
condition;
tlecturer. $Assistant Professor. 623
624
S. C. MISHRA and B. K. PARIDA
for da/u > 0 under displacement controlled condition. In eqns (I) and (2), G, denotes fracture toughness. G : strain energy release rate GSF : Geometric Stability Factor. P and u are respectively load and displacement at the point of application of load. The equations indicate that the stability of cracking depends on material property, G, as well as GSF, which is related to specimen geometry and mode of loading.
FINITE ELEMENT FORMULATION The application of FEM to crack instability problem can be made by two different approaches; namely, the SIF approach and the strain energy approach. The major drawback of the stress intensity approach is the necessity for incorporating in the finite element structural idealization an extremely fine grid near the crack tip in order to achieve a satisfactory accuracy. The strain energy approach on the other hand has gained steady ~pularity in recent years due to the fact that the contribution of strain energy from the region close to the crack tip is negligible compared to that of the rest of the structure. The method, however, requires the computation of several strain energy values at different crack lengths in order to accurately evaluate the strain energy release rate. The strain energy release rate G is given by the expression
where system system system
64 = increase in crack surface area, dU = the change in the strain energy stored in the = V, - U,, G = strain energy release rate per crack tip, U, = strain energy stored in the for a semi-crack length a, under constant load system and tJ, = strain energy stored in the for a semi-crack length (a + a), under constant load system.
PROCEDURE FOR EVALUATING THE CRACK INSTABILITY CRITERION According to the quasi-static energy principle, the fracture instability criterion can be represented as
for load controlled condition. This geomet~c normalized geometric stability factor as:
stability factor can be no~alized
to give the
(3) where W = width of the panel, t = thickness of the panel, G = strain energy release rate and A = crack surface area. For any particular crack geometry, the strain energy release rate may be computed with the help of finite element method, assuming the material to be linearly elastic. Then the derivative of the strain energy release rate with respect to increase in crack surface area, i.e. aG/aA can be found out either under constant load condition or constant displa~ment condition. W and t are known from the specimen dimensions. So, once 8GjaA is evaluated, the NGSF can be computed, which determines the stability for the problem under investigation.
625
Prediction of fracture instability in thin sheets by finite element method Table I. Calculated normalized geometric stability factors (FEM results) Crack No. of observation 1 2 3 4 5 6 7
Calculated
2a
length
‘G’
aG
W
in mm
from FEM
aA
= NGSF
0.1 0.2 0.25 0.3 0.4 0.5 0.6
14 28 35 42 56 70 84
0.3420 0.6136 0.8032 1.0029 1.466 3.7728 4.0472
0.0388 0.0439 0.0472 0.0535 0.1225 0.1058
4.4263 3.828 3.294 2.555 2.273 1.830
In the present investigation, a finite element elastic analysis has been carried out for predicting the fracture instability in a thin 2024-T3 aluminium alloy specimen (280 mm x 140 mm x 1.Omm), with a centre crack, loaded under uniaxial tension of magnitude 10 kgf/mm2 (98.1 MPa). Different crack length to sheet width ratios have been considered for this instability analysis. The method is based on quasi-static energy analysis and the NGSF for different centre cracks have been obtained by using finite element technique based on energy principle. For different crack lengths corresponding to the crack length to sheet width ratios of 0.1, 0.2, 0.25, 0.3, 0.4, 0.5 and 0.6 the strain energy release rate, G have been evaluated. In the present analysis, plane eight noded quadratic quadrilateral iso-parametric elements were used with displacement formulation. A quarter of the panel was discretized into number of elements varying from 50 to 84 and two degrees of freedom were considered at each node. The partial derivative of the strain energy release rate, G, with respect to increase in crack surface area can be expressed in difference form between any two convenient crack lengths. For example dG/BA for two different crack lengths can be written as: aG G2 - G, - = ___ = GSF corresponding
aA A,-A,
to second crack length.
where G2 = strain energy release rate for second crack length, Gi = strain energy release rate for first crack length, A, = crack surface area for second crack length and A, = crack surface area for the first crack length. In the present computation for NGSF at all the crack lengths mentioned above, 2a/W = 0.1 was taken as the reference crack length with respect to which aG/aA for other crack lengths were found out, which have been listed in Table 1. A graph plotted with NGSF vs 2a/W gave the critical 2a/W corresponding to the crack instability condition as 2aJ W = 0.69. In the present analysis for the panel of 140 mm width and an average stress level of 10 kgf/mm2 (98.1 MPa) considered, the critical crack length at fracture, 2u, ought to be 96.6 mm. From this, one could as well predict the plane stress fracture toughness value theoretically for such a narrow thin panel. It amounts to saying that the fracture instability criterion based on NGSF is akin to fracture toughness criterion, since both predict a certain critical crack length for a given applied stress level. However, the plane stress fracture toughness value of 2024-T3 for relatively narrow and thin panels is not very well defined. Broek [ 131has presented graphs showing variations of plane stress fracture toughness, Ki, with respect to panel width and thickness separately. However, from these graphs it is not possible to obtain an exact value of the plane stress fracture toughness for the panel considered. An experimental verification was therefore made to check the accuracy of above prediction. EXPERIMENTAL VERIFICATION With a view to verify the theoretical predictions concerning the critical crack length, corresponding to a given stress level, an experimental study was-made following the standard practice for plane stress fracture toughness testing. Two identical thin sheet specimens (2024-T3 Alclad) of the same panel size (420 mm x 140 mm x 1 mm) were tested. Initially at the centre of the plate a circular hole of 3 mm
626
S. C. MISHRA and 8. K. PARIDA Table 2. Experimental determination
of plane stress fracture toughness
(1)
Initial crack length 2a (2)
Critical crack length at failure 2a, (3)
Average stress at failure kgf/mm* (4)
It
36
38.9
2t
35
34.2
3t
96.6
98.552
44
84
94.1
5$.
90
100.5
26.0554 (255.603) 26.032 (255.373) 9.0328 (88.611) 10.42 (102.22) 9.866 (96.785)
Specimen NO.
Y* = 1.77 f 0.227 (2a/W) - 0.51 (2~jW)* + 2.7 (2a/W’)’ tRefers to tests carried without anti-buckling guide. *Refers to tests carried with anti-buckling guide.
. .
(0 2 2a/w
KI,
kgf . mm-312
(““(F2) 1.8515 1.8497 2.6187 2.51 2.569
212.756 (65.996) 207.934 (64.505) 166.648 (51.697) i85.32 ’ (57.49) 179.679 (55.74)
I: 0.7)
diameter was made, then two precision sawcuts of 1 mm length were made on both sides of the circular concentrator in a direction normal to the loading direction. These specimens with proper gripping a~angements were mounted in the fatigue testing machine SUP-50). Real fatigue cracks were developed, using small amplitude dynamic loading o,~, = 8.0 kgf/mm* (78.48 MPa}, R = 0.625. On both sides of the central con~ntrator, the total fatigue crack length developed was about 25% of the width of the plate. The generation of the fatigue crack was restricted to around 35 mm in accordance with the ASTM recommendation[ 131,which stipulates the initial crack length for fracture toughness testing of narrow thin panels to be less than one third of the panel width, i.e. 20 I W/3. After generating the cracks to the desired value the dynamic loading was discontinued and static tensile load was very slowly increased starting from zero, while measuring the further crack growth at both ends of the crack carefully. The final load and the corresponding crack length pertaining to ultimate fracture were noted and the plane stress fracture toughness values computed, considering ASTM r~ommended width correction factor, Y. These have been noted in Table 2. It was found that the average value of K,, was about 210 kgf/mmm3’* (65.142 MN m-312). However, if a fracture toughness computation is made based on the critical crack length obtained from quasi-static energy principle, Ki, is found to be equal to 178.665 kgf mm-3’2 (55422 MN m-312). This clearly shows a large discrepancy with respect to the experimentally obtained values. One possible reason for this could be the widely different initial crack lengths considered. With a view to further verify the accuracy of FEM prediction with regard to the crack-instability, a few identical specimens were tested with an initial crack length of around 96.6mm (corresponding to 2a,/ W = 0.69) or less which was developed under a similar low amplitude fatigue loading and tested for plane-stress fraeture toughness without and with antibuckling guides. The quasi-static load was gradually increased and increase in crack length was carefuhy noted as in the previous fracture toughness testing. The critical crack length at failure, 2a,, co~esponding average stress, Ok, finite width correction factor, Y, and computed values of Ki, are shown in Table 2. RESULTS AND DISCUSSIONS Table 1 shows the strain energy release rate, G and aG/aA values for different crack lengths, with the help of which the normalized geometric stability factor for the different centre cracks have been calculated. It is obvious that to get the accurate value of NGSF, the strain energy release rate, G must be calculated accurately as far as possible. The calculation of G, depends upon the finite element idealization of the continuum and the choice of displacement function. An important point taken note of during the analysis of result was that for f&ding the aG/&4 value for any particular crack length, always the numerical ~fferentiation was made with respect to a crack fength smaller than the former, because in this case the direction of crack propagation
Prediction of fracture i~stabi~ty in thin sheets by finite element method PREDICTIONS
BY
CHOW AND FUNG( 11 CLAUSING CHOW
5.0
A
1 -------
AN0
20/W Fig, 1. Prediction
(I
) FEM RESULTS
LAU (REPORTED
IN11
1 a
-
of fracture instability from FEM analysis.
were already known, In this case the incremental crack surface area at a particular crack length was the actual area. Figure I shows a plot of NCSF vs 2aj W. The NGSF values corresponding to different crack lengths, &t&n& in the present investigation by the finite element technique, have been plotted in this figure. It is observed that almost all the points lie on a curve which indicates that the stability decreases in a non-linear manner as the crack length is increased for a given stress level. BY extending the curve to the line corresponding to NGSF = 1.0, it is seen that the expected crack instability begins at the crwk length to sheet width ratio, 2alW = 0.69. In Fig. 1, the results Of instability studies by other investigators pertaining to centre cracks have also been shown, based on ex~~mental as well as numerical computations. Clausing’s[4] results have been indicated by a continuous dotted curve, where as the results of Chow and FungEl l], Chow and Lau[l2] have been shown as discrete points. However, none of the above results predict the exact crack instability condition. Nevertheless, the results obtained by these investigators show a similar trend, i.e. NGSF values decrease in a non-linear manner as crack length increases. In Table 2, it may be noted that the finite width.~orre~tion factor, Y, has been used for specimen NOS. 3 to 5 since, its limit of applicability for centre cracks (as per ASTM-STP 410) extends up to 2a/W = 0.7. From the observations for the specimen No. 3, which was tested without any anti-buckling guide, the value of Kr, was found to be slightly lower than the value predicted theoretically. The other two specimens, namely Nos. 4 and 5 were tested with antibnc~ing guides. It is seen from Table 2 that Ki, values obtained for both these specimens are quite close to each other and are also close to the Ki, value theoretic~ly predicted by FEM analysis employing quasi-static energy approach, i.e. K,, = 178.665 kgf mm-3’t (55.422 MN m-3’2). The error in the Kt, value, obtained without any anti-bucking guide with respect to the average K,, value obtained with antibudging guides happens to be around 9 percent.
A study of fracture instability for a narrow thin panel based on the quasi-static energy principle has been made, which is found to have dose agreement with the experimental results, in terms of the critical crack length at failure for a given stress level, The value of plane-stress fracture toughness, Ki, computed using the data from these fracture tests with anti-buckling guide also agrees well with that predicted theoretically by the FEM-analysis based on quasi-static energy
628
S. C. MISHRA and B. K. PARIDA
principle, although it differs considerably from that obtained recommended plane stress fracture toughness test practice.
following the standard
ASTM
REFERENCES [I] Cl. R. Irwin, Fracture mechanics. lsf Symp. on Naval Structural Mechanics. Pergamon Press, New York (1958). [2] J. M. Krafft, A. M. Sullivan and R. V. Boyle, Effect of dimensions on fast fracture instability of notched sheets. Proc. Cruck Propagation Symp., Cranfield, Vol. 1. pp. 8-28, 1961. [3] J. E. Srawley and W. P. Borwn, Fracture toughness testing and its applications. ASTM STP 381, 133-193 (1965). [4] D. P. Clausing, Crack instability in linear elastic fracture mechanics. Int. J. Fract. Mech. 5, 21 l-227 (1969). [5] J. Glucklich, On crack stability in some fracture tests. Engng Fracture Mech. 3, 333-344 (1971). [6] C. Gurney and I. Hunt, Quasi-static crack propagation. Proc. Roy. Sot. A 299, 508-524 (1967). [7] B. K. Parida, Crack edge instability-a criterion for safe crack propagation limit in thin sheets. Proc. 3rd Colloquium on Fracture, London, 8-10 Sept. 1980. [8] C. Wheeler, J. N. Eastabrook, D. P. Rooke, K. H. Schwalbe, W. Setz and A. U. De Koning, Recommendations for the measurement of R-curves using centre-cracked panels. J. Strain Analysis 17(4), 205213 (1982). [9] H. Anderson, A finite element representation of stable crack growth. J. Mech. Phys. Solids 21, 337-366 (1973). [IO] D. M. Parks, A stiffness derivative finite element technique for determination of crack tip SIFs. Int. J. Fracture, 10, 437-502 (1974). [I 11 C. L. Chow and W. C. Fung, Prediction of fracture instability by a finite element method. Fracture Mech. Tech. II, 1537-1545. [l2] C. L. Chow and P. M. Lau, Stability conditions in quasi-static crack propagation
for constant strain energy release rate. J. Engng Muter. Tech. 96, 41-48 (1974). [13] D. Broek, Elementary Engineering Fracture Mechanics, Chap. VIII, pp. 177-185. Noordhoff, Groningen, Netherlands (1974). (Received 15 August 1983; received for publication 29 November 1983)
PREDICTION OF FRACTURE INSTABILITY IN THIN SHEETS BY FINITE ELEMENT METHOD S. C. MISHRAt Department of Applied Mechanics, R.E.C., Rourkela, India and B. K. PARIDA Department of Aeronautical Engineering, Indian Institute of Technology, Kharagpur, India Abstract-A fracture instability study has been made for narrow thin panels with central cracks. The quasi-static energy approach in conjunction with finite element method has been employed for the prediction of critical crack length corresponding to a given applied stress level. The corresponding plane-stress fracture toughness, &< has been shown to differ considerably with regard to that obtained from tests following the standard practice. However, the plane stress fracture toughness value obtained experimentally based on the predicted critical crack length is found to be in close agreement with the theoretically obtained value.
RECENT advances in linear elastic fracture mechanics (LEFM) as a quantitative aid to fail safe design of structures has enabled the designers to focus their attention on crack initiation characteristics as a means of fracture control. This is because of the increasing awareness amongst the design engineers for the best utilization of the existing high strength materials to their full potential. Consequently the prediction of fracture instability has become important for predicting the occurence of catastrophic inservice failures of structural components. The fracture ins~bility problem was first studied by Irwin[l] through the concept of crack extension force. Subsequently Krafft et al.f2], Srawley et al. [3], Clausing[4], Glucklich[5], Gurney and Hunt[6], Parida[7] etc. have studied different aspects of crack instability problem. Irwin and Gurney’s findings of fracture instability involved the balance between the fracture surface energy of the material and strain energy stored in the system. Gurney’s analysis was based on quasi-static energy approach, which was quite valuable in solving linear and non-linear problems. Parida’s finding for the crack instability problem was that before the commencement of unstable crack growth leading to ultimate fracture, the crack-edges in thin sheets undergo local buckling, which can be construed as a safe limit for stable crack growth under a given loading condition, Wheeler et al.[8] have made extensive study of the fracture instability problem through experimental techniques for obtaining the R-Curve, which is a measure of fracture toughness used for selection and quality assurance of thin section materials. For solving the crack instability problem in any engineering structure with arbitrary crack configurations and loading one has to naturally rely upon one of the numerical method, amongst which FEM is well suited. The application of FEM to predict fracture instability was attempted by Anderson[9], who used the R-curve concept. Subsequently Parks[lO] and Chow er al.[ll] etc. used FEM in predicting fracture instability with a somewhat different approach to the problem. Among the stability criteria proposed by various investigators, the quasi-static energy analysis advocated by Gumey[6] is considered to be quite versatile in eharacterising the stability of crack propagation in structures. The stability criteria based on this analysis are expressed as
For dP/P > 0 under load-controlled
condition;
tlecturer. $Assistant Professor. 623
624
S. C. MISHRA and B. K. PARIDA
for da/u > 0 under displacement controlled condition. In eqns (I) and (2), G, denotes fracture toughness. G : strain energy release rate GSF : Geometric Stability Factor. P and u are respectively load and displacement at the point of application of load. The equations indicate that the stability of cracking depends on material property, G, as well as GSF, which is related to specimen geometry and mode of loading.
FINITE ELEMENT FORMULATION The application of FEM to crack instability problem can be made by two different approaches; namely, the SIF approach and the strain energy approach. The major drawback of the stress intensity approach is the necessity for incorporating in the finite element structural idealization an extremely fine grid near the crack tip in order to achieve a satisfactory accuracy. The strain energy approach on the other hand has gained steady ~pularity in recent years due to the fact that the contribution of strain energy from the region close to the crack tip is negligible compared to that of the rest of the structure. The method, however, requires the computation of several strain energy values at different crack lengths in order to accurately evaluate the strain energy release rate. The strain energy release rate G is given by the expression
where system system system
64 = increase in crack surface area, dU = the change in the strain energy stored in the = V, - U,, G = strain energy release rate per crack tip, U, = strain energy stored in the for a semi-crack length a, under constant load system and tJ, = strain energy stored in the for a semi-crack length (a + a), under constant load system.
PROCEDURE FOR EVALUATING THE CRACK INSTABILITY CRITERION According to the quasi-static energy principle, the fracture instability criterion can be represented as
for load controlled condition. This geomet~c normalized geometric stability factor as:
stability factor can be no~alized
to give the
(3) where W = width of the panel, t = thickness of the panel, G = strain energy release rate and A = crack surface area. For any particular crack geometry, the strain energy release rate may be computed with the help of finite element method, assuming the material to be linearly elastic. Then the derivative of the strain energy release rate with respect to increase in crack surface area, i.e. aG/aA can be found out either under constant load condition or constant displa~ment condition. W and t are known from the specimen dimensions. So, once 8GjaA is evaluated, the NGSF can be computed, which determines the stability for the problem under investigation.
625
Prediction of fracture instability in thin sheets by finite element method Table I. Calculated normalized geometric stability factors (FEM results) Crack No. of observation 1 2 3 4 5 6 7
Calculated
2a
length
‘G’
aG
W
in mm
from FEM
aA
= NGSF
0.1 0.2 0.25 0.3 0.4 0.5 0.6
14 28 35 42 56 70 84
0.3420 0.6136 0.8032 1.0029 1.466 3.7728 4.0472
0.0388 0.0439 0.0472 0.0535 0.1225 0.1058
4.4263 3.828 3.294 2.555 2.273 1.830
In the present investigation, a finite element elastic analysis has been carried out for predicting the fracture instability in a thin 2024-T3 aluminium alloy specimen (280 mm x 140 mm x 1.Omm), with a centre crack, loaded under uniaxial tension of magnitude 10 kgf/mm2 (98.1 MPa). Different crack length to sheet width ratios have been considered for this instability analysis. The method is based on quasi-static energy analysis and the NGSF for different centre cracks have been obtained by using finite element technique based on energy principle. For different crack lengths corresponding to the crack length to sheet width ratios of 0.1, 0.2, 0.25, 0.3, 0.4, 0.5 and 0.6 the strain energy release rate, G have been evaluated. In the present analysis, plane eight noded quadratic quadrilateral iso-parametric elements were used with displacement formulation. A quarter of the panel was discretized into number of elements varying from 50 to 84 and two degrees of freedom were considered at each node. The partial derivative of the strain energy release rate, G, with respect to increase in crack surface area can be expressed in difference form between any two convenient crack lengths. For example dG/BA for two different crack lengths can be written as: aG G2 - G, - = ___ = GSF corresponding
aA A,-A,
to second crack length.
where G2 = strain energy release rate for second crack length, Gi = strain energy release rate for first crack length, A, = crack surface area for second crack length and A, = crack surface area for the first crack length. In the present computation for NGSF at all the crack lengths mentioned above, 2a/W = 0.1 was taken as the reference crack length with respect to which aG/aA for other crack lengths were found out, which have been listed in Table 1. A graph plotted with NGSF vs 2a/W gave the critical 2a/W corresponding to the crack instability condition as 2aJ W = 0.69. In the present analysis for the panel of 140 mm width and an average stress level of 10 kgf/mm2 (98.1 MPa) considered, the critical crack length at fracture, 2u, ought to be 96.6 mm. From this, one could as well predict the plane stress fracture toughness value theoretically for such a narrow thin panel. It amounts to saying that the fracture instability criterion based on NGSF is akin to fracture toughness criterion, since both predict a certain critical crack length for a given applied stress level. However, the plane stress fracture toughness value of 2024-T3 for relatively narrow and thin panels is not very well defined. Broek [ 131has presented graphs showing variations of plane stress fracture toughness, Ki, with respect to panel width and thickness separately. However, from these graphs it is not possible to obtain an exact value of the plane stress fracture toughness for the panel considered. An experimental verification was therefore made to check the accuracy of above prediction. EXPERIMENTAL VERIFICATION With a view to verify the theoretical predictions concerning the critical crack length, corresponding to a given stress level, an experimental study was-made following the standard practice for plane stress fracture toughness testing. Two identical thin sheet specimens (2024-T3 Alclad) of the same panel size (420 mm x 140 mm x 1 mm) were tested. Initially at the centre of the plate a circular hole of 3 mm
626
S. C. MISHRA and 8. K. PARIDA Table 2. Experimental determination
of plane stress fracture toughness
(1)
Initial crack length 2a (2)
Critical crack length at failure 2a, (3)
Average stress at failure kgf/mm* (4)
It
36
38.9
2t
35
34.2
3t
96.6
98.552
44
84
94.1
5$.
90
100.5
26.0554 (255.603) 26.032 (255.373) 9.0328 (88.611) 10.42 (102.22) 9.866 (96.785)
Specimen NO.
Y* = 1.77 f 0.227 (2a/W) - 0.51 (2~jW)* + 2.7 (2a/W’)’ tRefers to tests carried without anti-buckling guide. *Refers to tests carried with anti-buckling guide.
. .
(0 2 2a/w
KI,
kgf . mm-312
(““(F2) 1.8515 1.8497 2.6187 2.51 2.569
212.756 (65.996) 207.934 (64.505) 166.648 (51.697) i85.32 ’ (57.49) 179.679 (55.74)
I: 0.7)
diameter was made, then two precision sawcuts of 1 mm length were made on both sides of the circular concentrator in a direction normal to the loading direction. These specimens with proper gripping a~angements were mounted in the fatigue testing machine SUP-50). Real fatigue cracks were developed, using small amplitude dynamic loading o,~, = 8.0 kgf/mm* (78.48 MPa}, R = 0.625. On both sides of the central con~ntrator, the total fatigue crack length developed was about 25% of the width of the plate. The generation of the fatigue crack was restricted to around 35 mm in accordance with the ASTM recommendation[ 131,which stipulates the initial crack length for fracture toughness testing of narrow thin panels to be less than one third of the panel width, i.e. 20 I W/3. After generating the cracks to the desired value the dynamic loading was discontinued and static tensile load was very slowly increased starting from zero, while measuring the further crack growth at both ends of the crack carefully. The final load and the corresponding crack length pertaining to ultimate fracture were noted and the plane stress fracture toughness values computed, considering ASTM r~ommended width correction factor, Y. These have been noted in Table 2. It was found that the average value of K,, was about 210 kgf/mmm3’* (65.142 MN m-312). However, if a fracture toughness computation is made based on the critical crack length obtained from quasi-static energy principle, Ki, is found to be equal to 178.665 kgf mm-3’2 (55422 MN m-312). This clearly shows a large discrepancy with respect to the experimentally obtained values. One possible reason for this could be the widely different initial crack lengths considered. With a view to further verify the accuracy of FEM prediction with regard to the crack-instability, a few identical specimens were tested with an initial crack length of around 96.6mm (corresponding to 2a,/ W = 0.69) or less which was developed under a similar low amplitude fatigue loading and tested for plane-stress fraeture toughness without and with antibuckling guides. The quasi-static load was gradually increased and increase in crack length was carefuhy noted as in the previous fracture toughness testing. The critical crack length at failure, 2a,, co~esponding average stress, Ok, finite width correction factor, Y, and computed values of Ki, are shown in Table 2. RESULTS AND DISCUSSIONS Table 1 shows the strain energy release rate, G and aG/aA values for different crack lengths, with the help of which the normalized geometric stability factor for the different centre cracks have been calculated. It is obvious that to get the accurate value of NGSF, the strain energy release rate, G must be calculated accurately as far as possible. The calculation of G, depends upon the finite element idealization of the continuum and the choice of displacement function. An important point taken note of during the analysis of result was that for f&ding the aG/&4 value for any particular crack length, always the numerical ~fferentiation was made with respect to a crack fength smaller than the former, because in this case the direction of crack propagation
Prediction of fracture i~stabi~ty in thin sheets by finite element method PREDICTIONS
BY
CHOW AND FUNG( 11 CLAUSING CHOW
5.0
A
1 -------
AN0
20/W Fig, 1. Prediction
(I
) FEM RESULTS
LAU (REPORTED
IN11
1 a
-
of fracture instability from FEM analysis.
were already known, In this case the incremental crack surface area at a particular crack length was the actual area. Figure I shows a plot of NCSF vs 2aj W. The NGSF values corresponding to different crack lengths, &t&n& in the present investigation by the finite element technique, have been plotted in this figure. It is observed that almost all the points lie on a curve which indicates that the stability decreases in a non-linear manner as the crack length is increased for a given stress level. BY extending the curve to the line corresponding to NGSF = 1.0, it is seen that the expected crack instability begins at the crwk length to sheet width ratio, 2alW = 0.69. In Fig. 1, the results Of instability studies by other investigators pertaining to centre cracks have also been shown, based on ex~~mental as well as numerical computations. Clausing’s[4] results have been indicated by a continuous dotted curve, where as the results of Chow and FungEl l], Chow and Lau[l2] have been shown as discrete points. However, none of the above results predict the exact crack instability condition. Nevertheless, the results obtained by these investigators show a similar trend, i.e. NGSF values decrease in a non-linear manner as crack length increases. In Table 2, it may be noted that the finite width.~orre~tion factor, Y, has been used for specimen NOS. 3 to 5 since, its limit of applicability for centre cracks (as per ASTM-STP 410) extends up to 2a/W = 0.7. From the observations for the specimen No. 3, which was tested without any anti-buckling guide, the value of Kr, was found to be slightly lower than the value predicted theoretically. The other two specimens, namely Nos. 4 and 5 were tested with antibnc~ing guides. It is seen from Table 2 that Ki, values obtained for both these specimens are quite close to each other and are also close to the Ki, value theoretic~ly predicted by FEM analysis employing quasi-static energy approach, i.e. K,, = 178.665 kgf mm-3’t (55.422 MN m-3’2). The error in the Kt, value, obtained without any anti-bucking guide with respect to the average K,, value obtained with antibudging guides happens to be around 9 percent.
A study of fracture instability for a narrow thin panel based on the quasi-static energy principle has been made, which is found to have dose agreement with the experimental results, in terms of the critical crack length at failure for a given stress level, The value of plane-stress fracture toughness, Ki, computed using the data from these fracture tests with anti-buckling guide also agrees well with that predicted theoretically by the FEM-analysis based on quasi-static energy
628
S. C. MISHRA and B. K. PARIDA
principle, although it differs considerably from that obtained recommended plane stress fracture toughness test practice.
following the standard
ASTM
REFERENCES [I] Cl. R. Irwin, Fracture mechanics. lsf Symp. on Naval Structural Mechanics. Pergamon Press, New York (1958). [2] J. M. Krafft, A. M. Sullivan and R. V. Boyle, Effect of dimensions on fast fracture instability of notched sheets. Proc. Cruck Propagation Symp., Cranfield, Vol. 1. pp. 8-28, 1961. [3] J. E. Srawley and W. P. Borwn, Fracture toughness testing and its applications. ASTM STP 381, 133-193 (1965). [4] D. P. Clausing, Crack instability in linear elastic fracture mechanics. Int. J. Fract. Mech. 5, 21 l-227 (1969). [5] J. Glucklich, On crack stability in some fracture tests. Engng Fracture Mech. 3, 333-344 (1971). [6] C. Gurney and I. Hunt, Quasi-static crack propagation. Proc. Roy. Sot. A 299, 508-524 (1967). [7] B. K. Parida, Crack edge instability-a criterion for safe crack propagation limit in thin sheets. Proc. 3rd Colloquium on Fracture, London, 8-10 Sept. 1980. [8] C. Wheeler, J. N. Eastabrook, D. P. Rooke, K. H. Schwalbe, W. Setz and A. U. De Koning, Recommendations for the measurement of R-curves using centre-cracked panels. J. Strain Analysis 17(4), 205213 (1982). [9] H. Anderson, A finite element representation of stable crack growth. J. Mech. Phys. Solids 21, 337-366 (1973). [IO] D. M. Parks, A stiffness derivative finite element technique for determination of crack tip SIFs. Int. J. Fracture, 10, 437-502 (1974). [I 11 C. L. Chow and W. C. Fung, Prediction of fracture instability by a finite element method. Fracture Mech. Tech. II, 1537-1545. [l2] C. L. Chow and P. M. Lau, Stability conditions in quasi-static crack propagation
for constant strain energy release rate. J. Engng Muter. Tech. 96, 41-48 (1974). [13] D. Broek, Elementary Engineering Fracture Mechanics, Chap. VIII, pp. 177-185. Noordhoff, Groningen, Netherlands (1974). (Received 15 August 1983; received for publication 29 November 1983)