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Finite element modeling of the three-point bend impact test
FINITE ELEMENT MODELING OF THE THREE-POINT BEND IMPACT TEST B. A. E. I.
CROUCH
du Pont de Nemours, Inc., Polymer Products Department, Wilmington, Delaware, U.S.A. (Received 26 June 1991)
Ahstraet-A dynamic finite element program has been applied to the analysis of the three-point bend impact specimen. First, the behavior of an unsupported specimen struck at its center was modeled. From this it was predicted that the best striker force response in an impact test would be obtained with an overhang at each end of 22% of the total length. This prediction was tested by modeling supported specimens of different lengths and by comparison with experiment. Reference curves of dynamic stress intensity for different material properties and impact speeds can be prepared from the numerical analysis. Providing the test geometry has been carefully chosen, a static displacement based stress intensity calculation appears valid even when pronouns oscillations are seen in the striker force.
INTRODUCI’ION The dynamic response of the three-point
bend speckmen under impact loading is not well understood. At high loading rates, an oscillatory striker force is obtained due to inertia effects. This makes it difficult to accurately determine the load at fracture in order to calculate the fracture toughness of the material. Williams and Adams [l] have proposed a massspring model to describe these vibration effects and an analysis that uses the loading time to failure instead of the striker force. The problems of impact toughness measurement were discussed by Kalthoff [2] in an excellent review article. The present work uses a finite element model to assess the severity of dynamic effects in three-point bend impact and to investigate ways of minimizing or overcoming these problems. A steady-state vibration analysis of a freely vibrating unsupported beam predicts that there will be stationary points (in the first mode) at points 22% along the span from each end 131.Therefore, we can speculate that in the case of a supported specimen the measured striker force response would be improved by locating the supports at these stationary points. Naim [4] has reported that experimentally the force versus time response of a three-point bend impact specimen is affected by the overhang at the ends of the specimen. However, he was considering situations where the loading time was many times greater than the natural frequency of the system. In many fracture tests this is not the case. It has been widely suggested that a static analysis of an impact test is acceptable if the loading time to failure is greater than three times the natural frequency of the specimen, but in polymers, for example, the failwe time may be significantIy shorter. One objective of the work reported here was to assess whether the striker force response in an impact test could be improved by careful CAS 48+-L
positioning of the supports. It is recognized that the gains will ultimately be limited by the available loading time for a given material. In modeling the three-point bend impact test it is important that realistic boundary conditions are used [5]. Nishioka et al. [6] have described a numerical model that allowed the specimen to bounce from the supports or the striker, as in the present work. Their interest was in modeling a propagating crack and they did not look at the effect of support position. Nakamura et ui. [7] have reported a numerical analysis of a ductile material under impact. Their objective was to assess the loading time necessary for a valid quasi-static J-integral analysis. Since it is clearly impractical to perform a full numerical analysis of routine impact test results in order to derive the fracture toughness, it is attractive to prepare dynamic response curves for different combinations of material properties or specimen dimensions using the finite element model. These could be used to extract fracture toughness from some measured failure parameter such as loading time to fracture, and this is also considered here. A similar approach has been suggested by Kalthoff using experimentally derived curves [2]. NUMERICAL. ANALYSIS
A two~imensional dynamic finite element program was used for this study. Constant strain triangular elements were used with an interactive time integration scheme (81. Loss of contact between the specimen and either the supports or the striker can be modeled. A small amount of viscous damping was included as this improved the fit of the numerical results to experimental data. Details of the numerical scheme are given in the Appendix. The crack tip stress intensity was approximated by statically calibrating the force on the node nearest to 167
168
B. A. CROUCH
the crack tip against known stress intensity solutions for a three-point bend specimen. This was done for all crack lengths before beginning the dynamic analysis. It is recognized that this calibration factor is geometry and mesh dependent. The mesh used was uniform across the specimen width (in the direction of crack propagation) with 16 nodes in this direction. Due to symmetry it was only necessary to simulate half of the length of the specimen. In the length direction 22 nodes were used, the smaller elements near to the crack plane. The ratio of the maximum to minimum nodal spacing was six. There was no practical change in the striker force predictions when this mesh was compared to a somewhat coarser mesh, nor when the orientation of the triangular elements was reversed relative to the striker contact point. Unless otherwise stated, the case simulated throughout was a beam of length (L) 1OOmm and depth (IV) 12.5 mm with a notch depth (a) of 6.25 mm. The material properties of nylon 66 were used, a Young’s modulus (E) of 2.54 GPa and density (p) of 1140 kg/m3. An impact speed of 1 m/set was used for the majority of the simulations. All overhangs are expressed as a percentage of the total specimen length. RESULTS AND DISCUSSION Free specimen vibration
An unsupported, notched beam was struck at its center at a constant speed and allowed to vibrate freely, including bouncing ahead of the striker. The predicted response, for a 1 m/set impact, is shown in Fig. 1 in terms of the vibration about the average (rigid body) displacement of the entire specimen. It is seen that the first mode of vibration is dominant, correctly predicting stationary points at 0.028 m from the center of the specimen. This corresponds to 22% of the total length from the end of the specimen. The same behavior is predicted for high speed impacts of 3 and 5 m/set, ls.o
with no apparent
mode change.
The
I
Fig. 1. Free resonance of a notched specimen after a 1 m/set impact, o/W = 0.4.
3.5
1
-1.5 -0.01
0.00
0.0,
Distance
0.02
from
Center
0.03
Cl.04
of Specimen
0.05
0.06
cm)
Fig. 2. Short time free vibration response of a notched specimen, a/W = 0.4.
frequency of free/free vibration for an unnotched specimen agrees well with the theoretical value of 1870Hz. The short time response of the notched specimen, up to 100psec after impact, is shown in Fig. 2. Up to 60 psec, the end of the specimen has not yet begun to move. Therefore, unless the loading time is substantially greater than 60 psec there will not be any improvement in the force versus time response, for the supported specimen, from positioning the supports at the stationary points. However, three times the period of steady-state vibration of the unnotched beam is around 1500/1sec, so that it is possible that some improvement could be obtained at loading times between these two extremes. Supported specimen of varying span, overhang and length
SimuIations were performed for the same specimen geometry as used for the free vibration analysis but with the specimen resting on supports as in the actual impact test. The specimen length was kept constant and the span varied so that the supports were either side of the theoretical stationary point. The results are shown in Fig. 3, where the striker force and the force at the crack tip node are shown. Where the striker force goes to zero, bouncing off the striker is predicted. The primary frequency of vibration is much higher than the natural frequency of the whole specimen and is controlled by the contact stiffness [ 11. Deviations from this high frequency, sinusoidal, response are due to either bouncing (from the supports or striker) or the appearance of the whole specimen vibration mode. At the smaller overhangs (Fig. 3a, b) the force response is more irregular due to bouncing. At the largest overhang (Fig. 3d) there is some non-linearity in the mean loading line which is probably due to the whole specimen vibration mode. The best compromise is the 22% overhang case (Fig. 3c) where the mean loading line is straight with regular sinusoidal oscillations superimposed. The above comments assume a common loading time to fracture. In practice, if the material fractures
169
Finite element modeling of the three-point bend impact test
(a) 300
-50
lh.ory.Strlkw
1
0
200
400
600
Yima
I)00
-,W000
1000
(4
-..c----_-, I
0
600
100
200
1
600
Time
(micmsec)
Time
(mkros6-c)
“““1
-100h4b0
,000
600
600
400
200
(microsac)
Time (mierosec)
’
600
-
600
1000
Fig. 3. Impact of a 100 mm long specimen at 1m/set, a/W
= 0.4: (a) 4% overhang, (II) 12% overhang, (c) 22% overhang, stationary point, (d) 30% overhang.
at the same stress intensity, then the tip force at fracture is the common factor. For example, if this is taken as 250 N in the above plots, then the number of cycles to failure reduces from nine to four as the span is reduced. This may be more important than the improvement in the quality of the signal for accurate inte~re~tion of the load at failure. To consider the effect of overhang alone, the span was fixed at 56.25mm and the overhang of the
r6
(a)
V
100
specimen first increased to 30% (Fig. 4a) and then reduced to 15% (Fig. lob). The results can be compared to Fig. 3c. Here, the number of oscillations in the force trace is the same in all cases and the slopes of the force curves are comparable. For the increased overhang case, there is greater oscillation in the striker force with greater non-linearity in the mean loading curve. The reduced overhang case shows less regular striker force oscillations, due to the trend
400
v,/-*v
J
,_-’ ,’
r’ _-’
,/
50
,’
,,-’ 0
-50
~ ; 0
.,-’
;’ 200
400 Time
!o 600 (microsac)
600
loo0 Tima
(microsec)
Fig. 4. Impact of a 56.25nun span specimen at 1 mfsec, o/W = 0.4: (a) 30% overhang, (b) 15% overhang,
after an initial delay while a stress wave travels out to the support point. The striker bounces off the specimen once. To reproduce the details of a force trace obtained experimentally it is important to include bouncing in the model. In contrast, the shorter specimen does not bounce off either the supports or the striker, according to the simulation, improving the quality of the force trace. Comparison of theory and experiment
-10&__t_
_v--r-
200
0
400
600
Time (micmsec)
oI’ ,/. ,’ -266s” 0
200 “--‘.----_ 100
600
If
Time (microsee)
Fig. 5. The effect of bouncing at the supports and striker, =0.4: (a) 143.75mm long, MOmm span, 15% overhang, (b) 8 1.25mm long, 37.5 mm span, 27% overhang.
a/W
towards more support bouncing as the support is moved away from the stationary point. Overall, therefore, for a given span, it is beneficial to choose a specimen length so that the supports can be positioned at the stationary points. This gives the most regular pattern in the force oscillations, making it easier to determine the mean loading line. In Fig. 4, and sub~uent plots, the crack tip force was calibrated against analytical solutions to estimate the stress intensity (X). The oscillation in crack tip force, and, therefore, in stress intensity, is insignificant in both of these cases. Kalthoff [2] has reported greater oscillation in K when the overhang was small. However, he increased both the span and specimen width while m~~taining the same span to width ratio so that the effect of increasing span may well have been dominant, Here, the same depth has been used throughout. Figure 5 shows specimens of different lengths, both with 43.75 mm total overhang, which were used to compare the relative tendencies for bouncing from the striker or supports. For the longer specimen, where the span is large and the overhang a smaller proportion of the total length, more bouncing occurs. The specimen bounces off the supports three times
fn Fig. 6, force versus time data from an actual test on Nylon 66 (“Zytel’ 101) is compared to theory. The specimen was 100 mm long, 12.5 mm deep, 6.25 mm thick, with an initial notch depth of 5 mm and a span of 90 mm. Therefore, the span was outside the 22% stationary point and it is clear that no one frequency dominates the response. Agreement between the model and experiment is good, with the majo~ty of the experimental oscillations reproduced. However, the theoretical force is generally a little higher. This may indicate some inaccuracy in the material property data assumed. A series of impact specimens were tested with different spans and specimen lengths to attempt to verify the numerical predictions. Sawn blunt notches, of a/W’ = 0.4, were chosen to maximize the loading time to failure allowing the force traces to be studied. Therefore, the time to failure is variable according to the bluntness of the notch and does not have any physical significance. Again, Nylon 66 was used with a constant depth of 12.5 mm and thickness 6.25 mm. Three impacts were performed for each case and a representative example for each condition is shown. The cases shown in Fig. 7 can be compared to the equivalent numerical predictions in Fig. 3. The impact velocity was I m/see, so that a displacement of 1 mm corresponds to a loading time of 1000 ,usec in Fig. 3. For the low overhang cases (Fig. 7a, b) the bouncing is reproduced well, both in terms of the number of bounces and the times at which they occur. At the largest overhang (Fig. 7d) the non-linearity in the force trace predicted numerically is seen. In all
Time (microsec)
Fig. 6. Experimental and theoretical striker forces for nylon 66, 1 m/set impact, 1OOmmlong, 5% overhang, a/W = 0.4.
171
Finite element modehag of the thres-point bead impact test
Disptacement, (mm)
0.0
0.8
0.4
1.2
1.6
8
0.8
2.0
1.2
1.8
I
2.0
Diipiacement, (mm)
Displacement, (mm)
Fig. 7. Experimental striker forces for a 100mm long specimen at 1 m/set, a/W = 0.4: (a) 4% overhang, (b) 12% overhang, (c) 22% overhang, (d) 30% overhang. cases the mean slopes of the experimental and theoretical force traces are in good agreement. The 22% overhang case does appear to be the best of these four examples, with no bouncing but a linear trend. Figure 8 looks at the effect of overhang at a constant span and can be compared to Fig. 4. The non-linearity at large overhangs and the single bounce obtained both ex~~rn~~lly and nume~~lly can be seen. Effect
of impact speed
The numerical results described have been based on a 1 m/set impact speed throughout. As the impact speed increases, the striker force becomes increasingly oscillatory. This is shown in Fig. 9a where the 22% overhang, 100 mm span optimum geometry case was
(a)
m1
modeled for different impact speeds. The oscillations scale linearly with impact speed. For a common displacement to failure, it is clear that no accurate mean trend in striker force can be drawn through the higher speed loading curves. In contrast, the crack tip stress intensity shows virtually no oscillation, as shown in Fig. 9b. Also, the curves for different speeds are identical, for all practical purposes. This is not always the case. For example, a similar set of simulations was performed for a longer specimen (143.75 mm long, 100 mm span, less than 22% overhang) as shown in Fig. 10. The force and stress intensity curves are again magnified images of the low speed result, but the stress intensity plot is no longer linear.
(b)
w”l
120
0.0
0.4
0.8
1.2
Displacement, (mm)
1.8
20
momo Displacement, (mm)
Fig. 8. Experimental striker forces for a 56.25mm span specimen at 1 m/set, a/W -0.4: (a) 30% overhang, (b) 15% overhang.
172
B.A. CROUCH
Dynamic response curves
(a)
5o0
One approach to the analysis of dynamic impact tests would be to perform numerical simulations of the type described above to predict the best span/ geometry combination. Then, if a series of tests were to be routinely performed using this geometry, with a range of material properties, response curves of K versus loading time to failure could be prepared for different striker speeds. A similar approach, based on experimental measurements, has been suggested by Kalthoff [Z]. An example using just one geometry and crack length (100 mm specimen, 22% overhang, a/W = 0.4) is shown in Fig. 11. Here, since the stress intensity is proportional to modulus, the results are presented divided by modulus. Three different materials are shown, with the impact speed scaled according to the stress wave speed in the material. The stress intensity plots for the three materials are the same, suggesting that dynamic calibration master curves may be prepared by considering the effect of modulus and impact speed. More importantly, providing the other guidelines reported here have been followed, it seems that such response curves will be basically linear, greatly simplifying the analysis. A static displacement based stress intensity calculation gives a slope of the
(4
“““1
0.0
0.t
0.2
0.3
0.1
0.5
0.6
Displacement
0.7
0.1)
0.9
A
(mm)
Fig. 10. Effect of impact speed, 143.75mm long, 15% overhang, a/W = 0.4: (a) striker force, (b) stress intensity. (K/E)
versus displacement line about 20% less than the slope of Fig. 11. However, given the static calibration of tip force used to calculate the dynamic K, it seems reasonable to assume that a static displacement based K calculation accurately describes Fig. 11. CONCLUSIONS 0.6
(b)
Displacement
(mm)
Displacement
(mm)
0.1
1.0
The vibration response of an impact specimen has been modeled numerically. Such a vibration analysis may be used to predict the most suitable combination
"1
Fig. 9. Effect of impact speed, 100 mm long, 22% overhang, a/W = 0.4: (a) striker force, (b) stress intensity.
Displacement (mm)
Fig. 11. Comparison of different material properties, 100mm long, 22% overhang, a/W = 0.4.
Finite element modeling of the three-point bend impact test of span to specimen length in order to obtain a good
force response for the calculation of fracture toughness. The effect of bouncing from the support and striker has also been considered. The development of dynamic response curves for routine testing using numerical analysis is practical. Further, it seems that, for carefully chosen specimen dimensions that minimize dynamic effects, these response curves reduce to a single line equivalent to a displacement-based static analysis. REFERENCES 1. J. G. Williams and G. C. Adams, The analysis of instrumented impact tests using a mass-spring model. Int. J. Fracture 33, 209-222 (1987). 2. J. F. Kalthoff, On the measurement of dynamic fracture toughness-a review of recent work. Int. J. Fracture 27, 277-298 (1985). 3. A. H. Church, ~ee~ical Vibrations. John Wiley (1963). 4. J. A. Naim, Measurement of polymer viscoelastic response during an impact experiment. Polymer Ertgng Sci. 29, 654-663 (1989). 5. B. A. Crouch, High speed crack growth in polymers. Ph.D. thesis, University of London (1986). 6. T. Nishioka, M. Per1 and S. N. Atluri, An analysis of dynamic fracture in an impact test specimen. .I. Pressure Vessel Tecbnol. 105, 124-131 (1983). 7. T. Nakamura, C. F. Shih and L. B. Freund, Analysis of
a dynamically loaded three-point-bend ductile fracture specimen. Engng Fracture Mech. 25, 323-339 (1986). 8. N. M. Newmark, A method for computation of structural dynamics. Proc. Am. Sot. Civil Engrs. 85, 67-94 (1959).
173
APPENDM: PINITE ELEMENT PROGRAM DETAILS A number of features of the finite element program used in this work are described below. (a) Ztzitialsolution. Prior to the dynamic analysis, a static solution is obtained to cafibrate the crack tip stresses against stress intensity solutions. This uses a inventions Gauss-Seidel iterative scheme. (b) Solution methodfor time btfegration. The equation of motion [K]u + [M]II + [C]ri = [F]
(1)
is solved repeatedly during the dynamic analysis, where [K] is the stiffness matrix, f&f] is a simple lumped mass matrix, [C] is a damping term and u is displacement. This is combined with Newmark expressions for future displacement and velocity u,,+ i = u” + At&, + At *[(O.S- j?)t&+ @t&+ ,]
where /I = 0.28, y = 0.55, giving a small amount of numerical damping, and At is the timestep. The solution at time n + 1 is obtained by iterating eqns (1) and (2) until adequate convergence is obtained. (c) Boumkvy conditions during time integration. Boundary conditions at the supports and the impact point are applied by resetting the di~la~~t, velocity and acceleration estimates obtained from equations (1) and (2) within the iterative loop. (d) Viscous akmping. Damping is based on the difference between the velocity of any one node and the average velocity of the surrounding nodes, multiplied by a damping coefficient chosen to fit experimental data.
CROUCH
du Pont de Nemours, Inc., Polymer Products Department, Wilmington, Delaware, U.S.A. (Received 26 June 1991)
Ahstraet-A dynamic finite element program has been applied to the analysis of the three-point bend impact specimen. First, the behavior of an unsupported specimen struck at its center was modeled. From this it was predicted that the best striker force response in an impact test would be obtained with an overhang at each end of 22% of the total length. This prediction was tested by modeling supported specimens of different lengths and by comparison with experiment. Reference curves of dynamic stress intensity for different material properties and impact speeds can be prepared from the numerical analysis. Providing the test geometry has been carefully chosen, a static displacement based stress intensity calculation appears valid even when pronouns oscillations are seen in the striker force.
INTRODUCI’ION The dynamic response of the three-point
bend speckmen under impact loading is not well understood. At high loading rates, an oscillatory striker force is obtained due to inertia effects. This makes it difficult to accurately determine the load at fracture in order to calculate the fracture toughness of the material. Williams and Adams [l] have proposed a massspring model to describe these vibration effects and an analysis that uses the loading time to failure instead of the striker force. The problems of impact toughness measurement were discussed by Kalthoff [2] in an excellent review article. The present work uses a finite element model to assess the severity of dynamic effects in three-point bend impact and to investigate ways of minimizing or overcoming these problems. A steady-state vibration analysis of a freely vibrating unsupported beam predicts that there will be stationary points (in the first mode) at points 22% along the span from each end 131.Therefore, we can speculate that in the case of a supported specimen the measured striker force response would be improved by locating the supports at these stationary points. Naim [4] has reported that experimentally the force versus time response of a three-point bend impact specimen is affected by the overhang at the ends of the specimen. However, he was considering situations where the loading time was many times greater than the natural frequency of the system. In many fracture tests this is not the case. It has been widely suggested that a static analysis of an impact test is acceptable if the loading time to failure is greater than three times the natural frequency of the specimen, but in polymers, for example, the failwe time may be significantIy shorter. One objective of the work reported here was to assess whether the striker force response in an impact test could be improved by careful CAS 48+-L
positioning of the supports. It is recognized that the gains will ultimately be limited by the available loading time for a given material. In modeling the three-point bend impact test it is important that realistic boundary conditions are used [5]. Nishioka et al. [6] have described a numerical model that allowed the specimen to bounce from the supports or the striker, as in the present work. Their interest was in modeling a propagating crack and they did not look at the effect of support position. Nakamura et ui. [7] have reported a numerical analysis of a ductile material under impact. Their objective was to assess the loading time necessary for a valid quasi-static J-integral analysis. Since it is clearly impractical to perform a full numerical analysis of routine impact test results in order to derive the fracture toughness, it is attractive to prepare dynamic response curves for different combinations of material properties or specimen dimensions using the finite element model. These could be used to extract fracture toughness from some measured failure parameter such as loading time to fracture, and this is also considered here. A similar approach has been suggested by Kalthoff using experimentally derived curves [2]. NUMERICAL. ANALYSIS
A two~imensional dynamic finite element program was used for this study. Constant strain triangular elements were used with an interactive time integration scheme (81. Loss of contact between the specimen and either the supports or the striker can be modeled. A small amount of viscous damping was included as this improved the fit of the numerical results to experimental data. Details of the numerical scheme are given in the Appendix. The crack tip stress intensity was approximated by statically calibrating the force on the node nearest to 167
168
B. A. CROUCH
the crack tip against known stress intensity solutions for a three-point bend specimen. This was done for all crack lengths before beginning the dynamic analysis. It is recognized that this calibration factor is geometry and mesh dependent. The mesh used was uniform across the specimen width (in the direction of crack propagation) with 16 nodes in this direction. Due to symmetry it was only necessary to simulate half of the length of the specimen. In the length direction 22 nodes were used, the smaller elements near to the crack plane. The ratio of the maximum to minimum nodal spacing was six. There was no practical change in the striker force predictions when this mesh was compared to a somewhat coarser mesh, nor when the orientation of the triangular elements was reversed relative to the striker contact point. Unless otherwise stated, the case simulated throughout was a beam of length (L) 1OOmm and depth (IV) 12.5 mm with a notch depth (a) of 6.25 mm. The material properties of nylon 66 were used, a Young’s modulus (E) of 2.54 GPa and density (p) of 1140 kg/m3. An impact speed of 1 m/set was used for the majority of the simulations. All overhangs are expressed as a percentage of the total specimen length. RESULTS AND DISCUSSION Free specimen vibration
An unsupported, notched beam was struck at its center at a constant speed and allowed to vibrate freely, including bouncing ahead of the striker. The predicted response, for a 1 m/set impact, is shown in Fig. 1 in terms of the vibration about the average (rigid body) displacement of the entire specimen. It is seen that the first mode of vibration is dominant, correctly predicting stationary points at 0.028 m from the center of the specimen. This corresponds to 22% of the total length from the end of the specimen. The same behavior is predicted for high speed impacts of 3 and 5 m/set, ls.o
with no apparent
mode change.
The
I
Fig. 1. Free resonance of a notched specimen after a 1 m/set impact, o/W = 0.4.
3.5
1
-1.5 -0.01
0.00
0.0,
Distance
0.02
from
Center
0.03
Cl.04
of Specimen
0.05
0.06
cm)
Fig. 2. Short time free vibration response of a notched specimen, a/W = 0.4.
frequency of free/free vibration for an unnotched specimen agrees well with the theoretical value of 1870Hz. The short time response of the notched specimen, up to 100psec after impact, is shown in Fig. 2. Up to 60 psec, the end of the specimen has not yet begun to move. Therefore, unless the loading time is substantially greater than 60 psec there will not be any improvement in the force versus time response, for the supported specimen, from positioning the supports at the stationary points. However, three times the period of steady-state vibration of the unnotched beam is around 1500/1sec, so that it is possible that some improvement could be obtained at loading times between these two extremes. Supported specimen of varying span, overhang and length
SimuIations were performed for the same specimen geometry as used for the free vibration analysis but with the specimen resting on supports as in the actual impact test. The specimen length was kept constant and the span varied so that the supports were either side of the theoretical stationary point. The results are shown in Fig. 3, where the striker force and the force at the crack tip node are shown. Where the striker force goes to zero, bouncing off the striker is predicted. The primary frequency of vibration is much higher than the natural frequency of the whole specimen and is controlled by the contact stiffness [ 11. Deviations from this high frequency, sinusoidal, response are due to either bouncing (from the supports or striker) or the appearance of the whole specimen vibration mode. At the smaller overhangs (Fig. 3a, b) the force response is more irregular due to bouncing. At the largest overhang (Fig. 3d) there is some non-linearity in the mean loading line which is probably due to the whole specimen vibration mode. The best compromise is the 22% overhang case (Fig. 3c) where the mean loading line is straight with regular sinusoidal oscillations superimposed. The above comments assume a common loading time to fracture. In practice, if the material fractures
169
Finite element modeling of the three-point bend impact test
(a) 300
-50
lh.ory.Strlkw
1
0
200
400
600
Yima
I)00
-,W000
1000
(4
-..c----_-, I
0
600
100
200
1
600
Time
(micmsec)
Time
(mkros6-c)
“““1
-100h4b0
,000
600
600
400
200
(microsac)
Time (mierosec)
’
600
-
600
1000
Fig. 3. Impact of a 100 mm long specimen at 1m/set, a/W
= 0.4: (a) 4% overhang, (II) 12% overhang, (c) 22% overhang, stationary point, (d) 30% overhang.
at the same stress intensity, then the tip force at fracture is the common factor. For example, if this is taken as 250 N in the above plots, then the number of cycles to failure reduces from nine to four as the span is reduced. This may be more important than the improvement in the quality of the signal for accurate inte~re~tion of the load at failure. To consider the effect of overhang alone, the span was fixed at 56.25mm and the overhang of the
r6
(a)
V
100
specimen first increased to 30% (Fig. 4a) and then reduced to 15% (Fig. lob). The results can be compared to Fig. 3c. Here, the number of oscillations in the force trace is the same in all cases and the slopes of the force curves are comparable. For the increased overhang case, there is greater oscillation in the striker force with greater non-linearity in the mean loading curve. The reduced overhang case shows less regular striker force oscillations, due to the trend
400
v,/-*v
J
,_-’ ,’
r’ _-’
,/
50
,’
,,-’ 0
-50
~ ; 0
.,-’
;’ 200
400 Time
!o 600 (microsac)
600
loo0 Tima
(microsec)
Fig. 4. Impact of a 56.25nun span specimen at 1 mfsec, o/W = 0.4: (a) 30% overhang, (b) 15% overhang,
after an initial delay while a stress wave travels out to the support point. The striker bounces off the specimen once. To reproduce the details of a force trace obtained experimentally it is important to include bouncing in the model. In contrast, the shorter specimen does not bounce off either the supports or the striker, according to the simulation, improving the quality of the force trace. Comparison of theory and experiment
-10&__t_
_v--r-
200
0
400
600
Time (micmsec)
oI’ ,/. ,’ -266s” 0
200 “--‘.----_ 100
600
If
Time (microsee)
Fig. 5. The effect of bouncing at the supports and striker, =0.4: (a) 143.75mm long, MOmm span, 15% overhang, (b) 8 1.25mm long, 37.5 mm span, 27% overhang.
a/W
towards more support bouncing as the support is moved away from the stationary point. Overall, therefore, for a given span, it is beneficial to choose a specimen length so that the supports can be positioned at the stationary points. This gives the most regular pattern in the force oscillations, making it easier to determine the mean loading line. In Fig. 4, and sub~uent plots, the crack tip force was calibrated against analytical solutions to estimate the stress intensity (X). The oscillation in crack tip force, and, therefore, in stress intensity, is insignificant in both of these cases. Kalthoff [2] has reported greater oscillation in K when the overhang was small. However, he increased both the span and specimen width while m~~taining the same span to width ratio so that the effect of increasing span may well have been dominant, Here, the same depth has been used throughout. Figure 5 shows specimens of different lengths, both with 43.75 mm total overhang, which were used to compare the relative tendencies for bouncing from the striker or supports. For the longer specimen, where the span is large and the overhang a smaller proportion of the total length, more bouncing occurs. The specimen bounces off the supports three times
fn Fig. 6, force versus time data from an actual test on Nylon 66 (“Zytel’ 101) is compared to theory. The specimen was 100 mm long, 12.5 mm deep, 6.25 mm thick, with an initial notch depth of 5 mm and a span of 90 mm. Therefore, the span was outside the 22% stationary point and it is clear that no one frequency dominates the response. Agreement between the model and experiment is good, with the majo~ty of the experimental oscillations reproduced. However, the theoretical force is generally a little higher. This may indicate some inaccuracy in the material property data assumed. A series of impact specimens were tested with different spans and specimen lengths to attempt to verify the numerical predictions. Sawn blunt notches, of a/W’ = 0.4, were chosen to maximize the loading time to failure allowing the force traces to be studied. Therefore, the time to failure is variable according to the bluntness of the notch and does not have any physical significance. Again, Nylon 66 was used with a constant depth of 12.5 mm and thickness 6.25 mm. Three impacts were performed for each case and a representative example for each condition is shown. The cases shown in Fig. 7 can be compared to the equivalent numerical predictions in Fig. 3. The impact velocity was I m/see, so that a displacement of 1 mm corresponds to a loading time of 1000 ,usec in Fig. 3. For the low overhang cases (Fig. 7a, b) the bouncing is reproduced well, both in terms of the number of bounces and the times at which they occur. At the largest overhang (Fig. 7d) the non-linearity in the force trace predicted numerically is seen. In all
Time (microsec)
Fig. 6. Experimental and theoretical striker forces for nylon 66, 1 m/set impact, 1OOmmlong, 5% overhang, a/W = 0.4.
171
Finite element modehag of the thres-point bead impact test
Disptacement, (mm)
0.0
0.8
0.4
1.2
1.6
8
0.8
2.0
1.2
1.8
I
2.0
Diipiacement, (mm)
Displacement, (mm)
Fig. 7. Experimental striker forces for a 100mm long specimen at 1 m/set, a/W = 0.4: (a) 4% overhang, (b) 12% overhang, (c) 22% overhang, (d) 30% overhang. cases the mean slopes of the experimental and theoretical force traces are in good agreement. The 22% overhang case does appear to be the best of these four examples, with no bouncing but a linear trend. Figure 8 looks at the effect of overhang at a constant span and can be compared to Fig. 4. The non-linearity at large overhangs and the single bounce obtained both ex~~rn~~lly and nume~~lly can be seen. Effect
of impact speed
The numerical results described have been based on a 1 m/set impact speed throughout. As the impact speed increases, the striker force becomes increasingly oscillatory. This is shown in Fig. 9a where the 22% overhang, 100 mm span optimum geometry case was
(a)
m1
modeled for different impact speeds. The oscillations scale linearly with impact speed. For a common displacement to failure, it is clear that no accurate mean trend in striker force can be drawn through the higher speed loading curves. In contrast, the crack tip stress intensity shows virtually no oscillation, as shown in Fig. 9b. Also, the curves for different speeds are identical, for all practical purposes. This is not always the case. For example, a similar set of simulations was performed for a longer specimen (143.75 mm long, 100 mm span, less than 22% overhang) as shown in Fig. 10. The force and stress intensity curves are again magnified images of the low speed result, but the stress intensity plot is no longer linear.
(b)
w”l
120
0.0
0.4
0.8
1.2
Displacement, (mm)
1.8
20
momo Displacement, (mm)
Fig. 8. Experimental striker forces for a 56.25mm span specimen at 1 m/set, a/W -0.4: (a) 30% overhang, (b) 15% overhang.
172
B.A. CROUCH
Dynamic response curves
(a)
5o0
One approach to the analysis of dynamic impact tests would be to perform numerical simulations of the type described above to predict the best span/ geometry combination. Then, if a series of tests were to be routinely performed using this geometry, with a range of material properties, response curves of K versus loading time to failure could be prepared for different striker speeds. A similar approach, based on experimental measurements, has been suggested by Kalthoff [Z]. An example using just one geometry and crack length (100 mm specimen, 22% overhang, a/W = 0.4) is shown in Fig. 11. Here, since the stress intensity is proportional to modulus, the results are presented divided by modulus. Three different materials are shown, with the impact speed scaled according to the stress wave speed in the material. The stress intensity plots for the three materials are the same, suggesting that dynamic calibration master curves may be prepared by considering the effect of modulus and impact speed. More importantly, providing the other guidelines reported here have been followed, it seems that such response curves will be basically linear, greatly simplifying the analysis. A static displacement based stress intensity calculation gives a slope of the
(4
“““1
0.0
0.t
0.2
0.3
0.1
0.5
0.6
Displacement
0.7
0.1)
0.9
A
(mm)
Fig. 10. Effect of impact speed, 143.75mm long, 15% overhang, a/W = 0.4: (a) striker force, (b) stress intensity. (K/E)
versus displacement line about 20% less than the slope of Fig. 11. However, given the static calibration of tip force used to calculate the dynamic K, it seems reasonable to assume that a static displacement based K calculation accurately describes Fig. 11. CONCLUSIONS 0.6
(b)
Displacement
(mm)
Displacement
(mm)
0.1
1.0
The vibration response of an impact specimen has been modeled numerically. Such a vibration analysis may be used to predict the most suitable combination
"1
Fig. 9. Effect of impact speed, 100 mm long, 22% overhang, a/W = 0.4: (a) striker force, (b) stress intensity.
Displacement (mm)
Fig. 11. Comparison of different material properties, 100mm long, 22% overhang, a/W = 0.4.
Finite element modeling of the three-point bend impact test of span to specimen length in order to obtain a good
force response for the calculation of fracture toughness. The effect of bouncing from the support and striker has also been considered. The development of dynamic response curves for routine testing using numerical analysis is practical. Further, it seems that, for carefully chosen specimen dimensions that minimize dynamic effects, these response curves reduce to a single line equivalent to a displacement-based static analysis. REFERENCES 1. J. G. Williams and G. C. Adams, The analysis of instrumented impact tests using a mass-spring model. Int. J. Fracture 33, 209-222 (1987). 2. J. F. Kalthoff, On the measurement of dynamic fracture toughness-a review of recent work. Int. J. Fracture 27, 277-298 (1985). 3. A. H. Church, ~ee~ical Vibrations. John Wiley (1963). 4. J. A. Naim, Measurement of polymer viscoelastic response during an impact experiment. Polymer Ertgng Sci. 29, 654-663 (1989). 5. B. A. Crouch, High speed crack growth in polymers. Ph.D. thesis, University of London (1986). 6. T. Nishioka, M. Per1 and S. N. Atluri, An analysis of dynamic fracture in an impact test specimen. .I. Pressure Vessel Tecbnol. 105, 124-131 (1983). 7. T. Nakamura, C. F. Shih and L. B. Freund, Analysis of
a dynamically loaded three-point-bend ductile fracture specimen. Engng Fracture Mech. 25, 323-339 (1986). 8. N. M. Newmark, A method for computation of structural dynamics. Proc. Am. Sot. Civil Engrs. 85, 67-94 (1959).
173
APPENDM: PINITE ELEMENT PROGRAM DETAILS A number of features of the finite element program used in this work are described below. (a) Ztzitialsolution. Prior to the dynamic analysis, a static solution is obtained to cafibrate the crack tip stresses against stress intensity solutions. This uses a inventions Gauss-Seidel iterative scheme. (b) Solution methodfor time btfegration. The equation of motion [K]u + [M]II + [C]ri = [F]
(1)
is solved repeatedly during the dynamic analysis, where [K] is the stiffness matrix, f&f] is a simple lumped mass matrix, [C] is a damping term and u is displacement. This is combined with Newmark expressions for future displacement and velocity u,,+ i = u” + At&, + At *[(O.S- j?)t&+ @t&+ ,]
where /I = 0.28, y = 0.55, giving a small amount of numerical damping, and At is the timestep. The solution at time n + 1 is obtained by iterating eqns (1) and (2) until adequate convergence is obtained. (c) Boumkvy conditions during time integration. Boundary conditions at the supports and the impact point are applied by resetting the di~la~~t, velocity and acceleration estimates obtained from equations (1) and (2) within the iterative loop. (d) Viscous akmping. Damping is based on the difference between the velocity of any one node and the average velocity of the surrounding nodes, multiplied by a damping coefficient chosen to fit experimental data.