To simulate impact situations, data at high strain rate is also needed. The UTM is usually too slow for this kind of test. This article presents a 3 point bending test method using an impact tower at much greater velocities. Solutions are presented to reduce noise due to vibrations in the system. The set up is tested using a Montell Polypropylene and the results show good consistency for modulus, yield stress and strain at yield.

The impact tower

The impact tower is generally used to obtain load deflection curves from disk specimens [6]. A heavy crosshead is dropped from a variable height, falling along 2 guiding rods (figure 1). A load cell at the tip of the tup records the load during the impact. When the crosshead goes through a flag, the acquisition of the data from the load cell starts and the velocity is recorded. After the failure of the specimen, two shock absorbers stop the fall of the crosshead.

For the high velocity 3 point bending test, a loading nose is attached to the tup. The specimen rests on 2 cylindrical supports. It is placed dirtectly below the crosshead, so that the loading nose strikes the beam in its center. During the impact, the load is recorded with the load cell. The load-time curve is processed to obtain a force-deflection curve. The momentum conservation applied to the crosshead only gives:

(1)

Where:
m_c is the mass of the crosshead

a(t) is the acceleration of the crosshead
g is the gravity
F(t) is the force
Integrating once gives the velocity:

(2)

Where v_o is the velocity at the flag.

Integrating a second time gives the displacement of the crosshead from the flag:

(3)

The deflection d at the center of the beam is then obtained by removing the distance flag-specimen from x(t). It can be argued that the force measured is not the actual force experienced by the specimen, because of the inertia of the loading nose. The conservation of momentum applied to the loading nose assuming that it sees the same acceleration as the crosshead gives:

(4)

Where:
m_ln is the mass of the loading nose
P(t) is the force exerted by the specimen on the loading nose.

(5)

In the set up, m_ln is very small compared to m_c, so that P(t)~F(t).

Stress and Strain calculation

From the load-deflection curve, the stresses and strains can be determined. Assuming small deformations, the engineering strain at the outer fiber of the beam is given by:

(6)

Where d is the deflection of the beam at its center, t is the thickness of the specimen and L is the span. The strain rate at the outer fiber is therefore given by:

(7)

Where v is the velocity of the crosshead. Assuming linear elasticity (figure 2a), the stress at the outer fiber is given by:

Where v is the velocity of the crosshead. Assuming linear elasticity (figure 2a), the stress at the outer fiber is given by:

(8)

Where P is the load at the center of the beam and w is the width of the specimen. If we assume perfectly and totally plastic deformation at yield in the center of the beam (figure 2b), the stress should be written [2]:

(9)

Experimental set up

The first experiments are performed on a Polypropylene specimen. The span used is 50.8mm. The stress-strain curves are calculated using a linear elastic approach. The resulting stress-strain curve is showed on figure 4. Low frequency vibrations heavily perturb the overall shape of the response. It is then difficult to obtain material properties from the curve. The goal is therefore to find a set up so that the period of the oscillations is much smaller than the duration of the test. If necessary, filters can be uses to smooth the curve without altering its overall shape. A closer look at the mechanics of the test shows that the parts of the beam which are outside the bending region (gray areas on figure 3) increase the inertia of the system. By removing them, the natural frequencies of the beam should increase. The same test is therefore performed with a “reduced length” beam using the same span. The resulting stress strain curve is shown on figure 4. Removing the sides of the specimen clearly improves the quality of the signal.


Another way to reduce the increase the natural frequencies of the beam is to reduce the span. A 25.4mm span beam with reduced length is tested, adjusting the velocity so that the strain rate at the outer fiber is the same as when using a 50.8mm span (100 s^-1) according to equation 7. Figure 4 shows that the quality of the signal is again enhanced. The curves show another effect: The reduction of the span increases the calculated yield stress. A more appropriate model for the stress calculation, currently in development, should be used to remove this discrepancy.

Results

Three point bending experiments are performed at different strain rates, using a 25.4mm span on a Montell polypropylene. An Instron UTM is used for low strain rates, and a Dynatup impact tower is used for high rates of strain.

The stress-strain curves are obtained using a linear elasticity approximation. The resulting stress-strain curves are shown on figure 5. The lowest 3 curves were obtained using the UTM with strain rates ranging from 0.0025
s^-1 to 0.25 s^-1 at the outer fiber. The highest curves (thinner) were obtained using the impact tower with strain rates ranging from 25 s^-1 to 115 s^-1.

The weight of the crosshead should insure a minimum velocity slow down during the test. Using a 12kg crosshead weight, the velocity actually increases during the impact. The modulus, yield stress and yield strain are also plotted as a function of strain rate (figure 6).

-The modulus increases with the strain rate, which is consistent with a viscoelastic approach. -The yield stress is calculated assuming a total and perfect plastic state at yield, using equation 9. The yield stress increases with strain rate and follows the Eyring theory for the yielding of polymers: it increases linearly with the logarithm of the strain rate [4].

-The strain at yield increases at very low strain rates and does not significally vary for higher rates of strain.

Conclusion

A methodology for 3 point bending tests at high rate of strain has been presented. Specific set ups have been shown to reduce the vibrations in the system, which has been a major problem in the past attempts. A family of stress-strain curves has been obtained for a Polypropylene. The results showed encouraging results, consistent with viscoelastic and yielding theories. The experiments also showed a dependence of calculated yield stress with the span used.

The model currently used to determine the stress at yield at the center of the beam should be improved to take the effect of the span in account. Different aspect ratios for the specimen will also be tested in the future.

References

1. H. Lobo and J. Lorenzo: “High Speed Stress-Strain Material Properties As Inputs For The Simulation Of Impact Situations”. IBEC Proceedings, 1997.

2. G. Trantina and P. Oehler: “Standardization, Is It Leading To More Relevant Data For Design Engineers?”. SPE ANTEC Proceedings, 1994.

3. L.E. Nielsen and R.F. Landel: “Mechanical Properties Of Polymers And Composites” (2^nd edition). L.L. Faulkner, 1994.

4. N. G. McCrum, C. P. Buckley and C. B. Bucknall: “Principle Of Polymer Engineering” 2^nd edition. Oxford Science Publications, 1997.

5. ASTM D 790 (Plastics): “Standard Tests Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials”.

6. ASTM D 3763 (Plastics): “Standard Tests Method for High Speed Puncture Properties of Plastics Using Load and Displacement Sensors”

Acknowledgments

The authors wish to thank Eric Dunbar from Dynatup for his technical support on the impact tower, and Jim Lorenzo from Montell for providing the specimens.