84 Notch sensitivity in static and fatigue loading

A practical concern in designing with metallic foams is the issue of damage tolerance: in the presence of a notch or hole, does it fail in a notch-insensitive, ductile manner when the net section stress equals the uniaxial strength? Or does it fail in a notch-sensitive, brittle manner, when the local stress at the edge of the hole equals the uniaxial strength? We can answer this question immediately for the case of static compression of a foam containing a circular hole: the large compressive ductility in a uniaxial compression test makes the foam notch insensitive. Experimental results confirm this assessment: a panel of width W, containing a hole of diameter D fails when the net section stress ans equals the uniaxial compressive strength apl of the foam. On noting that ans is related to the applied stress a1 by ans = (1 — (D/W))a 1 the net section failure criterion can be rewritten as a1 = api(1 — (D/W)) (8.2)

Figure 8.7 confirms that this relation is obeyed for Alporas, Alulight and Alcan aluminum foams, for panels cut in a variety of orientations with respect to the direction of foaming.

O Alporas,

♦ Alcan, 5 A Alcan, 5. A Alcan, 5 V Alulight, ▼ Alulight,

11 %, L, Compression W = 70mm 11 %, T, Compression W = 70mm 5.7%, L, Tension W = 70mm 5.7%, T, Tension W = 70mm 7%, L, Compression W = 115mm 7%, L, Compression W = 115mm 7%, L, Tension W = 115mm 7%, L, Tension W = 115mm 25-35%, L, Compression W = 20mm 25-35%, L, Tension W = 20mm

Notch sensitive, c" i

Notch lnsensitive,

Cpl W

Figure 8.7 Notch strength of foams, showing behavior to be notch insensitive in both monotonie tension and compression

Notch sensitive, c" i

Notch lnsensitive,

Cpl W

tttt<

Figure 8.7 Notch strength of foams, showing behavior to be notch insensitive in both monotonie tension and compression pl

For the case of compression-compression fatigue an analogous notched strength criterion applies as follows. Define the endurance limit for a un-notched panel or a notched panel as the gross-section value of amax in a fatigue test for which progressive shortening does not occur (N/ > 107 cycles). Then the net section stress criterion reads

where the subscript n refers to notched, for a hole of diameter D, and the subscript un refers to un-notched specimens. It is reasonable to expect that the net section stress criterion holds in compression-compression fatigue since metal foams progressively shorten under compressive fatigue loading, as Figure 8.4 showed. Notched fatigue data confirm this expectation, as shown in Figure 8.8: aluminum foams are notch-insensitive, and the endurance limit follows the net section stress criterion, given by equation (8.3).

1 i 1 1 1 i 1 1 1 i 1 1 1 i 1 O Alporas, (11%) : W = 70 mm # Alulight, (24-42%) : W = 10 mm liMA

Notch insensitive

1 i 1 1 1 i 1 1 1 i 1 1 1 i 1 O Alporas, (11%) : W = 70 mm # Alulight, (24-42%) : W = 10 mm

Notch insensitive

Notch sensitive

Figure 8.8 Compression-compression notch fatigue strength of foams, at infinite life; R = 0.1

TTTF

Notch sensitive

Figure 8.8 Compression-compression notch fatigue strength of foams, at infinite life; R = 0.1

Now consider a notched metallic foam under monotonic tension. Two types of failure mechanism can be envisaged: ductile behavior, whereby plasticity in the vicinity of the hole is sufficient to diffuse the elastic stress concentration and lead to failure by a net section stress criterion, as given by equation (8.2). Alternatively, a brittle crack can develop from the edge of the hole when the local stress equals the tensile strength of the foam Of & apl. In this case, a notch-sensitive response is expected, and upon assuming a stress concentration Kt for the hole, we expect brittle failure to occur when the remote stress o 1 satisfies

The following fracture mechanics argument suggests that a transition hole size Dt exists for the foam: for hole diameters D less than Dt the behavior is ductile, with a notched tensile strength a1 given by equation (8.2). Alternatively, for D greater than Dt the behavior is brittle, with a1 given by (8.4). A simple micro-mechanical model of failure assumes that the plasticity and tearing of the foam adjacent to the hole can be mimicked by a crack with a constant tensile bridging stress of magnitude api across its flanks, as illustrated in Figure 8.9. This approach follows recent ideas on 'large-scale bridging' of composites, see, for example, the recent review by Bao and Suo (1992). Assume that this bridging stress drops to zero when the crack flanks separate by a critical value U0. Measurements of U0 using deeply notched specimens reveal that U0 is approximately equal to the cell size l for Alporas foam, and we shall make this assumption. This physical picture of the tensile failure of the notched panel is consistent with the notion that the foam has a long-crack toughness of

and a tensile strength of apl. The transition size of hole, Dt, at which the notched strength drops smoothly from apl to a value of apl/3 is given by

api t t

Figure 8.9 A bridge crack model for yielding and cracking adjacent to an open hole

The transition hole size is plotted as a function of relative density for a large number of aluminum foams in Figure 8.10. We note that Dt is large, of the order of 1 m. This implies that, for practical problems, the net section stress criterion suffices for notched tensile failure.

The effect of a notch on the tension-tension fatigue strength has not yet been fully resolved. Experiments to date suggest that the net section stress upi (m)

Relative density

Figure 8.10 Predicted value of the transition hole size, Dt = Ei/api, plotted against relative density

0.01

Relative density

Figure 8.10 Predicted value of the transition hole size, Dt = Ei/api, plotted against relative density

criterion gives an accurate measure of the drop in fatigue strength due to the presence of a hole. But these tests were done on small holes (maximum diameter equals 30 mm). At the endurance limit, the extent of cyclic plasticity is expected to be much less than the tensile ductility in a static tensile test, and so U0 is expected to be much less than the cell size i. Consequently, the magnitude of Dt is expected to be significantly smaller for fatigue loading than for monotonic loading. This effect has been studied for fully dense solids (Fleck et al., 1994) but experiments are required in order to determine the appropriate value for tensile notched fatigue of metallic foams.

References

Ashby, M.F. and Jones, D.R.H. (1997) Engineering Materials, Vol. 1, 2nd edition, ButterworthHeinemann, Oxford. Banhart, J. (1997) (ed.) Metallschäume, MIT Verlag, Bremen, Germany. Banhart, J., Ashby, M.F. and Fleck, N.A. (eds) (1999) Metal Foams and Foam Metal Structures,

Proc. Int. Conf. Metfoam'99, 14-16 June 1999, Bremen, Germany, MIT Verlag. Bao, G. and Suo, Z. (1992) Remarks on crack-bridging concepts. Applied Mechanics Reviews 45, 355-366.

Fleck N.A., Kang K.J. and Ashby M.F. (1994) Overview No. 112: The cyclic properties of engineering materials. Acta Materialia 42(2), 365-381. Fuchs, H.O. and Stephens, R.I. (1980) Metal Fatigue in Engineering, Wiley, New York.

Harte, A.-M., Fleck, N.A. and Ashby, M.F. (1999) Fatigue failure of an open cell and a closed cell aluminum alloy foam. Acta Materialia 47(8), 2511-2524.

McCullough, K.Y.G., Fleck, N.A. and Ashby, M.F. (1999). The stress-life behavior of aluminum alloy foams. Submitted to Fatigue and Fracture of Engng. Mats. and Structures.

Olurin, O.B., Fleck, N.A. and Ashby, M.F. (1999) Fatigue of an aluminum alloy foam. In Banhart, J., Ashby., M.F. and Fleck, N.A. (eds), Metal Foams and Foam Metal Structures, Proc. Int. Conf. Metfoam'99, 14-16 June 1999, MIT Verlag, Bremen, Germany.

Schultz, O., des Ligneris, A., Haider, O. and Starke, P (1999) Fatigue behavior, strength and failure of aluminum foam. In Banhart, J., Ashby, M.F. and Fleck, N.A. (eds), Metal Foams and Foam Metal Structures, Proc. Int. Conf. Metfoam'99, 14-16 June 1999, MIT Verlag, Bremen, Germany.

Shwartz, D.S., Shih, D.S., Evans, A.G. and Wadley, H.N.G. (eds) (1998) Porous and Cellular Materials for Structural Application, Materials Reseach Society Proceedings, Vol. 521, MRS, Warrendale, PA, USA.

Sugimura, Y., Rabiei, A., Evans, A.G., Harte, A.M. and Fleck, N.A. (1999) Compression fatigue of cellular Al alloys. Mat. Sci. and Engineering A269, 38-48.

Suresh, S. (1991) Fatigue ofMaterials, Cambridge University Press, Cambridge.

Zettel, B. and Stanzl-Tschegg, S. (1999) Fatigue of aluminum foams at ultrasonic frequencies. In Banhart, J., Ashby, M.F. and Fleck, N.A. (eds), Metal Foams and Foam Metal Structures, Proc. Int. Conf. Metfoam'99, 14-16 June 1999, MIT Verlag, Bremen, Germany.

Chapter 9

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