Fatigue-related damping in some cellular metallic materials

Materials Science and Engineering A 370 (2004) 537–541

Fatigue-related damping in some cellular metallic materials I.S. Golovin a,∗,1 , H.-R. Sinning a , J. Göken b,2 , W. Riehemann b b

a Institut für Werkstoffe, Technische Universität Braunschweig, Langer Kamp 8, D-38106 Braunschweig, Germany Institut für Werkstoffkunde und Werkstofftechnik, Technische Universität Clausthal, Agricolastraße 6, D-38678 Clausthal-Zellerfeld, Germany

Received 12 July 2002

Abstract Damping in cellular metallic materials is enhanced by localised stresses in comparison with the corresponding dense materials. A pronounced increase in damping with the amplitude of vibration interferes with an additional time dependence due to fatigue processes. This effect, attributed to growth and movement of cracks, is observed in Al- and Zn-based foams, porous steel and titanium. A model of this contribution to amplitude dependent damping is discussed using a simulation of experimental data for a porous 316L (FeNiCr) steel. © 2003 Elsevier B.V. All rights reserved. Keywords: Internal friction; Fatigue; Cellular metallic materials

1. Introduction Experimental data and different mechanisms of damping (thermoelastic effect, sound absorption, dislocation related damping) in some cellular metallic materials (CMM): foams, sponges and sintered metals are discussed in [1]. Damping due to local acts of microplastic deformation in CMM is analysed in [2]. The accumulation of microplastic deformation and damage due to fatigue (increase in number of vibrations) and its influence on damping level and internal friction mechanisms is the subject of this paper. 2. Materials and methods Porous 316L steel (produced by Forschungszentrum Jülich FZJ, Dr. M. Bram: porosity [3] P = 1 − ρ∗ /ρs from 14 to 78%, and by GKN Sinter Metals Filters GmbH, P from 22 to 52%) and titanium (FZJ production: P from 47 to 72%) are studied. For the abbreviation of steel specimens, we use PM (for FZJ) or GKN plus porosity in percents, PT means porous titanium. GKN specimens have one level of porosity produced by direct sintering, while FZJ specimens ∗ Corresponding author. Tel.: +49-531-3913066; fax: +49-531-3913058. E-mail address: [email protected] (I.S. Golovin). 1 On leave Tula State University, Tula 300600, Russia. 2 Present address: Institut für Werkstofforschung, GKSS Forschungszentrum, Max-Plack-Straße 1, D-21502 Geesthacht, Germany.

0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.08.090

have two levels of porosity due to space holder method used: macrocells (350–500 ␮m) surrounded by walls with a porosity ≈14% due to microvoids (≈10 ␮m). More information on their structure and properties is given in [1–4]. Al based foam with trade name Alporas® (Shinko Wire, Japan, Dr. T. Miyoshi) with a density of 0.40 Mg/m3 (P ≈ 85%) and an average pore size of 4 mm (2–3 wt.% calcium metal and 1.6 wt.% titanium hydride TiH2 [4]) with rather homogeneous structure [1] is studied. For measurements of amplitude dependent internal friction (ADIF) with decreasing strain amplitude ε = 10−3 to 10−5 , starting from values ε0 = 10−3 to 10−4 , we used a free-clamped vibrating reed technique [5] with electromagnetic excitation and detection. The damping was measured as the logarithmic decrement δ of free bending vibrations from successive amplitudes in the frequency range of 20–80 Hz. The specimens cut by spark erosion were (2–10) mm × (5–30) mm ×(80–120) mm. Internal friction (δ), modulus of elasticity (E) and modulus defect (E/E) were determined as explained in detail in [5–7]. Cyclic tension–compression tests were done using a Zwick universal testing machine and Alporas® specimens with a cross-section of 30 mm×30 mm.

3. Fatigue dependent instability of damping The instability of damping is observed in CMM beyond a certain amplitude [1,2,7,8]. By instability we mean that internal friction measured in a first test differs from that one

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Fig. 1. Alporas® Al foam, density ≈0.4 Mg/m3 : (a) mechanical hysteresis loops in 1, 2 and 2700 cycles; (b) ADIF curves after different number of vibration N for: (I) rectangular specimen in as received state (two samples) (N from 300 to 5.5 × 104 ), (II) specially mechanically shaped specimen to avoid stress concentration near clamping (N from 300 to 7.6 × 104 ), (III) similar specimen after annealing at 200 ◦ C (N from 300 to 3.3 × 104 ), (IV) 50% compressed specimen (N from 300 to 107 ). Surface with fatigue lines after cycling tests, 2700×.

measured in the following tests. Even after the first cycle of deformation σ versus ε has a well defined hysteresis between loading and unloading curves [2,7]. At the same time a zero point drift was observed in the most cases [2]. A σ–ε hysteresis loop is also observed by push-pull mechanical tests with R = −1 (Fig. 1a). Asymmetric deformation occurs in the first cycle due to hardening, from the second cycle the loop is symmetrical. Increase in number of vibrations leads to increase in area between loading and unloading curves (dissipated energy), to decrease in averaged modulus of elasticity (determined as a slope of a middle line of hysteresis loop) and to shift of “zero” point to compression (“cyclic creep”). These effects relate to a change of the microstructure and are important in respect to fatigue, i.e. number of cycles (N). Two schemes [1] of ADIF curves with N change are observed in this case. First tendency is typical for Alporas® Al foams (made from nearly technically pure Al; Fig. 1b). Simple shift of δ versus ε curves to higher δ with N is observed. Absolute δ values for Alporas® Al foams are about one order higher than that is in porous steel and titanium. This tendency is controlled by deformation accumulated by dislocation [7]. Absolute values of damping are sensitive to: • value of starting amplitude, e.g. lower curve I (sample 1) is comparable with curves II, while curves I (sample 2) measured from higher amplitude differ from curves II; • specimen shape and clamping (special shape was used to decrease stress in clamped part of specimen); • mechanical and heat treatment of the specimen. Second tendency: Increase in number of vibrations leads to increase in damping at low amplitudes or to formation of a peak at a certain amplitude below ε0 (Fig. 2a–c). Contrary to Alporas® Al foam, increase in number of vibrations of Al-based foams with significant Si content (AlSi1Mg1, AlSi7—structures in Fig. 1 [1]), 50% com-

pressed Alporas® Al foam (Fig. 1b, compression leads to multiply cracks formation) and porous sintered titanium leads to formation of a peak at ADIF, while sintered by different methods 316L steels exhibit both: peak or simple increase in δ value dependently on their porosity. Decrease in modulus of elasticity accompanies fatigue for all above-mentioned CMM independently on internal friction tendency. The main emphasis in this paper is made on porous 316L steels produced by different methods [2] (porosity 14, 29, 45, 62, 64 (with different pore size), 72 and 78%) and porous titanium (porosity 47, 67 and 72%)—Fig. 2a. These materials exhibit the second tendency for IF with increase in N. As rule of thumb: an increase in porosity enhances this effect (Table 1), while the only weak dependence on size of spacer produced pore size (Fig. 2b) was observed. Additional annealing (1000 ◦ C for 20 min for porous 316L steel, 520 ◦ C for 20 min for porous Ti, 200–300 ◦ C for Alporas® foams) leads to a certain increase in damping and degree of modulus degradation with N. While increasing porosity, the ADIF peak shifts to lower vibration amplitudes ε, e.g. the peak is observed for PM-29 and PM-45 steels (Fig. 3), while at higher porosity (PM-64 Table 1 Increase in damping (δ at ε ≈ 10−4 ) with increase in N for porous 316L steel Porosity

N (×105 )

δN (×10−3 )

0 14 29 45 62 64 72

400 170 20 3 3 1 0.7

1 7 20 45 65 50 50

δcr (×10−3 ) +0.1 −0.2 15 40 48 40 35

N is the number of forced vibrations at amplitude ε0 ≈ 0.001; δcr = δN − δdisl , where δN is damping after N vibrations.

I.S. Golovin et al. / Materials Science and Engineering A 370 (2004) 537–541

0,020

δ

539

0,05

PM 29

ε03

0,015

ε02

δ 0,04

N=500; 11500; 96100; 177100;

700 42100 132100 206680 276280

0,03

ε01

0,010

0,02 6

0,005 N=700

0,0000

0,0003

N=2x10 εo3 εo2 εo1

0,01

ε0=0.001

ε

0,0006

PM 45

0,00 0,0000

0,0004

0,0008

ε

Fig. 3. ADIF curves in PM-29 and PM-45 as a function of number of vibrations and starting amplitude of vibrations.

(Fig. 2b) and PM-62 (Fig. 4a)) only the high amplitude shoulder of the peak is observed. The contribution of fatigue to damping can be separated from other contributions by subtracting from the ADIF curves after N vibrations the first measured ADIF curve with minimum Nmin (usually Nmin ≈ 300 used for the calibration). Like in the example of Fig. 4 for PM-62, a nearly linear decrease in the resulting crack-related damping δcr with ε is observed for 316L steels with a porosity of more than 45%. For PM-29 and PM-45 all the ADIF peak is within measuring interval. The behaviour as a function of N is seen more clearly in Fig. 5, where the changes of the starting amplitude ε0 of the individual ADIF curves (at constant excitation power), of the modulus defect E/E (absolute values of E are in [2]), and of the slope αN = −dδcr /dε are compared to each other. All these quantities increase with N in a very similar way and show first a non-linear and then a linear range on the log(N) scale in Fig. 5. Both αN and ε0 are almost proportional to E/E (inset in Fig. 5), which is expected if they are controlled by the same mechanism. The situation is slightly different in the case of porous titanium: a well defined peak in the ADIF curves appears and increases with N (Fig. 2a). Damping in porous Ti for

Fig. 2. Change of ADIF curve shape with increase in number of vibrations: (a) porous Ti (porosity ≈47%, different starting amplitude of vibrations ε0 ); (b) two porous 316L steel specimens PM-64 with different pore sizes 275 and 425 ␮m; (c) specially shaped GKN-33 specimen (surface with fatigue lines, 2300×).

δ 0,02

0,00 0,0000

(a)

0,04

increasein N

0,04

0,06 N max

last curve, N=338000

0,06

PM 62 fo=23,5 Hz εo=0.001

δcr

first curve

0,0005

ε

first curve

0,02

Nmin

0,00 0,0000

0,0010

(b)

0,0002

0,0004

ε

0,0006

Fig. 4. ADIF curves as a function of N for PM-62 (a) and the same curves after the first curve subtraction from all other curves (b).

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Fig. 5. Change in ε0 , αN and E/E vs. N and structure of PM-62.

comparable porosities is lower than in steel, which is in agreement with damping in dense Ti and 316L steel. The ADIF curves at the right side of this peak are similar to that observed in porous steels. Change in ε0 and E/E values with N is more intensive and jump-like in the first cycles of vibrations to be compared with porous steel. The dependence of αN versus E/E for porous Ti is not linear as in 316L steel and will be discussed elsewhere. This effect is stronger if porosity is higher. Similar peak in the ADIF takes place with increase in N in AlSi foams even at higher amplitudes of deformation [7]) and might be also characterised by an αN value.

ment. Then, the dependence of the internal friction of the strain for this elementary process is: IFe ∝

εc for ε ≥ εc , εc + ε

and

IF = 0 for ε < εc ,

(1)

where εc is the critical strain which is connected with the critical stress by σc ≈ Er εc . The assumption of only one crack or many cracks with only one critical stress is not realistic. The samples cycled to fatigue will have a lot of cracks resulting in a distribution of critical strains. It is assumed that a realistic

4. Crack induced damping—simulations There are many experimental observations indicating the crack origin and other crack-like defects of the high damping measured at low strains in the order of 10−4 not only in CMM [1,7] but also in crack-sensitive magnesium alloys [8,9]. Therefore, a simple model taking into account the crack origin of damping has been developed (Fig. 6a), which is explained in more details in [9]. It will be shown that this model also supports the assumption of cracks in the investigated foams and that the results of the model can be used to interpret the experimental findings reasonably in a more quantitative way. One elementary crack is assumed to be represented rheologically by a lock which is attached in series to a spring representing the modulus (Ee ) loss when the crack is opened (see Fig. 6a). Another spring attached parallel to this arrangement represents the relaxed modulus (Er ) of the solid with opened cracks. In this process, the elementary crack opens at the critical stress (σ cr ) and the mechanical energy is converted to heat by the displacement of the two crack surfaces, by emission of dislocations at the crack tips or by crack growth. The modulus decreases in the regime of tensile stresses due to the missing contribution of the cracked parts of the specimen, in compression the crack walls get into contact again resulting in a constant stiffness (Fig. 6a). It is assumed that the cracks do not change during measure-

Fig. 6. (a) Rheological model for crack related damping (two springs and lock) with corresponding hysteresis loop and amplitude dependent internal friction. (b) Logarithmic decrement of crack induced damping vs. strain of the bending vibrations for various numbers of cycles. Experiment and simulations according to Eq. (3) for PM-62.

I.S. Golovin et al. / Materials Science and Engineering A 370 (2004) 537–541 Table 2 Fit parameters according to Eq. (3) for various numbers of cycles to fatigue N

A (×10−3 )

γ

εm (×10−6 )

33540 86340 145740 213120 338000

47 96 166 174 202

2.32 2.20 1.70 2.18 2.12

68 68 91 68 75

crack collective is represented by the log-normal distribution function: 
(ln εc − ln εm )2 D(εc ) ∝ exp − (2) 2(ln γ)2 where εm is the mean average value of the critical strains and γ their geometrical standard deviation. This assumption is more reasonable than a normal (Gauss) distribution function because the log-normal distribution function does not permit negative critical strains. Then, for the internal friction caused by cracks the following equation holds: 
 ε εc (ln εc − ln εm )2 δcr (ε) = A dεc (3) exp − 2(ln γ)2 0 εc + ε where δ is the crack induced logarithmic decrement for bending vibrations and A the damping strength taking into account the number of cracks and their specific contribution to damping. In Fig. 6b the crack induced damping (from Fig. 4b) together with the fitted damping according Eq. (3) is plotted versus strain of the bending vibrations in logarithmic scale. For this the background damping produced by dislocation motion and other effects (“first measured curve”) has been subtracted from the corresponding measured values plotted in Fig. 4. It can be seen that the fitted curves represent the measured values fairly good. Some small but systematic deviations can be attributed to the more arbitrary choice of the distribution function, which cannot consider the distribution of cracks and their contribution to damping in detail. Nevertheless, some reasonable conclusions of the microstructure development in the foam samples with increasing number of cycles to fatigue can be drawn from the fit parameters being listed in Table 2. It can be seen that the parameter A denoting the damping strength increases monotonously with increasing number of cycles to fatigue. From this an increasing number of cracks or crack growth with increasing number of cycles can be concluded. On the other hand, neither the geometric standard deviation nor the mean average value of the critical strain varies monotonously or significantly with the number of cycles

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to fatigue. The small and non-systematic variations can be mainly attributed to small errors in the fit which are due to the fact, that only the descending part of the maximum has to be fitted and the maximum itself lies below the measured range of stains. So within a certain range of uncertainty both the geometric standard deviation and the mean average value of the critical strain are nearly constant or do not vary very much with the number of cycles. This finding for metallic foams is contrary to the situation in bulk metal specimens [9], where the maxima of the damping versus strain curves are shifted to lower strains due to increasing N, which is explained by corresponding crack growth. Nevertheless, the nearly constant maxima for foams are reasonable and can be explained by their cellular structure which restricts the crack length to the thickness of the cell walls. Contrary to many other CMM, the δ–ε hysteresis loops for Alporas® Al foam (Fig. 1a) are symmetrical in tensile and compressive parts under the used test conditions. This is an additional argument to say that the appearance of cracks is not the dominating factor under cycling for this material.

5. Conclusions Porous metallic materials exhibit unstable damping if a certain amplitude of deformation is exceeded. The fatiguerelated accumulation of this effect is reported quantitatively for different porous materials. The development of (micro-)cracks with increasing N produces a damping increase at low amplitudes ε, in many cases with a maximum as a function of ε. An analytical model is proposed on the base of a phenomenological model in which a log-normal distribution for cracks lengths is introduced. The proposed model reasonably describes the experimental data. Some quantitative parameters of cracks are computed.

References [1] I.S. Golovin, H.-R. Sinning, Mater. Sci. Eng. A 370 (2004) 504– 511. [2] I.S. Golovin, H.-R. Sinning, I.K. Arhipov, S.A. Golovin, M. Bram, Mater. Sci. Eng. A 370 (2004) 531–536. [3] L.G. Gibson, M.F. Ashby, Cellular Solids, Cambridge University Press, Cambridge, 1997, p. 510. [4] J. Banhart, Prog. Mater. Sci. 46 (2001) 559–632. [5] J. Göken, W. Riehemann, Mater. Sci. Eng. A 323 (2002) 134–140. [6] J. Göken, W. Riehemann, Technisches Messen 68 (2001) 535–545. [7] I.S. Golovin, H.-R. Sinning, J. Göken, W. Riehemann, Cellular Metals and Metal Foaming Technology, MIT Press, Cambridge, 2001, pp. 323–327. [8] J. Göken, W. Riehemann, Mater. Sci. Eng. A 323 (2002) 127–133. [9] J. Göken, W. Riehemann, Mater. Sci. Eng. A, in press.