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Creep of sandwich beams with metallic foam cores
Materials Science and Engineering A341 (2003) 264 /272 www.elsevier.com/locate/msea
Creep of sandwich beams with metallic foam cores O. Kesler, L.K. Crews, L.J. Gibson * Department of Materials Science and Engineering 8-135, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Received 22 January 2002; received in revised form 5 April 2002
Abstract The steady state creep deflection rates of sandwich beams with metallic foam cores were measured and compared with analytical and numerical predictions of the creep behavior. The deflection rate depends on the geometry of the sandwich beam, the creep behavior of the foam core and the loading conditions (stress state, temperature). Although there was a considerable scatter in the creep data (both of the foams and of the sandwich beams made using them), the data for the sandwich beams were fairly well described by the analysis. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Sandwich beams; Metallic foams; Creep behavior
1. Introduction Metallic foams are well suited to a variety of applications. Their capacity to undergo large deformations at roughly constant load makes them ideal energy absorbers in vehicles and in packaging. Their high surface area and large volume of interconnected porosity make open-cell metallic foams attractive for use as heat sinks in electronic devices. Their low density can be exploited in lightweight structural sandwich panels. Relative to polymer foams, they have higher melting points, allowing them to be used at higher temperatures. At temperatures above approximately 0.3 times the melting temperature, time-dependent creep deformation becomes important for metallic components carrying a sustained load over an extended time. Thus, an understanding of the high-temperature deformation of sandwich beams with metallic foam cores is of practical interest.
* Corresponding author. Tel.: /1-617-253-7107; fax: /1-617-2586275 E-mail address: [email protected] (L.J. Gibson).
Previous studies have examined the creep behavior of metallic foams, through a variety of experimental, analytical, and computational methods. The creep behavior of open- and closed-cell foams can be modeled by analyzing the bending of the cell edges and the stretching of the cell faces; the analysis describes the data for the creep of open- and closed-cell aluminum foams well [1,2]. While bending is the dominant mechanism of creep deformation at low stresses, at high stresses, in compression, creep buckling becomes important [3]. Finite element analysis of the role of cell geometry on the creep rate of two-dimensional cellular solids has shown that structures with random cells or with curved cell walls creep faster than a regular structure [4]. The creep behavior of a sandwich beam with a metallic foam core has been calculated numerically, giving the deflection rate of the center of the beam over time for a specific beam geometry and set of face sheet and foam core properties [5]. No experimental studies of the creep behavior of sandwich beams with metallic foam cores are available. This work seeks to provide a better understanding of the creep behavior of sandwich structures with metallic foam cores through a series of experimental studies.
0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 2 3 9 - 3
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
2. Analysis
ˆ o˙ij f (s)
The power creep law describing the steady state creep rate of a solid can be written as: ns s o˙s o˙0s (1) s0s where Qs
(2)
RT
and A /1 s 1. In the above expressions, ns, Qs, and s0s are material properties, s is the applied stress, T is the absolute temperature, and R is the universal gas constant, 8.314 J mol K 1. For an open-cell foam with relative density r =rs ; the steady state creep rate is [1]: (3ns 1) 2 o ˙ 0:6 1:7(2ns 1) s ns r o˙0s (ns 2) ns s0s rs
(3)
The steady state creep rate of a closed cell foam is [1]
o ˙ o˙0s
1
a 3
ns 2 1=ns
1:7
0:6
nP
1=2
2 1 nP
To solve for the expression f (s); ˆ we note that in uniaxial tension, se /s11 and sm /s11/3, giving s s ˆ 11 : For uniaxial stress, @ s=@s ˆ 11 1 and n s o˙11 o˙0 11 (7) s0 giving f (s) ˆ o˙0
n sˆ s0
A sandwich beam of span l and width b loaded in
ns
s 0s (3ns 1)=2ns r 2 r f (1 f) rs 3 rs 2ns 1
(4)
ns
where f is the volume fraction of solid in the cell edges. When f /1, Eq. (4) reduces to the open cell foam expression in Eq. (3). The model states that the creep constants ns and Qs in Eqs. (3) and (4) are identical to the creep constants of the solid material from which the foam is made; this has been confirmed experimentally [1,2].
three point bending with a concentrated load P is shown in Fig. 1. The thicknesses of the face and core are t and c , respectively. The distance between the centers of the two face sheets is d and the distance between the top and bottom of the beam is h /c/2t . The faces have a Young’s modulus Ef, and the core has a Young’s modulus Ec and a shear modulus Gc. The stresses in the face and core are given by Allen [8]. The normal
2.2. Multiaxial stress states When a foam is subjected to a multiaxial stress state, it becomes necessary to determine the resultant stress, s: ˆ The resultant stress and corresponding strain-rates can be calculated from the equivalent stress, se and the mean stress, sm, as follows [6,7]: sˆ 2 and
1 (1 (a=3)2 )
[s2e a2 s2m ]
(8)
2.3. Stresses and deflections in sandwich beams
s 1
(6)
@sij
Here, the constant a is related to the plastic Poisson’s ratio, nP, by
2.1. Power law creep
o˙0s A exp
@ sˆ
265
(5) Fig. 1. Sandwich panel in three point bending.
266
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
stress in the face at a cross section with a bending moment, M , is: sf
M d Ef (EI)eq 2
(9a)
Ef bt3 Ec bc3 Ef btd 2 Ef btd 2 : 6 12 2 2
The maximum normal stress in the core at a section with a bending moment, M, is: sc
M
c
(EI)eq 2
Ec
(9b)
The normal stress in the core is relatively small compared with that in the face, since Ec /Ef. The shear stress in the core at a section with a shear force, V , is: tc
V
(9c)
bd
The elastic deflection of such a sandwich beam is the sum of the bending and shear deflections [8]: d
Pl3 48(EI)eq
Pl 4(AG)eq
(10)
bd 2 c
Gc
(11)
The creep deflection rate of the beam is found in a similar fashion by adding the bending and shearing creep deflection rates.
2.4. Creep deflection rate of a sandwich beam in bending The bending creep rate is related to the power law creep response of the faces: o˙ Af snf
(12)
where Af and nf are creep properties of the face material. The bending deflection rate of the beam is then calculated by assuming that plane sections remain plane and that the axial stresses in the core are much smaller than the axial stresses in the faces. Ashby et al. [6] find the bending deflection rate at the center of the beam to be: P nf (l=2)nf 2 1 d˙b max 1 (13) Af 4bB nf 1 nf 2 where
where o˙0c ; s0c, and nc are the creep parameters of the core material. The core carries both normal stress s11 and shear stress s12; for metallic foam cores, both are significant. Using Eqs. (5) and (6) Ashby et al. [6] find the shear deflection rate at the center of the beam to be: nc sˆ cl 3s12 d˙s max o˙0c s0c 2d 1 (a=3)2 1=2 3 2 s211 s (16) 12 1 (a=3)2 Noticing that nc sˆ o˙0c s0c is an expression for the observed strain rate o˙c of the foam by itself, and using the definition of sˆ in Eq. (5), we can simplify Eq. (16) to:
and (AG )eq, the equivalent shear rigidity is: (AG)eq
(14)
The shear deflection rate is calculated from the creep rate of the core. The power law creep of the foam core under uniaxial stress is given by: nc sˆ o˙c o˙0c (15) s0c
where (EI )eq, the flexural rigidity of the section, is: (EI)eq
2 1 2 1 nf nf h c B 1 2 2 2 nf 1
d˙s max o˙c
cl 3 s12 2 2d 1 (a=3) sˆ
(17)
Eq. (17) was used to calculate the expected deflection rates for the sandwich beams tested in this study. The values of o˙c were those measured previously in uniaxial loading for specimens of the foam core materials [1,2]. For a sandwich beam in three point bending, the value of s12 is constant throughout the core. The value of s11 varies along the length of the beam and from the neutral axis to the top or bottom of the core. For the beams tested in this study, described below, the maximum ratio of s11 to s12 was 0.26. We find that the effect of the s11 on the value of sˆ is negligible (less than 2%) for the maximum ratio of s11 to s12 in the cores in this set of experiments. We note that if s11 can be ignored in calculating the resultant stress, s; ˆ then sˆ 2 :
3s212 2 a 1 3
giving d˙s max : o˙c
cl
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
2d
1 (a=3)2
(18)
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
where o˙c is the power law creep rate of the foam under a uniaxial stress equal to the resultant stress in the foam core of the sandwich beam and at the same temperature as the beam. In addition, the contribution to the total deflection rate of the bending of the faces was found to be negligible relative to the contribution from the shear of the core, as is described in Section 4. The total deflection rates of the beams in this study were calculated from the equations for shear deflection rates only (Eq. (17)).
3. Methods To assess the validity of the above analysis, an experimental program was undertaken to measure the creep deflection rates of sandwich beams with aluminum foam cores. The sandwich beams were manufactured by bonding aluminum foam cores to type 316 stainless steel face sheets using an aluminum-based metallic composite adhesive (trade name Durabond 950, Cotronics, Brooklyn, NY). Two types of commercially available aluminum foams were employed: an open-cell foam (trade name Duocel, ERG, Oakland, CA), and a closed-cell foam (trade name Alporas, Shinko Wire, Amagasaki, Japan). 3.1. Materials The Duocel foam is made from 6101-T6 aluminum via a casting process. The nominal density of the foam is 240 kg m 3 (relative density 0.09), and the cells are elliptical in shape, with major and minor axes of approximately 2.5 and 1.5 mm, respectively. Because of the elongated shape of the cells, the mechanical properties differ from one axis to the other. Parallel to the major cell axis, the uniaxial compressive strength of the foam is 2.17 MPa and the Young’s modulus is 0.634 GPa. Perpendicular to the major axis, the strength and modulus are 1.39 MPa and 0.236 GPa, respectively [9]; previous data suggests that the Duocel material is roughly transversely isotropic [10]. The plastic Poisson’s ratio np for the Duocel foam is 0.052, [11], resulting in a
Table 1 Uniaxial creep data for Duocel foam [1] Stress (MPa)
Temperature (8C)
0.42 0.42 0.42 0.53 0.68
250 275 300 275 275
o˙min (s 1)
/
1.10 10 8 1.10 10 8 4.20 10 7 1.20 10 8 4.10 10 8
o˙max (s 1)
/
1.00 10 7 4.20 10 7 9.00 10 6 4.90 10 8 2.60 10 7
267
Table 2 Uniaxial creep data for Alporas foam [2] Stress (MPa)
Temperature (8C)
0.25 0.32 0.42 0.42 0.42
300 300 300 280 260
o˙min (s 1)
/
5.50 10 10 1.67 10 9 5.39 10 9 1.69 10 9 7.76 10 10
o˙max (s 1)
/
1.91 10 9 3.58 10 9 1.72 10 8 1.18 10 8 4.70 10 9
value for a of 1.96. The activation energy, Q , and power law constant, n for the Duocel foam have been measured in tension and compression, for loading along the major axis [1]: Q /214 kJ mol 1 and n /4.6. These values are slightly higher than those of solid 6101 aluminum (Q /173 kJ mol 1, n /4.0). At each stress and temperature, there was roughly an order of magnitude spread in the creep data; the range of creep rates measured in the uniaxial tests on the Duocel foam at the stresses and temperatures used in this study are listed in Table 1. The Alporas foam is made from molten aluminum mixed with titanium hydride (1 /3 wt.%) and calcium (0.2 /8 wt.%) via a liquid-state process. The nominal density of the Alporas foam used in the creep experiments is 235 kg m 3 (relative density 0.09). The cells are equiaxed with a cell size of 4.5 mm. Mechanical property tests [9,11] indicate that the strength of the foam is 1.46 / 1.84 MPa, the modulus is 1.00 /1.14 GPa and the plastic Poisson’s ratio is 0.024, resulting in a value for a of 2.05. The average values of the activation energy, Qc, and power law creep exponent, nc, of the Alporas foam in tension and compression have been found to be [2]: Q / 97.7 kJ mol 1 and n /4.65 for temperatures less than or equal to 300 8C and stresses less than or equal to 0.42 MPa. The measured values of Q and n are close to those for solid pure aluminum, Q /70 kJ mol 1 and n /4.4. As for the Duocel foam, at each stress and temperature, there was roughly an order of magnitude spread in the creep data; the range of creep rates measured in the uniaxial tests on the Alporas foam at the stresses and temperatures used in this study are listed in Table 2. Type 316 stainless steel, used to make the sandwich faces, has a tensile yield strength of 205 MPa and a modulus of 216 GPa [12]. Frost and Ashby [13] report the creep constants of the material to be Q /280 kJ mol 1 and n /7.9, with s0 /33.5 MPa; the melting temperature is 1680 K. The creep behavior of the Durabond 950 metallic adhesive used to bond the core to the face sheets was evaluated in this study. Three flat steel bars, each measuring 25 mm wide, 50 mm tall, and 3.5 mm thick, were stacked in a double lap joint configuration and cemented together. The area of overlap in the joint
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O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
measured 25 mm wide /33 mm tall. The adhesive was cured using the standard cycle of two hours at 100 8C followed by 2 h at 200 8C. The specimen was then loaded in compression at 325 8C for 24 h. The shear stress carried by the adhesive during the test was 1.7 MPa, at least three times higher than the shear stress expected to be carried by the adhesive during the sandwich beam creep tests. 3.2. Manufacturing and testing Sandwich beams were made by bonding precut aluminum foam cores to stainless steel faces. The foam cores were cut to size using a diamond blade Isomet 2000 precision saw (Buehler, Lake Bluff, IL). The stainless steel face sheets were cut to size in a sheet metal cutter. The sandwich beams with ERG foam cores were 12 /18 mm wide, with a core thickness of 12 /15 mm and a face thickness of 0.72 or 0.88 mm; beam geometry details for each specimen are given in Appendix A. The sandwich beams with Alporas foam cores were 189/0.5 mm wide, with a core thickness of 149/ 0.5mm and face thicknesses of 0.72 mm (specimens T1414 to T1423) or 0.88 mm (specimens T1401to T1413; S1405; S1406). The mass of each of the foam cores was measured using a balance and the dimensions were measured with digital calipers allowing the density of the foam cores to be calculated. A thin coat of Durabond 950 adhesive was applied to each face sheet and the foam core was sandwiched in between. The beams were then weighted down, allowed to dry overnight, and fired in an oven for 2 h at 100 8C and 2 h at 200 8C to fully cure the adhesive. The testing apparatus consisted of a three-zone furnace (Applied Test Systems, Butler, PA) mounted on a creep load frame. The temperature of the furnace was maintained at a constant value within 1 8C by a control system (Omega, Stamford, CT). Constant dead weight loads were applied using steel plates. A threepoint bending apparatus with loading points made of 8 mm-wide blocks of alumina was supported in the center of the furnace. The centers of the lower supports were placed 80 mm apart. The geometry of the setup is shown in Fig. 1. When the test specimen was in place, the furnace was turned on and the temperature allowed to equilibrate, at which point the beam was loaded. The vertical displacement of the top face sheet at the center of the beam was measured using an LVDT (Sensotec, Columbus, OH) attached to a testing apparatus extension outside the furnace. The LVDT has an accuracy of 9/0.0025 mm and a span of 12.7 mm. The steady state displacement rate of the center of the beam, associated with secondary creep, was assessed for each specimen tested. Tests were continued until the specimens entered tertiary creep, characterized by an increasing displacement rate, or until at least 80 h had elapsed.
Actual test durations ranged from 12 to 304 h. The steady state displacement rate was quantified by fitting a line to the steady state portion of the data and determining the slope. The test temperatures and beam loads were selected to obtain a data set within the same range as the experimental conditions in creep studies of Duocel [1] and Alporas foams [2]. For tests of Duocel-core beams, temperatures ranged from 250 to 300 8C and loads were selected such that the stress resultant in the center of the core ranged from 0.42 to 0.68MPa. Two different orientations of Duocel foam were tested: with the major cell axis parallel to the plane of the faces, and with the major cell axis perpendicular to the plane of the faces. Tests were completed for 20 Duocel foam specimens. Of these, seven had the major cell axis perpendicular to the plane of the faces, and 13 had the major cell axis of the pores parallel to the plane of the faces. For tests of Alporas core beams, temperatures ranged from 260 to 300 8C and the core stress ranged from 0.25 to 0.45 MPa. Tests were completed for 21 specimens with Alporas foam cores.
4. Results and discussion The average densities of the Duocel and Alporas foams tested in the sandwich beams in this study were almost identical to those of the foams tested uniaxially in previous studies. The mean density of the Duocel foam used in the cores of the sandwich beams was 244 kg m 3 (standard deviation 9.9 kg m 3), compared with the nominal density of the Duocel foam tested uniaxially of 243 kg m 3 [1]. The mean density of the Alporas foam used in the cores of the sandwich beams was 233 kg m 3, (S.D. 20.3 kg m 3), compared with the mean density of the Alporas foam tested uniaxially of 233 kg m 3 [2]. Tests to evaluate the creep behavior of the adhesive used to bond the face sheets to the core indicated no
Fig. 2. Typical displacement vs. time curve for Alporas foam core sandwich panel during a creep test (T /280 8C, sˆ/ /0.42 MPa).
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
detectable creep (/g˙/ B/9.3 /10 10 s 1) of the adhesive at a temperature of 325 8C and a shear stress of 1.7 MPa.. Since this temperature and stress level are higher than those encountered by the adhesive in a sandwich beam test, it can be inferred that the adhesive will not contribute to the creep deflection rate of the beam. The creep of the stainless steel face sheets was negligible compared with that of the core. The deformation-mechanism map for 316 stainless steel can be used to estimate the order of magnitude of the creep rate of the faces [13]. The maximum temperature used in our creep tests was 300 8C, corresponding to a homologous temperature, T /Tm, /0.34, and the maximum normal stress in the faces was about 25 MPa, corresponding to a equivalent shear stress normalized with respect to the shear modulus, ss/m /1.9 /10 4. At these conditions, the deformation is well within the elastic range. We expect a creep rate several orders of magnitude less than the lowest rate of 10 10 s 1 given in the neighboring diffusional flow regime. Using a creep rate of 10 14 s 1 and assuming diffusional flow (nf /1) from the deformation mechanism map to obtain Af in Eq. (12), we calculate the bending deflection rate using Eq. (13) to be about 10 13 mm s 1. The calculated deflection rate from shearing of the core (Eq. (17)) is of the order of 10 5 /10 7 mm s 1. The contribution of the face to the overall creep of the beam is negligible. A typical plot of the central beam deflection against time is shown in Fig. 2 for one of the Alporas foam core sandwich beams, with a test temperature of T /280 8C, and sˆ/ /0.42 MPa. The creep behavior seen in the sandwich beams indicates an initial elastic response upon loading, followed by the primary creep phase with a decreasing displacement rate. The specimen then enters secondary or steady-state creep, with a constant displacement rate over an extended period of time. For many of the specimens, this region of secondary creep lasted for over 80 h of testing, and those tests were typically ended while the specimen was still in the secondary creep stage. For some of the specimens tested at higher loads and temperatures, the specimen also entered the tertiary creep stage, with a rapidly increasing displacement rate. Some of these specimens reached the maximum displacement allowed by the geometry of the loading apparatus, or approximately 6 mm, before the
Fig. 3. A sandwich beam specimen which was tested to tertiary creep, showing failure by shearing and indentation of the metallic foam core. The test was stopped when the creep displacement reached the maximum displacement possible in the testing fixture.
269
testing time was complete. One such sandwich beam is shown in Fig. 3. It can be seen that the beam has failed by the shearing of the foam core combined with indentation. The top face has formed a plastic hinge at the outer edges of the 8 mm loading plate in the center of the beam (not shown in the figure) to accommodate the shearing and indentation of the core at large deflections. The measured creep deflection rates for the sandwich beams with the Duocel foam cores are compared with the analysis in Table 3 and in Figs. 4 and 5. The analytical values are obtained by substituting the measured range of creep rates in uniaxial loading of the Duocel foam for each resultant stress sˆ and temperature, T , from Table 1 in Eq. (17) The upper and lower bounds of the analytical results are indicated by two dashes for each stress and temperature on Figs. 4 and 5. For all the tests except those at sˆ/ /0.42 MPa and T /300 8C, the deflection rates for the sandwich beams lie within or very close to the range expected from Eq. (17). Most of the data for sˆ/ /0.42 MPa and T /300 8C lie below the expected range, with two data points well below the low end of the range. Of the 20 specimens tested, all but five have deflection rates within or close to the expected range. Sandwich beams with Duocel foam cores were tested with the major cell axis parallel and perpendicular to the faces of the beam, in the plane of shearing. For constant test conditions, the deflection rates for specimens with foam cores in both orientations were similar, as would be expected. The data for the creep of the Duocel foam core under uniaxial stress, used in calculating the expected deflection rates of the sandwich beams, was obtained with the load direction parallel to the major cell axis. The analysis of the sandwich beam deflection rate does not account for anisotropy in the foam core. There are no data in the literature for the creep of the anisotropic Duocel foam in shear loading. The measured creep deflection rates for the sandwich beams with the Alporas foam cores are compared with the analysis in Table 4 and in Figs. 6 and 7. The analytical results are presented in the same way as for the sandwich beams with Duocel foam cores, using Eq. (17) and the data in Table 2. We note that several data points listed in Table 4 lie so close together that they cannot be separately distinguished on Figs. 6 and 7. For all the tests except those at sˆ/ /0.42 MPa and T / 300 8C, the deflection rates for the sandwich beams lie within or very close to the range expected from Eq. (17). The data for sˆ/ /0.42 MPa and T /300 8C all lie below the expected range, although three of the seven specimens have deflection rates close to the lower bound. Of the 21 specimens tested, all but four are within or close to the range expected from the analysis. Among the sandwich beams with Alporas foam cores four specimens had values of sˆ that were about 10%
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
270
Table 3 Creep deflection rates for sandwich beams with Duocel foam cores sˆ (MPa)
Temperature (8C)
Calculated d˙min (mm s 1)
Calculated d˙max (mm s 1)
V1201 V1202 V1203 H1502 H1503
0.426 0.426 0.426 0.420 0.421
275 275 275 275 275
5.94 10 7 5.94 10 7 5.96 10 7 6.09 10 7 6.08 10 7
2.27 10 5 2.27 10 5 2.28 10 5 2.33 10 5 2.32 10 5
9.38 10 7 1.71 10 6 1.37 10 6 6.94 10 7 2.02 10 6
V1205 H1504 H1401 H1402
0.529 0.530 0.530 0.530
275 275 275 275
6.48 10 7 6.63 10 7 6.60 10 7 6.61 10 7
2.65 10 6 2.71 10 6 2.69 10 6 2.70 10 6
1.53 10 6 1.53 10 6 4.72 10 6 3.33 10 6
V1204 H1505 H1403 H1404
0.679 0.681 0.680 0.670
275 275 275 275
2.22 10 6 2.27 10 6 2.26 10 6 2.26 10 6
1.41 10 5 1.44 10 5 1.43 10 5 1.43 10 5
6.14 10 6 1.17 10 5 3.06 10 5 1.67 10 5
V1206 H1506 H1405 H1406 H1407 H1408
0.420 0.420 0.420 0.420 0.421 0.420
300 300 300 300 300 300
2.28 10 5 2.32 10 5 2.31 10 5 2.31 10 5 2.31 10 5 2.31 10 5
4.88 10 4 4.97 10 4 4.96 10 4 4.96 10 4 4.95 10 4 4.94 10 4
1.61 10 6 2.12 10 4 1.39 10 5 3.00 10 6 9.75 10 6 3.00 10 6
V1207
0.420
250
5.96 10 7
5.42 10 6
2.43 10 7
Specimen ID
/
/
d˙measured (mm s 1)
d˙min and d˙max are calculated values using the creep data for the Duocel foam (Table 1) and Eq. (17).
/
experimental error. The calculated values of d˙min and d˙max were estimated using the foam data for sˆ/ /0.42 MPa (Table 2) and then applying a correction factor of sˆ 4:65 0:42
Fig. 4. Deflection rate of the center point of the sandwich beams with Duocel foam cores plotted against stress, sˆ at constant temperature (T /275 8C).
higher or lower than the target value of 0.42 MPa due to
Fig. 5. Deflection rate of the center point of the sandwich beams with Duocel foam cores plotted against inverse absolute temperature, 1/T , at constant stress (/sˆ/ /0.42 MPa).
since the foam obeys power law creep with n /4.65 [2]. Corrected values are reported in Table 4 for specimens T1410, T1413, S1405 and S1406. The measured deflection rate of one of the sandwich beam designs with an Alporas foam core was compared with that obtained from a finite element study on a sandwich beam of the same geometry (L /80 mm, b / 18.5 mm, t /0.88 mm, c /14.0 mm) and material properties (/o˙0c/ /0.987 s 1, s0c /12.6 MPa and nc / 5.4, Eq. (15), fit to the compressive data of [2]), at the same test conditions (P /96.5 N, s 0:25 ˆ MPa, T / 300 8C) [5]. The average measured creep deflection rate of the three specimens was 3.9 /10 8 mm s 1 while the finite element analysis predicted a rate of 4.1 /10 8 mm s 1. The analysis described in this paper gives a creep rate of 3.4 /10 8 mm s 1 (Eq. (16) with o˙0c/ / 0.987 s 1, s0c /12.6 MPa and nc /5.4), 16% lower than the finite element analysis. Using Eq. (17) and the range of creep data obtained by Andrews et al. [2] for the Alporas foam, the range of expected deflection rates for the sandwich beam is: 2.96 /10 8 to 1.03 /10 7 mm s 1. The finite element analysis and the analytical method (Eqs. (16) and (17)) give consistent results; both give a good description of the measured creep deflection rate of the sandwich beam.
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
271
Table 4 Creep deflection rates for sandwich beams with Alporas foam cores sˆ (MPa)
Temperature (8C)
Calculated d˙min (mm s 1)
Calculated d˙max (mm s 1)
T1402 T1404 T1406
0.251 0.250 0.251
300 300 300
2.96 10 8 2.96 10 8 2.96 10 8
1.03 10 7 1.03 10 7 1.03 10 7
4.14 10 8 3.32 10 8 4.17 10 8
T1417 T1418 T1419
0.320 0.320 0.320
300 300 300
9.09 10 8 9.09 10 8 9.09 10 8
1.95 10 7 1.95 10 7 1.95 10 7
2.15 10 7 7.96 10 8 1.82 10 7
T1401 T1405 T1403 T1420 T1421 T1422 T1423
0.416 0.421 0.417 0.420 0.420 0.419 0.421
300 300 300 300 300 300 300
2.90 10 7 2.90 10 7 2.90 10 7 2.93 10 7 2.93 10 7 2.93 10 7 2.93 10 7
9.26 10 7 9.26 10 7 9.26 10 7 9.36 10 7 9.36 10 7 9.36 10 7 9.36 10 7
1.28 10 7 1.00 10 7 1.34 10 7 2.43 10 7 2.13 10 7 2.90 10 7 1.54 10 7
T1410 T1415 T1416 S1405 S1406
0.447 0.420 0.420 0.391 0.381
280 280 280 280 280
1.26 10 7 9.20 10 8 9.20 10 8 6.41 10 8 6.41 10 8
8.75 10 7 6.40 10 7 6.40 10 7 4.49 10 7 4.49 10 7
1.96 10 7 1.76 10 7 1.83 10 7 8.30 10 8 8.30 10 8
T1411 T1412 T1413
0.420 0.409 0.450
260 260 260
4.18 10 8 4.19 10 8 5.78 10 8
2.53 10 7 2.53 10 7 3.49 10 7
1.01 10 7 1.10 10 7 2.17 10 7
Specimen ID
/
/
d˙measured (mm s 1)
d˙min and d˙max/are calculated values using the creep data for the Alporas foam (Table 2) and Eq. (17).
/
Fig. 6. Deflection rate of the center point of the sandwich beams with Alporas foam cores plotted against stress, sˆ at constant temperature (T /300 8C).
There is roughly an order of magnitude scatter in the creep data, both for the foams tested uniaxially (Tables 1 and 2) and for the sandwich beams (Tables 3 and 4). There are several sources of scatter in the data. The creep of foams is most sensitive to the relative density, varying with relative density raised to the power /(3n/ 1)/2 (Eq. (3)). For aluminum foams, with n: / /4.5, the creep rate depends on relative density raised to the power 7.25. This alone can lead to order of magnitude variation in creep rates. For example, the relative densities of the closed-cell foams tested by Andrews et al. [2] varied from 7.4 to 10.6%: Eq. (3) indicates that the lower density foam should creep at a rate of over 13 times that of the denser foam at the same stress and temperature. Defects, such as curved cell walls or missing cell walls within the foam can also have a substantial effect on the creep rate. Cell wall curvatures typical of the Alporas foam increase the creep rate of an analogous honeycomb 50% over that of the identical honeycomb with straight walls [4]. Cell walls with preexisting cracks can act like missing cell walls: removal of just 2% of the walls almost doubles the creep rate of a honeycomb over that of an identical, intact one [2].
5. Conclusions Fig. 7. Deflection rate of the center point of the sandwich beams with Alporas foam cores plotted against inverse absolute temperature, 1/T , at constant stress (/sˆ/ /0.42 MPa).
Experimental tests of sandwich beams with open- and closed-cell metallic foam cores were performed under a range of temperatures and applied stresses. Although
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272
the scatter in the data is large, the creep deflection rates of the sandwich beams were found to be within or close to the expected range with the exception of specimens tested at the highest temperature. A finite element analysis of one particular beam geometry and one set of loading conditions was available [5]. The analysis described in this paper gave similar results; both the finite element analysis as well as our analytical approach gave a good estimate of the measured deflection rate.
Acknowledgements The authors would like to thank Dr Erik Andrews for assistance with the design of the experimental setup and Professor Norman Fleck of Cambridge University Engineering Department and Professor John Hutchinson of the Division of Engineering and Applied Sciences, Harvard University for helpful discussions. Dr Antonio Makiyama assisted in producing the figures. This work was supported by the ARPA-MURI Ultralight Metal Structures Program (ONR Contract N00014-1-96-1028).
Appendix A: Table A1. Beam geometries for sandwich beams with Duocel foam core Specimen ID
b (mm)
c (mm)
t (mm)
V1201/V1207 H1502 /H1506 H1401 /H1410
18 12.5 18
12.5 15 14
0.88 0.72 0.72
Specimen ID beginning with V indicates that the core was oriented with the major cell axis in the vertical direction (parallel to the applied load).
Specimen ID beginning with H indicates that the core was oriented with the major cell axis in the horizontal direction (perpendicular to the applied load).
References [1] E.W. Andrews, L.J. Gibson, M.F. Ashby, The creep of cellular solids, Acta Mater. 47 (10) (1999) 2853 /2863. [2] E.W. Andrews, J.-S. Huang, L.J. Gibson, Creep behavior of a closed-cell aluminum foam, Acta Mater. 47 (10) (1999) 2927 / 2935. [3] A.C.F. Cocks, M.F. Ashby, Creep-buckling of cellular solids, Acta Mater. 48 (2000) 3395 /3400. [4] E.W. Andrews, L.J. Gibson, The role of cellular structure in creep of two-dimensional cellular solids, Mater. Sci. Eng. A303 (2001) 120 /126. [5] C. Chen, N.A. Fleck, M.F. Ashby, Numerical prediction of the creep response of sandwich beams with a metallic foam core, Proceedings of 2nd International Conference on Cellular Metals and Metal Foaming Technology. Bremen, Germany, 18 /20 June, 2001, p. 369. [6] M.F. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, H.N.C. Wadley, Metal Foam: A Design Guide, Butterworth Heinemann, Oxford, 2000. [7] V.S. Deshpande, N.A. Fleck, Isotropic constitutive models for foams, J. Mech. Phys. Solids 48 (2000) 1253 /1283. [8] H.G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon Press, New York, 1969. [9] E.W. Andrews, W. Sanders, L.J. Gibson, Compressive and tensile behaviour of aluminum foams, Mater. Sci. Eng. A270 (1999) 113 /124. [10] T.C. Triantafillou, J. Zhang, T.L. Shercliff, L.J. Gibson, M.F. Ashby, Failure surfaces for cellular materials under multiaxial loads II: comparison of models with experiment, Int. J. Mech. Sci. 31 (1989) 665 /678. [11] G. Gioux, T.M. McCormack, L.J. Gibson, Failure of aluminum foams under multiaxial loads, Int. J. Mech. Sci. 42 (2000) 1097 / 1117. [12] American Society for Metals, Properties and Selection, Stainless Steels, Tool Materials, and Special Purpose Metals. American Society for Metals, Metals Park, OH, 1980. [13] H.J. Frost, M.F. Ashby, Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics, Pergamon Press, New York, 1982.
Creep of sandwich beams with metallic foam cores O. Kesler, L.K. Crews, L.J. Gibson * Department of Materials Science and Engineering 8-135, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Received 22 January 2002; received in revised form 5 April 2002
Abstract The steady state creep deflection rates of sandwich beams with metallic foam cores were measured and compared with analytical and numerical predictions of the creep behavior. The deflection rate depends on the geometry of the sandwich beam, the creep behavior of the foam core and the loading conditions (stress state, temperature). Although there was a considerable scatter in the creep data (both of the foams and of the sandwich beams made using them), the data for the sandwich beams were fairly well described by the analysis. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Sandwich beams; Metallic foams; Creep behavior
1. Introduction Metallic foams are well suited to a variety of applications. Their capacity to undergo large deformations at roughly constant load makes them ideal energy absorbers in vehicles and in packaging. Their high surface area and large volume of interconnected porosity make open-cell metallic foams attractive for use as heat sinks in electronic devices. Their low density can be exploited in lightweight structural sandwich panels. Relative to polymer foams, they have higher melting points, allowing them to be used at higher temperatures. At temperatures above approximately 0.3 times the melting temperature, time-dependent creep deformation becomes important for metallic components carrying a sustained load over an extended time. Thus, an understanding of the high-temperature deformation of sandwich beams with metallic foam cores is of practical interest.
* Corresponding author. Tel.: /1-617-253-7107; fax: /1-617-2586275 E-mail address: [email protected] (L.J. Gibson).
Previous studies have examined the creep behavior of metallic foams, through a variety of experimental, analytical, and computational methods. The creep behavior of open- and closed-cell foams can be modeled by analyzing the bending of the cell edges and the stretching of the cell faces; the analysis describes the data for the creep of open- and closed-cell aluminum foams well [1,2]. While bending is the dominant mechanism of creep deformation at low stresses, at high stresses, in compression, creep buckling becomes important [3]. Finite element analysis of the role of cell geometry on the creep rate of two-dimensional cellular solids has shown that structures with random cells or with curved cell walls creep faster than a regular structure [4]. The creep behavior of a sandwich beam with a metallic foam core has been calculated numerically, giving the deflection rate of the center of the beam over time for a specific beam geometry and set of face sheet and foam core properties [5]. No experimental studies of the creep behavior of sandwich beams with metallic foam cores are available. This work seeks to provide a better understanding of the creep behavior of sandwich structures with metallic foam cores through a series of experimental studies.
0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 2 3 9 - 3
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
2. Analysis
ˆ o˙ij f (s)
The power creep law describing the steady state creep rate of a solid can be written as: ns s o˙s o˙0s (1) s0s where Qs
(2)
RT
and A /1 s 1. In the above expressions, ns, Qs, and s0s are material properties, s is the applied stress, T is the absolute temperature, and R is the universal gas constant, 8.314 J mol K 1. For an open-cell foam with relative density r =rs ; the steady state creep rate is [1]: (3ns 1) 2 o ˙ 0:6 1:7(2ns 1) s ns r o˙0s (ns 2) ns s0s rs
(3)
The steady state creep rate of a closed cell foam is [1]
o ˙ o˙0s
1
a 3
ns 2 1=ns
1:7
0:6
nP
1=2
2 1 nP
To solve for the expression f (s); ˆ we note that in uniaxial tension, se /s11 and sm /s11/3, giving s s ˆ 11 : For uniaxial stress, @ s=@s ˆ 11 1 and n s o˙11 o˙0 11 (7) s0 giving f (s) ˆ o˙0
n sˆ s0
A sandwich beam of span l and width b loaded in
ns
s 0s (3ns 1)=2ns r 2 r f (1 f) rs 3 rs 2ns 1
(4)
ns
where f is the volume fraction of solid in the cell edges. When f /1, Eq. (4) reduces to the open cell foam expression in Eq. (3). The model states that the creep constants ns and Qs in Eqs. (3) and (4) are identical to the creep constants of the solid material from which the foam is made; this has been confirmed experimentally [1,2].
three point bending with a concentrated load P is shown in Fig. 1. The thicknesses of the face and core are t and c , respectively. The distance between the centers of the two face sheets is d and the distance between the top and bottom of the beam is h /c/2t . The faces have a Young’s modulus Ef, and the core has a Young’s modulus Ec and a shear modulus Gc. The stresses in the face and core are given by Allen [8]. The normal
2.2. Multiaxial stress states When a foam is subjected to a multiaxial stress state, it becomes necessary to determine the resultant stress, s: ˆ The resultant stress and corresponding strain-rates can be calculated from the equivalent stress, se and the mean stress, sm, as follows [6,7]: sˆ 2 and
1 (1 (a=3)2 )
[s2e a2 s2m ]
(8)
2.3. Stresses and deflections in sandwich beams
s 1
(6)
@sij
Here, the constant a is related to the plastic Poisson’s ratio, nP, by
2.1. Power law creep
o˙0s A exp
@ sˆ
265
(5) Fig. 1. Sandwich panel in three point bending.
266
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
stress in the face at a cross section with a bending moment, M , is: sf
M d Ef (EI)eq 2
(9a)
Ef bt3 Ec bc3 Ef btd 2 Ef btd 2 : 6 12 2 2
The maximum normal stress in the core at a section with a bending moment, M, is: sc
M
c
(EI)eq 2
Ec
(9b)
The normal stress in the core is relatively small compared with that in the face, since Ec /Ef. The shear stress in the core at a section with a shear force, V , is: tc
V
(9c)
bd
The elastic deflection of such a sandwich beam is the sum of the bending and shear deflections [8]: d
Pl3 48(EI)eq
Pl 4(AG)eq
(10)
bd 2 c
Gc
(11)
The creep deflection rate of the beam is found in a similar fashion by adding the bending and shearing creep deflection rates.
2.4. Creep deflection rate of a sandwich beam in bending The bending creep rate is related to the power law creep response of the faces: o˙ Af snf
(12)
where Af and nf are creep properties of the face material. The bending deflection rate of the beam is then calculated by assuming that plane sections remain plane and that the axial stresses in the core are much smaller than the axial stresses in the faces. Ashby et al. [6] find the bending deflection rate at the center of the beam to be: P nf (l=2)nf 2 1 d˙b max 1 (13) Af 4bB nf 1 nf 2 where
where o˙0c ; s0c, and nc are the creep parameters of the core material. The core carries both normal stress s11 and shear stress s12; for metallic foam cores, both are significant. Using Eqs. (5) and (6) Ashby et al. [6] find the shear deflection rate at the center of the beam to be: nc sˆ cl 3s12 d˙s max o˙0c s0c 2d 1 (a=3)2 1=2 3 2 s211 s (16) 12 1 (a=3)2 Noticing that nc sˆ o˙0c s0c is an expression for the observed strain rate o˙c of the foam by itself, and using the definition of sˆ in Eq. (5), we can simplify Eq. (16) to:
and (AG )eq, the equivalent shear rigidity is: (AG)eq
(14)
The shear deflection rate is calculated from the creep rate of the core. The power law creep of the foam core under uniaxial stress is given by: nc sˆ o˙c o˙0c (15) s0c
where (EI )eq, the flexural rigidity of the section, is: (EI)eq
2 1 2 1 nf nf h c B 1 2 2 2 nf 1
d˙s max o˙c
cl 3 s12 2 2d 1 (a=3) sˆ
(17)
Eq. (17) was used to calculate the expected deflection rates for the sandwich beams tested in this study. The values of o˙c were those measured previously in uniaxial loading for specimens of the foam core materials [1,2]. For a sandwich beam in three point bending, the value of s12 is constant throughout the core. The value of s11 varies along the length of the beam and from the neutral axis to the top or bottom of the core. For the beams tested in this study, described below, the maximum ratio of s11 to s12 was 0.26. We find that the effect of the s11 on the value of sˆ is negligible (less than 2%) for the maximum ratio of s11 to s12 in the cores in this set of experiments. We note that if s11 can be ignored in calculating the resultant stress, s; ˆ then sˆ 2 :
3s212 2 a 1 3
giving d˙s max : o˙c
cl
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
2d
1 (a=3)2
(18)
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
where o˙c is the power law creep rate of the foam under a uniaxial stress equal to the resultant stress in the foam core of the sandwich beam and at the same temperature as the beam. In addition, the contribution to the total deflection rate of the bending of the faces was found to be negligible relative to the contribution from the shear of the core, as is described in Section 4. The total deflection rates of the beams in this study were calculated from the equations for shear deflection rates only (Eq. (17)).
3. Methods To assess the validity of the above analysis, an experimental program was undertaken to measure the creep deflection rates of sandwich beams with aluminum foam cores. The sandwich beams were manufactured by bonding aluminum foam cores to type 316 stainless steel face sheets using an aluminum-based metallic composite adhesive (trade name Durabond 950, Cotronics, Brooklyn, NY). Two types of commercially available aluminum foams were employed: an open-cell foam (trade name Duocel, ERG, Oakland, CA), and a closed-cell foam (trade name Alporas, Shinko Wire, Amagasaki, Japan). 3.1. Materials The Duocel foam is made from 6101-T6 aluminum via a casting process. The nominal density of the foam is 240 kg m 3 (relative density 0.09), and the cells are elliptical in shape, with major and minor axes of approximately 2.5 and 1.5 mm, respectively. Because of the elongated shape of the cells, the mechanical properties differ from one axis to the other. Parallel to the major cell axis, the uniaxial compressive strength of the foam is 2.17 MPa and the Young’s modulus is 0.634 GPa. Perpendicular to the major axis, the strength and modulus are 1.39 MPa and 0.236 GPa, respectively [9]; previous data suggests that the Duocel material is roughly transversely isotropic [10]. The plastic Poisson’s ratio np for the Duocel foam is 0.052, [11], resulting in a
Table 1 Uniaxial creep data for Duocel foam [1] Stress (MPa)
Temperature (8C)
0.42 0.42 0.42 0.53 0.68
250 275 300 275 275
o˙min (s 1)
/
1.10 10 8 1.10 10 8 4.20 10 7 1.20 10 8 4.10 10 8
o˙max (s 1)
/
1.00 10 7 4.20 10 7 9.00 10 6 4.90 10 8 2.60 10 7
267
Table 2 Uniaxial creep data for Alporas foam [2] Stress (MPa)
Temperature (8C)
0.25 0.32 0.42 0.42 0.42
300 300 300 280 260
o˙min (s 1)
/
5.50 10 10 1.67 10 9 5.39 10 9 1.69 10 9 7.76 10 10
o˙max (s 1)
/
1.91 10 9 3.58 10 9 1.72 10 8 1.18 10 8 4.70 10 9
value for a of 1.96. The activation energy, Q , and power law constant, n for the Duocel foam have been measured in tension and compression, for loading along the major axis [1]: Q /214 kJ mol 1 and n /4.6. These values are slightly higher than those of solid 6101 aluminum (Q /173 kJ mol 1, n /4.0). At each stress and temperature, there was roughly an order of magnitude spread in the creep data; the range of creep rates measured in the uniaxial tests on the Duocel foam at the stresses and temperatures used in this study are listed in Table 1. The Alporas foam is made from molten aluminum mixed with titanium hydride (1 /3 wt.%) and calcium (0.2 /8 wt.%) via a liquid-state process. The nominal density of the Alporas foam used in the creep experiments is 235 kg m 3 (relative density 0.09). The cells are equiaxed with a cell size of 4.5 mm. Mechanical property tests [9,11] indicate that the strength of the foam is 1.46 / 1.84 MPa, the modulus is 1.00 /1.14 GPa and the plastic Poisson’s ratio is 0.024, resulting in a value for a of 2.05. The average values of the activation energy, Qc, and power law creep exponent, nc, of the Alporas foam in tension and compression have been found to be [2]: Q / 97.7 kJ mol 1 and n /4.65 for temperatures less than or equal to 300 8C and stresses less than or equal to 0.42 MPa. The measured values of Q and n are close to those for solid pure aluminum, Q /70 kJ mol 1 and n /4.4. As for the Duocel foam, at each stress and temperature, there was roughly an order of magnitude spread in the creep data; the range of creep rates measured in the uniaxial tests on the Alporas foam at the stresses and temperatures used in this study are listed in Table 2. Type 316 stainless steel, used to make the sandwich faces, has a tensile yield strength of 205 MPa and a modulus of 216 GPa [12]. Frost and Ashby [13] report the creep constants of the material to be Q /280 kJ mol 1 and n /7.9, with s0 /33.5 MPa; the melting temperature is 1680 K. The creep behavior of the Durabond 950 metallic adhesive used to bond the core to the face sheets was evaluated in this study. Three flat steel bars, each measuring 25 mm wide, 50 mm tall, and 3.5 mm thick, were stacked in a double lap joint configuration and cemented together. The area of overlap in the joint
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O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
measured 25 mm wide /33 mm tall. The adhesive was cured using the standard cycle of two hours at 100 8C followed by 2 h at 200 8C. The specimen was then loaded in compression at 325 8C for 24 h. The shear stress carried by the adhesive during the test was 1.7 MPa, at least three times higher than the shear stress expected to be carried by the adhesive during the sandwich beam creep tests. 3.2. Manufacturing and testing Sandwich beams were made by bonding precut aluminum foam cores to stainless steel faces. The foam cores were cut to size using a diamond blade Isomet 2000 precision saw (Buehler, Lake Bluff, IL). The stainless steel face sheets were cut to size in a sheet metal cutter. The sandwich beams with ERG foam cores were 12 /18 mm wide, with a core thickness of 12 /15 mm and a face thickness of 0.72 or 0.88 mm; beam geometry details for each specimen are given in Appendix A. The sandwich beams with Alporas foam cores were 189/0.5 mm wide, with a core thickness of 149/ 0.5mm and face thicknesses of 0.72 mm (specimens T1414 to T1423) or 0.88 mm (specimens T1401to T1413; S1405; S1406). The mass of each of the foam cores was measured using a balance and the dimensions were measured with digital calipers allowing the density of the foam cores to be calculated. A thin coat of Durabond 950 adhesive was applied to each face sheet and the foam core was sandwiched in between. The beams were then weighted down, allowed to dry overnight, and fired in an oven for 2 h at 100 8C and 2 h at 200 8C to fully cure the adhesive. The testing apparatus consisted of a three-zone furnace (Applied Test Systems, Butler, PA) mounted on a creep load frame. The temperature of the furnace was maintained at a constant value within 1 8C by a control system (Omega, Stamford, CT). Constant dead weight loads were applied using steel plates. A threepoint bending apparatus with loading points made of 8 mm-wide blocks of alumina was supported in the center of the furnace. The centers of the lower supports were placed 80 mm apart. The geometry of the setup is shown in Fig. 1. When the test specimen was in place, the furnace was turned on and the temperature allowed to equilibrate, at which point the beam was loaded. The vertical displacement of the top face sheet at the center of the beam was measured using an LVDT (Sensotec, Columbus, OH) attached to a testing apparatus extension outside the furnace. The LVDT has an accuracy of 9/0.0025 mm and a span of 12.7 mm. The steady state displacement rate of the center of the beam, associated with secondary creep, was assessed for each specimen tested. Tests were continued until the specimens entered tertiary creep, characterized by an increasing displacement rate, or until at least 80 h had elapsed.
Actual test durations ranged from 12 to 304 h. The steady state displacement rate was quantified by fitting a line to the steady state portion of the data and determining the slope. The test temperatures and beam loads were selected to obtain a data set within the same range as the experimental conditions in creep studies of Duocel [1] and Alporas foams [2]. For tests of Duocel-core beams, temperatures ranged from 250 to 300 8C and loads were selected such that the stress resultant in the center of the core ranged from 0.42 to 0.68MPa. Two different orientations of Duocel foam were tested: with the major cell axis parallel to the plane of the faces, and with the major cell axis perpendicular to the plane of the faces. Tests were completed for 20 Duocel foam specimens. Of these, seven had the major cell axis perpendicular to the plane of the faces, and 13 had the major cell axis of the pores parallel to the plane of the faces. For tests of Alporas core beams, temperatures ranged from 260 to 300 8C and the core stress ranged from 0.25 to 0.45 MPa. Tests were completed for 21 specimens with Alporas foam cores.
4. Results and discussion The average densities of the Duocel and Alporas foams tested in the sandwich beams in this study were almost identical to those of the foams tested uniaxially in previous studies. The mean density of the Duocel foam used in the cores of the sandwich beams was 244 kg m 3 (standard deviation 9.9 kg m 3), compared with the nominal density of the Duocel foam tested uniaxially of 243 kg m 3 [1]. The mean density of the Alporas foam used in the cores of the sandwich beams was 233 kg m 3, (S.D. 20.3 kg m 3), compared with the mean density of the Alporas foam tested uniaxially of 233 kg m 3 [2]. Tests to evaluate the creep behavior of the adhesive used to bond the face sheets to the core indicated no
Fig. 2. Typical displacement vs. time curve for Alporas foam core sandwich panel during a creep test (T /280 8C, sˆ/ /0.42 MPa).
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
detectable creep (/g˙/ B/9.3 /10 10 s 1) of the adhesive at a temperature of 325 8C and a shear stress of 1.7 MPa.. Since this temperature and stress level are higher than those encountered by the adhesive in a sandwich beam test, it can be inferred that the adhesive will not contribute to the creep deflection rate of the beam. The creep of the stainless steel face sheets was negligible compared with that of the core. The deformation-mechanism map for 316 stainless steel can be used to estimate the order of magnitude of the creep rate of the faces [13]. The maximum temperature used in our creep tests was 300 8C, corresponding to a homologous temperature, T /Tm, /0.34, and the maximum normal stress in the faces was about 25 MPa, corresponding to a equivalent shear stress normalized with respect to the shear modulus, ss/m /1.9 /10 4. At these conditions, the deformation is well within the elastic range. We expect a creep rate several orders of magnitude less than the lowest rate of 10 10 s 1 given in the neighboring diffusional flow regime. Using a creep rate of 10 14 s 1 and assuming diffusional flow (nf /1) from the deformation mechanism map to obtain Af in Eq. (12), we calculate the bending deflection rate using Eq. (13) to be about 10 13 mm s 1. The calculated deflection rate from shearing of the core (Eq. (17)) is of the order of 10 5 /10 7 mm s 1. The contribution of the face to the overall creep of the beam is negligible. A typical plot of the central beam deflection against time is shown in Fig. 2 for one of the Alporas foam core sandwich beams, with a test temperature of T /280 8C, and sˆ/ /0.42 MPa. The creep behavior seen in the sandwich beams indicates an initial elastic response upon loading, followed by the primary creep phase with a decreasing displacement rate. The specimen then enters secondary or steady-state creep, with a constant displacement rate over an extended period of time. For many of the specimens, this region of secondary creep lasted for over 80 h of testing, and those tests were typically ended while the specimen was still in the secondary creep stage. For some of the specimens tested at higher loads and temperatures, the specimen also entered the tertiary creep stage, with a rapidly increasing displacement rate. Some of these specimens reached the maximum displacement allowed by the geometry of the loading apparatus, or approximately 6 mm, before the
Fig. 3. A sandwich beam specimen which was tested to tertiary creep, showing failure by shearing and indentation of the metallic foam core. The test was stopped when the creep displacement reached the maximum displacement possible in the testing fixture.
269
testing time was complete. One such sandwich beam is shown in Fig. 3. It can be seen that the beam has failed by the shearing of the foam core combined with indentation. The top face has formed a plastic hinge at the outer edges of the 8 mm loading plate in the center of the beam (not shown in the figure) to accommodate the shearing and indentation of the core at large deflections. The measured creep deflection rates for the sandwich beams with the Duocel foam cores are compared with the analysis in Table 3 and in Figs. 4 and 5. The analytical values are obtained by substituting the measured range of creep rates in uniaxial loading of the Duocel foam for each resultant stress sˆ and temperature, T , from Table 1 in Eq. (17) The upper and lower bounds of the analytical results are indicated by two dashes for each stress and temperature on Figs. 4 and 5. For all the tests except those at sˆ/ /0.42 MPa and T /300 8C, the deflection rates for the sandwich beams lie within or very close to the range expected from Eq. (17). Most of the data for sˆ/ /0.42 MPa and T /300 8C lie below the expected range, with two data points well below the low end of the range. Of the 20 specimens tested, all but five have deflection rates within or close to the expected range. Sandwich beams with Duocel foam cores were tested with the major cell axis parallel and perpendicular to the faces of the beam, in the plane of shearing. For constant test conditions, the deflection rates for specimens with foam cores in both orientations were similar, as would be expected. The data for the creep of the Duocel foam core under uniaxial stress, used in calculating the expected deflection rates of the sandwich beams, was obtained with the load direction parallel to the major cell axis. The analysis of the sandwich beam deflection rate does not account for anisotropy in the foam core. There are no data in the literature for the creep of the anisotropic Duocel foam in shear loading. The measured creep deflection rates for the sandwich beams with the Alporas foam cores are compared with the analysis in Table 4 and in Figs. 6 and 7. The analytical results are presented in the same way as for the sandwich beams with Duocel foam cores, using Eq. (17) and the data in Table 2. We note that several data points listed in Table 4 lie so close together that they cannot be separately distinguished on Figs. 6 and 7. For all the tests except those at sˆ/ /0.42 MPa and T / 300 8C, the deflection rates for the sandwich beams lie within or very close to the range expected from Eq. (17). The data for sˆ/ /0.42 MPa and T /300 8C all lie below the expected range, although three of the seven specimens have deflection rates close to the lower bound. Of the 21 specimens tested, all but four are within or close to the range expected from the analysis. Among the sandwich beams with Alporas foam cores four specimens had values of sˆ that were about 10%
O. Kesler et al. / Materials Science and Engineering A341 (2003) 264 /272
270
Table 3 Creep deflection rates for sandwich beams with Duocel foam cores sˆ (MPa)
Temperature (8C)
Calculated d˙min (mm s 1)
Calculated d˙max (mm s 1)
V1201 V1202 V1203 H1502 H1503
0.426 0.426 0.426 0.420 0.421
275 275 275 275 275
5.94 10 7 5.94 10 7 5.96 10 7 6.09 10 7 6.08 10 7
2.27 10 5 2.27 10 5 2.28 10 5 2.33 10 5 2.32 10 5
9.38 10 7 1.71 10 6 1.37 10 6 6.94 10 7 2.02 10 6
V1205 H1504 H1401 H1402
0.529 0.530 0.530 0.530
275 275 275 275
6.48 10 7 6.63 10 7 6.60 10 7 6.61 10 7
2.65 10 6 2.71 10 6 2.69 10 6 2.70 10 6
1.53 10 6 1.53 10 6 4.72 10 6 3.33 10 6
V1204 H1505 H1403 H1404
0.679 0.681 0.680 0.670
275 275 275 275
2.22 10 6 2.27 10 6 2.26 10 6 2.26 10 6
1.41 10 5 1.44 10 5 1.43 10 5 1.43 10 5
6.14 10 6 1.17 10 5 3.06 10 5 1.67 10 5
V1206 H1506 H1405 H1406 H1407 H1408
0.420 0.420 0.420 0.420 0.421 0.420
300 300 300 300 300 300
2.28 10 5 2.32 10 5 2.31 10 5 2.31 10 5 2.31 10 5 2.31 10 5
4.88 10 4 4.97 10 4 4.96 10 4 4.96 10 4 4.95 10 4 4.94 10 4
1.61 10 6 2.12 10 4 1.39 10 5 3.00 10 6 9.75 10 6 3.00 10 6
V1207
0.420
250
5.96 10 7
5.42 10 6
2.43 10 7
Specimen ID
/
/
d˙measured (mm s 1)
d˙min and d˙max are calculated values using the creep data for the Duocel foam (Table 1) and Eq. (17).
/
experimental error. The calculated values of d˙min and d˙max were estimated using the foam data for sˆ/ /0.42 MPa (Table 2) and then applying a correction factor of sˆ 4:65 0:42
Fig. 4. Deflection rate of the center point of the sandwich beams with Duocel foam cores plotted against stress, sˆ at constant temperature (T /275 8C).
higher or lower than the target value of 0.42 MPa due to
Fig. 5. Deflection rate of the center point of the sandwich beams with Duocel foam cores plotted against inverse absolute temperature, 1/T , at constant stress (/sˆ/ /0.42 MPa).
since the foam obeys power law creep with n /4.65 [2]. Corrected values are reported in Table 4 for specimens T1410, T1413, S1405 and S1406. The measured deflection rate of one of the sandwich beam designs with an Alporas foam core was compared with that obtained from a finite element study on a sandwich beam of the same geometry (L /80 mm, b / 18.5 mm, t /0.88 mm, c /14.0 mm) and material properties (/o˙0c/ /0.987 s 1, s0c /12.6 MPa and nc / 5.4, Eq. (15), fit to the compressive data of [2]), at the same test conditions (P /96.5 N, s 0:25 ˆ MPa, T / 300 8C) [5]. The average measured creep deflection rate of the three specimens was 3.9 /10 8 mm s 1 while the finite element analysis predicted a rate of 4.1 /10 8 mm s 1. The analysis described in this paper gives a creep rate of 3.4 /10 8 mm s 1 (Eq. (16) with o˙0c/ / 0.987 s 1, s0c /12.6 MPa and nc /5.4), 16% lower than the finite element analysis. Using Eq. (17) and the range of creep data obtained by Andrews et al. [2] for the Alporas foam, the range of expected deflection rates for the sandwich beam is: 2.96 /10 8 to 1.03 /10 7 mm s 1. The finite element analysis and the analytical method (Eqs. (16) and (17)) give consistent results; both give a good description of the measured creep deflection rate of the sandwich beam.
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Table 4 Creep deflection rates for sandwich beams with Alporas foam cores sˆ (MPa)
Temperature (8C)
Calculated d˙min (mm s 1)
Calculated d˙max (mm s 1)
T1402 T1404 T1406
0.251 0.250 0.251
300 300 300
2.96 10 8 2.96 10 8 2.96 10 8
1.03 10 7 1.03 10 7 1.03 10 7
4.14 10 8 3.32 10 8 4.17 10 8
T1417 T1418 T1419
0.320 0.320 0.320
300 300 300
9.09 10 8 9.09 10 8 9.09 10 8
1.95 10 7 1.95 10 7 1.95 10 7
2.15 10 7 7.96 10 8 1.82 10 7
T1401 T1405 T1403 T1420 T1421 T1422 T1423
0.416 0.421 0.417 0.420 0.420 0.419 0.421
300 300 300 300 300 300 300
2.90 10 7 2.90 10 7 2.90 10 7 2.93 10 7 2.93 10 7 2.93 10 7 2.93 10 7
9.26 10 7 9.26 10 7 9.26 10 7 9.36 10 7 9.36 10 7 9.36 10 7 9.36 10 7
1.28 10 7 1.00 10 7 1.34 10 7 2.43 10 7 2.13 10 7 2.90 10 7 1.54 10 7
T1410 T1415 T1416 S1405 S1406
0.447 0.420 0.420 0.391 0.381
280 280 280 280 280
1.26 10 7 9.20 10 8 9.20 10 8 6.41 10 8 6.41 10 8
8.75 10 7 6.40 10 7 6.40 10 7 4.49 10 7 4.49 10 7
1.96 10 7 1.76 10 7 1.83 10 7 8.30 10 8 8.30 10 8
T1411 T1412 T1413
0.420 0.409 0.450
260 260 260
4.18 10 8 4.19 10 8 5.78 10 8
2.53 10 7 2.53 10 7 3.49 10 7
1.01 10 7 1.10 10 7 2.17 10 7
Specimen ID
/
/
d˙measured (mm s 1)
d˙min and d˙max/are calculated values using the creep data for the Alporas foam (Table 2) and Eq. (17).
/
Fig. 6. Deflection rate of the center point of the sandwich beams with Alporas foam cores plotted against stress, sˆ at constant temperature (T /300 8C).
There is roughly an order of magnitude scatter in the creep data, both for the foams tested uniaxially (Tables 1 and 2) and for the sandwich beams (Tables 3 and 4). There are several sources of scatter in the data. The creep of foams is most sensitive to the relative density, varying with relative density raised to the power /(3n/ 1)/2 (Eq. (3)). For aluminum foams, with n: / /4.5, the creep rate depends on relative density raised to the power 7.25. This alone can lead to order of magnitude variation in creep rates. For example, the relative densities of the closed-cell foams tested by Andrews et al. [2] varied from 7.4 to 10.6%: Eq. (3) indicates that the lower density foam should creep at a rate of over 13 times that of the denser foam at the same stress and temperature. Defects, such as curved cell walls or missing cell walls within the foam can also have a substantial effect on the creep rate. Cell wall curvatures typical of the Alporas foam increase the creep rate of an analogous honeycomb 50% over that of the identical honeycomb with straight walls [4]. Cell walls with preexisting cracks can act like missing cell walls: removal of just 2% of the walls almost doubles the creep rate of a honeycomb over that of an identical, intact one [2].
5. Conclusions Fig. 7. Deflection rate of the center point of the sandwich beams with Alporas foam cores plotted against inverse absolute temperature, 1/T , at constant stress (/sˆ/ /0.42 MPa).
Experimental tests of sandwich beams with open- and closed-cell metallic foam cores were performed under a range of temperatures and applied stresses. Although
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the scatter in the data is large, the creep deflection rates of the sandwich beams were found to be within or close to the expected range with the exception of specimens tested at the highest temperature. A finite element analysis of one particular beam geometry and one set of loading conditions was available [5]. The analysis described in this paper gave similar results; both the finite element analysis as well as our analytical approach gave a good estimate of the measured deflection rate.
Acknowledgements The authors would like to thank Dr Erik Andrews for assistance with the design of the experimental setup and Professor Norman Fleck of Cambridge University Engineering Department and Professor John Hutchinson of the Division of Engineering and Applied Sciences, Harvard University for helpful discussions. Dr Antonio Makiyama assisted in producing the figures. This work was supported by the ARPA-MURI Ultralight Metal Structures Program (ONR Contract N00014-1-96-1028).
Appendix A: Table A1. Beam geometries for sandwich beams with Duocel foam core Specimen ID
b (mm)
c (mm)
t (mm)
V1201/V1207 H1502 /H1506 H1401 /H1410
18 12.5 18
12.5 15 14
0.88 0.72 0.72
Specimen ID beginning with V indicates that the core was oriented with the major cell axis in the vertical direction (parallel to the applied load).
Specimen ID beginning with H indicates that the core was oriented with the major cell axis in the horizontal direction (perpendicular to the applied load).
References [1] E.W. Andrews, L.J. Gibson, M.F. Ashby, The creep of cellular solids, Acta Mater. 47 (10) (1999) 2853 /2863. [2] E.W. Andrews, J.-S. Huang, L.J. Gibson, Creep behavior of a closed-cell aluminum foam, Acta Mater. 47 (10) (1999) 2927 / 2935. [3] A.C.F. Cocks, M.F. Ashby, Creep-buckling of cellular solids, Acta Mater. 48 (2000) 3395 /3400. [4] E.W. Andrews, L.J. Gibson, The role of cellular structure in creep of two-dimensional cellular solids, Mater. Sci. Eng. A303 (2001) 120 /126. [5] C. Chen, N.A. Fleck, M.F. Ashby, Numerical prediction of the creep response of sandwich beams with a metallic foam core, Proceedings of 2nd International Conference on Cellular Metals and Metal Foaming Technology. Bremen, Germany, 18 /20 June, 2001, p. 369. [6] M.F. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, H.N.C. Wadley, Metal Foam: A Design Guide, Butterworth Heinemann, Oxford, 2000. [7] V.S. Deshpande, N.A. Fleck, Isotropic constitutive models for foams, J. Mech. Phys. Solids 48 (2000) 1253 /1283. [8] H.G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon Press, New York, 1969. [9] E.W. Andrews, W. Sanders, L.J. Gibson, Compressive and tensile behaviour of aluminum foams, Mater. Sci. Eng. A270 (1999) 113 /124. [10] T.C. Triantafillou, J. Zhang, T.L. Shercliff, L.J. Gibson, M.F. Ashby, Failure surfaces for cellular materials under multiaxial loads II: comparison of models with experiment, Int. J. Mech. Sci. 31 (1989) 665 /678. [11] G. Gioux, T.M. McCormack, L.J. Gibson, Failure of aluminum foams under multiaxial loads, Int. J. Mech. Sci. 42 (2000) 1097 / 1117. [12] American Society for Metals, Properties and Selection, Stainless Steels, Tool Materials, and Special Purpose Metals. American Society for Metals, Metals Park, OH, 1980. [13] H.J. Frost, M.F. Ashby, Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics, Pergamon Press, New York, 1982.