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APPENDIX A

DESIGN FAMILY PICTORIALS

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APPENDIX H

REVIEW DRAFT OF:

"NONLINEAR PROPERTIES OF METALLIC CELLULAR

MATERIALS WITH A NEGATIVE POISSON'S RATIO"

(

Nonlinear Materials Properties of Metallic Cellular with a Negative Poisson's Ratio J.B. Chol', M.S., Graduate Student and R.S. Lakes§, Ph.D., Professor §Deparlment of Mechanical Engineering "§Department of Biomedical Engineering §Center for Laser Science and Engineering Unive_ty of Iowa, Iowa City, IA 52242 Department of Biomedical Engineering University of Iowa iowa City, Iowa 52242 May 23,1991 25H-1

Abstract

NegativePoisson'sratio copper foam was preparedand characterized

experimentally. Thetransformation intore-entrant foamwasaccomplished by applying

sequential permanent compressions abovethe yield point to achieve a triaxlal

compression. ThePoisson's ratioofthere-entrant foam depended onstrainandattained a

relative minimum at strainsnearzero. Poisson's ratioas smallas -0.8 was achieved.

The strain dependence of propertiesoccurredover a narrower range of strain than in the polymer foams studied earlier. Annealing of the foam resulted in a slightly greater magnitude of negative Poisson'sra_ and greater toughness at the expense of a decrease in the Young'smodulus.

25H-2 jl., 1. Introduction Cellular solids are lightweight materials consisting of a network of solid ribs or plates. Man-made cellular solids have been widely utilized in the form of structural honeycombs (two dimensional cellular solids) in aircraft and in the form of foams (three dimensional cellular solids) for packing, cushioning, energy absorption applications, sandwich panel cores, structuralpurposes and thermal protection systems.

Natural materials such as wood, cancellous bone, coral and leaves have a cellular structure. All these 'conventional' cellular materials have a convex cell shape and exhibit a positive Poisson's ratio. Recently, isotropio foam structures with negative Poisson's ratios have been fabricated by one of the authors[I]. The fabrication was achieved through a transformationof the cell structure from a convex polyhedral shape to a concave or "re-entrant" shape. Increase in some.material properties such as flexural rigidity and plane strain fracture toughness was reported[I,2,3].

The range of Poisson's ratio for isotmpic matedal is -1 to 0.5, as demonstrated by energy arguments[4]. Negative Poisson's ratios are rare but not unknown. Negative Poisson's ratios have been reported in single crystal pydtes[5] and in some rocks[6].

Synthetic anisotropic microstructures['/] were found to give such an effect. One of the authore[8] suggested that non-affine deformation kinematics are essential for the produntlon of negative Poisson's ratios in isotropic materials. A recent study of polymeric re-entrant foams with negative Poisson's ratio[9] reported other enhanced material properties: an increase in the toughness, in the shear modulus and in the resilience In the sense of a wide range of linear stress-strain behavior. Conventional and re-entrant polymer foams also differ in their stress strain behavior and deformation

(

25H-3 L mechanism maps [9]. In copper foam an increase in the indentation resistance was demonstrated experimentally [10], which is consistent with theory{Ill.

Mechanical properties of a conventional foam material depend on the physical properties of the solid material making up the cell dbs, relative density (the ratio of

i

the bulk density of the foam matedal to the density of the solid from which It is made), cell shape, cell size and loading conditions. The simplest and most comprehensive treatment of conventional fGam properties is that of Gibson and Ashby [12], which includes extensive comparisons between analytical results and experiment. In re- entrant foam; relative density is increased and the cell size is reduced slightly by the transformation process; the cell shape is dramatically altered. Mechanical behavior of a re-enVant foam matedal differs from that of a conventional foam in ways not addressed by existing theoretical treatments; most of the difference is attributed to the change in cell shape.

In this study, the mechanical properties of beth conventional and re-entrant copper foam matedais are examined at small strain by optical methods and at larger strains by sefvohydraulic (MTS) machine tests.

2. Experimental procedures 2.1. Specimen preperation: optical study A large block of copper foam (Astromet Corporation) with relative density between 0.08 and 0.1 was cut into smaller 18 x 18 x 36 mm blocks; another large with relative density 0.04 was cut into 25 x 25 x 50 mm blocks. Bar shaped specimens were cut from these blocks using the procedure described below. The blocks were not perfectly uniform: specimens of relative density of 0.04, 0.08, 0.09, 0.1 had deviations of 12%, 6%, 5.5%, 5%, respectively in relative density.

25H-4

Blockspecimens werecut with a highspeed sawto minimize surfaceplastic

deformation. The foamwas thentransformed into a re-entrant structure by applyir_

smallsequential increments (lessthan 2% stra!:_) of plasticdeformation in three

orthogonal directions using a visefittedwithPMMA (Plexiglas®) endpieces to provide

anevensurface. Thecompressed foams were cutagain intoslender bars(approx. 7 by 7

by 30 mm) following the above cutting method. The lengths of the cell ribs were measured with a microscope in order to compare the cell size for each specimen.

The re-entrant foam specimens have irregular surfaces which were polished with graded abrasives (silicon carbide grits 120, 320, 600 and 1000), under water Irrigation, in order to obtain a smooth surface for optical tests. The water Irrigation served to minimize t3mperature increases due to fdction and improved the surface finish. High grinding speed and low pressure were used to avoid plastic deformation of the surface. The finest polishing medium grains of 1000 mesh had a size of about 18 wn[13]. This provided sufficient smoothness to the surface so that the shadow moir_ method could be applied to measure deformation.

2.2. Specimen preparation - MTS machine test Rectangular spedmens from a large block of initial relative density 0.08:1: 5%.

were made with volumetric compression ratio of 1.2.0, 2.5, 3.0, according to the ASTM standard for subsize specimens[14] for both tension and compression tests. The dimensions of the specimens were 6.4x6.4x25.4(mm) for tension, 19x19xg.5(mm) for compression; the compression specimens were made shorter to prevent budding.

These specimens were porshed as _ above.

Specimen ends were cast in dental grade polymelhyl methac_late(PMMA) which is much more dgid (E,,3GPa) than the copper foam (E-200MPa). This procedure

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25H-5 avoided the madline grips crushing the ends. The mold for casting was threaded and was made of Teflon ® and was sprayed with Teflon ® lubricant to prevent adhesion of the polymerizing PMMA. The viscosity of the polymerizing PMMA was adjusted to minimize leakage from the mold and to be sufficiently low to mold adequate threads. The curing temperature of PMMA was about 80°C, too low to affect the matedal properties of the copper foam.

2.3 Annealing The re-entrant copper foam has some cold work due to the trl-axtal compressions, so an annealing process was used to evaluate the effects of co',d work. The polished specimen was dried for two days in room conditions and heated under nitrogen purge at a temperature of 500°C for lhr[15] and cooled slowly under nitrogen to minimize oxidation.

2.4. Optical test - Shadow moir6 The moird methods are based on fdnge patterns arising from overlap between gratings consisting of straight parallel opaque bars. In the shadow moir_ method, a Ronchi ruling is placed close to the epedmen surface and is illuminated with collimated light. Ovedap occurs between the ruling and its reflected (shadow) image, generating fringes which represent a contour map of the deformed surface. The displacement 8 of the surface perpendicular to itself is given by [16].

(1) 8 = Np/(tana + tanl3 ) 25H-6 where N is the fringe order, p Is the pitch of the grating, ,_ is angle of the incident light and 13is the viewing angle with respect to the normal to the grating.

Specimens of initial relative densities of 0.08,0.09 and 0.1 were used, however conventional foam of initial relative density of 0.04 is was not used in the optical test because of the poor fdnge pattern due to the large cell size.

Bending experiments were performed by applying known weights to an aluminum arm cemented to the top end of a vertically oriented copper foam specimen; the lower end was clamped. The aluminum arm was sufficiently long that specimen strain due to bending substantially exceeded compressional strain, approximating 'pure' bending. The Young's modulus E and Poisson's ratio v were extracted from the displacement field by first using simple beam theory to obtain an approximate val,;9 for E from the end defleclJon 6: 6 - ML2/2Ei ( 2 ) where M is the bending moment, L is length, and I is the area moment of inertia. Then the three dimensional solution was used to infer v and obtain a corrected value for E: 5 - ML 2 ( z2 + vx 2 - vy 2 )/2El (3) in which z is a longitudinal coordinate along the length of the bar, x is a lateral coordinate, and y is a transverse coordinate. The con'ec_n to E was small in all cases.

Positive Poisson's ratios give dse to hyperbolic fdnge contours, and negative Poisson's ratios give dee to elliptic contours. In either case the Poisson's ratio was found from the observed shape of the contours.

In this experiment, the grating was located a small distance, within a millimeter, away from the specimen, parallel to the surface and odantad so that no fdnge pattern occurred. Light from a mercury vapor lamp was collimated by a lens and directed upon the grating at an angle 48 ° from the normal to the grating plane; the viewing angle was

t

25H-7 19 o. A 300 linetinch (11.8 line/ram) grating was used for determining the elastic modulus and a 1000 line/inch (39.4 line/mm) grating for obtaining the Poisson's ratio, since the latter was too sensitive for ready determination of fringe order at the specimen end. For Poisson's ratio determination, the grating was tilted so that the center of the fringe pattern appeared near the center of the bar. Measurements were performed at a specimen surface strain of 0.05% to 0.1%. Theoretical contour lines based on equation (3) were prepared for a variety of Poisson's ratio were drawn with an Apollo computer using 'Promatlab' graphics software to facilitate the analysis of tha observed fringe patterns appearing on the specimen surface.

2.5. MTS machine test A servohydraulic testing machine (MTS corp) with a 2.5 kN load cell was used to apply loads for the characterization of nonlinear properties of the foams. All t.'-e experiments were performed at equal strain rates of 0.006 sec -1 . To determine lateral deformation for the purpose of determining Poisson's ratio, dial gages (No. 25-109, Starrett company, resolution 1.2 p.m, range 0.38 ram)were applied to each lateral surface near the center portion of the specimen. A fiat tip was used in contact with the spedmen in the case of lateral bulging, while a round one was used to evaluate lateral contraction. A universal joint was _sed if'. ;he te_!on test and a small ball socket joint was used in the compression test to assure alignment. In compression tests, Teflon tape was used between the specimen and the loading surface to minimize the friction force on the contact surfaces.

25H-8 3. Results and Discussion In the conventional copper foam, the moird fdnge patterns were observed to b_ hyperbolas corresponding to a positive Poisson's ratio. Accuracy in the Poisson's ratio determination was limited to ¢0.1 by uncertainty (about :1:5%) in measuring the slope of the asymptote. For homogeneous materials, accuracy could be Improved by increasing the strain hence the fringe density; however in these materials the fringes became blurred beyond a certain strain[17]. In the re-entrant copper foams, the moir6 fringes had elliptical contours corresponding to a negative Poisson's ratio. It was possible to achieve better accuracy in this case as a result of the different fdnge shape and the greater compliance of the transformed material.

Fig. 1 and 2 show optically determined Poisson's ratio at small strain as it depends on the initial and final relative densities. The Poisson's ratio of conventional foam with initial relative density of 0.09 is 0.2 ¢0.1. In the re-entrant foam, the smallest Poisson's ratio observed was -0.8 ¢0.05 at a strain of 0.1%; the permanent volumetric compression ratio was 2.13 and the Initial relative density was 0.1.

The negative Poisson's ratio of re.entrant foam attains a minimum for an optimal value of permanent volumetric compression ratio; the optimal compression is smaller for higher values of Initial relative density, as shown in Fig. 1. For copper foam with Initial relative density of 0.04 the optimal compression ratio, 3.6, is comparable to that of the polymer foams, 3.3 to 3.7, which have a similar relative density of 0.03. This con'_oondence between two different metedais suggests a geometrical cause associated with the cell structure. For example, the optimum volumetric compression ratio arises in part from the fact that too much compression causes the dbs to come in contact, which hinders the unfolding which gives dse to the negative Poisson's ratio. In such a simple view, one would expect the optimum to occur for a particular final relative _ 25H-9

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density (determined as the product of the Initial relative density and the volumetric compression ratio). Fig. 2, however, shows the optimum to depend on both the initial and final relative densities, so the above interpretation is incomplete.

The elastic modulus decreases monotonically with permanent volumetric compression as shown in Fig. 3, so that re-entrant copper foam is less stiff than the conventional foam from which it was derived. Polymer foams behaved similarly, except that for sufficiently large permanent volumetric compression, the stiffness began to increase again.

The effect of annealing on the stress-strein r?latlonships for foams with volumetric compression ratios of t, 2.0, 2.5, 3.0 is shown in Fig. 4. In tension, most of the specimens failed near the cast polymer end piece due to stress concentration; the end points on the tensikJside of the graphs represent this fracture. In compression, the tests were terminated at the beginning of unstable compression due to buckling. The conventional foam has some strain-hardeningtendency, but the re-entrant foam does not exhibit this effect, thus, the latter can be modeled as a elastic-perfectly plastic material. In both tension and compression, the con,:gntlonal and re-entrant foams exhibit a rather long plateau above the proportionalIlmiL This behavior is attributed to the plastic hinge formation of cell n'bs which occurs when the moment exerted on the cell dbs exceed the fully plastic momenL By contrast, the fracture behavior of both conventional and re-entrant copper foam in tension was brittle-like: failure occurred abruptly without necking or ¢_rawing.The foams were bdttle even though the solid copper from which the foam was made is ductile; such behavior has been observed in other foams and has been analyzed in view of the alignment of cell dbs which occurs under tension [12].

251-1-10 As for the compressive properties, errors can arise due to the effect of friction with the loading surface combined with the Poisson effect. An apparent yield stress ¢ry,app can be expressed [18]in terms of the true yield stress for a spedmen with a ratio of 2:1 in width to height as Gy,app-Gy, true(l+m ), where m is the friction ¢oeffident.

Since m is less than 0.1 for copper and Teflon, the maximum experimental deviation through the whole load history due to the frio_on force is within 10%.

For the conventional copper foams the propertlss in tension and compression are similar, as seen in Fig. 4. Young's modulus and the ultimate strength are lower by about 33% and 25%, respectively, for annealed material. The inset graph shows that the foams are nonlinear even for strains below 1%; by contrast to the polymer foams studied earlier. The difference adses from the yield of the ribs in copper foam at small strain. The m-entrant spedmens as shown in Fig. 4 show very dissimilar properties in tension and compression. Increase in the permanent volumetric compression ratio results in reduced tensile stiffness and Increased compressive saffness, as seen in Fig. 5 which shows the effective Young's modulusat 0.5% strain as it depends upon permanent volumetric compression ratio. As for ultimate strength, it is reduced by about 15% in both tension and compmsdon by the annealing Wocsss.

The experiments disciosad that the toughness, defined as _e energy per unit volume to fracture, of the re-entrant foam compared to that of conventional foam, inc_asad by factors of 1.4, 1.5, 1.7 with increases of volumetric compression ratio of 2.0, 2.5, 3.0, respectively. Annealing effects can further increase the toughness. This effect was most evident at a volumeffio compression ratio of 2.0, as shown in Fig. 6.

However, the annealed m-antrant foams at a volumetric compression ratio of 2.5 did not show as much Increase, and the toughnessactuallydecreased in the foam at a volume0tc compression ratio of 3.0. In case of an elaatomedc foam such as polyurethanefoam, the

(

25H-11

experimental resultsshowedincreases by factorsof 1.7,2.1, 2.3, 2.6, 3.2 in the

toughness of the re-entrant foamcompared to that of the conventional foam, at

volumetric compression ratios of 2.0, 2.8, 3.2, 3.7, 4.2, respectively[9]. Elastomertc foam (which can be deformed at large strain)exhibited a somewhat greater toughness increase with re-entrant transformation in comparison to copper foam (which is elasto-plastic).

The Poisson's ratio of the conventional foam shows highly nonlinear dependence upon the strain, as shown in Fig. 7. The Poisson'sratiosof the conventional copper foam are near 0.35 for small strain and approach 0.5 in tension and 0 in compression; similar behavior was observed in conventional polymer foam {9]. This approach of Poisson's ratio to zero at large strain In compression also agrees with other results: Poisson's ratio for the plastic compression of foam matedal was typically 0.03- 0.05119,20] and Poleson's ratio beyond the yield point was 0.04 {21] in compression.

As for small-strain behavior of conventional foam, a strong dependence of Poisson's ratio on the bulk density at a strain of not more than 0.5% was suggested [22].

Other authors showed a constant Poisson's ratio of 0.33 [12,23,24] or 0.23 [25] Independent of relative density. Even though the cell morphology changes with the relative density [25] any effect may be difficult to extract from the considerable scatter in experimental values for Poisson's ratio [12]. The density changes In the re-entrant foam are accompanied by cell shape changes, so the above considerations are not applicable.

Annealed specimens gave a slightly better negative effect in the Poisson's ratio (Fig. 7,8,9) at the expense of a decrease in the Young's modulus for each volumetric compression ratio. The lowest Poisson's ratio obtained in this mechanical testing (MTS) experiment was -0.49 at a strain of 0.2% as indicated in Fig. 8. In comparison, a 25H-12

Poisson's ratioof -0.6:L-0.05 wasdetermined optically at a strainof 0.1%for specimens

havingthe sameinitialrelative density of 0.08 and the same volumetric compression

ratio of 2.5. A minimum Poisson's ratio of -0.8¢0.05 was determined op_cally at a strain of 0.1% for a different specimen as described above. Fdis eL al. [3] obtained a minimum Potsson's ratio of -0.39 at a compressive strain of 0.013 in mechanical tests on copper foam with a permanent volumetflc compression ratio of 2.0 and a relative density of 0.053. The difference may be attributed to the highly nonlinear behavior of the metal foam as indicated by the cusp in Poisson's ratio vs strain. This contrasts with the polymericfoam which exhibited a smoother dependence of Poisson's ratio upon slrein and which exhibited a broader minimum in Poisson's ratio vs strain, in comparison with the metal foams. The difference in behavior between the metal and polymer foams is attributedto the fact that the dbs of the copper foam can yield and form plastic hinges at relatively small strain, in contrast to the polymer which is an elastomer. The nonlinearities in compression are attdbutad in part to regions of contact between cell dbs in the re-entrant foam; no such contacts occur in the conventionaJfoam (Fig. 10).

4. Conclusions 1. The Poisson's ratio of both conventional and re-enVant foam,depends on sualn, Poisson's ratio of rHntrant foam attains a relative minimum as small as -0.8 for zero strain.

2. Optimum permanent compression to achieve the best negative Poisson's ratio depends on the initial foam relative density: higher density foam requires less compression for transforma*Jon.This optimum appears to have a purely geometriceJ origin, since the same value was observed for metal and polymer foam of the same relative density.

25H-13 3. The re-entrant copper foam in compressiondoes not exhibit strain han:lening.

4. The toughness of the re-enCant foam increases with volumetric compression ratio and annealing effects further increase the tougl_ness.

Acknowledgment Support of this research by the NSF and by the NASA/Boeing ATCAS program under contract #NAS1-18889, and by a University Faculty Scholar Award (to RSL) is gratefully acknowledged.

25H-14 5. References 1. R.S. Lakes, Science, 235 (1987) 1038 2. R.S. Lakes, Science, 238 (1987) 551 3. E.A. Fdis, R. S. Lakes and J. B. Park, J. Mat. Sd., 23 (1988) 4406 4. Y.C. Fung, "Foundation of Solid Mechanics" (Prentice-Hall c.ngiewood, NJ, (1968) p.353 i, ° 5. A. F_ H. Love, in "A Trea_se on the Mathematical Theory of Elasticity" /!

, !

(Dover Pub., NY, 1944) p.163 6. O.G. Ingles, I. K. Lee, R. C. Nell, Rock Mechanics, 5 (1973) 203 7. K.E. Evans, B. Caddock, J. Phys. D: Appl. Phys., 22 (1989) 1883 .p 8. R.S. Lakes, J. Mat. Soi.,26 (1991) 2287 9. J.B. Choi, R. S. Lakes, J. MaL Sci., submitted 10. K. F_Jms and R. S. Lakes, in preparation 1 1. S.P. Timoshenko and J. N. Goodler, "Theory of Elas_ (McGraw- HHI, NY, 1969) 1 2. L J. Gibson, M. F. Ashby, "Cellular Solids" (Pergamon Press, 1988) 13. L E. Samuels, in "Metaliographic Polishingby Mechanical Methods" (American Society for Metals,1982) p 80-81 14. "Annual Book of ASTM Standards" (Designatfon:ES-85b,1986) 1 5. "Metal Handbook, Heat treating,cleaning & Finishing" (American Sodaty for Metals) p 285 1 6. A.S. Kobayashi, in "Manual of EngineeringStress Analysis" (Prentice Hall, 1978) p 62-63 1 7. C.P. Chan and R. S. Lakes, "Holographic study of conventionaland negative Poisson's ratio metallic foams: elasticity, yield, and micro-deformation',

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25H-15 J. Materials _,Ccience, in press.

18. R. Hill, in "The Mathematical Theory of Plasticity" (Oxford University Press,1983) p 236 19. M.C. Shaw and T. Sara, Int. J. Mech. Scl., 8 (1966) 469 20. J.A. Rinde, J. Appl. Polym. Sd., 14 (1970) 1913 21. M. Wilsea, K. L Johnson and M. F. Ashby, Int. J. Mech. Sd., 17 (1975) 22. A.G. Demant'ev, P. I. Seliverstov and O. G. Tarakanov, Mekhan. Polim., No.

1 (1973) 45 23. A.N. Gent and A. G. Thomas, Rubb. Chem. Tech., 36 (1963) 597 24. J.M. Lederman, J. A_opl. Polim. Sd., 15 (1971) 693 25. A. Mclntyre, G. E. Anderton, Polimer 20 (1979) 247 25H-16

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List of Figures Fig. 1 Polsson's ratio vs volumetric compression ratio, Initial relative density; El ,0.1; ,0.09; # ,0.08; a ,0.04 Fig. 2 Polsson's ra_o vs final relative density; Initial relative density; _ ,0.1; 't ,o.og; • ,0.08; a ,0.04 Fig. 3 Young's modulus as it depends upon initial relative density and upon volumetric compression ratio; Initial relative density; I_ ,0.1; _ ,0.09; X ,0.08; o ,0.C4 Fig. 4 Stress- strain relationships for conventional and m-entrant foams. Initial relative density:.0.08. Solid symbols:annealed. Open symbols:not annealed.

I_ Squares: conventional foam, volumetric compression 1.

A Tdangles: re-entrant foam, volumetric compression 2.0.

0 Circles: re-entrant foam, volumetflc compression 2.5.

0 Diamonds: re-entrant foam, volumetric compression 2.0.

Fig. 5 Young's modulus vs permanent volumetric compression ratio for annealed and non annealed copper foam at a s_'ainof 0.5%; _ ,non-annealed; • ,annealed Fig. 6 Toughness vs permanent volumetric compression of the conventional and re- entrant copper foams: e, non-annealed; IF, annealed r t Fig. 7 Potsson's ratio vs longitudinal strain for copper foam, initial reiaUve density: 0.08. Solid t, conventional foam; O, re-entrant foam, not annealed, volumetric compression ratio of 2.0 Fig. 8 Poissons ratio vs longitudinal strain for volumetric comprsssion ratio of 2.5, Initial re_stivedensity: 0.08; _ ,non-annealed; ¢I ,annealed; c] _optlcal result f.

25H-17

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Fig. 9 Poisson's ratio vs longitudinal strain for volumetric compression ratio of 3.0, Initial relative density: 0.08; o ,non.annealed; • ,annealed Fig. 10 Scanning electron micrographs of conventional copper foam (top), relaIive density 0.08; re-entrant copper foam (bottom) initla; relative density 0.03, volumetric compression ratio 2.5.

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251-I-18 O.4 -0.0 m -O.2 -0.4 W m -0.6 L -0.8 -1.0 Volumetric Compression Ratio

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m -,s m "0 _00 .= C 5O ),,, I '_ 3 4 Volumetric Compression Ratio 25H-21 A q a.

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L 25H-25 -0.1 m (B e" el i M M m n_ -0.8 -O.7 -0.10 -0 05 0.00 0.05 0.10 0.15 Longitudinal Strain 25H-26 0.1 -0.0 m m o m iI ,,0.3' o n_ -O.4 -0.5 -0.10 -0.05 0.00 0.0G 0.10 0.15 Longitudinal Strain r* i t 25H-27 25H-28