non-stochastic ti-6al-4v foam structures with negative poisson’s ratio

8/13/2019 Non-Stochastic Ti-6Al-4V Foam Structures With Negative Poissons Ratio

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Authors Accepted Manuscript

Non-Stochastic Ti-6Al-4V Foam Structures With

Negative Poissons Ratio

Li Yang, Denis Cormier, Harvey West, Ola Harrys-

son, Kyle Knowlson

PII: S0921-5093(12)01174-4

DOI: http://dx.doi.org/10.1016/j.msea.2012.08.053

Reference: MSA29004

To appear in: Materials Science & Engineering A

Received date: 19 March 2012

Revised date: 13 August 2012

Accepted date: 13 August 2012

Cite this article as: Li Yang, Denis Cormier, Harvey West, Ola Harrysson and Kyle

Knowlson, Non-Stochastic Ti-6Al-4V Foam Structures With Negative Poissons Ratio,

Materials Science & Engineering A, http://dx.doi.org/10.1016/j.msea.2012.08.053

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Non-Stochastic Ti-6Al-4V Foam Structures With Negative Poissons Ratio

Li Yanga([email protected]), *Denis Cormier

b([email protected]), Harvey West

a

([email protected]), Ola Harryssona([email protected]), and Kyle Knowlson

1

([email protected])

aEdward P. Fitts Department of Industrial & Systems Engineering

North Carolina State University400 Daniels Hall, 111 Lampe Dr. Raleigh, NC27695

bDepartment of Industrial & Systems Engineering

Rochester Institute of Technology

81 Lomb Memorial Dr. Rochester, NY14623-5603

Phone: (+1) 585-475-2713Fax: (+1) 585-475-2713

* To whom the correspondence should be addressed

Abstract

This paper details the design, fabrication, and testing of non-stochastic auxetic lattice

lattice structures. All Ti-6Al-4V samples were created via the Electron Beam Melting (EBM)

additive manufacturing process. It was found that the Poissons ratio values significantly influence

the mechanical properties of the structures. The bending properties of the auxetic samples were

significantly higher than those of currently commercialized metal foams. The compressive

strength was moderately higher than available metal foams. These results suggest that metallic

auxetic structures have considerable promise for use in a variety of applications in which tradeoffs

between mass and mechanical properties are crucial.

Keywords: Cellular materials, porous materials, titanium alloys, failure


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1. IntroductionFor a specimen subjected to uniaxial stress in the elastic region, Poisson's ratio is defined as the

negative ratio of the transverse strain to the axial strain, and can theoretically range from -1.0 to 0.5

for isotropic structures and materials. Although materials with negative Poisson's ratios are notcommonplace, the emergence of additive manufacturing processes with few geometric limitations

has generated considerable interest in this class of materials.

Materials with negative Poisson's ratios (auxetic materials) were first reported by Lakes in

1987 [1]. He described a straightforward method to produce polymer foam structures with

negative Poisson's ratios. The technique involves first compressing a regular foam structure in

three orthogonal directions and then holding it there at an elevated temperature for a period of time.

After allowing it to cool to room temperature, it is then decompressed. The technique typically

starts with a low relative density foam in order to accommodate the volume of materials

compressed [2], and it is primarily used with polymer materials. A number of researchers have

successfully employed this process to produce structures with negative Poisson's ratio [3-5].

According to the theory of elasticity for isotropic materials, the shear modulus Gand bulk

moduli Kcan be determined by Eq. (1) and Eq.(2):

=

2(1 )

EG

+

(1)

=3(1 2 )

EK

(2)

where is the Poisson's ratio of the material.

From Eq.(1) and Eq.(2), it is clear that when Poisson's ratio is negative, the material will

have G>>Kwhich indicates significantly higher shear modulus than bulk modulus. Materials that

possess superior shear modulus are expected to possess high indentation resistivity [6], high

torsional rigidity [7], high bending stiffness and shear resistance [8], and high energy absorption

efficiency [4]. Furthermore, auxetic materials are also appealing candidates for use as cores in

sandwich panels [9]. During the bending of a sandwich panel, the inner half of the bending section

is subjected to compressive stress while the outer half is subjected to tensile stress. For regular

materials with a positive Poisson's ratio, the compressed sections cross sectional area tends to

expand while the elongated sections area tends to shrink. This produces internal shear stresses and


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buckling of the cellular structure. For auxetic materials, the compressed section exhibits transverse

shrinkage, therefore these parts are expected to have better overall mechanical properties during

bending.

Although there have been theoretical analyses of auxetic structures, challenges with

manufacturing these geometries have limited their use in practical applications until very recently

[10]. Furthermore, the majority of research dealing with auxetic materials has been focused on

polymer materials. As new techniques for synthesis of these materials become available,

applications in areas such as medical implants and energy absorbers will be explored. A

prerequisite for actual implementation, however, is that the mechanical properties must be

consistent and predictable. In this paper, the Electron Beam Melting (EBM) additive

manufacturing process is employed to produce foam structures with predetermined auxetic

behavior using Ti-6Al-4V. The EBM process is capable of fabricating complicated open cell metal

foam structures from CAD models whose mechanical properties are comparable to those obtained

from theoretical calculations [11].

2. Auxetic Structure DesignA number of auxetic structures have been proposed in the literature [1, 2, 12-15], the

majority of which are two-dimensional. A number of these geometries are easily interpreted as 3D

structure, while others show significant chirality and therefore cannot be readily translated into 3D

with the same symmetry [16]. In this study, the re-entrant lattice auxetic structure designed by

Warren [12] is adopted and expanded into a 3D geometry.

For materials with high elastic modulus such as Ti-6Al-4V, deformation of the re-entrant

lattice structure is mostly attributed to the deflection of ribs in the structure. Wan et al. [17] showed

that for the 2D re-entrant lattice structure, the Poisson's ratios are largely determined by the size of

the struts as well as the re-entrant angle. For structures whose struts have a square cross section,

the unit cell design can be described in terms of length of the diagonal ribs (L), the length of thevertical ribs (H), and the angle () between the diagonal ribs and the horizontal planes. Their work

shows that for 2D re-entrant lattice, the H/Lratio and have an opposite effect on the Poisson's

ratios of the two principal directions. Under a small deflection assumption, the relationship

between Poisson's ratio and the parameters is approximated by Eq.(3) and Eq. (4) [17].


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2cos

=( / sin )sin

xH L

(3)

2

( / sin )sin=

cosy

H L

(4)

The expanded 3D structure studied by the current research is shown in Fig.1. Since the 2D

structure is anisotropic, the corresponding expanded 3D structure is also anisotropic. The design

parameters are shown in Fig.1 in the 2D demonstration. Assuming an ideal situation in which the

number of unit cell repetitions is infinite in all three principal directions, the relationship between

Poissons ratio and the parameters also happens to be expressed as indicated by Eq.(3) and Eq.(4).

In order to estimate the effect of Poisson's ratio on the mechanical properties of the structure, two

different designs are used in this experiment as indicated in Table 1. Design variation 1 (DV1) has

a small H/Lratio and relatively small and therefore is expected to exhibit a less negative y. On

the other hand, design variation 2 (DV2) has large H/Lratio and and is therefore expected to

exhibit a greater negative y. Using Eq.(3) and Eq.(4), the theoretical Poisson's ratios yfor DV1

and DV2 are listed in Table 1. This assumes a sufficiently large number of unit cells so that the

edge effects could be ignored. Due to the finite number of unit cells in the actual samples, it is to be

expected that the actual Poisson's ratios might be influenced by the edge effects.

The cross section of the strut for both design variations is square with width of 1mm. Two

different types of specimens were fabricated for bending and compressive testing respectively.

Fig.2 shows the actual parts built in Ti-6Al-4V via the EBM process. For both the bending and

compressive test specimens, the repetition of unit cell are 2232 and 664 respectively.

3. Experimental ApproachSix bending specimens for each design variation were built in a single batch in an Arcam

A-2 EBM system. Two DV1 and four DV2 specimens were built in a second batch using the same

processing parameters. Following fabrication, the specimens were left in the chamber to cool

down overnight in vacuum.

Bending and compression tests were performed on an Applied Test Systems model 1620C

which has a maximum loading force of 100kN. For the bending tests, the support span was 4"

(101.6mm), the loading nose span was 2" (50.8mm), and the support and load rollers had a


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diameter of 12.7mm.

For the compression tests, two parallel plates were used to avoid stress concentration on the

structures, and the loading speed was 1.27mm/min. Poisson's ratio for each sample was obtained

by pausing the ATS machine and measuring the sample size in the transverse directions using

digital calipers. Due to the relatively high modulus of the specimens, the transverse dimension

contraction was on the order of 0.01mmscale which was close to the resolution of the caliper. The

measurement of Poisson's ratio was therefore only an approximation. Provided Poisson's ratio is

negative and there are significant differences between two design variations, the discussion can be

qualitatively justified.

Although the EBM process fully melts the titanium powder, there is always a boundary

between powder that is melted during fabrication and the surrounding powder that is not melted.

At this interface, some particles will sinter and become partially attached to the struts. Due to this

sintering effect, the actual dimensions of each given EBM part slightly differs from the specified

CAD model dimensions. The actual dimensions and weights of the specimens were therefore

measured and compared with the theoretical designs using digital calipers and an electronic

balance. The measurement indicated that the dimension and density variation was negligible for

this experiment. For both bending and compression samples, the standard deviation of the

dimension measurements were


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Structures having negative Poisson's ratios were expected to possess higher compressive

strength (C) and flexural strength (F). In order to estimate the strength values, Eq.(7) and Eq.(8)

were used.

= y

F

y

I (7)

=CF

A (8)

The results obtained in this study were compared with the corresponding mechanical

properties of other metal foams. Values were normalized by the properties of the bulk materials in

order to estimate the effect of the lattice geometry versus the bulk material. Due to the loading

limit of the testing machine, the compressive strength for the specimens of DV1 was not reached.The compressive stress value was therefore conservatively taken as the value of compressive stress

up to the machines limit. During compressive testing, the strain values reached approximately 17%

and 46% respectively for DV1 and DV2. The total energy absorption (W) values at 50% strain for

compressive testing were interpolated based on the testing curve patterns. In addition, the specific

energy absorption (w) was also obtained by calcualting the energy absorption per unit weight of

the structure.

4. Results and AnalysisTables 2 and 3 show the dimensions of the actual samples, and compare the actual and

predicted relative densities. In the tables L1, L2and L3are the averaged length, width and height of

the samples, respecitvely. The dimensional variations between specimens are quite small

(0.5mm), and are therefore neglected for the future discussion. In Table 2 and 3 the TRD and

ARD stand for the theoretical relative denstiy and the actual relative density, respectively.

Not surprisingly, there exist slight differences between theoretical and actual relative

density. This is to be expected based on the aforementioned sintering effect. For both bending and

compressive specimen sets, the difference between actual and predicted density was less than

10%.

The force-displacement curves for the bending test are shown in Fig.4. In some of the


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curves, a plateau stage near the start of elastic deformation can be seen which deviates from the

regular pattern. This phenomenon was likely to be caused by the slack of the crosshead at the start

of compression. Therefore, for these samples, the maximum displacement values were offset by

the slack values. The lack of significant yielding prior to failure indicates that the structures

underwent sudden and catastrophic fracture which is typical for this type of metal foam.

From Eq.(7), the modulus of elasticity E and the flexural strength BS of DV1 and DV2

can be determined as shown in Table 4. In Table 4 Wand wrepresent the total energy absorption

and specific energy absorption during the bending, respectively. Again the values are averaged

over the 6 specimens. The standard deviation is indicated in the parentheses. With a target +/-5%

margin of error for modulus and flexural strength respectively, a sample size of six specimens, and

the sample standard deviations shown in Table 4, the computed confidence intervals for the

modulus of DV1 and DV2 are 99.8% and 99.6% respectively. Likewise, the computed confidence

intervals for the flexural strength of DV1 and DV2 are 99.9% and 92.0% respectively. For

purposes of preliminary scientific experimentation using a relatively new fabrication technology,

these confidence values were deemed to be quite reasonable.

From the results, DV1 shows significantly higher average modulus and average flexural

strength than DV2. DV2 has higher total energy absorption before fracture, which is attributable to

the greater strut thickness.

The strain-stress curves for the compressive tests are shown in Fig. 5, and the

corresponding data is shown in Table 5. The test results for DV2 show a cyclic pattern throughout

the compression which was also observed by Cansizoglu during testing of hexagonal lattice

structures made by EBM [19]. The tests ended at strain values of approximately 47% for DV2. The

energy absorption values up to that point were taken to be the value for 50% strain for comparison

with other published work. The DV1 specimens did not crush under the machines limit of 100kN

during the test, therefore the compressive strength for these samples exceeds 130MPa. Also, it was

expected that DV1 specimens would have the same cyclic loading pattern. The energy absorbed is

therefore estimated by assuming that the curve shown in Fig.5(a) represents one complete cycle

and can be repeated up to 50% strain.


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Poisson's ratios for the DV1 and DV2 specimens were about -0.052 and -0.097,

repsectively. The measured values for both design configurations were negative, with DV2 being

more negative than DV1, as expected due to the larger re-entrant angle as well as the H/Lratio for

DV2.

5. Discussion5.1 Flexural properties

To date, the majority of research involving flexural performance of foam materials has

involved foam core sandwich panels with face skins [20-23]. Auxetic materials show promise for

improved flexural performance with respect to conventional foam structures. Figs.6(a) and 6(b)

compare flexural performance of auxetic structures with other metal foam sandwich panels

reported in the literature. In order to allow comparison of properties for lattices built in different

materials, the lattice strength and modulus values were normalized by dividing them by the bulk

material strength and modulus, respecitvely.

Note that the foam materials used for comparison have solid face skins, whereas the

auxectic lattice materials presented here do not. Although sandwich structures with face skins are

expected to outperform structures without face skins, the auxetic lattice structures presented here

compared very favorably with non-auxetic foams having face skins. In relation to published results,

DV1 had relatively high bending modulus as well as flexural strength.

According to Eq.(1), however, auxetic structures will exhibit higher shear modulus as the

Poisson's ratio becomes increasingly negative. As a consequence, the auxetic structures can be

expected to exhibit higher flexural strength compared to regular foam structures. During bending,

the normal stress direction is along the xaxis. DV1 would be expected to exhibit higher flexural

strength than DV2. The experimental results supported this hypothosis.

Not surprisingly, both design variations exhibited relatively low ductility during flexural

testing. This has also been observed in other EBM fabricated Ti6Al4V lattice structures and can be

attributed to two factors. First, Ti-6Al-4V does not have high plastic deformability after yield.

Second, lattice structures with small diameter sloped struts inevitably have some fabrication

defects resulting from the well known stair stepping effect common to all additive processes [11].

The stair stepping effect reduces the mechanical strength by reducing the effective strut diameter.


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In this case, the strut diameter reduction is estimated as 19% and 28% for DV1 and DV2

respectively. Therefore, it is reasonable to predict that under improved fabrication conditions, the

bending modulus and strength will significantly improve.

5.2 Compressive propertiesDue to the loading limit of the testing machine, the maximum compressive strength and

energy absorption of DV1 could not be determined. This result was partly associated with the fact

that the re-entrant joints move inwards during compression. The small distance between the

re-entrant nodes within each DV1 unit cell means that nodes actually come into contact with each

other during compression and tend to brace the structure. Greater force is therefore needed to

achieve further compression, thus the significant increase in apparent strength relative to the more

open DV2 unit cell geometry created by its 45 reentrant angle. The cyclic stress-strain pattern

shown in Fig.5 is due to the fact that failure always occured on the ribs within the same layer

normal to the compressive direction. In other words, the part is compressed until one entire layer

collapses. Then the next layer starts to take up the load until it collapses. Unlike most metal foams,

non-stochastic auxetic lattice materials have a regular layered structure. When an individual strut

fails, the load supported by that layer is carried by a smaller numer of struts. This places stress

concentration on the other struts in the same layer, and collapse of the entire layer quickly follows.

Not surprisingly, the total number of cycles seen in the stress-strain curve is equal to the number of

layers in the specimen. Fig.7 shows a sample during testing in which two distinct layers havecollapsed.

The compressive modulus and strength of the specimens are shown in Figs.8(a) and 8(b)

respectively. Again, the normalized properties were compared in order to take the solid material

properties into account. The auxetic structures exhibited higher compressive strength and average

modulus compared with most other metal foams. Although DV1 has higher relative density than

DV2, it exhibited smaller modulus. This conflicts with Eq.(12). This could be partly contributed to

the Poisson's ratio values. Since the axis of compression lies along the parts ydirection, DV2 is

expected to exhibit greater negative Poisson's ratio compared to DV1. During the compression

tests, the upper and lower surface of the lateral movement of specimens were restricted by contact

with the parallel platens. On the other hand, the center section of the specimens tended to shrink

transversely. Consequently, additional shear stress is generated in the cross-section plane normal

to the compressive direction. As previously discussed, auxetic structures with greater negative


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Poisson's ratio are expected to be more resistant to shear deformation.

Energy absorption is another property of interest for many applications of metal foams.

Energy absorption during the compression test was compared with regular foam structures as

shown in Fig.9. In order to take the solid material properties into consideration, the energy

absorption per unit weight (kJ/kg) was used. From Fig.9, it is clear that the energy absorption per

unit weight of both DV1 and DV2 were significantly greater than most other metal foams. It is also

seen that with greater negative Poisson's ratio, the energy absorption ability of the auxetic lattice

structure also increases. Due to the stair stepping effect common to layer-based manufacturing

processes, the tested structures were expected to be actually weaker than the ideal case, which

suggested that the auxetic lattice would exhibit even better mechanical properties. Overall, the

discussion showed that the auxetic lattice structure holds great potential for future energy

absorption applications.

6. ConclusionsNon-stochastic auxetic lattice structures were manufactured with two different geometric

parameter variations. Bending and compression tests were conducted for both design variations.

The improvement of bending properties was so significant that auxetic structures without solid

skins on the upper and lower faces were able to match the strength of the regular sandwich panel

structures. On the other hand, the compressive test showed less significant, though still promising,

improvement. Due to the stair stepping effect and other fabrication induced defects, the actual

structure did not exhibit as high compressive strength and energy absorption as expected. However,

structures with negative Poisson's ratios were experimentally proven to possess excellent

mechanical properties that warrant further study and process development.

References

[1] R. Lakes, Sci. 235 (1987) 1038-1040.

[2] R.S. Lakes, R. Witt, Int. J. Mech. Eng. Educ. 30 (2002) 50-58.

[3] F. Scarpa, F.C. Smith, J. Intellect. Mat. Syst. Struct. 15 (2004) 973-979.

[4] A. Bezazi, F. Scarpa, Int. J. Fatigue 29 (2007) 922-930.

[5] F. Scarpa, P. Pastorino, A. Garelli, S. Patsias, M. Ruzzene, Phys. Status Solidi (b) 242 (2005)

681-694.


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[6] R.S. Lakes, K. Elms, J. Comp. Mat. 27 (1993) 1193-1202.

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[10] W. Yang, Z.-M. Li, W. Shi, B.-H. Xie, M.-B. Yang, J. Mat. Sci. 39 (2004) 3269-3279.

[11] O. Cansizoglu,O. Harrysson, D. Cormier, H. West, T. Mahale, Mat. Sci. Eng. A 492 (2008)468-474.

[12] T.L. Warren, J. Appl. Phys. 67 (1990) 7591-7594.

[13] U.D. Larsen, O. Sigmund, S. Bouwstra, J. Microelectromechanical Syst. 6 (1997) 99-106.

[14] C.W. Smith, J.N. Grima, K.E. Evans, Acta Mat. 48 (2000) 4349-4356.

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[16] D. Prall, R.S. Lakes, Int. J. Mech. Sci. 39 (1996) 305-314.

[17] H. Wan, H. Ohtaki, S. Kotosaka, G. Hu, Euro. J. Mech. A/Solids 23 (2004) 95-106.

[18] D. Roylance, Beam Displacements. Massachussetts Institute of Technology OpenCourseware: Mechanics of Materials, 10, 2000.

[19] O. Cansizoglu, Mesh Structures with Tailored Properties and Applications in Hip Stems.

Ph.D. Dissertation. North Carolina State University, Raleigh, 2008

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[21] C. Chen, A.-M. Harte, N.A. Fleck, Int. J. Mech. Sci. 43 (2001) 1483-1506.

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[24] C.-H. Lim, I. Jeon, K.-J. Kang. Mat. Des. 30 (2009) 3082-3093.

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September 2007. Montreal, Canada, 2007, pp. 7-10.

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[27] L.J. Vendra, A. Rabiei, Mat. Sci. Eng. A 465 (2007) 59-67.

[28] Duocel Aluminum Foam. ERG Materials and Aerospace Corporation Product Specification

Data Sheet, 2011.

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Figure Captions

Fig.1 3D unit cell for auxetic lattice structure

Fig.2 Specimens built by EBM (a) bending specimens; (b) compressive specimens

Fig.3 Calculation of the modulus values

Fig.4 Force-displacement curves for four-point bending tests (a) Design variation 1; (b) Design

variation 2

Fig.5: Strain-stress curves of compression experiments (a) Design variation 1; (b) Designvariation 2

Fig.6 Comparison of bending properties between auxetic lattice and regular sandwich panels; (a)Normalized bending modulus; (b) Normalized flexural strength

Fig.7 Layer collapse during compressive test

Fig.8 Comparison of compressive properties of various structures; (a) Normalized modulus; (b)

Normalized Strength

Fig.9 Comparison of energy absorption per unit weight of various metal foams


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Fig. 1


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Fig. 2

Fig. 3


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Fig. 4(a)

Fig. 4(b)


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Fig. 5(a)

Fig. 5(b)


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Fig. 6(a)

Fig. 6(b)


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Fig. 7


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Fig. 8(a)


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Fig. 8(b)


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Fig. 9


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Table 1 Design parameters and theoretical Poissons ratio values in y

Design H(mm) L (mm) (deg) Theoretical PR in y

DV1 2.38 2.714 20 -0.351

DV2 5.685 1.591 45 -1.970

Table 2 Parameters for bending specimens

Design1

L (mm)2

L (mm)3

L (mm) TRD (%) ARD (%)

DV1 143.610.05 10.090.01 8.170.04 34.47 36.40

DV2 143.710.06 10.120.03 16.780.03 30.22 27.24

Table 3 Parameters for compressive specimens

Design1

L (mm)2

L (mm)3

L (mm) TRD (%) ARD (%)

DV1 26.750.04 26.840.05 15.710.01 34.47 36.04

DV2 26.740.03 26.790.08 33.290.06 30.22 26.51

Table 4 Mechanical properties from bending test

Design E (GPa) BS(MPa) W (J) w (J/kg)

DV1 11.25(0.43) 143.54(7.30) 1.71(0.18) 126.54(12.99)

DV2 1.88(0.08) 54.30(3.80) 3.05(0.53) 146.72(25.41)

Table 5 Mechanical properties from compressive tests

Design S(MPa) E (GPa) 50%W (J) 50%w (J/kg)

DV1 130.16* 1.21 300 16.67

DV2 58.61(5.38) 1.63(0.13) 285.13(32.63) 10.18(1.17)

* Reaches maximum loading capacity of the tester

non-stochastic ti-6al-4v foam structures with negative poisson’s ratio

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