Materials Science and Engineering A 422 (2006) 176–183

Mechanical behaviors of porous metals under biaxial tensile loads

P.S. Liu ∗The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Department of Materials Science and Engineering,

Beijing Normal University, Beijing 100875, China

Received 13 May 2005; received in revised form 4 January 2006; accepted 17 January 2006

Abstract

Based on a new simplified structural model of three-dimensional isotropic reticulated foamed porous metals, a practical analytical model isestablished for these materials under biaxial loading. From this new mechanical model, an effective relationship between biaxial nominal stressesand porosity at the beginning of failure is derived for metallic foams under biaxial tension. With this relationship, the biaxial load bearing capacitycan be roughly predicted for metallic foams under biaxial loading; hence this equation can be used as the strength-design criterion for metallicfoams under biaxial loads. When the biaxial nominal stresses are equal to each other, the relationship between loading strength and porosity willbe then conveniently obtained for metallic foams under equi-biaxial tensile loading. Experiments indicate that these equations have a quite goodpractical effect. In addition, some related mechanical behaviors were investigated for these porous materials under biaxial tensile loading.©

K

1

iimtttraassbimttnnb

0d

2006 Elsevier B.V. All rights reserved.

eywords: Metallic foams; Foamed metals; Porous metals; Porous materials; Cellular solids

. Introduction

The mechanical properties of porous metals have been draw-ng researchers’ extensive attentions [1–48], and plenty of mean-ngful results have been attained for porous metals, such as defor-

ation mechanisms, compressive stress–strain maps, constitu-ive models, failure behaviors, creep behaviors, elastic proper-ies, the relationship between relative Young’s modulus and rela-ive density, the relationship between relative shear modulus andelative density, the relationship between relative Poisson’s rationd relative density, the relationship between fracture toughnessnd relative density, the relationship between full densificationtrain and relative density, the relationship between compres-ive and tensile strength and relative density, the relationshipetween compressive strength and temperature, fatigue behav-ors, bending behaviors, yield behaviors, modulus–temperature

aps, energy absorption behaviors, size effects upon proper-ies, and so on. Among these mechanical properties studied, theensile behavior is also a basic section. Materials used in engi-eering design often suffer a complex stress state, so it is alsoecessary to study the tensile properties of metallic foams under

have conducted many excellent researches for formed materials,such as on deformation behaviors, various failure surfaces, yieldsurfaces, fracture surface, and so forth [2,5,6,8–10,16,17,32,35].In the present paper, we further investigate the biaxial tensileproperties of metallic foams with three-dimensional reticulatedhigh-porosity structure, based on our previous studies on theuniaxial tensile behavior of high-porosity metals and combin-ing some other relevant research achievements [8,9,16], andeduce the mathematical relationship between two applied nom-inal stresses and porosity at beginning of failure for open-cellmetallic foams under biaxial tensile loading. From this relationequation, the biaxial loading level at which these porous mate-rials are able to bear can be calculated with regard to variousvalues of porosity. That is to say, this relation can be used asthe strength-design criterion for metallic foams under biaxialloading states.

2. Basic hypotheses of the mechanical model(“octahedron model”)

Metallic foams with three-dimensional isotropic reticulated

iaxial or multiaxial loading. In this aspect, some researchers

∗ Tel.: +86 10 62205350; fax: +86 10 62231765.E-mail address: liu996@263.net.

structure are composed of metallic struts and continuous pores(see Fig. 1). For simplifying the analyses of the tensile behavior,we might as well consider that those metallic struts construct astructure with the struts regularly connected in the way of diag-

921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2006.01.083

P.S. Liu / Materials Science and Engineering A 422 (2006) 176–183 177

Fig. 1. SEM morphology of a representative three-dimensional reticulatedmetal—a nickel foam.

onals of cubes [22,23,31,45,46,49], and thereby form a greatdeal of octahedron units in the form of closed-stacked BCC (seeFig. 2). All those struts are fully equivalent or identical becausethey form uniformly by connecting between the vertex and thebody center of the cube. In addition, the resultant octahedronunits may have their four-fold axis of symmetry in any of threeperpendicular directions (for example, the marked octahedronunit in Fig. 2 has its four-fold axis of symmetry in the verti-cal direction) with every strut shared by three units, of whichfour-fold axes of symmetry are perpendicular to one another.Such octahedron units are space filling, and makes up the entiremetallic foamed bodies. Such model configuration can makeevery strut entirely equivalent or identical to one another, and theoctahedron units can be arbitrarily constructed in three perpen-dicular directions with the equivalent struts that are connected inthe way of diagonals of cubes. As a result, this model can quitesatisfactorily realize the regular extension of three-dimension,thereby present a regular reticulated structure with isotropy at

Fm

three perpendicular dimensions. Since the bearing capacity ofnodes is commonly higher than that of struts, the rupture fail-ure always occurs in struts. When the maximum normal stressat any position within the metallic strut reaches the magnitudeof the tensile strength (σ0) of the corresponding dense materialspecies, rupture which can cause the failure of the porous bodyas a whole will be brought about.

A real isotropic three-dimensional reticulated structure mayhave many unidentical pore-units, and the morphologies of strutsmay be very various from one to another (e.g. Fig. 1). As a result,the fine mechanical behavior of struts is different to one anotherto some extent, resulting in a complicated mechanical status.For simplifying the mechanical analyses, this complex struc-ture may be treated as an equivalent simple one [8–10,16,17].In the present “octahedron model”, the mechanical behaviorof the simplified specific structure just represents the statisticalequivalence of the real three-dimensional isotropic reticulatedstructure, and the reliability is further guaranteed by imputingthe material constant.

Of course, this octahedron-model may not exhibit sufficientsymmetries, but we know there is almost no theoretical modelperfect in every way, and we mean to do our best to consider boththe macroscopic isotropy and the simplicity simultaneously withthe present model, which has been found having advantage overthe Gibson–Ashby model [8–10,16,17] in the present applica-tion.

3m

3

osa

sorat

σ

wlp

3

is

σ

ig. 2. Octahedron unit of three-dimensional isotropic reticulated open-celletallic foams.

. Resultant relation equation of the present analyticalodel

.1. Maximum normal stress within the porous body

Within the above-described foamed metallic body, the unit-ctahedrons make a regular crisscross-orthogonal arrangement,o they will be identically loaded when the porous body suffersn external load.

While the porous body bears a biaxial tension, the metallictruts are meant to be loaded correspondingly, and will deflectr tend to deflect. For simplifying, the struts might as well beegarded as beams for the force analysis. Via relative geometricnd mechanical analyses (see Appendix A for detail), we havehe maximum normal stress within the strut:

max ≈ 2

√6√

3π(1 − θ)−3/2√

σ21 + σ2

2 − σ1σ2

+ (1 − θ)−1(σ1 + σ2) (1)

here σ1 and σ2 are two nominal stresses of the biaxial externaload on the porous body in two tensile directions, and θ is theorosity of the porous body.

.2. Rupture failure condition of the strut

When the maximum normal stress within the metallic struts equal to or larger than the tensile strength (σ0) of the corre-ponding dense material species of the porous body:

max ≥ σ0

178 P.S. Liu / Materials Science and Engineering A 422 (2006) 176–183

namely

σmax ≈ 2

√6√

3π(1 − θ)−3/2√

σ21 + σ2

2 − σ1σ2

+ (1 − θ)−1(σ1 + σ2) ≥ σ0 (2)

the porous body under biaxial tension will fail in this situation.While

σmax < σ0

namely

σmax ≈ 2

√6√

3π(1 − θ)−3/2√

σ21 + σ2

2 − σ1σ2

+ (1 − θ)−1(σ1 + σ2) < σ0 (3)

the porous body under biaxial tension will be safely loaded.

4. Modification of the relation equation

The developments of the above model are conducted in theideal simplified condition (see the Appendix A), but the realcase is more complicated. Thus, the maximum stress practi-cally produced within the porous body under biaxial tensionshould be achieved by the revision of σmax, which is theoret-ically derived from the above model. There into the simplestmtbi

σ

Tib

k

⇒√

wst

sg[

√

where m is the plastic–brittle index of the corresponding compactmaterial, with the value between 1 and 1.5 (i.e., 3/2) [23,31]:when the corresponding compact material is very brittle, m tendsto 1.5; when it is plastic to a very great extent, m tends to 1; andwhen it is the common metal or alloy species with the moderateplasticity, such as nickel for an example, we may approximatelytake m ≈ 1.25 [45,46]. That is to say, as for metallic foams madefrom common metals or alloys, we roughly have:

√σ2

1 + σ22 − σ1σ2 +

√2√

3π

12π(1 − θ)1/2(σ1 + σ2)

≈ K(1 − θ)1.25σ0 (8)

While the biaxial nominal stresses on the porous body areequal, i.e., σ1 = σ2 = σ, the above equation will becomes:[

1 +√

2√

3π

6π(1 − θ)0.5

]σ ≈ K(1 − θ)1.25σ0 (9)

where σ is the loading strength of the foamed metal under biaxialequal-load tension.

From Section 2 in this paper, it can be seen that the modeland corresponding assumptions are commonly suitable to three-dimensional isotropic reticulated metals. Therefore, the resultantequations can be expected to meet the general applications,aFmm

Fig. 3. PLS-S100 biaxial tensile tester used for the biaxial tension.

odification is setting a revised factor k, which characterizeshe material species and the preparation technique of the porousody, before σmax. That is to say, the practical maximum stresss

MAX ≈ kσmax (4)

hen we can find the following relationships between two nom-nal principle stresses and porosity for the porous body underiaxial tension at the beginning of its failure:[

2

√6√

3π(1 − θ)−3/2√

σ21 + σ2

2 − σ1σ2

+ (1 − θ)−1(σ1 + σ2)

]≈ σ0 (5)

σ21 + σ2

2 − σ1σ2 +√

2√

3π

12π(1 − θ)1/2(σ1 + σ2)

≈ K(1 − θ)3/2σ0 (6)

here K =√

2√

3π/(12πk), which is in fact a material con-tant and depends on the material species and the preparationechnique of the porous body.

The above development is based on the brittle failure. Foruiting manifold rupture ways, Eq. (6) can be rewritten as moreeneral form on the basis of the relevant contents of Refs.23,31]:

σ21 + σ2

2 − σ1σ2 +√

2√

3π

12π(1 − θ)1/2(σ1 + σ2)

≈ K(1 − θ)mσ0 (7)

lthough the results obtained depend on those assumptions.or previous excellent example, the results of Gibson–Ashbyodel [8,9,16] also depend strongly on a series of assumptionsade throughout the derivation, but they still have gotten many

P.S. Liu / Materials Science and Engineering A 422 (2006) 176–183 179

good applications. Analogous to this, the results of “octahedronmodel” can be also believed to achieve success, and the follow-ing experiment is just an example, but this example is a goodrepresentative case.

5. Application of relevant relation equations

5.1. Introduction of classical equations

Some authors [5,6,8,9,15,16] studied the mechanical proper-ties of porous materials under multiaxial loads, and from theircubic structural model they derived the typical equations forisotropic foamed bodies as follows [8,9,15,16]:

(i) Plastic failure relation

σd

σys= ±γ

(ρ∗

ρs

)3/2

⎧⎪⎨⎪⎩1 −

⎡⎣ 3σm

σys

(ρ∗ρs

)⎤⎦

2⎫⎪⎬⎪⎭ (10)

where σd is the deviated stress on the porous body, σys theyield stress of solid cell wall material, γ a constant, ρ* theapparent density of the porous body, ρs the density of solidcell wall material, and σm is the mean nominal principalstress on the porous body. In addition:

σd =√

12 [(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2] (11)

σm = 13 (σ1 + σ2 + σ3) (12)

where σ1, σ2 and σ3 are three principal stresses applied onthe porous body.

(ii) Brittle failure relation

σd

σfs= ±γ ′

(ρ∗

ρs

)3/2⎡⎣1 − 3σm

σfs

(ρ∗ρs

)⎤⎦ (13)

where σfs is the modulus of rupture of solid cell wall mate-rial, and the other symbols mean the same as Eq. (10),respectively.

Fig. 4. The morphologies of representative failed specimens: (a) under eq

ual-speed biaxial tension; (b) under unequal-speed biaxial tension.

180 P.S. Liu / Materials Science and Engineering A 422 (2006) 176–183

Fig. 5. The distribution line of maximum stress (dotted line) in specimens underbiaxial tension.

In the case of porous bodies under biaxial tension, we have:σ3 = 0. Consequently, using Eqs. (8)–(12) and the fact thatρ*/ρs = (1 − θ), Eqs. (10) and (13) can be rewritten as follows,respectively:

√σ2

1 + σ22 − σ1σ2 = KP(1 − θ)3/2

{1 −

[σ1 + σ2

σys(1 − θ)

]2}

σys

(14)

√σ2

1 + σ22 − σ1σ2 = KB(1 − θ)3/2

[1 − σ1 + σ2

σfs(1 − θ)

]σfs (15)

where KP and KB are both constants, and: KP = ±γ , KB = ±γ ′.

5.2. Experimental verification and results

The nickel foam plates used as the experimental material wereproduced by electrodeposition with the thickness of 2–3 mm andporosity of 89–99% [50], and their biaxial tensile specimenswere prepared having the shape of simple straight cruciform withthe clamped width of 60 mm [45,46]. The tensile test was car-ried out using a PLS-S100 biaxial tensile test machine (Fig. 3) atroom temperature of about 25 ◦C, with the computer controllingand recording system. The specimens were biaxially tensionedat the speed of 6.5–9.0 mm/min with the loading error below1wisttsoFblfit

Fig. 6. Relative fluctuation amplitude of the constant in different equations.

will not be essentially changed although the stress-concentrationat the round corner of specimens may takes an effect. Even ifthe stress-concentration plays a role, the basic regularity of therelationship will not fail.

Using the data of metal nickel in Ref. [51], i.e., σ0 = 317 MPa,σys = 59 MPa, σfs ≈ 317 MPa, and substituting the relativeexperimental data into Eqs. (8), (14) and (15) respectively, thecorresponding constants can be solved or determined. Resul-tantly, the relative fluctuations of the values of the constants inEqs. (8), (14) and (15) can be obtained for different specimensto their mean values each. For convenience of direct observationand visual comparison, these relative fluctuation amplitudes aredisplayed in Fig. 6. From the calculated results it can be seen that,the mean relative fluctuation amplitude is 6.6% for the constantK in Eq. (8), which is a small value. This shows this equationwell expresses the relative magnitude relationship displayed bymetal foams under biaxial tension. While, the mean values ofrelative fluctuation amplitude of constants KP and KB in Eqs.(14) and (15) are 34.6% and 16.5%, respectively. Thus thesetwo constants can be also regarded as the high-stability ones. Ofcourse, the structural pattern and intrinsic feature of real foamedbodies are much more complicated than that of the ideal model,so it can be said that the above equations are all much successfulto get such an agreement with the experimental results, thereinto especially for Eq. (8) (see Fig. 6).

What needs to be pointed out is that, the fluctuation of theccmOctGfi

%. The biaxial forces with different groups of the tensile speedere measured during tension, and the nominal loading stresses

n two principal directions at the time of failure were furthercaled out. Test observation and computer record both showedhat the tensile forces in two orthogonal directions almost simul-aneously reached the maximum value at the instant when thepecimen begun its rupture, and decreased with the enlargementf the crack. The representative failed specimens are shown inig. 4. From the morphologies of failed specimens, the distri-ution line of maximum stress can be summarized as the dottedine shown in Fig. 5. Because the regular pattern of the stress-eld distribution in all tested specimens is similar to each other,

he relative mathematical relationship from the present models

onstants in Eqs. (8), (14) and (15) may result mainly from theomplication and change in specific structural statuses of realetallic foams and from the unavoidable errors of experiments.f course, this fluctuation may be also an indication that some

haracteristic is missed in the present theoretical models, andhis relative work will be further done deepgoingly in the future.enerally speaking, most theoretical models cannot be fully per-

ect to factual applications, and will be allowed to constantlymprove.

P.S. Liu / Materials Science and Engineering A 422 (2006) 176–183 181

Indeed, fracture criteria for ductile solids (including metals)are typically strain based as loading history plays an importantrole. However, the struts of three-dimensional reticulated high-porosity metals can be regarded and treated as “thin beams”, sothey will fail in compliance with the maximum normal-stressrule [33–35,52], and this treatment have been proved to be rea-sonable and practical with satisfying results [22,23,31,45,46].To this point, the work in the present paper also illustrates well.

6. Conclusions

(1) From the “octahedron model” which is established on thebasis of the structural feature of isotropic three-dimensionalreticulated high-porosity metallic foams, the approximatemathematical relationship between biaxial nominal failurestresses and porosity is derived for these materials underbiaxial tension:

√σ2

1 + σ22 − σ1σ2 +

√2√

3π

12π(1 − θ)1/2(σ1 + σ2)

≈ K(1 − θ)mσ0

where σ1 and σ2 are biaxial tensile nominal principalstresses on the bulk material at failure, θ the porosity ofthe porous body, σ0 the uniaxial tensile strength of the

(

(

Acknowledgements

This work is from Project 50201003 supported by NationalNatural Science Foundation of China (NSFC).

Appendix A

While the porous body bears a biaxial tension, the metallicstruts are meant to be loaded correspondingly, and will deflector tend to deflect sequentially. For simplifying the deduction,the strut might as well be firstly regarded as the brittle cantileverbeam for the force analysis. Indeed, this treatment is just ananalyzing way to solve the present question, and it should bechecked by the relative experimental results (see Section 5 inthis paper) whether this way is practical or not. Because thestruts in the present model are all equivalent both in structureand in loading situation, any one can represent all of the others(see Section 2 in this paper). As shown in Fig. A1, one strut(AB) may be isolated for this analysis (Fig. A2), with its sidenode (A) thought to be stable and its top node (B) suffering theexternal loading. As for the closed-stacked unit octahedrons,

Fc

Fig. A2. A diagram of the isolated strut (AB) for force analysis.

corresponding fully dense material, K a material constantdetermined by the material species and the producing pro-cess, and m is the plastic–brittle index of the correspondingcompact material, with the value between 1 and 1.5: tak-ing 1.5 for brittle materials and 1 for pliable ones. Thehigher brittleness of the materials, the nearer to 1.5 of m. Asfor metallic foams made from the common metal or alloyspecies with the moderate plasticity such as nickel for anexample, we may approximately take m ≈ 1.25.

2) Based on the above equation and Section 3, the bearingcapability criterion for porous bodies under biaxial tensileload can be obtained as follows: when√

σ21 + σ2

2 − σ1σ2

+√

2√

3π

12π(1 − θ)1/2(σ1 + σ2) ≥ K′(1 − θ)mσ0

the porous body will rupture to fail; and when

√σ2

1 + σ22 − σ1σ2

+√

2√

3π

12π(1 − θ)1/2(σ1 + σ2) < K′(1 − θ)mσ0

the porous body will be safely loaded. Here K′ is a safetycoefficient less than 1, which is also a characterization factorreflecting the material species and the producing process forthe porous body.

3) As to the nickel foam made by electrodeposition, m ≈ 1.25,K ≈ 0.329.

ig. A1. Octahedron unit under biaxial loading for isotropic reticulated open-ell metallic foams.

182 P.S. Liu / Materials Science and Engineering A 422 (2006) 176–183

each strut is shared by three octahedron units. When the biaxialtensile forces are analyzed in Figs. A1 and A2, only the unitswith axial-tensile force are considered. More precisely, as forthe strut (AB) in Figs. A1 and A2, just the main unit shown inthe figure and the vice-unit on the left above the main unit areconsidered, with the former having its axial in the direction ofthe external load σ1 and the latter having its axial in the directionof the external load σ2. The following deduction is based on theassumption of “brittle failure” for the struts, and the influence ofshear stresses is disregarded from the “thin-beam” theory, whichshould be an acceptable approximation.

Let the side length of the cube containing the unit-octahedronbe a, we have the outside force on the top node (B) of the strut(AB) as follows:

Vertical force:

p1 = 14a2σ1 (A1)

Horizontal force:

p2 = 14a2σ2 (A2)

Resultant force:

P =√

p21 + p2

2 =√(

1

4a2σ1

)2

+(

1

4a2σ2

)2

√

mttlc

b

c

t

ao

A

C

a

B

Using Eqs. (A7) and (A8) with the cosine theorem (seeFig. A1):

BD2 = BC

2 + CD2 − 2BC CD cos 45◦

=(√

2

2a

)2

+(

a

2

σ2

σ1

)2

−2

(√2

2a

)(a

2

σ2

σ1

) √2

2

= a2

4

[2 +

(σ2

σ1

)2

− 2

(σ2

σ1

)](A9)

In addition:

AB = 1

2

√a2 + a2 + a2 =

√3

2a (A10)

Let the included angle between AD and AB be α2, the followingequations can be then obtained from Eqs. (A6), (A9) and (A10)with the cosine theorem (see Fig. A1):

cos α2 = AB2 + AD

2 − BD2

2AB AD

=

(√3

2a

)2+(

a2

√σ2

1+σ22

σ1

)2

− a2

4

[2 +

(σ2σ1

)2−2(

σ2σ1

)]

2(√

32 a)(

a2

√σ2

1+σ22

σ1

)

= σ1 + σ2√3(σ2

1 + σ22 )

(A11)

⇒

sin α2 =√

1 − cos2 α2 =√√√√1 −

(σ1 + σ2√3(σ2

1 + σ22 )

)

=√

2(σ21 + σ2

2 − σ1σ2)

3(σ21 + σ2

2 )(A12)

From Eqs. (A3), (A11) and (A12) combining Fig. A1, the exter-nal loading force on the strut (AB) can be expressed as

Orthogonal force:

p1 = P sin α2 = a2

4

√σ2

1 + σ22

√2(σ2

1 + σ22 − σ1σ2)

3(σ21 + σ2

2 )

=√

6

12a2√

σ21 + σ2

2 − σ1σ2 (A13)

Axial force:

p2 = P cos α2

= a2

4

√σ2

1 + σ22

σ1 + σ2√3(σ2

1 + σ22 )

=√

3

12a2(σ1 + σ2) (A14)

= a2

4σ2

1 + σ22 (A3)

The real nodal force should have also included bendingoments, but we treat the node as an abstract mathematical point

o act a role through which only transfers the external force. Thisreatment could not bring an obvious deviation because the ana-ytical system should not generate new forces and the bearingapacity of real nodes is commonly higher than that of real struts.

Let the included angle between the resultant force P and σ1e α1, then from Eqs. (A1)–(A3) we have:

os α1 = p1

P= σ1√

σ21 + σ2

2

(A4)

g α1 = σ2

σ1(A5)

Through point A the parallel line of BP is made and intersectst point D with the upper strut of the cuboid containing thectahedron (Fig. A1), then from Eqs. (A4) and (A5) we have:

D = AC

cos α1=

a2σ1√σ2

1+σ22

= a

2

√σ2

1 + σ22

σ1(A6)

D = AC tg α1 = a

2

σ2

σ1(A7)

nd

C =√(a

2

)2 +(a

2

)2 =√

2

2a (A8)

P.S. Liu / Materials Science and Engineering A 422 (2006) 176–183 183

Besides, the bending sectional modulus of the strut can beobtained by the relevant geometrical operation [23,31]:

Z ≈ 1

96

√√3

π(1 − θ)3/2a3 (A15)

where θ is the porosity of the porous body.Using the linear elastic bending theory for brittle materials

with Eqs. (A13) and (A15), the maximum normal stress resultingfrom the bending moment can be written as [23,31]:

σ01 ≈ p1

√3

4 a

Z≈

(√6

12 a2√

σ21 + σ2

2 − σ1σ2

) √3

4 a

196

√√3

π(1 − θ)3/2a3

= 2

√6√

3π(1 − θ)−3/2√

σ21 + σ2

2 − σ1σ2 (A16)

Moreover, from Refs. [23,31] it gives the sectional area of thestrut:

S0 =√

3

12(1 − θ)a2 (A17)

so the axial tensile stress caused by the axial force p2 is (fromEqs. (A14) and (A17)):

σ ≈√

312 a2(σ1 + σ2)√ = 1

(σ + σ ) (A18)

m

σ

p

R

[

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(2000) 301–322.[28] A.C.F. Cocks, M.F. Ashby, Acta Mater. 48 (2000) 3395–3400.[29] S. Jemiolo, S. Turteltaub, J. Appl. Mech. 67 (2000) 248–254.[30] R.E. Miller, Int. J. Mech. Sci. 42 (2000) 729–754.[[

[

[[[

[

[[[[

[[

[

[[[[[[[

[

023

12 (1 − θ)a2 1 − θ1 2

It can be known from Eqs. (A16) and (A18) that, the maxi-um normal stress within the strut is

max = σ01 + σ02 ≈ 2

√6√

3π(1 − θ)−3/2√

σ21 + σ2

2 − σ1σ2

+(1 − θ)−1(σ1 + σ2) (A19)

From the above context, it will be clear that the axial com-ression of the struts need not be taken into account additionally.

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