- Email: [email protected]
On the elastic properties of three-dimensional honeycomb lattices
Journal Pre-proof On the elastic properties of three-dimensional honeycomb lattices V.H. Carneiro PII:
S2452-2139(19)30162-7
DOI:
https://doi.org/10.1016/j.coco.2019.11.005
Reference:
COCO 274
To appear in:
Composites Communications
Received Date: 7 August 2019 Revised Date:
30 October 2019
Accepted Date: 2 November 2019
Please cite this article as: V.H. Carneiro, On the elastic properties of three-dimensional honeycomb lattices, Composites Communications, https://doi.org/10.1016/j.coco.2019.11.005. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
On the elastic properties of three-dimensional honeycomb lattices V.H. Carneiroa* a
MEtRiCS - Mechanical Engineering and Resource Sustainability Center, Portugal
* [email protected]
Abstract Cellular solids are composed by an interconnection of solid struts to form edges in foam shaped materials. Due to their high specific mechanical properties, they are very desirable in both industrial and academic fields. Even though, these materials have been widely used in the last decades, there is still no unified theory to predict their elastic behavior, especially in non-stochastic configurations. Due to the current development in the manufacturing of three-dimensional cellular structures by additive manufacturing and investment casting, there is a need to predict their behavior in practical applications. This study presents a geometrical and base material dependent formulation that predicts the elastic properties of honeycomb lattices. Such theory is based in flexure deformations using the classic beam theory, while the novelty of its formulation includes a rib angle dependent hinging correction. The suggested formulation that is able to show an enhanced correlation to experimental results, relatively to other current analytical models.
Keywords: Elasticity; Honeycombs; Modulus; Poisson’s ratio; Non-Stochastic.
Cellular solids are characterized by an interconnection of solid struts and/or plates forming edges/faces of cells [1]. These solid scaffolds are frequently impregnated by a fluid and assume the shape of a foam [2]. Due to their inherent low density, they possess high specific mechanical properties [3,4] (e.g. elastic modulus [5,6], strength [7,8] and vibration damping [9,10]). The referred properties make cellular solids very attractive for applications in transportation [11,12], medical [13,14] and thermal [15,16] industries. These properties may be optimized and/or tailored if the unitary cells of the cellular solids are nonstochastic, i.e. they have a specific periodic geometry. Commonly, these geometries are inspired in nature and most assume a honeycomb shape [1]. Even though there are developments in the manufacturing of these composites [3,17] and studies on their mechanical properties, there still is no unified theory that describes their elastic behavior. In the early 1980’s Gibson et al [1,18] determined equations 1 and 2 (Table 1) to calculate the fundamental elastic properties of honeycombs. It is known that the latter
have a good correlation with flexure dominated deformation, i.e. honeycombs with reduced wall thickness and low specific densities. To address these limitations, in the 1990’s Masters et al [19] included the effect of rib/wall stretching and hinging, interpreting these effects on the overall cell stiffness. The fundamental elastic properties are determined by equations 3 and 4 (Table 1), being supported by the calculation of the stretching, hinging and flexure stiffness constants (respectively Ks, Kh and Kf – equations 5 to 7 in Table 1). Recently, Malek et al determined equations 8 and 9 (Table 1) [20] to compensate the effect of thick walled honeycombs and further approximated the values of the theoretical models to the experimental results. A similar approach was performed by Hedayati et al [21], in which equations 11 and 12 (Table 1) are deduced to model the elastic properties of thick honeycombs. Table 1 Models for honeycomb elastic properties. Author
Ref.
Gibson et al
[1]
Masters et al
[19]
ℎ
+ sin =
Malek et al
[20] -
ℎ
$
cos
+ sin
E* Apparent Modulus cos 1 ℎ + sin sin
sin cos 5 sin
+
sin cos
OBS: &$ = 6
+
cos =
!
$
1 . 1 + 2.4 + 1.50 + cot
3
ℎ
7
-
2 8
ℎ
ν Poisson’s ratio cos 2 ℎ + sin sin 1 1 1 − sin cos * + − +
+ sin
cos
+ sin
*
sin cos
+
sin cos
+
1 + 1.4 + 1.50 . 1 + 2.4 + 1.50 + cot
sin
cos
-
+
4
-
2 9
OBS:
Hedayati et al
[21]
-
ℎ cos
= −
2 cos l sin
+ 1 sin
10
+ cos
11
sin
sin
+
cos cos
− ℎ + cos
12
These models are frequently adopted in two-dimensional honeycombs (Fig.1 (a)). Given the recent advances in the manufacturing of three-dimensional honeycombs (Fig.1 (b)), however, there is a need to further adapt the referred models to this context.
Fig.1. Honeycombs in (a) two- and (b) three-dimensional configurations.
Considering Fig. 1, it may be seen that they possess symmetry in the Cartesian planes. Thus, each cell in two- or three-dimensions may be divided into an Euler-Bernoulli beam system, as displayed in Fig. 2. It may be observed that the deformation behavior is dependent on the geometry of the cell, i.e. rib horizontal length (h), vertical length (l),
thickness (t) and angle (α). In two dimensional approaches, the width (b) of the lattice can be also considered to determine the elastic properties. However, this aspect may be simplified in three-dimensional cells if the width is equal to the rib thickness (b=t).
Fig.2. Honeycomb deformation mechanism: (a) initial dimensions and loading; (b) axial deformation; (c) flexural deformation.
According to this model, if a load (F) is applied in the system (Fig. 2 (a)), axial (Fa – Fig. 2 (b)) and flexural (Fb – Fig. 2 (c)) components are imposed in the ribs. The vertical (∆Ya and ∆Yb) and horizontal (∆Xa and ∆Xb) deformations in the ribs are proportional to the loading by axial and flexural stiffness coefficients (respectively, Ka and Kb – equations 13 and 14). Given that these stiffness coefficients are not dimensionless magnitudes, they are dependent not only on the geometry of the cell, but also on the fundamental elastic modulus of the base material (E0). 7 -
=
=
8 2 cos
38 : 8 sin
9 9
ℎ
ℎ
13
14
According to Masters et al [19], it is fundamental to consider the rib hinging effect to determine their elastic properties. According to the referred model, this hinging effect may be compensated by the inclusion of a hinging stiffness coefficient (Kh - equation 6), that is dependent on the cell with (b), thickness (l), length (l) and base material shear modulus (G0). The novelty of the presented study considers that this hinging effect is also dependent on the rib angle (α). According to this hypothesis, in designs with negative (Fig. 3 (a)), zero (Fig. 3 (b)) and positive (Fig. 3 (c)) rib angles, the hinging effect is substantially different. Fig. 3 shows that in negative rib angles, there is a double closing effect, while
in positive values, there seems to be a single closing effects. Thus, honeycombs with negative rib angles (i.e. auxetic honeycombs [3]) display a higher hinging stiffness coefficient.
Fig.3. Hinging deformation mechanism in honeycombs with: (a) negative, (b) zero and (c) positive rib angles.
To compensate this rib angle dependent hinging stiffness variation, a hinging coefficient (q – equation 15) is introduced in the reformulated hinging stiffness coefficient (KH – equation 16). ; = tan =
=
15
2 &9 16 ℎ ;
According to equation 17, all the stiffness coefficients (Ka, Kb and Kh) may be correlated by a simple addition operation of springs in series. >
=
1
7
+
1 1
-
1
+
=
17
The apparent elastic modulus of the modeled honeycombs (E*) may be determined using Hooke’s law (equation 18). In which A* (equation 19) and L* (equation 20) are, respectively, the initial honeycomb apparent resistant area and height. ∗
=
∗ > @ A∗
A∗ = BC 1 + sin @∗ = C cos
18
D 19
20
According to Fig. 2, it is possible to observe that the Poisson’s ratio (ν) of the honeycombs is generated by the referred axial and horizontal deformations (∆Ya, ∆Yb, ∆Xa and ∆Xb). It is suggested that the imposed deformations may be determined by equations 21 to 24. EF7 =
EF- =
EH7 =
EH- =
2 G cos 8
8 G sin 38 :
C
9 9
2 G cos sin 8 9
8 G cos sin 38 : 9
C
21
22
C
C
23
24
Considering the novel analytical model, the Poisson’s ratio (ν) of the honeycombs may be determined using equation 25, in which p (equation 26) is a novel correcting factor to compensate hinging effect in the ribs. EH7JL7M + EH-NOPLOQ IJ 0 = − = − IK A∗ p = cos
−
@∗ 25 2EF-NOPLOQ EF7JL7M + R
8 + 1 26 2
Fig. 4 displays plots of the novel analytical model that is suggested in this study and other honeycomb deformation models (Table 1) [1,18–21]. All analytical models are compared with experimental results [3] on three-dimensional aluminum honeycombs (h=4 mm, l=2 mm, t=0.6 mm, E0=71 GPa and G0=27 GPa).
Fig.4. Comparison between analytical models with experimental results.
It may be observed that the novel model is able display a good description of the experimental apparent modulus (E*) and Poisson’s ratio (ν) in comparison with other analytical models [1,18–21]. While most models are fairly symmetrical between negative and positive rib angles, the suggested model is able to adapt itself to the changes in hinging stiffness. Conclusions The presented study shows a formulation intended to describe the elastic properties of two- and three-dimensional honeycomb lattices. It is shown that the correction of rib angle dependent hinging stiffness is fundamental to enhance the correlation between analytical models with experimental results. The suggested model and formulation are suggested as a good approximation to determine the elastic properties of this structures based on their initial geometry and base material.
Acknowledgements This research was supported by the project iRAIL Innovation in Railway Systems and Technologies Doctoral Programme funds and by national funds through FCT Portuguese Foundation for Science and Technology and was developed on the aim of the Doctoral grant PD/ BD/114096/2015.
References [1]
L.J. Gibson, M.F. Ashby, Cellular solids: structure and properties, Cambridge university press, 1999. [2] P. Patel, P.P. Bhingole, D. Makwana, Manufacturing, characterization and applications of lightweight metallic foams for structural applications: Review, Mater. Process. Charact. 16th – 18th March 2018. 5 (2018) 20391–20402. [3] V.H. Carneiro, H. Puga, J. Meireles, Positive, zero and negative Poisson’s ratio non-stochastic metallic cellular solids: Dependence between static and dynamic mechanical properties, Compos. Struct. 226 (2019) 111239. [4] J. Banhart, Light-Metal Foams—History of Innovation and Technological Challenges, Adv. Eng. Mater. 15 (2013) 82–111. [5] L. Zorzetto, D. Ruffoni, Re-entrant inclusions in cellular solids: From defects to reinforcements, Compos. Struct. 176 (2017) 195–204. [6] L. Vendra, A. Rabiei, Evaluation of modulus of elasticity of composite metal foams by experimental and numerical techniques, Mater. Sci. Eng. A. 527 (2010) 1784–1790. [7] J. Maurath, N. Willenbacher, 3D printing of open-porous cellular ceramics with high specific strength, J. Eur. Ceram. Soc. 37 (2017) 4833–4842. [8] I. Duarte, L. Krstulović-Opara, J. Dias-de-Oliveira, M. Vesenjak, Axial crush performance of polymer-aluminium alloy hybrid foam filled tubes, Thin-Walled Struct. 138 (2019) 124–136. [9] K. Yang, X. Yang, C. He, E. Liu, C. Shi, L. Ma, Q. Li, J. Li, N. Zhao, Damping characteristics of Al matrix composite foams reinforced by in-situ grown carbon nanotubes, Mater. Lett. 209 (2017) 68–70. [10] G. Petrone, V. D’Alessandro, F. Franco, S. De Rosa, Numerical and experimental investigations on the acoustic power radiated by Aluminium Foam Sandwich panels, Compos. Struct. 118 (2014) 170–177. [11] C. Meola, S. Boccardi, G. maria Carlomagno, Chapter 1 - Composite Materials in the Aeronautical Industry, in: C. Meola, S. Boccardi, G. maria Carlomagno (Eds.), Infrared Thermogr. Eval. Aerosp. Compos. Mater., Woodhead Publishing, 2017: pp. 1–24.
[12] H. Heo, J. Ju, D.-M. Kim, Compliant cellular structures: Application to a passive morphing airfoil, Compos. Struct. 106 (2013) 560–569. [13] V.H. Carneiro, H. Puga, Deformation behaviour of self-expanding magnesium stents based on auxetic chiral lattices, Ciênc. Tecnol. Mater. 28 (2016) 14–18. [14] R. Parai, S. Bandyopadhyay-Ghosh, Engineered bio-nanocomposite magnesium scaffold for bone tissue regeneration, J. Mech. Behav. Biomed. Mater. 96 (2019) 45–52. [15] K.S. Al-Athel, S.P. Aly, A.F.M. Arif, J. Mostaghimi, 3D modeling and analysis of the thermomechanical behavior of metal foam heat sinks, Int. J. Therm. Sci. 116 (2017) 199–213. [16] X. Shenming, Y. Ran, S. Limei, W. Yupeng, X. Junlong, C. huanxin, The experimental study of a novel metal foam heat pipe radiator, Innov. Solut. Energy Transit. 158 (2019) 5439–5444. [17] L. Yang, O. Harrysson, H. West, D. Cormier, Mechanical properties of 3D re-entrant honeycomb auxetic structures realized via additive manufacturing, Int. J. Solids Struct. 69–70 (2015) 475–490. [18] Gibson L. J., Ashby Michael Farries, Schajer G. S., Robertson C. I., The mechanics of twodimensional cellular materials, Proc. R. Soc. Lond. Math. Phys. Sci. 382 (1982) 25–42. [19] I. Masters, K. Evans, Models for the elastic deformation of honeycombs, Compos. Struct. 35 (1996) 403–422. [20] S. Malek, L. Gibson, Effective elastic properties of periodic hexagonal honeycombs, Mech. Mater. 91 (2015) 226–240. [21] R. Hedayati, M. Sadighi, M. Mohammadi Aghdam, A.A. Zadpoor, Mechanical Properties of Additively Manufactured Thick Honeycombs, Materials. 9 (2016).
Table 1 Models for honeycomb elastic properties. Author
Ref.
Gibson et al
[1,18]
Masters et al
[19]
ℎ
+ sin =
Malek et al
[20] -
ℎ
$
cos
+ sin
E* Apparent Modulus cos 1 ℎ + sin sin
sin cos 5 sin
+
sin cos
OBS: &$
=
+
6
cos =
!
$
1 . 1 + 2.4 + 1.50 + cot
3
ℎ
7
-
2 8
ℎ
ν Poisson’s ratio cos 2 ℎ + sin sin 1 1 1 − sin cos * + − +
+ sin
cos
+ sin
*
sin cos
+
sin cos
+
1 + 1.4 + 1.50 . 1 + 2.4 + 1.50 + cot
sin
cos
-
+
4
-
OBS:
Hedayati et al
[21]
-
ℎ cos
= −
2 cos l sin
+ 1 sin
10
+ cos
11
sin
sin
+
cos cos
− ℎ + cos
12
2 9
Highlights - A formulation is proposed to describe the elastic properties of honeycomb lattices; - A rib angle dependent hinging coefficient is deducted to correct the lattice stiffness; - The analytical displays a good agreement with experimental results.
Conflict of Interest and Authorship Conformation Form Please check the following as appropriate:
o
All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.
o
This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.
o
The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript
o
The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript:
Author’s name
Affiliation
Vitor Carneiro - MEtRiCS - Mechanical Engineering and Resource Sustainability Center, Portugal
S2452-2139(19)30162-7
DOI:
https://doi.org/10.1016/j.coco.2019.11.005
Reference:
COCO 274
To appear in:
Composites Communications
Received Date: 7 August 2019 Revised Date:
30 October 2019
Accepted Date: 2 November 2019
Please cite this article as: V.H. Carneiro, On the elastic properties of three-dimensional honeycomb lattices, Composites Communications, https://doi.org/10.1016/j.coco.2019.11.005. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
On the elastic properties of three-dimensional honeycomb lattices V.H. Carneiroa* a
MEtRiCS - Mechanical Engineering and Resource Sustainability Center, Portugal
* [email protected]
Abstract Cellular solids are composed by an interconnection of solid struts to form edges in foam shaped materials. Due to their high specific mechanical properties, they are very desirable in both industrial and academic fields. Even though, these materials have been widely used in the last decades, there is still no unified theory to predict their elastic behavior, especially in non-stochastic configurations. Due to the current development in the manufacturing of three-dimensional cellular structures by additive manufacturing and investment casting, there is a need to predict their behavior in practical applications. This study presents a geometrical and base material dependent formulation that predicts the elastic properties of honeycomb lattices. Such theory is based in flexure deformations using the classic beam theory, while the novelty of its formulation includes a rib angle dependent hinging correction. The suggested formulation that is able to show an enhanced correlation to experimental results, relatively to other current analytical models.
Keywords: Elasticity; Honeycombs; Modulus; Poisson’s ratio; Non-Stochastic.
Cellular solids are characterized by an interconnection of solid struts and/or plates forming edges/faces of cells [1]. These solid scaffolds are frequently impregnated by a fluid and assume the shape of a foam [2]. Due to their inherent low density, they possess high specific mechanical properties [3,4] (e.g. elastic modulus [5,6], strength [7,8] and vibration damping [9,10]). The referred properties make cellular solids very attractive for applications in transportation [11,12], medical [13,14] and thermal [15,16] industries. These properties may be optimized and/or tailored if the unitary cells of the cellular solids are nonstochastic, i.e. they have a specific periodic geometry. Commonly, these geometries are inspired in nature and most assume a honeycomb shape [1]. Even though there are developments in the manufacturing of these composites [3,17] and studies on their mechanical properties, there still is no unified theory that describes their elastic behavior. In the early 1980’s Gibson et al [1,18] determined equations 1 and 2 (Table 1) to calculate the fundamental elastic properties of honeycombs. It is known that the latter
have a good correlation with flexure dominated deformation, i.e. honeycombs with reduced wall thickness and low specific densities. To address these limitations, in the 1990’s Masters et al [19] included the effect of rib/wall stretching and hinging, interpreting these effects on the overall cell stiffness. The fundamental elastic properties are determined by equations 3 and 4 (Table 1), being supported by the calculation of the stretching, hinging and flexure stiffness constants (respectively Ks, Kh and Kf – equations 5 to 7 in Table 1). Recently, Malek et al determined equations 8 and 9 (Table 1) [20] to compensate the effect of thick walled honeycombs and further approximated the values of the theoretical models to the experimental results. A similar approach was performed by Hedayati et al [21], in which equations 11 and 12 (Table 1) are deduced to model the elastic properties of thick honeycombs. Table 1 Models for honeycomb elastic properties. Author
Ref.
Gibson et al
[1]
Masters et al
[19]
ℎ
+ sin =
Malek et al
[20] -
ℎ
$
cos
+ sin
E* Apparent Modulus cos 1 ℎ + sin sin
sin cos 5 sin
+
sin cos
OBS: &$ = 6
+
cos =
!
$
1 . 1 + 2.4 + 1.50 + cot
3
ℎ
7
-
2 8
ℎ
ν Poisson’s ratio cos 2 ℎ + sin sin 1 1 1 − sin cos * + − +
+ sin
cos
+ sin
*
sin cos
+
sin cos
+
1 + 1.4 + 1.50 . 1 + 2.4 + 1.50 + cot
sin
cos
-
+
4
-
2 9
OBS:
Hedayati et al
[21]
-
ℎ cos
= −
2 cos l sin
+ 1 sin
10
+ cos
11
sin
sin
+
cos cos
− ℎ + cos
12
These models are frequently adopted in two-dimensional honeycombs (Fig.1 (a)). Given the recent advances in the manufacturing of three-dimensional honeycombs (Fig.1 (b)), however, there is a need to further adapt the referred models to this context.
Fig.1. Honeycombs in (a) two- and (b) three-dimensional configurations.
Considering Fig. 1, it may be seen that they possess symmetry in the Cartesian planes. Thus, each cell in two- or three-dimensions may be divided into an Euler-Bernoulli beam system, as displayed in Fig. 2. It may be observed that the deformation behavior is dependent on the geometry of the cell, i.e. rib horizontal length (h), vertical length (l),
thickness (t) and angle (α). In two dimensional approaches, the width (b) of the lattice can be also considered to determine the elastic properties. However, this aspect may be simplified in three-dimensional cells if the width is equal to the rib thickness (b=t).
Fig.2. Honeycomb deformation mechanism: (a) initial dimensions and loading; (b) axial deformation; (c) flexural deformation.
According to this model, if a load (F) is applied in the system (Fig. 2 (a)), axial (Fa – Fig. 2 (b)) and flexural (Fb – Fig. 2 (c)) components are imposed in the ribs. The vertical (∆Ya and ∆Yb) and horizontal (∆Xa and ∆Xb) deformations in the ribs are proportional to the loading by axial and flexural stiffness coefficients (respectively, Ka and Kb – equations 13 and 14). Given that these stiffness coefficients are not dimensionless magnitudes, they are dependent not only on the geometry of the cell, but also on the fundamental elastic modulus of the base material (E0). 7 -
=
=
8 2 cos
38 : 8 sin
9 9
ℎ
ℎ
13
14
According to Masters et al [19], it is fundamental to consider the rib hinging effect to determine their elastic properties. According to the referred model, this hinging effect may be compensated by the inclusion of a hinging stiffness coefficient (Kh - equation 6), that is dependent on the cell with (b), thickness (l), length (l) and base material shear modulus (G0). The novelty of the presented study considers that this hinging effect is also dependent on the rib angle (α). According to this hypothesis, in designs with negative (Fig. 3 (a)), zero (Fig. 3 (b)) and positive (Fig. 3 (c)) rib angles, the hinging effect is substantially different. Fig. 3 shows that in negative rib angles, there is a double closing effect, while
in positive values, there seems to be a single closing effects. Thus, honeycombs with negative rib angles (i.e. auxetic honeycombs [3]) display a higher hinging stiffness coefficient.
Fig.3. Hinging deformation mechanism in honeycombs with: (a) negative, (b) zero and (c) positive rib angles.
To compensate this rib angle dependent hinging stiffness variation, a hinging coefficient (q – equation 15) is introduced in the reformulated hinging stiffness coefficient (KH – equation 16). ; = tan =
=
15
2 &9 16 ℎ ;
According to equation 17, all the stiffness coefficients (Ka, Kb and Kh) may be correlated by a simple addition operation of springs in series. >
=
1
7
+
1 1
-
1
+
=
17
The apparent elastic modulus of the modeled honeycombs (E*) may be determined using Hooke’s law (equation 18). In which A* (equation 19) and L* (equation 20) are, respectively, the initial honeycomb apparent resistant area and height. ∗
=
∗ > @ A∗
A∗ = BC 1 + sin @∗ = C cos
18
D 19
20
According to Fig. 2, it is possible to observe that the Poisson’s ratio (ν) of the honeycombs is generated by the referred axial and horizontal deformations (∆Ya, ∆Yb, ∆Xa and ∆Xb). It is suggested that the imposed deformations may be determined by equations 21 to 24. EF7 =
EF- =
EH7 =
EH- =
2 G cos 8
8 G sin 38 :
C
9 9
2 G cos sin 8 9
8 G cos sin 38 : 9
C
21
22
C
C
23
24
Considering the novel analytical model, the Poisson’s ratio (ν) of the honeycombs may be determined using equation 25, in which p (equation 26) is a novel correcting factor to compensate hinging effect in the ribs. EH7JL7M + EH-NOPLOQ IJ 0 = − = − IK A∗ p = cos
−
@∗ 25 2EF-NOPLOQ EF7JL7M + R
8 + 1 26 2
Fig. 4 displays plots of the novel analytical model that is suggested in this study and other honeycomb deformation models (Table 1) [1,18–21]. All analytical models are compared with experimental results [3] on three-dimensional aluminum honeycombs (h=4 mm, l=2 mm, t=0.6 mm, E0=71 GPa and G0=27 GPa).
Fig.4. Comparison between analytical models with experimental results.
It may be observed that the novel model is able display a good description of the experimental apparent modulus (E*) and Poisson’s ratio (ν) in comparison with other analytical models [1,18–21]. While most models are fairly symmetrical between negative and positive rib angles, the suggested model is able to adapt itself to the changes in hinging stiffness. Conclusions The presented study shows a formulation intended to describe the elastic properties of two- and three-dimensional honeycomb lattices. It is shown that the correction of rib angle dependent hinging stiffness is fundamental to enhance the correlation between analytical models with experimental results. The suggested model and formulation are suggested as a good approximation to determine the elastic properties of this structures based on their initial geometry and base material.
Acknowledgements This research was supported by the project iRAIL Innovation in Railway Systems and Technologies Doctoral Programme funds and by national funds through FCT Portuguese Foundation for Science and Technology and was developed on the aim of the Doctoral grant PD/ BD/114096/2015.
References [1]
L.J. Gibson, M.F. Ashby, Cellular solids: structure and properties, Cambridge university press, 1999. [2] P. Patel, P.P. Bhingole, D. Makwana, Manufacturing, characterization and applications of lightweight metallic foams for structural applications: Review, Mater. Process. Charact. 16th – 18th March 2018. 5 (2018) 20391–20402. [3] V.H. Carneiro, H. Puga, J. Meireles, Positive, zero and negative Poisson’s ratio non-stochastic metallic cellular solids: Dependence between static and dynamic mechanical properties, Compos. Struct. 226 (2019) 111239. [4] J. Banhart, Light-Metal Foams—History of Innovation and Technological Challenges, Adv. Eng. Mater. 15 (2013) 82–111. [5] L. Zorzetto, D. Ruffoni, Re-entrant inclusions in cellular solids: From defects to reinforcements, Compos. Struct. 176 (2017) 195–204. [6] L. Vendra, A. Rabiei, Evaluation of modulus of elasticity of composite metal foams by experimental and numerical techniques, Mater. Sci. Eng. A. 527 (2010) 1784–1790. [7] J. Maurath, N. Willenbacher, 3D printing of open-porous cellular ceramics with high specific strength, J. Eur. Ceram. Soc. 37 (2017) 4833–4842. [8] I. Duarte, L. Krstulović-Opara, J. Dias-de-Oliveira, M. Vesenjak, Axial crush performance of polymer-aluminium alloy hybrid foam filled tubes, Thin-Walled Struct. 138 (2019) 124–136. [9] K. Yang, X. Yang, C. He, E. Liu, C. Shi, L. Ma, Q. Li, J. Li, N. Zhao, Damping characteristics of Al matrix composite foams reinforced by in-situ grown carbon nanotubes, Mater. Lett. 209 (2017) 68–70. [10] G. Petrone, V. D’Alessandro, F. Franco, S. De Rosa, Numerical and experimental investigations on the acoustic power radiated by Aluminium Foam Sandwich panels, Compos. Struct. 118 (2014) 170–177. [11] C. Meola, S. Boccardi, G. maria Carlomagno, Chapter 1 - Composite Materials in the Aeronautical Industry, in: C. Meola, S. Boccardi, G. maria Carlomagno (Eds.), Infrared Thermogr. Eval. Aerosp. Compos. Mater., Woodhead Publishing, 2017: pp. 1–24.
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Table 1 Models for honeycomb elastic properties. Author
Ref.
Gibson et al
[1,18]
Masters et al
[19]
ℎ
+ sin =
Malek et al
[20] -
ℎ
$
cos
+ sin
E* Apparent Modulus cos 1 ℎ + sin sin
sin cos 5 sin
+
sin cos
OBS: &$
=
+
6
cos =
!
$
1 . 1 + 2.4 + 1.50 + cot
3
ℎ
7
-
2 8
ℎ
ν Poisson’s ratio cos 2 ℎ + sin sin 1 1 1 − sin cos * + − +
+ sin
cos
+ sin
*
sin cos
+
sin cos
+
1 + 1.4 + 1.50 . 1 + 2.4 + 1.50 + cot
sin
cos
-
+
4
-
OBS:
Hedayati et al
[21]
-
ℎ cos
= −
2 cos l sin
+ 1 sin
10
+ cos
11
sin
sin
+
cos cos
− ℎ + cos
12
2 9
Highlights - A formulation is proposed to describe the elastic properties of honeycomb lattices; - A rib angle dependent hinging coefficient is deducted to correct the lattice stiffness; - The analytical displays a good agreement with experimental results.
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The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript
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Author’s name
Affiliation
Vitor Carneiro - MEtRiCS - Mechanical Engineering and Resource Sustainability Center, Portugal