Large Deformation and Energy Absorption Behaviour of Perforated Hollow Sphere Structures under Quasi-Static Compression - MDPI

materials
Article
Large Deformation and Energy Absorption Behaviour of
Perforated Hollow Sphere Structures under
Quasi-Static Compression
Meiling Dai 1 , Junping Liang 1 , Cheng Cheng 1 , Zhiwen Wu 1 , Jiexun Lu 1 and Jiyu Deng 2, *

 1 School of Civil and Transportation Engineering, Guangdong University of Technology,
 Guangzhou 510006, China; meiling-dai@163.com (M.D.); supclot@163.com (J.L.);
 1216416395cchh@gmail.com (C.C.); 15088195692@163.com (Z.W.); a18819161330@163.com (J.L.)
 2 School of Architecture and Urban Planning, Guangdong University of Technology, Guangzhou 510090, China
 * Correspondence: jiyudeng@gmail.com

 Abstract: Hollow sphere structures with perforations (PHSSs) in three different arrangements (sim-
 ple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC)) were fabricated through
 three-dimensional (3D) printing, and the mechanical behaviours of these PHSSs under quasi-static
 compression were investigated experimentally and numerically. The results indicated that under
 uniaxial compression, the PHSSs mainly undergo three stages, i.e., a linear elastic stage, a large
 deformation or plateau stage, and a densification stage. During the stage of large deformation, the SC
 and BCC PHSSs experience a preliminary compaction sub-stage after layer-by-layer buckling, while
 for the FCC PHSS, layer-by-layer collapse and compaction are the dominant deformation behaviours.
 


 A numerical simulation was employed to study the mechanical properties of PHSSs with different
 

 geometric parameters under quasi-static compression and to explore the effect of the wall thickness,
Citation: Dai, M.; Liang, J.; Cheng,
 hole diameter, and sphere arrangement on the first peak stress, plateau stress, and specific energy
C.; Wu, Z.; Lu, J.; Deng, J. Large
 absorption (SEA) of the PHSSs. The results reveal that the geometric parameters have a significant
Deformation and Energy Absorption
 impact on the large deformation behaviour and energy absorption capacity of PHSSs. The presented
Behaviour of Perforated Hollow
 PHSS is also proven to be much lighter than traditional metallic hollow sphere structure (MHSS) and
Sphere Structures under Quasi-Static
Compression. Materials 2021, 14, 3716.
 has higher specific strength and SEA.
https://doi.org/10.3390/ma14133716
 Keywords: cellular material; hollow sphere structure; compression test; numerical simulation;
Academic Editor: large deformation; energy absorption
A. Javier Sanchez-Herencia

Received: 3 June 2021
Accepted: 30 June 2021 1. Introduction
Published: 2 July 2021
 With abundant pores in the matrix material and an internal microstructure, cellular
 metal or metal foam has unique functional and structural properties. This material is ultra-
Publisher’s Note: MDPI stays neutral
 lightweight and is characterised by good sound absorption, high heat resistance, energy
with regard to jurisdictional claims in
 absorption, and high specific stiffness and strength [1]. It has been used extensively in
published maps and institutional affil-
 engineering fields, such as aerospace, military, and transportation engineering. A metallic
iations.
 hollow sphere structure (MHSS) is a high-quality cellular material that is usually manufac-
 tured by filling thin-walled hollow spheres into a polymer matrix, sintering the spheres by
 applying heat and pressure and bonding spheres using a liquid phase [2]. In addition to
 the advantages of conventional cellular metals, MHSSs are characterised by fewer defects
Copyright: © 2021 by the authors.
 and more uniform geometry, resulting in more consistent mechanical properties [3], and
Licensee MDPI, Basel, Switzerland.
 have drawn the attention of a large number of researchers.
This article is an open access article
 Based on the regular atomic arrangement concept, Sanders and Gibson performed a
distributed under the terms and
 numerical simulation analysis of the elasticity modulus and initial yield stress of MHSSs
conditions of the Creative Commons
 in different packing patterns, such as simple cubic (SC), body-centred cubic (BCC) and
Attribution (CC BY) license (https://
 face-centred cubic (FCC) patterns, and found that MHSSs have better mechanical properties
creativecommons.org/licenses/by/
4.0/).
 than cellular materials with random porous structures [4,5]. Gasser et al. used a similar

Materials 2021, 14, 3716. https://doi.org/10.3390/ma14133716 https://www.mdpi.com/journal/materials

Materials 2021, 14, 3716 2 of 16

 method to determine the tensile elastic constant of a regularly stacked (FCC) brazed MHSS;
 they also analysed the effects of the thickness-sphere radius ratio and the weld radius-
 sphere radius ratio on the overall elastic modulus and established an empirical formula for
 the nominal elasticity modulus [6,7]. Fiedler and Aöchsner et al. studied the mechanical
 properties of composite MHSSs and partially bonded MHSSs in an SC arrangement as
 well as the mechanical properties and size effect of sintered MHSSs [8–10]. Shufrin et al.
 derived a three-dimensional (3D) continuum model to analyse the negative Poisson’s ratio
 (NPR) behaviour of hollow sphere structures and evaluated the effect of the arrangement
 of hollow spheres on the NPR behaviour through numerical simulations [11].
 Many experimental and numerical studies have also been performed to reveal the large
 deformation process and quasi-static and dynamic responses of hollow sphere structures
 in detail. Gao et al. numerically studied the static and dynamic deformation behaviours
 and mechanical properties by using simplified geometric models of sintered MHSSs and
 proposed formulas for the elastic modulus, yield strength, and plateau stress of the FCC
 packing material under quasi-static compression [12]. Vesenjak et al. studied the dynamic
 mechanical behaviours of composite MHSSs and partially bonded MHSSs in different
 packing patterns under impact loading through numerical simulation [13]. Cunda et al.
 analysed the differences in the mechanical properties of MHSSs under static and impact
 loadings using the finite element method and evaluated the effect of the loading rate on the
 dynamic mechanical behaviours and energy absorption capacity [14]. Marcadon et al. stud-
 ied the compressive mechanical properties and creep behaviours of MHSSs under quasi-
 static loading and the effect of geometric defects on these mechanical properties [15,16].
 Most studies used representative volume element (RVE) analyses to avoid considerable
 computing time for the simulation; however, these models cannot reveal the deformation
 mechanism of a real material due to the localization of deformation at large strains.
 With ping-pong balls as the experimental object, Yu et al. carried out a series of static
 and dynamic compression experiments on single-ball and multi-ball arrays, theoretically
 derived the compressive deformation pattern of a stacking of spheres, and verified the
 theories through various two-dimensional (2D) ping-pong ball stackings; some findings
 were beneficial to understanding the deformation mechanism of a real MHSS [17–19].
 They also conducted experimental studies through quasi-static tests and dynamic tests to
 investigate the large deformation behaviour of sintered MHSSs, and a long plateau length
 (up to 67% of the nominal strain) and localisation of deformation were observed [20]. Luo
 et al. studied the crushing performance of thin-walled tubes filled with MHSs under axial
 compression; the results indicated that the MHSs significantly improved the axial bearing
 capability of thin-walled tubes [21]. Zeng et al. conducted experimental studies on the
 dynamic mechanical properties of MHSS with density gradients [22]. However, due to
 the limitation of the specimen, the influence of the arrangement of hollow spheres on the
 mechanical properties could not be examined experimentally.
 Although some understanding of the static and dynamic mechanical properties of hol-
 low sphere structures/materials has been gained through these studies, which were mostly
 conducted under ideal stacking and connection conditions by using the finite element
 method, the obtained theoretical formulas are not consistent with the experimental results
 in some cases. Although not impossible, it is at least difficult for traditional manufacturing
 technology to ensure that the stacking pattern of hollow spheres is kept in line as well
 as the connection mode between spheres; moreover, the mechanical properties of hollow
 sphere materials are extremely sensitive to the arrangement pattern of the spheres and the
 connection geometry between the spheres.
 Additive manufacturing/3D printing, as an emerging manufacturing technology, can
 be used to directly produce more complex components regardless of the structure or shape,
 and this technique offers a new approach for fabricating hollow sphere materials. Recently,
 researchers have used stainless steel, photosensitive resin, polylactic acid (PLA) plastics,
 and nylon to fabricate different hollow sphere structures through additive manufactur-
 ing techniques, such as fusion selective laser melting (SLM) [23,24], stereolithography

Materials 2021, 14, 3716 3 of 16

 (SLA) [25], fused deposition modelling (FDM) [26], and multi-jet fusion (MJF) [27], and car-
 ried out a series of experiments to investigate the mechanical properties of these structures,
 most of which introduced perforations or open porosities in spherical walls to remove the
 excess material inside the structure during post-processing of 3D printing; however, these
 perforations in hollow spheres change the physical and mechanical properties of hollow
 sphere structures to some degree. Parameter studies have also been conducted by using
 finite element models (FEMs) to quantitatively analyse the effect of geometric parameters
 on the structural mechanical properties [25,27]. However, these studies focused on the
 elastic and initial yielding stages because of the dramatic increase in the computational
 time in the simulation of large plastic deformations.
 Understanding the mechanism of large deformation or plastic behaviour of perforated
 hollow sphere structures (PHSSs) is important for designers to more fully utilise the en-
 ergy absorption characteristics. In this work, multi-cell PHSSs in different arrangements
 were prepared by using the MJF 3D printing technology and the large deformation be-
 haviour of PHSSs under quasi-static compression was then investigated experimentally
 and numerically. The experimental results verified the effectiveness of the associated FEMs.
 Furthermore, the effect of geometric parameters, such as wall thickness, hole diameter and
 packing pattern, on the structural deformation mode, first peak stress, plateau stress and
 specific energy absorption (SEA) was analysed by using the finite element method, which
 offers a reference for further lightweight engineering applications of 3D printed hollow
 sphere structures.

 2. Materials and Methods
 For the traditional soldered or adhesive bonded metallic hollow-sphere structure
 (Figure 1a), the connection part between two spheres adopts a certain geometry, as shown
 in Figure 1b. For the PHSSs, the connection cavity shown in Figure 1c was used to link two
 perforated spheres, and the wall thickness of the connection cavity was identical to that
 of a hollow sphere; this geometric model can be divided into appropriate shell elements
 to reduce the simulation time significantly [12]. The main geometric parameters include
 the hollow sphere radius Rm , wall thickness t, opening or hole diameter dm , and distance
 d between adjacent hollow spheres. Since the perforations are located between spheres
 or in connections, for different sphere spatial arrangements, the numbers of openings
 in a single hollow sphere are different: there are 6, 8 and 12 openings corresponding to
 6, 8 and 12 connections for one single hollow sphere in the SC, BCC and FCC packing
 structures, respectively. Figure 2a–c shows the 3D printed 3 × 3 × 3-cell PHSSs in different
 arrangements, where Rm = 9.75 mm, d = 20.5 mm, dm = 5.5 mm, and t = 0.5 mm, and
 the relative density is 0.0654, 0.0833, and 0.0882, respectively. The relative density can be
 calculated by Equation (1) [25].
 V
 ρrel = So (1)
 VUC
Materials 2021, 14, x FOR PEER REVIEW
 where 4 of
 VSo is the volume of the solid material and VUC is the volume of the unit cell or 16
 the
 entire specimen. VSo and VUC can be obtained from the 3D design software Rhino.

 Figure1.
 Figure 1. (a)
 (a) Traditional
 TraditionalMHSS;
 MHSS;structure
 structureofoftwo
 twoconnected
 connectedspheres
 spheres(b)
 (b)without
 withoutand
 and(c)(c)with
 withperfora-
 perfo-
 rations.
 tions.

Materials 2021, 14, 3716 4 of 16
 Figure 1. (a) Traditional MHSS; structure of two connected spheres (b) without and (c) with perfo-
 rations.

 Figure2.2.AA3D
 Figure 3Dprinted
 printedPHSS:
 PHSS:(a)
 (a)SC;
 SC;(b)
 (b)BCC;
 BCC;(c)
 (c)FCC.
 FCC.

 In
 Inthis
 thispaper,
 paper,HP HPMJF MJF3D 3Dprinting
 printingtechnology
 technologyisisused usedtotoproduce
 producethe theexperimental
 experimental
 samples,
 samples, andand HP nylon P12 powder is used as the base material.
 nylon P12 powder is used as the base material. Under the standard Under the standard
 print-
 printing process,
 ing process, the MJFthe MJF 3D printing
 3D printing accuracy
 accuracy is 0.1ismm,
 0.1 mm,
 and and the density
 the density of theofprinted
 the printed
 solid
 solid material
 material is 1.01
 is 1.01 g/cmg/cm3; this3 ;material
 this material
 can be can be simplified
 simplified as anas an ideal
 ideal elastoplastic
 elastoplastic material
 material with
 with a Young’s
 a Young’s modulus
 modulus of 1700
 of 1700 MPaMPa and and a yield
 a yield stress
 stress of 48ofMPa
 48 MPa [27,28].
 [27,28].
 To
 Toensure
 ensurethe thereliability
 reliabilityofofthe theexperimental
 experimentalresults,results,three
 threesamples
 samplesofofeach eachpacking
 packing
 pattern were prepared and tested in this study. After printing,
 pattern were prepared and tested in this study. After printing, all the hollow sphere all the hollow sphere
 struc-
 structures stood for 10 days in a dry environment at room temperature (25 ◦ C) and were
 tures stood for 10 days in a dry environment at room temperature (25 °C) and were then
 then subjected
 subjected to a to a quasi-static
 quasi-static compression
 compression experiment.
 experiment. TheThe samples
 samples werewere placed
 placed between
 between the
 the upper
 upper and and lower
 lower platesofofa ahydraulic
 plates hydraulicuniversal
 universal testing
 testing machine
 machine (Changchun
 (Changchun DDL100, DDL100,
 China)
 China)withwithan anaccuracy
 accuracyofof0.5% 0.5%in inthe
 theexperiment,
 experiment,and andthethedownward
 downwardloading loadingrate rateofofthe
 the
 upper plate was set to 10
 upper plate was set to 10 mm/min. mm/min.
 Numerical
 Numericalsimulations
 simulationswere wereperformed
 performedfor forthe
 thestructures
 structuresunder
 underthe thecorresponding
 corresponding
 experimental
 experimental conditions using ABAQUS 6.14 finite element software. The The
 conditions using ABAQUS 6.14 finite element software. geometric
 geometric mod-
 models of PHSSs in different arrangements were simulated
 els of PHSSs in different arrangements were simulated with automatically generated with automatically generated
 four-
 four-node
 node shellshell element
 element (S4R) (S4R) meshes,
 meshes, which
 which significantly
 significantly reduced
 reduced thethe modelling
 modelling time.
 time. AnAn av-
 average length of the elements 0.8 mm, approximately 4% of the
 erage length of the elements 0.8mm, approximately 4% of the sphere diameter, was used sphere diameter, was used
 following a mesh convergence analysis. For the SC, BCC, and FCC PHSSs with 3 × 3 × 3
 following a mesh convergence analysis. For the SC, BCC, and FCC PHSSs with 3 × 3 × 3
 cells, the FEMs contained 53,352, 96,336, and 189,216 elements, respectively. The FEM of
 cells, the FEMs contained 53,352, 96,336, and 189,216 elements, respectively. The FEM of
 the BCC PHSS is shown in Figure 3 as an example. The bottom plate was subjected to fixed
 the BCC PHSS is shown in Figure 3 as an example. The bottom plate was subjected to
 restraints, a vertical downward displacement load was arranged at the top plate, and the
 fixed restraints, a vertical downward displacement load was arranged at the top plate,
 contact between the rigid plates and the structure was defined as surface-to-surface contact.
 and the contact between the rigid plates and the structure was defined as surface-to-sur-
 The hard contact algorithm was used for normal behaviour, and a friction coefficient of 0.1
 face contact. The hard contact algorithm was used for normal behaviour, and a friction
 was considered for tangential behaviour. In consideration of the self-contact behaviour in
 coefficient of 0.1 was considered for tangential behaviour. In consideration of the self-
 the large deformation stage, general contact was used for all structures.
 contact behaviour in the large deformation stage, general contact was used for all struc-
 The ABAQUS/Explicit method was then employed to simulate the mechanical be-
 tures.
 haviour of different structures under quasi-static compression, and a comparison was made
 The ABAQUS/Explicit method was then employed to simulate the mechanical be-
 between the numerical simulations and the experimental results. For ease of analysis, the
 haviour of different
 nominal stress σ was structures
 defined as underthe platequasi-static
 pressure compression,
 divided by theand a comparison
 projected was
 area of the
 made between the numerical simulations and the experimental
 structure on the plane, and the nominal strain ε was defined as the ratio of the loading results. For ease of analy-
 sis, the nominal
 displacement to thestress σ was
 initial heightdefined
 of the as the plateIt pressure
 structure. is necessarydivided
 to notebythatthe all
 projected area
 the “stress”
 of the
 and structure
 “strain” on the plane,
 mentioned and thesections
 in following nominal strain εthe
 indicate was defined
 nominal as the
 stress and ratio of the
 nominal
 strain respectively, if not indicated otherwise.

Materials 2021, 14, x FOR PEER REVIEW 5 of 16

Materials 2021, 14, 3716
 loading displacement to the initial height of the structure. It is necessary to note that all
 5 of 16
 the “stress” and “strain” mentioned in following sections indicate the nominal stress and
 nominal strain respectively, if not indicated otherwise.

 Figure3.3.(a)
 Figure (a)Experimental
 Experimentalmodel
 modeland
 and(b)
 (b)FEM
 FEMofofthe
 theBCC
 BCCstructure.
 structure.

 AAparameter
 parameterstudy studywas wasconducted
 conductedto toassess
 assessthe theeffect
 effectofofwall thicknesst tand
 wallthickness andhole
 hole
 diameterddmm on the compressive
 diameter compressive mechanical
 mechanicalproperties
 propertiesand andenergy
 energyabsorption
 absorptioncapacity
 capacityof
 of the
 the structures.
 structures. TheThe relationships
 relationships between
 between thethe geometric
 geometric parameters
 parameters andand the relative
 the relative den-
 density
 sity are are
 shownshown in Table
 in Table 1. all
 1. For Forgeometric
 all geometricmodels,models, the sphere
 the sphere radius radius
 Rm was Rmfixed
 was to
 fixed
 9.75
 to 9.75and
 mm, mm, theand the distance
 distance between between
 adjacent adjacent
 hollowhollow
 spheresspheres
 was fixedwastofixed to 20.5
 20.5 mm. Asmm. As
 t varied
 t(tvaried (t = 0.195~0.975
 = 0.195~0.975 mm), dm was fixed mm), d was fixed to 5.5 mm.
 m to 5.5 mm. As dm varied (d As d varied (d = 3.5~7.5
 mm = 3.5~7.5 mmm), t was fixedmm),
 ttowas
 0.5 fixed
 mm. As to 0.5 mm.inAs
 shown shown
 Table 1, theint Table
 value has1, the t value has
 a significant a significant
 effect effect density,
 on the relative on the
 relative density, while
 while the dm value has a small the d value has a small effect on the density. In
 m effect on the density. In addition to the relative density, addition to the
 the
 relative density, the ratio
 thickness-to-radius thickness-to-radius
 (t/Rm) is also an ratio (t/Rm parameter
 important ) is also anofimportant
 hollow spheres parameter of
 [12]. Ac-
 hollow
 cordingspheres
 to Table [12]. According
 1, when to Table
 dm = 5.5 1, when
 mm, there is andm approximately
 = 5.5 mm, therelinear is an relationship
 approximately be-
 linear
 tweenrelationship
 t/Rm and the between t/Rmdensity
 relative and theby relative
 usingdensity by using
 the linear themethod:
 fitting linear fitting
 method:
 1.275 ∙
 ρSC ≈ 1.275
 ⁄ ; ·( t/R m ) ; ≈ 1.622
 1.622 ∙ ⁄ ; and 
 ρ BCC ·( t/R m ) ; and ρ FCC ≈ ·( t/R
 1.712 ∙ ⁄ . The fitting accuracyaccuracy
 1.712 m ) . The fitting is above
 is98%.
 above 98%.

 Table 1. Geometrical parameters and relative densities.
 Table 1. Geometrical parameters and relative densities.
 Geometrical Parameter
 Geometrical Parameter Relative Density
 Relative ρrl ρrl
 Density
 t/mmt/mm dm /mmdm/mm SC SC BCCBCC FCC
 FCC
 0.195
 0.195 0.02553
 0.02553 0.03253
 0.03253 0.0344
 0.0344
 0.39 0.05105 0.06504 0.06875
 0.39 0.05105 0.06504 0.06875
 0.585 5.5 0.07654 0.0975 0.10303
 0.5850.78 5.5 0.07654
 0.10207 0.0975
 0.13 0.10303
 0.13731
 0.78
 0.975 0.10207
 0.12746 0.13
 0.16229 0.13731
 0.17129
 0.975 3.5 0.06756
 0.12746 0.08706
 0.16229 0.09434
 0.17129
 0.5 5.5 0.0654 0.08332 0.08822
 3.5 0.06756 0.08706 0.09434
 7.5 0.06274 0.07874 0.0807
 0.5 5.5 0.0654 0.08332 0.08822
 7.5
 3. Results and Discussion 0.06274 0.07874 0.0807
 3.1. Mechanical Behaviour of the PHSS
 3. Results and
 Figure Discussion
 4 shows the nominal stress–strain curves of the different PHSSs, and Figures
 3.1.
 5–7Mechanical
 show the Behaviour of the PHSS
 large deformation process of the SC, BCC, and FCC PHSSs, respectively.
 Figure 4toshows
 According the nominal results
 the experimental stress–strain curves of the
 and numerical different PHSSs,
 simulations, similarand Figures
 to the 5–7
 compres-
 show the large deformation
 sive mechanical behaviours process of thehollow
 of traditional SC, BCC, and materials,
 sphere FCC PHSSs, the respectively. Ac-
 deformation pro-
 cording
 cess cantogenerally
 the experimental
 be dividedresults
 into aand numerical
 linear simulations,
 elastic stage, a large similar to the stage
 deformation compressive
 (or plat-
 mechanical
 eau stage), behaviours of traditional
 and a densification stage. hollow
 As shownsphere materials,
 in Figure 4, thethe deformation
 elastic process
 stage is relatively
 can generally be divided into a linear elastic stage, a large deformation stage (or plateau
 stage), and a densification stage. As shown in Figure 4, the elastic stage is relatively short.
 When the deformation reaches the initial yield stage, for different PHSSs, the buckling of
 weak sphere walls in one layer leads to the first local collapse deformation, and the stress

Materials 2021, 14, x FOR PEER REVIEW 6 of 16

Materials 2021, 14, 3716 6 of 16

 short. When the deformation reaches the initial yield stage, for different PHSSs, the buck-
 ling of weak sphere walls in one layer leads to the first local collapse deformation, and the
 stress decreases
 decreases (① in Figures
 ( 1 in Figures 4–7). As4–7). As the displacement
 the displacement increases,increases,
 hollowhollow spheresspheres
 contactcontact
 each
 each other in the layer where buckling develops; as a result,
 other in the layer where buckling develops; as a result, the stress increases, and energy the stress increases, and en-
 ergy accumulates
 accumulates for the fornext the next collapse
 collapse deformation.
 deformation. This deformation
 This deformation mechanism mechanism
 causes the causes
 SC
 theBCC
 and SC and PHSSsBCCtoPHSSsundergo to aundergo a deformation
 deformation process inprocess in the
 the initial stage initial stage
 of large of large de-
 deformation,
 formation,
 which features which
 local,features local, layer-by-layer
 layer-by-layer continuous buckling.continuous Notebuckling.
 that plastic Noteringsthat plastic
 form in
 SC structures during this stage. When a preliminary compaction state ( in Figures 4–6)in
 rings form in SC structures during this stage. When a preliminary compaction
 2 state (②
 isFigures
 achieved 4–6)
 iniswhich
 achieved in which adjacent
 all vertically all vertically
 hollow adjacent
 spheres hollow
 are inspheres
 contactare in contact
 with each other,with
 each other, under continuous uniaxial loading, the hollow spheres
 under continuous uniaxial loading, the hollow spheres inside the structure are interactively inside the structure are
 interactively
 compressed compressed
 layer by layer, and layerthebystress
 layer,increases
 and the stress
 slowlyincreases
 ( 3 in Figuresslowly4–6).(③ Forin Figures
 the FCC4–
 6). Forthe
 PHSS, thelarge
 FCCdeformation
 PHSS, the large stagedeformation
 features local, stage features local,
 layer-by-layer layer-by-layer
 continuous collapse continu-
 and
 ous collapseand
 compaction, andin compaction,
 this stage, the andstress
 in thisfluctuates
 stage, thesmoothly
 stress fluctuates
 ( 1 2 3 smoothly
 in Figures (①②③
 4 and 7).in
 Figuresthe
 Finally, 4 and 7). Finally,
 different PHSSsthe are different
 substantiallyPHSSs are substantially
 squashed, and the space squashed,
 inside and the space
 the structure
 inside the structure is further squeezed; as a result, the sphere-wall
 is further squeezed; as a result, the sphere-wall contacts, especially the internal sphere-wall contacts, especially
 the internal
 contacts, sphere-wall
 expand, contacts,enters
 and the material expand, theand the material
 densification enters
 stage, the densification
 in which the stress beginsstage,
 toinincrease
 which the stress
 rapidly ( 4begins
 in Figuresto increase rapidly
 4–7). After (④ in Figures
 the compression test,4–7). After the
 the PHSSs were compression
 unloaded
 test,left
 and thestanding
 PHSSs were for unloaded
 half an hour. and left
 Thestanding
 BCC and forFCC
 halfPHSSs
 an hour. The BCC60–70%
 regained and FCC of PHSSs
 their
 original
 regained height,
 60–70% while the SC
 of their PHSS height,
 original was substantially
 while the SC crushed
 PHSS and wasfailed to rebound,
 substantially as
 crushed
 shown in Figure
 and failed 8.
 to rebound, as shown in Figure 8.
 InInthis
 thisstudy,
 study,compression
 compressiontests testswere
 wereconducted
 conductedfor forthree
 threegroups
 groupsofofstructures
 structureswith with
 different
 different arrangements, and the results of each group reveal favourable consistency.AA
 arrangements, and the results of each group reveal favourable consistency.
 comparison
 comparisonbetween betweenthe theexperimental
 experimentalresultsresultsandandthethenumerical
 numericalsimulation
 simulationresultsresultsshows
 shows
 certain
 certaindifferences
 differencesbetweenbetweenthe thetwo,
 two,mainly
 mainlybecause
 becausesome someerrors
 errorsand anddefects
 defectsinevitably
 inevitably
 occurred
 occurredduring during 3D 3D printing and and post-processing,
 post-processing,thereby therebyresulting
 resulting inin changes
 changes in the
 in the me-
 mechanical properties of the structure. The presence of errors and
 chanical properties of the structure. The presence of errors and defects is also a common defects is also a common
 problem
 problemtotobebesolved solvedininthe theengineering
 engineeringapplication
 applicationofof3D 3Dprinting
 printingtechnologies.
 technologies.Overall,
 Overall,
 the deformation patterns of the PHSSs in different
 the deformation patterns of the PHSSs in different compression stages compression stages and the andcorrespond-
 the corre-
 ing stress–strain
 sponding curves were
 stress–strain curves substantially consistent;
 were substantially therefore,therefore,
 consistent; the FEM the established
 FEM estab- in
 this study
 lished is reliable.
 in this study is reliable.

 Figure4.4.Nominal
 Figure Nominalstress–strain
 stress–straincurves
 curvesofofdifferent
 differentstructures.
 structures.

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 Figure 5. Deformed SC structures at different stages.
 Figure 5. Deformed SC structures at different stages.
 Figure 5. Deformed SC structures at different stages.
 Figure 5. Deformed SC structures at different stages.

 Figure 6. Deformed BCC structures at different stages.
 Figure 6. Deformed
 Figure 6. Deformed BCC
 BCC structures
 structures at
 at different
 differentstages.
 stages.
 Figure 6. Deformed BCC structures at different stages.

 Figure 7. Deformed FCC structures at different stages.
 Figure 7. Deformed FCC structures at different stages.
 Figure7.7.Deformed
 Figure DeformedFCC
 FCCstructures
 structuresatatdifferent
 differentstages.
 stages.

Materials 2021, 14, 3716 8 of 16
 Materials 2021, 14, x FOR PEER REVIEW 8 of 16

 Figure 8. Compressed hollow sphere structures.
 Figure 8. Compressed hollow sphere structures.

 3.2.Effect
 3.2. EffectofofGeometric
 GeometricParameters
 Parameters
 Accordingto
 According tothe
 theresults
 resultsmentioned
 mentionedabove, above,all alllarge
 largedeformation
 deformationbehaviours
 behavioursofofthe the
 PHSSsunder
 PHSSs underquasi-static
 quasi-staticuniaxial
 uniaxialcompression
 compressionbegan beganlocally.
 locally.To Tofacilitate
 facilitatediscussion
 discussionand and
 comparison,
 comparison,the thefirst
 firstpeak stressYYPPisisdefined
 peakstress definedas asthethefirst
 firstlocal
 localmaximum
 maximumon onthe
 thenominal
 nominal
 stress–strain
 stress–straincurvecurveand andserves
 servesas asoneoneof ofthe
 theindicators
 indicatorsfor forpredicting
 predictingthe thelarge
 largedeformation
 deformation
 behaviour;
 behaviour; the the mean
 meanstressstressin inthetheplateau
 plateaustage stage(also
 (alsoknown
 knownas as“plateau
 “plateaustress”)
 stress”)ininthe the
 nominal
 nominal stress (σ)–strain ( ) curve is also an important indicator for evaluatingthe
 stress (σ)–strain (ε) curve is also an important indicator for evaluating thelarge
 large
 deformation
 deformation behaviour
 behaviour of ofporous
 porousmaterials
 materialsor orstructures;
 structures;in inaddition,
 addition,the theSEA,
 SEA,i.e.,
 i.e.,the
 the
 energy
 energyabsorption
 absorptionper perunitunitof ofmass,
 mass,isisone oneofofthethemost
 mostimportant
 importantindicators
 indicatorsofofthe theenergy
 energy
 absorption
 absorptioncapacity
 capacityfor forlightweight
 lightweight engineering applications. The plateau stress σ pl and
 engineering applications. and
 SEA are defined as Equations (2) and
 SEA are defined as Equations (2) and (3), respectively. (3), respectively.

 R εD
 ε σ ( ε ) dε (2)
 σpl = = Y (2)
 ε D − ε Y 
 Rε
 SEA = = Fdδ =0 σ (ε)dε
 R
 (3)
 SEA = Wm = = (3)
 m ρ
 where and generally correspond to the strains at the yield stage and the densifica-
 where ε Y and
 tion stage, ε D generallyF correspond
 respectively. represents the to the strainsforce
 reacting at theofyield stage and
 the loading the densification
 plate; δ denotes the
 stage,
 displacement of the F
 respectively. represents
 loading end; m the reacting
 is the mass force
 of the of the loading
 porous material; plate;
 and ρδ isdenotes the
 the density.
 displacement of the loading end; m is the mass of the porous
 In this study, is determined as the strain corresponding to the stationary point in the material; and ρ is the density.
 In this study, ε D is determined as the strain corresponding to the stationary point in the
 efficiency-strain curve where the efficiency E( ) provides the last local maximum [29]:
 efficiency-strain curve where the efficiency E(ε) provides the last local maximum [29]:
 R / 
 (4)
 ) | = dε/σ ) | =0
 ε
 dE(ε d( 0 σ (ε) 
 |ε=ε D = |ε=ε D = 0 (4)
 dε dε
 In the calculation of SEA, because the densification strain varies with geometric pa-
 rameters,
 In thethe strain in
 calculation ofEquation
 SEA, because (3) was thesetdensification
 as 0.7 for comparison.
 strain varies with geometric
 parameters,
 Based on thefinite
 strainelement
 ε in Equation (3) wasthe
 simulations, seteffect
 as 0.7offor comparison.
 geometric parameters, such as wall
 Based on finite element simulations, the effect of
 thickness, hole diameter, and arrangement pattern, on the mechanical response, geometric parameters, such asdefor-
 wall
 thickness, hole diameter, and arrangement pattern, on the
 mation mode, and energy absorption properties of PHSSs under quasi-static compression mechanical response, deforma-
 tion
 wasmode, and energy
 investigated. Becauseabsorption properties ofofPHSSs
 of the localization under quasi-static
 large deformations compression
 in hollow was
 sphere struc-
 investigated. Because of the
 tures, the representative localization
 volume element of (RVE)
 large deformations
 under periodical in hollow sphere structures,
 or symmetric boundary
 the representative
 conditions volumeiselement
 [4,7,11,12,15] unsuitable (RVE) forunder periodical
 the study of theorwhole symmetric boundary condi-
 large deformation pro-
 tions [4,7,11,12,15] is unsuitable for the study of the whole large
 cess. Thus, simulation calculations were performed using the multi-cell structure numer- deformation process. Thus,
 simulation
 ical model calculations
 with five cells were performed
 at each dimension using the multi-cell
 according structure numerical
 to a convergence study on model
 the cell
 with
 number. All FEMs were established by using a similar method, described in number.
 five cells at each dimension according to a convergence study on the cell Section 2.All
 FEMs were established by using a similar method, described in Section 2.
 3.2.1. Wall Thickness
 3.2.1. Wall Thickness
 When investigating the influence of wall thickness t, the t value was set as 0.195 mm,
 When investigating the influence of wall thickness t, the t value was set as 0.195 mm,
 0.39 mm, 0.585 mm, 0.78 mm, and 0.975 mm; the dm value was fixed to 5.5 mm; other
 0.39 mm, 0.585 mm, 0.78 mm, and 0.975 mm; the dm value was fixed to 5.5 mm; other
 parameters were unchanged; and the corresponding t/Rm was 0.02, 0.04, 0.06, 0.08, and
 parameters were unchanged; and the corresponding t/Rm was 0.02, 0.04, 0.06, 0.08, and
 0.1. Figure 9a,c,e shows the nominal stress–strain curves of the SC, BCC, and FCC PHSSs,

Materials 2021, 14, 3716 9 of 16

Materials 2021, 14, x FOR PEER REVIEW 9 of 16
 0.1. Figure 9a,c,e shows the nominal stress–strain curves of the SC, BCC, and FCC PHSSs,
 respectively, with different wall thicknesses under quasi-static compression; Figure 9b,d,f
 shows thewith
 respectively, curves describing
 different the variations
 wall thicknesses in first peak
 under quasi-static stress YP , Figure
 compression; plateau stress σpl , and
 9b,d,f
 shows
 SEAthewithcurves describing
 different wallthe variations for
 thicknesses in first
 the peak stress and
 SC, BCC, YP, plateau stress σrespectively.
 FCC PHSSs, pl, and The
 SEA with different
 results wall the
 indicate that thicknesses for the
 YP , σpl and SEASC,values
 BCC, and FCC
 of the PHSSs,PHSSs
 different respectively.
 increasedThe as the wall
 results indicate
 thickness that the YPand
 increased, , σpl and
 the SEA values ofwere
 differences the different PHSSs
 as follows: forincreased as the wall
 the SC PHSS, YP was always
 thickness increased, and the differences were as follows: for the SC PHSS, Y P was always
 less than σpl ; for the BCC PHSS, YP was essentially equal to σpl ; and for the FCC PHSS, YP
 less than σpl; for the BCC PHSS, YP was essentially equal to σpl; and for the FCC PHSS, YP
 was always greater than σpl .
 was always greater than σpl.

 Figure 9. Effect of wall thickness on the nominal stress -strain curve and first peak stress YP, plat-
 Figure 9. Effect of wall thickness on the nominal stress -strain curve and first peak stress YP , plateau
 eau stress and SEA of the (a,b) SC; (c,d) BCC and (e,f) FCC structures.
 stress σpl and SEA of the (a,b) SC; (c,d) BCC and (e,f) FCC structures.
 Figure 10 shows the large deformation patterns of the PHSSs in different arrange-
 Figure
 ments, and 10 shows
 the dotted box the large
 in the deformation
 figures representspatterns of the
 the original PHSSsdimensions.
 structural in differentThearrangements,
 and the dotted box in the figures represents the original structural
 numerical simulation results also indicate that when the wall was very thin (the t/Rm val- dimensions. The
 uesnumerical simulation
 were 0.02 and resultswall
 0.04), the sphere also indicate
 near that when
 each opening in the the
 PHSS wall
 was was
 pronevery thin (the t/Rm
 to stress
 values were
 concentration; as 0.02 and 0.04), the
 the displacement spherefor
 increased, wall
 twonear eachhollow
 adjacent opening in the
 spheres, onePHSS
 could was prone
 to stress
 undergo concentration;
 unstable as the displacement
 buckling deformation, increased,
 and the other for two
 could collapse intoadjacent hollow
 it; and for the spheres,
 one could undergo unstable buckling deformation, and the other could collapse into it;
 and for the different PHSSs, such a local deformation mechanism could cause different
 changes in the lateral dimensions of the structure. For the SC PHSS, the hollow spheres

Materials 2021, 14, 3716 10 of 16
Materials 2021, 14, x FOR PEER REVIEW 10 of 16

 collapsed
 different PHSSs, suchinto each
 a local other in mechanism
 deformation the same direction
 could causeasdifferent
 the loading,
 changes and the hollow spheres
 in the
 squeezed each other vertically, which could increase the lateral
 lateral dimensions of the structure. For the SC PHSS, the hollow spheres collapsed into size. When the wall was
 each other in the same direction as the loading, and the hollow spheres squeezed eacht value increased
 very thin, the lateral dimension did not change significantly; as the
 (t/Rm reached
 other vertically, which could0.06), the deformation
 increase the lateral size.mechanism
 When the wall of the
 wasmutual depression
 very thin, the of the SC
 PHSS became
 lateral dimension did not stable,
 change and the lateral
 significantly; dimension
 as the after compression
 t value increased (t/Rm reachedgradually
 0.06), increased as
 the deformation mechanism
 t increased (Figure of 10a,b).
 the mutual
 Fordepression
 the BCC and of theFCC
 SC PHSS
 PHSSs, became
 when stable, and thickness t was
 the wall
 the lateralvery
 dimension after compression gradually increased as t increased (Figure
 small, under axial compression, the local buckling/depression due to the mutual 10a,b).
 For the BCC and FCC
 extrusion ofPHSSs, when spheres
 the hollow the wall thickness
 caused the t was very small,
 transverse under axial of
 contraction com-
 the entire structure
 pression, (t/R
 the local buckling/depression due to the mutual extrusion of the hollow spheres
 m was 0.02; Figure 10c,e); as t gradually increased, the stress concentration and local
 caused the transverse contraction of the entire structure (t/Rm was 0.02; Figure 10c,e); as t
 buckling/depression of the sphere wall around the opening decreased; and when the wall
 gradually increased, the stress concentration and local buckling/depression of the sphere
 was thick, the layer-by-layer bending deformation of the spherical wall dominated, while
 wall around the opening decreased; and when the wall was thick, the layer-by-layer bend-
 the transverse
 ing deformation contraction
 of the spherical of the overall
 wall dominated, whilestructure gradually
 the transverse disappeared
 contraction of the (t/Rm was 0.1;
 Figure 10d,f).
 overall structure gradually disappeared (t/Rm was 0.1; Figure 10d,f).

 Figure 10. Deformations of SC structure with the (a) smallest and (b) largest wall thicknesses at a
 strain of 43%; deformations of BCC structure with the (c) smallest and (d) largest wall thicknesses at
 a strain of 37%; deformations of FCC structure with the (e) smallest and (f) largest wall thicknesses at
 a strain of 44%.

Materials 2021, 14, x FOR PEER REVIEW 11 of 16

 Figure 10. Deformations of SC structure with the (a) smallest and (b) largest wall thicknesses at a
 strain of 43%; deformations of BCC structure with the (c) smallest and (d) largest wall thicknesses
Materials 2021, 14, 3716 11 of 16
 at a strain of 37%; deformations of FCC structure with the (e) smallest and (f) largest wall thick-
 nesses at a strain of 44%.

 On the whole,
 On the whole,for forthethedifferent
 differentPHSSs,
 PHSSs, when
 when thethe hollow
 hollow sphere
 sphere wallwallwaswas extremely
 extremely thin
 thin (t/R was 0.02 and 0.04), layer-by-layer deformation developed
 (t/Rm was 0.02 and 0.04), layer-by-layer deformation developed first to reach a preliminary
 m first to reach a pre-
 liminary
 compaction compaction
 state (i.e.,state (i.e., thehollow
 the adjacent adjacent hollow
 spheres spheres
 were were with
 in contact in contact with each
 each other), and
 other),
 then theand thenspheres
 hollow the hollow spheres
 entered anotherentered
 stageanother
 of largestage of largefeaturing
 deformation deformation featuring
 layer-by-layer
 layer-by-layer
 compaction (i.e., compaction
 the hollow (i.e., the hollow
 spheres squeezedspheres
 eachsqueezed each uniaxial
 other) under other) under uniaxial
 compression.
 compression. As the t value increased (t/R was 0.06, 0.08, and
 As the t value increased (t/Rm was 0.06, 0.08, and 0.1), such deformation mode for the SC
 m 0.1), such deformation
 mode
 and BCC for the
 PHSSsSC and
 tended BCCtoPHSSs
 be stable,tended
 whileto the
 be stable, while the large
 large deformation of thedeformation
 FCC PHSS ofwas
 the
 FCC PHSS wasbycharacterised
 characterised layer-by-layer bycontinuous
 layer-by-layer continuous
 collapse collapse and
 and compaction compaction
 (Figure (Fig-
 10f). Finally,
 ure
 the 10f).
 overallFinally, the overall
 structure structure
 was further was further
 compacted undercompacted
 continuousunderuniaxialcontinuous
 loading uniaxial
 (i.e., the
 loading (i.e., the
 densification densification
 stage), stage),
 and the stress and the sharply
 increased stress increased sharply (Figure 9a,c,e).
 (Figure 9a,c,e).
 For
 For the
 the comparison
 comparison of of the
 the compressive
 compressive mechanical
 mechanical properties
 properties and energy absorption
 characteristics
 characteristics of of the
 the PHSSs
 PHSSs in different arrangements, the numerical simulation results
 were fitted as a function of the relative density, as shown in Figure 11. The results show
 that the first peak stress YPP and plateau stress σpl
 that the of the
 pl of the PHSSs
 PHSSs increased
 increased as the relative
 density increased;
 density increased; the lower the relative relative density was, the closer the results of the different
 PHSSs were
 PHSSs were to one another. Overall, with the same relative density, density, thethe YYPP, σσpl
 pl,, and SEA
 values of
 values of the
 the SC PHSS were the lowest, the YPP value value ofof the
 the FCC
 FCC PHSS
 PHSS was was thethe highest,
 highest,
 and the values of the BCC PHSS were in between. The possible
 and the values of the BCC PHSS were in between. The possible causes are as follows: the causes are as follows:
 the individual
 individual hollowhollowspheresphere
 insideinside the PHSS
 the SC SC PHSSwaswas subject
 subject to the
 to the leastleast constraints;
 constraints; the the
 in-
 individual hollow sphere inside the FCC PHSS was under
 dividual hollow sphere inside the FCC PHSS was under the most constraints; and the the most constraints; and the
 constraints of
 constraints of the
 the BCC
 BCC PHSS PHSS were
 were in in between.
 between. The The more
 more constraints
 constraints aa hollow
 hollow sphere
 sphere is is
 under, the
 under, the harder
 harder it it is
 is to
 to buckle.
 buckle. WhenWhen the the relative
 relative density
 density was
 was between
 between 0.02 0.02 and
 and 0.1,
 0.1, the
 the
 andSEA
 σσplpland SEAvalues
 valuesof ofthe
 theBCC
 BCCPHSSPHSSwere wereextremely
 extremely close
 close to
 to those
 those ofof the
 the FCC
 FCC PHSS.
 PHSS. WhenWhen
 the relative density exceeded 0.1, the σ and SEA values of the
 the relative density exceeded 0.1, the σplpl and SEA values of the FCC PHSS progressively FCC PHSS progressively
 exceeded those
 exceeded those ofof the
 the BCC
 BCC PHSS.
 PHSS.

 Figure
 Figure 11.
 11. Effect
 Effectofofthe relative
 the density
 relative on on
 density thethe
 (a) first peakpeak
 (a) first stressstress
 YP, (b)
 YPplateau stress σstress
 , (b) plateau pl andσ(c) SEA for PHSSs in different
 pl and (c) SEA for PHSSs in
 arrangements.
 different arrangements.

 3.2.2. Hole
 3.2.2. Hole Diameter
 Diameter
 When investigating the effect of hole diameter dm
 When m,, the
 the dmm value
 valuewas
 was 3.5
 3.5 mm,
 mm, 5.55.5 mm,
 mm,
 and 7.5 mm, the t value was fixed to 0.5 mm, and the other parameters
 and 7.5 mm, the t value was fixed to 0.5 mm, and the other parameters were unchanged. were unchanged.
 Figure 12a,c,e
 Figure 12a,c,e shows
 shows thethe nominal
 nominal stress–strain
 stress–strain curves
 curves of of the
 the SC,
 SC, BCC,
 BCC, and
 and FCC
 FCC PHSSs,
 PHSSs,
 respectively, with different hole diameters under quasi-static compression; Figure 12b,d,f
 respectively,
 show the
 show the curves
 curves describing
 describing the
 the variations
 variations in
 in first
 first peak stressYYPP,, plateau
 peak stress plateau stress
 stress σσpl and SEA
 pl and SEA
 with the
 with the change
 change inin the
 the hole
 hole diameter
 diameter for
 for the
 the SC,
 SC, BCC,
 BCC, andand FCCFCC PHSSs,
 PHSSs, respectively.
 respectively. The
 The
 results show that for the SC and BCC PHSSs, the Y value increased as d
 results show that for the SC and BCC PHSSs, the YP value increased as dm increased; i.e.,
 P m increased; i.e.,
 Y was clearly positively correlated with
 YPP was clearly positively correlated with dm d , while for the FCC PHSS,
 m, while for the FCC PHSS, when dm when d increased
 m increased
 to a certain value (approximately 6 mm by fitting), YP began to decrease, and the curve
 reflecting the variation in YP of the FCC PHSS with the change in dm was parabolic, which
 is probably because the FCC PHSS had the most openings; therefore, the larger dm was,
 the greater the loss of mass was, and when dm reached approximately 6 mm, the strength

Materials 2021, 14, x FOR PEER REVIEW 12 of 16

 Materials 2021, 14, 3716 to a certain value (approximately 6 mm by fitting), YP began to decrease, and the curve 12 of 16
 reflecting the variation in YP of the FCC PHSS with the change in dm was parabolic, which
 is probably because the FCC PHSS had the most openings; therefore, the larger dm was,
 the greater the loss of mass was, and when dm reached approximately 6 mm, the strength
 begantotodecrease
 began decreasedueduetotoananexcessively
 excessivelyhigh highmass
 massloss.
 loss.Similar
 Similarconclusions
 conclusionshave havebeen
 been
 drawn for the mechanical properties in the elastic stage and the initial yield stage of PHSSsof
 drawn for the mechanical properties in the elastic stage and the initial yield stage
 PHSSs
 [27]. [27]. In addition,
 In addition, the holethe hole diameter
 diameter had a significant
 had a significant effect effect
 on theonYthe YP value
 P value
 of PHSSs
 of PHSSs in
 in different arrangements, but it had a relatively small effect
 different arrangements, but it had a relatively small effect on the σpl and on the σ pl SEA values in thein
 and SEA values
 the large
 large deformation
 deformation stagestage of various
 of various PHSSs PHSSs(due(due to the
 to the average
 average effect
 effect inin thestress–strain
 the stress–strain
 curves).The
 curves). TheSEA
 SEAresults
 resultsshow
 showthat
 thatfor
 forthe
 theSCSCPHSS,
 PHSS,the theSEA
 SEAvalue
 valuedecreased
 decreasedslightly
 slightlyasas
 d m increased, but the energy absorption did not fluctuate significantly.
 dm increased, but the energy absorption did not fluctuate significantly. For the BCC For the BCCand and
 FCC PHSSs, when d
 FCC PHSSs, when dm = 7.5 mm, the SEA value was significantly higher than the values ofof
 m = 7.5 mm, the SEA value was significantly higher than the values
 structureswith
 structures withsmaller
 smallerholeholediameters
 diameters(3.5 (3.5mm,
 mm,5.5 5.5mm).
 mm).ThisThismaymaybebebecause
 becausethetheBCC
 BCC
 and FCC PHSSs had more openings than the SC PHSS; when the
 and FCC PHSSs had more openings than the SC PHSS; when the dm value was very large, d m value was very large,
 the decrease in the density was greater, and according to Equation (3), the SEA value of the
 the decrease in the density was greater, and according to Equation (3), the SEA value of
 BCC and FCC PHSSs could be higher for large d values.
 the BCC and FCC PHSSs could be higher for largemdm values.

 Figure 12. Effect of hole diameter on the nominal stress–strain curve and first peak stress YP , plateau
 stress σpl and SEA of (a,b) SC; (c,d) BCC and (e,f) FCC structures.

Materials 2021, 14,
 Materials x FOR
 2021, 14,PEER
 3716 REVIEW 13 of 16 13 of 16

 Figure 12. Effect of hole diameter on the nominal stress–strain curve and first peak stress YP, plateau
 Figure
 stress and SEA 13 shows
 of (a,b) SC; (c,d)the
 BCCdeformation modes of PHSSs in different packing
 and (e,f) FCC structures. patterns with
 the smallest (dm = 3.5 mm) and largest (dm = 7.5 mm) hole diameters. The influence
 Figure 13 shows the
 mechanism deformation
 of hole diameter modes of PHSSs
 dm was in different
 similar to that packing
 of wallpatterns with
 thickness t, i.e., the sphere
 the smallest
 wall(dnear
 m = 3.5
 themm) and largest
 opening in the(ddifferent
 m = 7.5 mm) hole diameters.
 PHSSs was prone The
 toinfluence mech-
 stress concentration and local
 anism ofbuckling/depression
 hole diameter dm was similar
 when dtom that
 wasofvery
 wallsmall;
 thickness
 the t,minor
 i.e., the sphere wall
 changes near
 in the lateral dimensions
 the opening
 of theinSCthePHSS
 different PHSSs
 could almost was
 beprone
 ignoredto stress concentration
 (zero expansion) (dmand
 waslocal
 3.5 buck-
 mm; Figure 13a); and
 ling/depression when dm was very small; the minor changes in the lateral dimensions of
 obvious transverse contraction developed in the BCC and FCC PHSSs (dm was 3.5 mm;
 the SC PHSS could almost be ignored (zero expansion) (dm was 3.5 mm; Figure 13a); and
 Figure 13c,e). As d increased, the stress concentration and local buckling or inward
 obvious transverse contractionm developed in the BCC and FCC PHSSs (dm was 3.5 mm;
 depression
 Figure 13c,e). of spheres
 As dm increased, theinstress
 BCCconcentration
 and FCC structures
 and localwere significantly
 buckling or inward improved,
 de- and the
 major deformation was the layer-by-layer bending behaviour
 pression of spheres in BCC and FCC structures were significantly improved, and the ma- of the sphere walls; moreover,
 the transverse
 jor deformation was the contraction
 layer-by-layerofbending
 the entire structure
 behaviour gradually
 of the disappeared
 sphere walls; moreover,(dm was 7.5 mm;
 Figure 13d,f).
 the transverse contraction of the entire structure gradually disappeared (dm was 7.5 mm;
 Figure 13d,f).

 Figure 13. Deformations of SC structure with the (a) smallest and (b) largest hole diameters at a
 strain of 38%; deformations of BCC structure with the (c) smallest and (d) largest hole diameters at a
 strain of 28%; deformations of FCC structure with the (e) smallest and (f) largest hole diameters at a
 strain of 36%.

Materials 2021, 14, x FOR PEER REVIEW 14 of 16

 Figure 13. Deformations of SC structure with the (a) smallest and (b) largest hole diameters at a
Materials 2021, 14, 3716 strain of 38%; deformations of BCC structure with the (c) smallest and (d) largest hole diameters at
 14 of 16
 a strain of 28%; deformations of FCC structure with the (e) smallest and (f) largest hole diameters
 at a strain of 36%

 3.3.
 3.3.Comparison
 Comparisonwith
 withthe
 theTraditional
 TraditionalMHSS
 MHSS
 The
 Thespecific
 specificstrength
 strength(plateau
 (plateaustress/density,
 stress/density,σσplpl/ρ)
 /ρ) and energy absorption
 absorption character-
 character-
 istics
 istics(SEA)
 (SEA)of ofPHSS
 PHSSarearecompared
 comparedwithwiththose
 thoseof
 ofthe
 thetraditional
 traditionalMHSS
 MHSSmade madeof ofmild
 mildsteel,
 steel,
 asasshown
 shownin inFigure
 Figure14.
 14.The
 Thematerial
 materialisisdefined
 definedto
 tobe
 beelastoplastic
 elastoplasticwith
 withaaYoung’s
 Young’smodulus
 modulus
 ofof200
 200GPa
 GPaand
 andaayield
 yieldstress
 stressof
 of200
 200MPa.
 MPa.TheThedensity
 densityρρof ofthe
 theporous
 porousstructure
 structureor ormaterial
 material
 isisthe product of the relative density and solid material density. This comparison
 the product of the relative density and solid material density. This comparison focuses focuses on
 the results from FCC structures with different wall thicknesses according to
 on the results from FCC structures with different wall thicknesses according to Reference Reference [12].
 The 3 , and
 [12].presented 3D-printed
 The presented PHSSPHSS
 3D-printed has a has
 lower density
 a lower ranging
 density from 34–174
 ranging kg/mkg/m
 from 34–174 3, the
 and
 traditional MHSS has ahas
 higher density ranging fromfrom
 173 to1731600 3
 kg/mkg/m , while the specific
 the traditional MHSS a higher density ranging to 1600 3, while the spe-
 strength and SEA
 cific strength andcapacities of the 3D-printed
 SEA capacities PHSS are
 of the 3D-printed PHSS higher
 are than
 higher those
 thanofthose
 the traditional
 of the tra-
 MHSS. The reason can be attributed to two aspects: using lightweight
 ditional MHSS. The reason can be attributed to two aspects: using lightweight solid material
 solidwith
 ma-
 good mechanical properties and perforating appropriately in hollow sphere
 terial with good mechanical properties and perforating appropriately in hollow sphere structures for
 further dead weight reduction.
 structures for further dead weight reduction.

 Figure14.
 Figure 14.Comparison
 Comparisonofofσσpl/ρ
 /ρ and
 and SEA
 SEA of the PHSS
 of the PHSS in
 in this
 this work
 work with
 with the
 theMHSS
 MHSSin
 inReference
 Reference[12].
 [12].
 pl

 4.4.Conclusions
 Conclusions
 For
 Forthis
 thisstudy,
 study,PHSSs
 PHSSsininthree
 threedifferent
 differentarrangements
 arrangementswere werefabricated
 fabricatedusing
 usingMJF
 MJF3D 3D
 printing
 printing technology,
 technology,and andthe
 themechanical
 mechanicalbehaviours
 behavioursof ofthese
 thesePHSSs
 PHSSsunderunderquasi-static
 quasi-static
 compression
 compressionwere werestudied
 studiedthrough
 through experiments
 experiments and finite
 and element
 finite elementsimulations.
 simulations.TheThe
 exper-
 ex-
 imental
 perimentalresults verify
 results the effectiveness
 verify the effectiveness of theofFEM. The results
 the FEM. show that
 The results showthethat
 deformation
 the defor-
 process
 mationof the 3D-printed
 process PHSSs was
 of the 3D-printed similar
 PHSSs wasto similar
 that of traditional hollow sphere
 to that of traditional structures
 hollow sphere
 and
 structures and consisted of three stages, i.e., an elastic deformation stage, a large stage,
 consisted of three stages, i.e., an elastic deformation stage, a large deformation defor-
 and a densification
 mation stage, andstage. Special attention
 a densification stage. was given
 Special to the large
 attention was deformation
 given to thestage,
 largewhich
 defor-
 ismation
 directly related
 stage, whichto the energyrelated
 is directly absorbing capacity.
 to the energy Based on the
 absorbing numerical
 capacity. Basedsimulation,
 on the nu-
 the effects
 merical of the geometric
 simulation, parameters
 the effects of the PHSSs,
 of the geometric such asofthe
 parameters thesphere
 PHSSs, arrangement,
 such as the
 wall thickness, and hole diameter, on the compressive mechanical
 sphere arrangement, wall thickness, and hole diameter, on the compressive mechanical behaviours and energy
 absorption characteristics were investigated. The following conclusions
 behaviours and energy absorption characteristics were investigated. The following con- can be drawn:
 (1) Withcan
 clusions different geometric parameters, the SC and BCC PHSSs underwent a preliminary
 be drawn:
 compaction sub-stage
 (1) With different geometric afterparameters,
 layer-by-layer the buckling
 SC and BCC in the stage underwent
 PHSSs of large deformation;
 a prelimi-
 the FCC PHSS did not exhibit a similar deformation
 nary compaction sub-stage after layer-by-layer buckling in the stage behaviour unless the t or
 of large dm
 defor-
 value was extremely small; and the layer-by-layer collapse-compaction
 mation; the FCC PHSS did not exhibit a similar deformation behaviour unless the t behaviour
 was dominant for the large deformation of the FCC PHSS.
 or dm value was extremely small; and the layer-by-layer collapse-compaction behav-
 (2) Among the PHSSs with the same relative density, the FCC PHSS exhibited the best
 iour was dominant for the large deformation of the FCC PHSS.
 compressive mechanical properties and energy absorption capacity; the SC PHSS
 exhibited the lowest; and the BCC PHSS was in between.
 (3) The first peak stress, plateau stress and SEA of the PHSSs increased as the wall
 thickness t increased, and the hole diameter dm had a significant effect on the first