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A monolithic sandwich panel with microlattice core
Accepted Manuscript A monolithic sandwich panel with microlattice core Dae Han Choi, Yoon Chang Jeong, Kiju Kang PII:
S1359-6454(17)30988-6
DOI:
10.1016/j.actamat.2017.11.045
Reference:
AM 14216
To appear in:
Acta Materialia
Received Date: 10 August 2017 Revised Date:
8 November 2017
Accepted Date: 16 November 2017
Please cite this article as: D.H. Choi, Y. Chang Jeong, K. Kang, A monolithic sandwich panel with microlattice core, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.11.045. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
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A Monolithic Sandwich Panel with Microlattice Core Dae Han Choi, Yoon Chang Jeong, Kiju Kang
Gwangju 61186, Republic of Korea
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*Corresponding author ([email protected])
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School of Mechanical Engineering, Chonnam National University
ABSTRACT
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We introduced a novel technique to fabricate a Microlattice-cored miniature sandwich panel with an ultralow density. This sandwich panel was made of a monolithic Ni-P foil without any bonding between the core and the faces. As a preliminary study, the mechanical properties were evaluated under compression in two different loading directions, and the effects of the
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face sheets added on the top and bottom were investigated in comparison to the conventional Microlattice without the face sheets. Under the out-of-plane compression, almost no difference was evident between the conventional Microlattice and the Microlattice sandwich in their
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compressive behaviors, because of the high friction with the compression platens. However,
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under the in-plane compression, the addition of the face sheets to the conventional Microlattice significantly increased the strength and stiffness, when the wall thickness was higher than a certain limit. Otherwise, the reinforcing effect became much less significant, because the very thin face sheets were easily buckled across the entire area even at the beginning of the loading.
Keywords: cellular material; thin film; sandwich panel; UV lithography 1
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INTRODUCTION
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Recently, a novel material with a density less than 0.01 g/cc (here, we refer to it as ultralow density) and a periodic micro architecture garnered attention because of the exceptionally high strength and stiffness of the material at ultralow densities, which might be a technical
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breakthrough for the near future. The Microlattice introduced in 2012 [1] was the first realization, and was followed by Nanolattice [2, 3] and Mechanical metamaterial [4]. Nowadays,
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these three materials are often referred to as a type of mechanical metamaterial in a wide sense, whose mechanical properties are defined by its geometry rather than its composition. These three materials are common in terms of the fabrication principle and geometry. Namely, their fabrications start with the formation of a polymer template based on 3D UV lithography,
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which is followed by the deposition of a hard constituent material such as metal or ceramic, and the polymer template is then etched out, leaving a hollow structure as the final product. Also, these materials comprise hollow truss-like architectures. The hierarchical structures and
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stretching dominated deformations [5] result in superior mechanical properties. These materials differ in the dimension such as overall size or feature size and the specific techniques
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of the UV lithography. In fact, the different UV techniques dictate the architectures possible, and hence do play a significant role in the feasible material space. Among these three materials, the productivity of Microlattice seems to be the best owing to the mask-based lithography, i.e., self-propagating polymer wave guide (SPPW) [6]. However, the SPPW forms the truss elements of the template only in the out-of-plane directions, because it uses UV rays that are illuminated only through a mask on a single (upper) side of the photo-
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monomer. Consequently, the structure of the Microlattice lacks the truss elements in the inplane directions, and it is neither statically determinate nor kinematically determinate
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according to Maxwell’s criterion [7], which is a necessary and sufficient condition for the rigidity of a truss with a minimal number of elements. Fig. 1 shows a 2D schematic of expected deformation pattern of the Microlattice subjected to out-of-plane compression. Therefore, the
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strength and stiffness might be significantly lower than the expectations [1, 8].
In 2015, Shellular, another material with an ultralow density and a periodic micro
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architecture, was introduced [9]. Its fabrication principle is the same as the previous three, particularly, Microlattice, because the template for Shellular is also fabricated based on SPPW; however, its micro architecture is totally different. Namely, Shellular is composed of polyhedron shells, or more ideally a single smooth and continuous shell with a triply periodic
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minimal surface (TPMS) [10]. Han et al. [9] insisted that the Shellular in the TPMS configuration also provides stretching-dominated deformation, resulting in mechanical properties that are comparable with those of the predecessors with the hollow truss architectures [8, 11]. The
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template of Shellular, however, comprises a much larger volume and a lower porosity than the truss-shaped ones of the three predecessors, thereby limiting the access to the interior space,
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which is likely to hinder uniform deposition of the coating material. A sandwich panel is composed of a thick porous soft core and thin dense hard faces, which provides very high bending strength and stiffness at light weight. Consequently, sandwich panels are widely used in engineering applications. We believe that, if a miniature sandwich panel could be fabricated with an ultralow density, as can be observed in numerous natural life forms such as the leaf [12], shown in Fig. 2, it could be very useful for building the structures of
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micro robots or bio-mimic drones in the near future. Microlattice is noticeably a promising candidate that can serve as the ultralow density core of a miniature sandwich panel, because
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the other two, i.e., Nanolattice [2] and Mechanical metamaterial [3] are too tiny and/or inefficient for mass fabrication. Although the Microlattice architecture lacks the in-plane truss elements [1, 13], it would become rigid when faces are attached to its top and bottom;
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however, it should be challenging to attach the faces to the core using the conventional adhesive bonding to build a micro-sandwich, because the adhesive is likely to fail in provision of
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a very thin and uniform bonding without substantial increase of the total weight, as observed in the references [14, 15], which negatively affects the realization of an ultralow density. In this paper, we introduce a novel technique for fabrication of a Microlattice-cored miniature sandwich panel with an ultralow density. This sandwich panel is composed of a
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monolithic Ni-P foil without any bonding between the core and faces. First, the template was formed using the SPPW technique combined with 2D lithography. The rest of the process was pretty much the same as those of the previous ultralow density materials with micro
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architectures. Namely, the template was then coated with a hard constituent material before it was etched out, leaving the miniature sandwich panel. As a preliminary study, the mechanical
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properties were evaluated under compression in two different loading directions using experiments, and the effects of the face sheets that were added onto the top and bottom of the Microlattice core were investigated compared with the conventional Microlattice without the face sheets.
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SPECIMEN PREPARATION Fig. 3 depicts a schematic of the fabrication process of the miniature sandwich panel. First,
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the template was formed within a pool of Thiolene UV monomer resin. A glass slide was laid together with a microfiche mask on the top. Parallel and uniform UV rays were illuminated through the mask in four evenly distributed and constantly inclined directions, which produced
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a group of out-of-plane polymer beams in the resin pool owing to the SPPW effect [6]. The polymer beams intersected with each other at a constant interval, forming a truss structure
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under the top glass slide. After the truss structure was removed from the resin pool, it was washed out in toluene to remove the residual resin on its surface and its bottom side was polished to be flat. The top glass side was then carefully detached using a razor blade, leaving the template for a Microlattice.
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Separately, two glass slides were prepared with fresh UV monomer resin painted on. Thereafter, the glass slides were placed on the top and bottom of the template. Single UV ray was vertically illuminated from the bottom without any microfiche mask to solidify the resin.
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After the assembly of the template and glass slides was turned over upside down, the single UV ray was again illuminated from the bottom. The resin on the other glass slide was then
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solidified, forming a polymer sandwich template between the two glass slides. Thereafter, the assembly of the template and slides was fully hardened in an electric oven for 1 hr at 120oC. A uniform Ni-P layer was deposited on the surface using electroless plating, with the layer thickness controlled by the plating time. Lastly, the polymer template was etched out in a 60 oC solution of 3 mole NaOH and methanol at a ratio of 1:1 for 4 hr, after the lateral sides of the assembly were polished. As the
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etching was proceeded, the glass slides attached on the top and bottom surfaces of the polymer sandwich templates separated without any intervention. The etching solution was then slowly diluted with deionized (DI) water. For the specimens with a density lower than 10-3
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g/cc, the DI water was diluted again with acetone and then dried at 50 oC, leaving behind a final specimen comprising the hollow truss structure made of Ni-P walls. The technical details of the
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plating and etching are given in Han et al.’s paper [9].
Figs. 4(a) and 4(b) show photos for the UV-exposure system that was used to form the out-
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of-plane truss elements of the template and a template assembled with the glass slides, respectively. Fig. 5(a) shows a finished Microlattice sandwich specimen. The out-of-plane struts had a circular and hollow cross-section, and every four struts were joined at a point and connected to the face sheets with small fillets. The face sheets comprise holes with a diameter
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that was approximately four times the radius of the hollow struts. Another sets of specimens were prepared without the top and bottom faces, i.e., the conventional Microlattice. Fig. 5(b) shows the CAD images of the Microlattice sandwich specimen. The inset shows a unit cell of the
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Microlattice core. Both the conventional Microlattice and Microlattice sandwich specimens comprised the out-of-plane elements with an identical radius and length of a = 0.1125 mm and
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l = 2.23 mm, respectively. The distance between the holes on the face sheets was c = 1.6 mm. However, the inclination angle was almost constant at ω = 59.5o, regardless of the presence of the face sheets. The dimensions of the finished specimens were measured using an optical microscope. The overall dimensions of the Microlattice sandwich specimens were approximately width of 14.5mm, depth of 14.5mm, and height of 4mm.
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COMPRESSION TEST To measure the primary mechanical properties of the specimens of the Microlattice-cored
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miniature sandwich and the conventional Microlattice, compression tests were performed on an INSTRON 8872 frame, in which the displacement was controlled at a rate of 0.005 mm/s. A Lebow 3397–50 load cell was used to measure the load. The displacement that was measured
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using a built-in linear variable differential transformer (LVDT) was taken as that of the specimens, because the stiffness of the specimens was very low compared with that of the test
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frame. The specimens were placed between a pair of brass compression platens. The compression force was applied in two different directions, i.e., the out-of-plane and the inplane directions.
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ANALYTIC SOLUTION
For the original conventional Microlattice, analytic solutions of the relative density and the compressive strength under the out-of-plane compression are available in Valdevit and Godfrey
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[8]. For the Microlattice sandwich with the thin face sheets on the top and bottom, new equations were derived. First, the relative density is given by
(
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ρ 4πa + l cos2 ω t = 2 . 2 ρ s l cos ω ⋅ sin ω
--- (1)
Here, the subscript “s” means the property of the constituent material of the thin wall composing the hollow truss architecture, and t is the wall thickness. The first and second terms correspond to the core and the face sheets, respectively.
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In the out-of-plane compression tests, no failure was observed to occur on the face sheets, which will be discussed below. Therefore, assuming that the strength of the Microlattice
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sandwich is governed by failures occurring in the out-of-plane hollow struts on the core, the compressive strength is estimated to be exactly the same as those for the hollow octahedron truss, given in Lee, et al. [11]:
4π sin ω ⋅ at σc. cos 2 ω ⋅ l 2
--- (2)
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σo =
Here, the critical strength of the out-of-plane hollow struts,σ c, is taken as a minimum among
for plastic yielding.
for Euler buckling.
for local buckling.
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σ c = σ os π 2 a 2 σ c = min 2 Es 2l 1 t Es 2 3(1 − ν s ) a
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the values corresponding to the three failure modes as follows:
In the in-plane compression tests, the failure was observed to first occur on the face sheets,
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which will be discussed below. Therefore, we assume that the face sheets supported the entire load. So if the failure occurs due to the plastic yielding at the narrow regions of the face sheets
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that are divided by the holes joining the four hollow struts, the compressive strength is given by
σo =
( c − 4 a )t σ os . cl sin ω
--- (3)
If the failure occurs due to the plastic buckling of the face sheets, the strength under the inplane compression is given by:
σo =
t σ os . kl sin ω
--- (4)
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Here, k is the relative eccentricity, scaled by the thickness of the face sheets. If the failure occurs due to the elastic buckling of the face sheets, the strength is given by:
49 π 2 E s t3 36 (1 − ν 2 ) ( c − 4 a ) 2 ⋅ l sin ω .
--- (5)
See Appendix A for the derivations of Equations (3), (4), and (5).
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RESULTS
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σo =
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Out-of-plane compression
Figs. 6(a) and 6(b) show the stress–strain curves measured from the out-of-plane compression tests for the conventional Microlattices and Microlattice sandwiches with two different wall thicknesses. It is surprising that, for a given wall thickness, the stress-strain curve of the Microlattice sandwich was very similar to that of the conventional Microlattice. Namely, the
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addition of the face sheets to the bottom and top of the conventional Microlattices did not result in any reinforcement effects, contrary to our expectation. We suspect that the friction at
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the contact with the compression platens would exert a reinforcing effect on the strength and stiffness of the conventional Microlattice. Finite element analyses (FEAs) were carried out to
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simulate the effect of the friction. Figs. 7(a) and 7(b) depict the finite element models of the unit cells of the conventional Microlattice and Microlattice sandwich, respectively. Thin but perfectly rigid plates, simulating the compression platens, were placed above the top surface and under the bottom surface of the conventional Microlattice, and 10 different friction coefficients in the range of f = 0 to 0.9 were given to the contacts with the top and bottom surfaces. The vertical displacement of the top rigid plate was applied downward to simulate the
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compression test, while the bottom rigid plate was fixed. For the Microlattice sandwich, the compressive load was applied with and without the rigid plates placed above and below, as
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shown in Fig. 7(b). Namely, the vertical displacement was applied directly onto the face sheets (left) or on the top (while fixed on the bottom) via the rigid plates (right).
Fig. 8(a) shows the force versus the displacement curves that were estimated from the
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FEAs for the conventional Microlattice and the Microlattice sandwich. As mentioned above, because the architecture of the Microlattice lacks the in-plane elements, it was expected that
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its strength would be much lower than that of the one with the attached face sheets. The FEA results, however, revealed that the friction with the surface of the compression platens greatly affected the apparent strength of the Microlattice under compression. That is, when the friction coefficient was lower than f = 0.5, the strength of the Microlattice was much lower than
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that of the Microlattice sandwich, as expected. In contrast, when the friction coefficient was equal to or higher than f = 0.5, the strength of Microlattice was almost the same as that of the Microlattice sandwich, as observed in the experiments, shown in Fig. 6. Because the
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conventional Microlattice specimens tested in this work were fabricated of NiP foils with thicknesses of t = 1 to 10 µm, even the polished surfaces of the brass compression platens
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failed to provide a sufficiently smooth and hard contact, which would have resulted in the low friction. Hence, it seemed that the friction coefficient between the top (or bottom) end of the hollow tubular struts and the compression platens was higher than f = 0.5, and the high friction resulted in the barely discernible difference between the compressive behaviors of the conventional Microlattices and the Microlattice sandwich. This is why we carried out the in-
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plane compression tests to prove the strength enhancement through the attachment of the face sheets, because the high aspect minimized the effect of the friction [16].
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Fig. 8(b) shows side views of the unit cell models of the conventional Microlattice when it was compressed between the rigid platens with two different friction coefficients, f = 0.4 (left) and 0.5 (right). With f = 0.4, the Microlattice model revealed substantial horizontal slips at both
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the top and the bottom, which resulted in a low strength and stiffness, as shown in Fig. 8(a). In contrast, with the small increment of the friction coefficient to f = 0.5, the Microlattice model
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revealed no horizontal slip at all, which resulted in a strength and stiffness as high as those of the Microlattice sandwich models, as mentioned above. In contrast, for the Microlattice sandwich, the friction effect on the strength was negligible, and even with the low friction coefficient of f = 0.1, the strength was almost the same as that when only the vertical
the compression platens).
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In-plane compression
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displacement was directly applied on the face sheet surface without the rigid plates (simulating
Figs. 9(a) to 9(d) show the stress-strain curves that were measured from the in-plane
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compression tests. Each figure contains two curves for the conventional Microlattice and the Microlattice sandwich specimens of similar thicknesses in order to examine the effect of the face sheets that were attached to the Microlattice core. The stress-strain curves in Figs. 9(a) and 9(b) were measured from the two types of specimens with similar wall thicknesses of about t = 13 µm and t = 9 µm, respectively. In the figures, all of the stress-strain curves revealed
several fluctuations after the first peaks in both of the specimens with and without the face 11
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sheets. Movies S1 and S2 in Complementary Materials are the videos taken from the Microlattice sandwich specimens. Each video initially revealed the overall elastic deformation
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before the concentration of the plastic deformation (i.e., plastic buckling) at a specific section, leading to a local fracture, while the elastic deformation was recovered in the remaining area. Thereafter, the same behaviors were repeated in other sections, leaving the fluctuation and the
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large permanent deformations in the stress-strain curves, as shown in Figs. 9(a) and 9(b).
The stress-strain curves in Fig. 9(c) were measured from the specimens with and without
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the face sheets with a similar wall thickness of about t = 3 to 5 µm. Both stress-strain curves comprise only two fluctuations, and they left much lower permanent deformations than those measured from the specimens with the thicker walls that are shown in Figs. 9(a) and 9(b). Movie S3 for the Microlattice sandwich specimen initially revealed the overall elastic
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deformation before the plastic deformation was concentrated at a specific section without any elastic recovery in the remaining area, but it did not lead to the local fracture. Then, the same failure behavior occurred in another section, resulting in the lesser fluctuations and the lesser
curves in Figs. 9(c).
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permanent deformations in the stress-strain curves, as shown in the corresponding stress-strain
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The stress-strain curves in Fig. 9(d) were measured from the specimens with and without the face sheets with a similar wall thickness of about t = 1 µm. Both of the stress-strain curves showed no fluctuations and left almost no permanent deformations. Movie S4 for the Microlattice sandwich specimen revealed the overall elastic deformation that was maintained until the end without any local plastic deformation or fracture, as shown in the corresponding stress-strain curve in Fig. 9(d). 12
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Figs. 10(a) and 10(b) compare the images taken at four stages from the specimens with and without the face sheets with similar wall thicknesses of about t = 3 to 5 µm. Both of the images on the second row of the two figures revealed the local collapse at one section at the
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compressive strain of ε = 0.15, and both of the images on the third row revealed more local collapses at the highest compressive strain of ε = 0.3. After the unloading, shown on the fourth
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row, both of the images revealed some permanent deformations. For more details, see Movies S3 and S5, the latter of which is another video taken from the conventional Microlattice
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specimens with t = 3.14 µm. Overall, at a given (or similar) wall thickness, the behaviors of the conventional Microlattice specimens were pretty much the same as those of Microlattice sandwich specimens, and they varied according to the wall thickness, as shown in all of the stress-strain curves in Fig. 9. The only difference in the behaviors of the two types of the
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specimens was that the conventional Microlattices revealed much more compliant deformations, which were scaled by the initial linear portions of the stress-strain curves, because of their kinematically-indeterminate architecture, as mentioned above.
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Fig. 11(a) summarizes the effect of the added face sheets on the strengths and stiffness under the in-plane compression. For this purpose, the strengths or stiffness per the densities of
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the Microlattices sandwiches are divided by those of the conventional Microlattices, and the ratios are plotted according to the wall thickness. The figure shows that the addition of the face sheets to the conventional Microlattices increased the strength per density more than twice, if the wall thickness was higher than about t = 4 µm. Otherwise, the strength per density was not increased or even decreased. The addition of the face sheets also increased the stiffness per density up to 9 to 15 times, if the wall thickness was equal to or higher than about t = 4 µm. 13
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Otherwise, the stiffness per density was not increased by such an amount. Overall, the reinforcements owing to the addition of the face sheets were more prominent in terms of the
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stiffness than in terms of the strength. However, for both the strength and the stiffness, the reinforcing effect became much less significant, if the wall thickness was lower than about t = 4 µm. This finding means that the very thin face sheets were easily buckled over the entire area
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even at the beginning of loading and they continued to deform further as the load level was increased.
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Fig. 11(b) is a chart of the relative density versus the compressive strength that has been normalized by the yield strength of the constituent material, Ni-P foil. The measured values, denoted by the black circles, are plotted together with the estimations using the analytic solutions, denoted by the solid, dotted, and dashed lines, in the figure. In general, in the high
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density region, the measured values agreed soundly with the estimation from Equations (3) or (4), corresponding to the plastic yielding or buckling of the face sheets, respectively, whereas in the low density region, the measured values instead agreed with the estimation by Equation (5),
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corresponding to the elastic buckling of the face sheets. The transition between the elastic buckling and plastic buckling occurred at ρrel = 0.0074, which corresponded to the wall
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thickness, t = 4.13 µm. This wall thickness at the transition that was estimated by the analytic solutions agreed well with the measured one, t = 4.66 µm, that is mentioned above. In comparison with Fig. 11(a), the inferior properties observed at the thin wall were fairly well explained by the transition of the failure mode to the elastic buckling, which also agreed with deformation patterns, that are shown in Fig. 10(b) and Movies S1 to S4. Table 1 lists the measured properties including the thickness of Ni-P walls of each specimen. 14
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DISCUSSION
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As shown in Figs. 6 to 8, the high friction hindered the ends of the hollow tubular struts from horizontally sliding on the top and bottom, resulting in an overestimation of the strengths of the conventional Microlattice under the out-of-plane compression. This phenomenon seems
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to be dominant in failure of a Microlattice with hollow tubular struts of a high slenderness ratio like the specimens tested in this study. Generally, friction is not the only influencing factor. Unit
Microlattice structures [17].
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cell size and sample size (relative to unit cell size) may do play a role in node constraint for
The novel material with the ultralow density and the periodic micro architecture, which has been described in the present paper, was a type of monolithic sandwich. Also, it was shown
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that the addition of the face sheets substantially increased the strength and stiffness under the in-plane compression, if the wall was thicker than a certain limit. A sandwich panel is well known as an optimal structure, designed to support a bending load at light weights. If the
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Microlattice sandwich is to be used to support bending loads, the following issues must be addressed. First, it is likely that the face sheets composed of a foil, whose thickness is identical
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to that composed of the Microlattice core thickness, do not have a sufficient resistance against bending loads. Namely, one of the face sheets in a sandwich panel is supposed to support compressive stress, while the other supports tensile stress. The flat cross section of the face sheet in the compressed side provides a very low second moment of inertia, and it is never appropriate in the resistance against buckling. Second, the holes on the face sheets might cause a stress concentration around the fillets, as they are connected to the Microlattice core.
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Additionally, if a mass transfer is to be blocked across the sandwich panel, the face sheets with the regular-patterned holes do not provide a shield that is as effective as that of the ordinary
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panel. Therefore, the Microlattice sandwich, introduced in this study, is typically not a proper support for bending loads, especially if an ultralow density is to be achieved with a wall
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thickness of submicron order. The Microlattice sandwich could, however, be used as the core of the optimal sandwich panel, where the much higher strength and stiffness for bending loads
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are realized together with an ultra-lightness. That is, each face sheet of the Microlattice sandwich provides a large surface area that could be useful as a firm attached for an additional thin, dense, and hard face with a minimum amount of adhesive.
If it is to be used for a micro robot or bio-mimic drone as small as a beetle or a fly, the
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technique described in the present paper needs to be further improved to fabricate a sandwich with much thinner overall dimensions. Arias et al. [14] and Kolodziejska et al. [15] reported fabrication techniques of sandwich panels with the overall thicknesses of the 1 mm order;
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however, the densities of their sandwiches were too high to be used for this application. Another problem was that the facesheets and core in Arias et al.’s sandwich panel was joined
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using solder or polymer adhesive [14], and Kolodziejska et al. ’s sandwich panel was composed of polymers only [15]. Consequently, their use at a high temperature environment would be limited by melting or softening of the constituent materials or the bonding materials. On contrast, the Microlattice sandwich described in this study was monolithically composed of the Ni alloy foils without any bonding or joining with an additional material. Hence, the temperature range of its use must be much further expanded. In the authors’ research group, a
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new technique is under development for the fabrication of a novel sandwich, which is optimized to support bending loads, with an overall thickness and density of about 1 mm and
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0.01g/cc, respectively, based on this very Microlattice sandwich technique.
CONCLUSIONS
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We introduced a novel technique for the fabrication of a Microlattice-cored miniature sandwich panel with an ultralow density. This sandwich panel was made of a monolithic Ni-P
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foil without any bonding between the core and the faces. As a preliminary study, the mechanical properties were evaluated under compression in two different loading directions using experiments, and the effects of the face sheets that were added on the top and bottom were investigated compared with the conventional Microlattice without the face sheets. The
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conclusions are as follows:
i) Because the specimens were fabricated of NiP foils with thicknesses from 1 to 10 µm, even the polished surfaces of the brass compression platens failed to provide a sufficiently
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smooth and hard contact, resulting in the friction higher than f = 0.4. ii) The high friction hindered the ends of the hollow tubular struts from horizontally sliding
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on the top and bottom, resulting in an overestimation of the strengths of the conventional Microlattice under the out-of-plane compression, and almost no difference was observed between the conventional Microlattices and the Microlattice sandwiches in terms of their compressive behaviors.
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iii) Under the in-plane compression, because of the high aspect ratio, the effect of the high friction was minimized, and the intrinsic strengths of the conventional Microlattices were measured, whereby the true effects of the face sheet additions could be evaluated.
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iv) If the wall thickness was higher than about t = 4 µm, the addition of the face sheets to the conventional Microlattices increased the strength per density more than twice and the
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stiffness per density up to 9 to 15 times. Otherwise, for both the strength and stiffness, the reinforcing effect became much less significant, because the very thin face sheets were easily
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buckled across the entire area even at the beginning of the loading.
v) In the high density region, the failure of the Microlattice sandwich under the in-plane compression was governed by the plastic buckling or yielding of the face sheets, whereas in the low density region, the failure was governed by the elastic buckling of the face sheets. The
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transition between the elastic and plastic buckling at the relative density of ρrel = 0.0074, which showed a fairly sound agreement with the deformation patterns observed in the experiments,
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was estimated using analytic solutions.
ACKNOWLEDGMENTS
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This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT and Future Planning (No.2015R1A2A1A01003702).
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[6] A.J. Jacobsen, W. Barvosa-Carter, S. Nutt, Compression behavior of micro-scale truss
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structures formed from self-propagating polymer waveguides, Acta Mater. 55 (2007) 67246733.
[7] J.C. Maxwell, On the calculation of the equilibrium and stiffness of frames, Philos. Mag. 27
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(1864) 294-299.
[8] L. Valdevit, S.W. Godfrey, Compressive strength of hollow microlattices: Experimental
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characterization, modeling, and optimal design, J. Mater. Res. 28 (2013) 2461-2473. [9] S.C. Han, J.W. Lee, K. Kang, A new type of low density material; Shellular, Adv. Mater. 27 (2015) 5506-5511.
[10] S. Hyde, Z. Blum, T. Landh, S. Lidin, B.W. Ninham, S. Andersson, K. Larsson, The Language of Shape, Elsevier, 1996.
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[11] M.G. Lee, J.W. Lee, S.C. Han, K. Kang, Mechanical analyses of “Shellular”, an ultralowdensity cellular metal, Acta Mater. 103 (2016) 595-607.
https://www.thoughtco.com/plant-tissue-systems-373615 .
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[12] R. Bailey, Plant Tissue Systems, ThoughtCo, Mar. 4, 2017.
[13] L. Valdevit, A.J. Jacobsen, J.R. Greer, W.B. Carter, Protocol for the optimal design of
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multifunctional structures: from hypersonics to micro-architected materials, J. Am. Ceram. Soc. 94 (2011) s15-s34.
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[14] F. Arias, P.J.A. Kenis, B. Xu, T. Deng, O.J.A. Schueller, G.M. Whitesides, Y. Sugimura , A.G. Evans, Fabrication and characterization of microscale sandwich beams, J. Mater. Res. 16 (2001) 597-605.
[15] J.A. Kolodziejska, C.S. Roper, S.S. Yang, W.B. Carter, A.J. Jacobsen, Research Update:
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Enabling ultra-thin lightweight structures: Microsandwich structures with microlattice cores. APL Materials (2015) 3, 050701.
[16] M. Zupan, V.S. Deshpande, N.A. Fleck, The out-of-plane compressive behaviour of woven-
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core sandwich plates. Eur. J. Mech. A Solid 23 (2004) 411–421. [17] R.E. Doty, J.A. Kolodziejska, A.J. Jacobsen, Hierarchical polymer microlattice structures. Adv.
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Eng. Mater. 14 (2012) 503-507.
[18] N. Wicks, J.W. Hutchinson, Optimal truss plates. International J. Solids and Structures. 38 (2001) 5165-5183.
[19] A.C. Ugural, Stresses in Plates and Shells, McGraw-Hill, New York, 1981, p. 158.
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APPENDIX A Fig. A1 shows a schematic of the Microlattice sandwich under the in-plane compression.
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Note that only the face sheets resist against the compressive load. Therefore, the force equilibrium in the x-direction of the unit cell gives
∑F
y
=0
σ o × c × l sin ω = Fcr
- - - (A1)
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Here, σo and Fcr are the compressive strength of the Microlattice sandwich and the failure load
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of a single face sheet itself, respectively. If the failure occurs due to the plastic yielding at the narrow regions of the face sheets that are connected to the four hollow struts (i.e., at the gray regions on the front view of Fig. A1), the failure load is simply given by Fcr = ( c − 4 a )t ⋅ σ os . Substituting this into Equation (A1), the compressive strength is given by
( c − 4 a )t σ os . cl sin ω
--- (3)
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σo =
When the face sheets were relatively thick such as t = 13 µm and t = 9 µm, they failed due to plastic buckling on the wide areas of the face sheets after the overall initial elastic
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deformation of bending, as shown in Movies S1 and S2 in Complementary Materials,
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respectively. The left of Fig. A2 shows the buckled shape of the face sheets of the first section from the top just prior to a local fracture. The right of the figure is a schematic of the failure mode, that is, longitudinal collapse of a thin plate with clamped ends, which accompanies the plastic hinges at three points of the top, middle, and bottom. On the plastic buckling, the external energy supplied by the critical force, Fcr, is given by (External Engery) = Fcr × δ ,
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while the internal energy consumed by rotations at the plastic hinges is given by (Internal Engery) = 4 M p × θ .
δ = 2×
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Here, θ is the rotation angle, and δ is the displacement along the critical force, given by 2 2 c (1 − cos θ ) ≈ c1 − 1 + θ = cθ . 2 2 2
And M p is the limit moment needed to yield a plastic hinge, given as a function of the thickness,
ct 2 σ os . 4
Hence, energy balance results in
Fcr =
2t 2
θ
σ os .
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Mp =
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t, the length, c, and the yield strength of the constituent material, σos, as follows;
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The value of Fcr is a function of the initial value of θ as well as t and σos. The initial θ is related to the eccentricity, e, (due to the initial curvature existing in the face sheets) as θ =2e/c. And if e
Fcr =
ct σ os . k
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is represented as e =k t, the critical force, Fcr, is given by
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Substituting this into Equation (A1), the compressive strength due to the plastic buckling is given by
σo =
t σ os . kl sin ω
--- (4)
Here, k is the relative eccentricity, scaled by the thickness of the face sheets. If the failure occurs due to elastic buckling on the wide areas of the face sheets, the failure stress in the face sheets at the buckling, σcr, is given in Wicks and Hutchinson [17] as follows: 22
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σ cr =
49 × 3 π 2 Es t 432 (1 −ν 2 ) c − 4a
2
Interestingly, Equation (A2) is very similar to the solution of Ugural [18] for the minimum
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buckling load of the lateral shell, as follows:
π 2 Es t . 3(1 −ν 2 ) c − 4a 2
σ cr =
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Therefore, the following equation applies: 49 × 3 π 2 Es t Fcr = ct. 432 (1 −ν 2 ) c − 4a 2
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- - - (A2)
The above equations are based on the assumption that the boundaries of a flat rectangular sheet are simply supported, allowing for a free rotation. However, the face sheets of the specimen are not flat because of the fillets that are connected to the hollow tubular struts.
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Consequently, the fillets played a constraining role at the boundaries: therefore, considering this constraint, Equation (A2) is slightly modified as follows: 49 × 3 × 4 π 2 Es t Fcr = ct. 432 (1 − ν 2 ) c − 4a 2
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- - - (A3)
By substituting Equation (A3) into (A1), the compressive strength is given by
49 π 2 E s t3 . 36 ( 1 − ν 2 ) ( c − 4 a )2 ⋅ l sin ω
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σo =
- - - (5)
Supplementary Material Movies S1-S5
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FIGURE CAPTIONS Fig. 1. A 2D schematic of deformation pattern of the Microlattice subjected to out-of-plane
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compression. Fig. 2. (a) a schematic of leaf anatomy, reproduced by courtesy of Richard Wheeler [12], and (b) optical image taken from a leaf of magnolia grandiflora (scale bar: 500 µm).
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Fig. 3. A schematic of the fabrication process of a Microlattice sandwich.
Fig. 4. Photos for: (a) A UV exposure system used to form the out-of-plane truss elements of a
sandwich specimen.
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template, and (b) a template (assembled with glass slides) along with a Microlattice
Fig. 5. (a) A photo of a Microlattice sandwich specimen, and (b) CAD images of a Microlattice sandwich with the dimensions.
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Fig. 6. Stress-strain curves measured from out-of-plane compression tests for the conventional Microlattice and the Microlattice sandwich with two different wall thicknesses: (a) about t = 3 µm and (b) about t = 1 µm.
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Fig. 7. Finite element models of the unit cells of: (a) a conventional Microlattice and (b) a Microlattice sandwich.
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Fig. 8. (a) Force versus displacement curves estimated from an FEA for a conventional Microlattice and a Microlattice sandwich, and (b) side views of the unit cell models of the conventional Microlattice compressed between the rigid plates with two different friction coefficients, f = 0.4 (left) and f = 0.5 (right).
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Fig. 9. Stress-strain curves measured from in-plane compression tests for the conventional Microlattice and Microlattice sandwich specimens with similar thicknesses: (a) about t =
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13 µm, (b) t = 9 µm, (c) about t = 3 to 5 µm, and (d) about t = 1 µm. Fig. 10. Images taken at four stages from specimens with similar wall thicknesses of about t = 3 to 5 µm under in-plane compression: (a) conventional Microlattice and (b) Microlattice
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sandwich specimens.
Fig. 11. (a) Ratio of the strengths or stiffness per the densities of Microlattices sandwiches to
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those of the conventional Microlattices plotted according to the wall thickness, and (b) a chart of the relative density versus the compressive strength normalized by the yield strength of the constituent material, Ni-P foil.
Fig. A1. A schematic of the unit cell of a Microlattice sandwich under in-plane compression.
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Fig. A2. Buckled shape of the face sheets of the first section from the top of the specimen with t = 9 µm under in-plane compression just prior to a local fracture, and a schematic of the failure mode, that is, longitudinal collapse of a thin plate with clamped ends,
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accompanying three plastic hinges.
CAPTIONS for Movies S1 to S5 Movie S1. In-situ video (played at 10x speed) of a Microlattice sandwich specimen under inplane compression with the wall thickness of t = 13.27 µm. Movie S2. In-situ video (played at 10x speed) of a Microlattice sandwich specimen under inplane compression with the wall thickness of t = 9.15 µm.
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Movie S3. In-situ video (played at 10x speed) of a Microlattice sandwich specimen under inplane compression with the wall thickness of t = 4.66 µm. Movie S4. In-situ video (played at 10x speed) of a Microlattice sandwich specimen under in-
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plane compression with the wall thickness of t = 1.37 µm.
Movie S5. In-situ video (played at 10x speed) of a conventional Microlattice specimen under in-
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plane compression with the wall thickness of t = 3.14 µm.
Table 1. Properties measured for the conventional Microlattice and Microlattice sandwich specimens. Wall Thickness Relative Density (µ µm)
Strength (MPa)
Young's Modulus (MPa)
Conventional
3.11
3.99E-03
0.2167
12.60
Sandwich
3.56
6.40E-03
0.2485
12.98
Conventional
1.14
1.46E-03
0.02835
2.109
Sandwich
1.00
1.80E-03
0.02801
2.002
Conventional
13.35
1.93E-02
0.2304
3.400
Sandwich
13.27
2.39E-02
0.6171
60.90
Conventional
8.78
1.27E-02
0.1322
1.706
Sandwich
9.15
1.65E-02
0.3719
23.69
Conventional
3.14
4.55E-03
0.01998
0.288
Sandwich
4.66
8.39E-03
0.09271
5.050
Conventional
1.28
1.87E-03
0.002924
0.048
Sandwich
1.37
2.47E-03
0.003742
0.467
Conventional
1.08
1.57E-03
0.001719
0.041
Sandwich
0.80
1.43E-03
0.001226
0.144
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Out-of-plane
Specimen Type
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Load Direction
In-plane
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Fig. 1. A 2D schematic of deformation pattern of the Microlattice subjected to out-of-plane compression.
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Fig. 2. (a) a schematic of leaf anatomy, reproduced by courtesy of Richard Wheeler [12], and (b) optical image taken from a leaf of magnolia grandiflora (scale bar: 500 µm).
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Fig. 3. A schematic of the fabrication process of a Microlattice sandwich.
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Fig. 4. Photos for: (a) A UV exposure system used to form the out-of-plane truss elements of a template, and (b) a template (assembled with glass slides) along with a Microlattice sandwich specimen.
(Scale bar: 2 mm)
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Fig. 5. (a) A photo of a Microlattice sandwich specimen, and (b) CAD images of a Microlattice sandwich with the dimensions. (Scale bar: 2 mm)
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Fig. 6. Stress-strain curves measured from out-of-plane compression tests for the conventional Microlattice and the Microlattice sandwich with two different wall thicknesses: (a) about t = 3 µm and (b) about t = 1 µm.
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Fig. 7. Finite element models of the unit cells of: (a) a conventional Microlattice and (b) a Microlattice sandwich.
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Fig. 8. (a) Force versus displacement curves estimated from an FEA for a conventional Microlattice and a Microlattice sandwich, and (b) side views of the unit cell models of the conventional Microlattice compressed between the rigid plates with two different friction factors, f = 0.4 (left) and f = 0.5 (right).
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Fig. 9. Stress-strain curves measured from in-plane compression tests for the conventional Microlattice and Microlattice sandwich specimens with similar thicknesses: (a) about t = 13 µm, (b) t = 9 µm, (c) about t = 3 to 5 µm, and (d) about t = 1 µm.
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Fig. 10. Images taken at four stages from specimens with similar wall thicknesses of about t = 3 to 5 µm under in-plane compression: (a) conventional Microlattice and (b) Microlattice sandwich specimens.
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Fig. 11. (a) Ratio of the strengths or stiffness per the densities of Microlattices sandwiches to those of the conventional Microlattices plotted according to the wall thickness, and (b) a chart of the relative density versus the compressive strength normalized by the yield strength of the constituent material, Ni-P foil.
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Fig. A1. A schematic of the unit cell of a Microlattice sandwich under in-plane compression.
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Fig. A2. Buckled shape of the face sheets of the first section from the top of the specimen with t = 9 µm under in-plane compression just prior to a local fracture, and a schematic of the failure mode, that is, longitudinal collapse of a thin plate with clamped ends, accompanying three plastic hinges.
S1359-6454(17)30988-6
DOI:
10.1016/j.actamat.2017.11.045
Reference:
AM 14216
To appear in:
Acta Materialia
Received Date: 10 August 2017 Revised Date:
8 November 2017
Accepted Date: 16 November 2017
Please cite this article as: D.H. Choi, Y. Chang Jeong, K. Kang, A monolithic sandwich panel with microlattice core, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.11.045. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
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A Monolithic Sandwich Panel with Microlattice Core Dae Han Choi, Yoon Chang Jeong, Kiju Kang
Gwangju 61186, Republic of Korea
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*Corresponding author ([email protected])
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School of Mechanical Engineering, Chonnam National University
ABSTRACT
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We introduced a novel technique to fabricate a Microlattice-cored miniature sandwich panel with an ultralow density. This sandwich panel was made of a monolithic Ni-P foil without any bonding between the core and the faces. As a preliminary study, the mechanical properties were evaluated under compression in two different loading directions, and the effects of the
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face sheets added on the top and bottom were investigated in comparison to the conventional Microlattice without the face sheets. Under the out-of-plane compression, almost no difference was evident between the conventional Microlattice and the Microlattice sandwich in their
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compressive behaviors, because of the high friction with the compression platens. However,
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under the in-plane compression, the addition of the face sheets to the conventional Microlattice significantly increased the strength and stiffness, when the wall thickness was higher than a certain limit. Otherwise, the reinforcing effect became much less significant, because the very thin face sheets were easily buckled across the entire area even at the beginning of the loading.
Keywords: cellular material; thin film; sandwich panel; UV lithography 1
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INTRODUCTION
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Recently, a novel material with a density less than 0.01 g/cc (here, we refer to it as ultralow density) and a periodic micro architecture garnered attention because of the exceptionally high strength and stiffness of the material at ultralow densities, which might be a technical
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breakthrough for the near future. The Microlattice introduced in 2012 [1] was the first realization, and was followed by Nanolattice [2, 3] and Mechanical metamaterial [4]. Nowadays,
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these three materials are often referred to as a type of mechanical metamaterial in a wide sense, whose mechanical properties are defined by its geometry rather than its composition. These three materials are common in terms of the fabrication principle and geometry. Namely, their fabrications start with the formation of a polymer template based on 3D UV lithography,
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which is followed by the deposition of a hard constituent material such as metal or ceramic, and the polymer template is then etched out, leaving a hollow structure as the final product. Also, these materials comprise hollow truss-like architectures. The hierarchical structures and
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stretching dominated deformations [5] result in superior mechanical properties. These materials differ in the dimension such as overall size or feature size and the specific techniques
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of the UV lithography. In fact, the different UV techniques dictate the architectures possible, and hence do play a significant role in the feasible material space. Among these three materials, the productivity of Microlattice seems to be the best owing to the mask-based lithography, i.e., self-propagating polymer wave guide (SPPW) [6]. However, the SPPW forms the truss elements of the template only in the out-of-plane directions, because it uses UV rays that are illuminated only through a mask on a single (upper) side of the photo-
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monomer. Consequently, the structure of the Microlattice lacks the truss elements in the inplane directions, and it is neither statically determinate nor kinematically determinate
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according to Maxwell’s criterion [7], which is a necessary and sufficient condition for the rigidity of a truss with a minimal number of elements. Fig. 1 shows a 2D schematic of expected deformation pattern of the Microlattice subjected to out-of-plane compression. Therefore, the
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strength and stiffness might be significantly lower than the expectations [1, 8].
In 2015, Shellular, another material with an ultralow density and a periodic micro
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architecture, was introduced [9]. Its fabrication principle is the same as the previous three, particularly, Microlattice, because the template for Shellular is also fabricated based on SPPW; however, its micro architecture is totally different. Namely, Shellular is composed of polyhedron shells, or more ideally a single smooth and continuous shell with a triply periodic
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minimal surface (TPMS) [10]. Han et al. [9] insisted that the Shellular in the TPMS configuration also provides stretching-dominated deformation, resulting in mechanical properties that are comparable with those of the predecessors with the hollow truss architectures [8, 11]. The
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template of Shellular, however, comprises a much larger volume and a lower porosity than the truss-shaped ones of the three predecessors, thereby limiting the access to the interior space,
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which is likely to hinder uniform deposition of the coating material. A sandwich panel is composed of a thick porous soft core and thin dense hard faces, which provides very high bending strength and stiffness at light weight. Consequently, sandwich panels are widely used in engineering applications. We believe that, if a miniature sandwich panel could be fabricated with an ultralow density, as can be observed in numerous natural life forms such as the leaf [12], shown in Fig. 2, it could be very useful for building the structures of
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micro robots or bio-mimic drones in the near future. Microlattice is noticeably a promising candidate that can serve as the ultralow density core of a miniature sandwich panel, because
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the other two, i.e., Nanolattice [2] and Mechanical metamaterial [3] are too tiny and/or inefficient for mass fabrication. Although the Microlattice architecture lacks the in-plane truss elements [1, 13], it would become rigid when faces are attached to its top and bottom;
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however, it should be challenging to attach the faces to the core using the conventional adhesive bonding to build a micro-sandwich, because the adhesive is likely to fail in provision of
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a very thin and uniform bonding without substantial increase of the total weight, as observed in the references [14, 15], which negatively affects the realization of an ultralow density. In this paper, we introduce a novel technique for fabrication of a Microlattice-cored miniature sandwich panel with an ultralow density. This sandwich panel is composed of a
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monolithic Ni-P foil without any bonding between the core and faces. First, the template was formed using the SPPW technique combined with 2D lithography. The rest of the process was pretty much the same as those of the previous ultralow density materials with micro
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architectures. Namely, the template was then coated with a hard constituent material before it was etched out, leaving the miniature sandwich panel. As a preliminary study, the mechanical
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properties were evaluated under compression in two different loading directions using experiments, and the effects of the face sheets that were added onto the top and bottom of the Microlattice core were investigated compared with the conventional Microlattice without the face sheets.
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SPECIMEN PREPARATION Fig. 3 depicts a schematic of the fabrication process of the miniature sandwich panel. First,
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the template was formed within a pool of Thiolene UV monomer resin. A glass slide was laid together with a microfiche mask on the top. Parallel and uniform UV rays were illuminated through the mask in four evenly distributed and constantly inclined directions, which produced
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a group of out-of-plane polymer beams in the resin pool owing to the SPPW effect [6]. The polymer beams intersected with each other at a constant interval, forming a truss structure
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under the top glass slide. After the truss structure was removed from the resin pool, it was washed out in toluene to remove the residual resin on its surface and its bottom side was polished to be flat. The top glass side was then carefully detached using a razor blade, leaving the template for a Microlattice.
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Separately, two glass slides were prepared with fresh UV monomer resin painted on. Thereafter, the glass slides were placed on the top and bottom of the template. Single UV ray was vertically illuminated from the bottom without any microfiche mask to solidify the resin.
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After the assembly of the template and glass slides was turned over upside down, the single UV ray was again illuminated from the bottom. The resin on the other glass slide was then
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solidified, forming a polymer sandwich template between the two glass slides. Thereafter, the assembly of the template and slides was fully hardened in an electric oven for 1 hr at 120oC. A uniform Ni-P layer was deposited on the surface using electroless plating, with the layer thickness controlled by the plating time. Lastly, the polymer template was etched out in a 60 oC solution of 3 mole NaOH and methanol at a ratio of 1:1 for 4 hr, after the lateral sides of the assembly were polished. As the
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etching was proceeded, the glass slides attached on the top and bottom surfaces of the polymer sandwich templates separated without any intervention. The etching solution was then slowly diluted with deionized (DI) water. For the specimens with a density lower than 10-3
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g/cc, the DI water was diluted again with acetone and then dried at 50 oC, leaving behind a final specimen comprising the hollow truss structure made of Ni-P walls. The technical details of the
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plating and etching are given in Han et al.’s paper [9].
Figs. 4(a) and 4(b) show photos for the UV-exposure system that was used to form the out-
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of-plane truss elements of the template and a template assembled with the glass slides, respectively. Fig. 5(a) shows a finished Microlattice sandwich specimen. The out-of-plane struts had a circular and hollow cross-section, and every four struts were joined at a point and connected to the face sheets with small fillets. The face sheets comprise holes with a diameter
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that was approximately four times the radius of the hollow struts. Another sets of specimens were prepared without the top and bottom faces, i.e., the conventional Microlattice. Fig. 5(b) shows the CAD images of the Microlattice sandwich specimen. The inset shows a unit cell of the
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Microlattice core. Both the conventional Microlattice and Microlattice sandwich specimens comprised the out-of-plane elements with an identical radius and length of a = 0.1125 mm and
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l = 2.23 mm, respectively. The distance between the holes on the face sheets was c = 1.6 mm. However, the inclination angle was almost constant at ω = 59.5o, regardless of the presence of the face sheets. The dimensions of the finished specimens were measured using an optical microscope. The overall dimensions of the Microlattice sandwich specimens were approximately width of 14.5mm, depth of 14.5mm, and height of 4mm.
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COMPRESSION TEST To measure the primary mechanical properties of the specimens of the Microlattice-cored
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miniature sandwich and the conventional Microlattice, compression tests were performed on an INSTRON 8872 frame, in which the displacement was controlled at a rate of 0.005 mm/s. A Lebow 3397–50 load cell was used to measure the load. The displacement that was measured
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using a built-in linear variable differential transformer (LVDT) was taken as that of the specimens, because the stiffness of the specimens was very low compared with that of the test
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frame. The specimens were placed between a pair of brass compression platens. The compression force was applied in two different directions, i.e., the out-of-plane and the inplane directions.
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ANALYTIC SOLUTION
For the original conventional Microlattice, analytic solutions of the relative density and the compressive strength under the out-of-plane compression are available in Valdevit and Godfrey
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[8]. For the Microlattice sandwich with the thin face sheets on the top and bottom, new equations were derived. First, the relative density is given by
(
)
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ρ 4πa + l cos2 ω t = 2 . 2 ρ s l cos ω ⋅ sin ω
--- (1)
Here, the subscript “s” means the property of the constituent material of the thin wall composing the hollow truss architecture, and t is the wall thickness. The first and second terms correspond to the core and the face sheets, respectively.
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In the out-of-plane compression tests, no failure was observed to occur on the face sheets, which will be discussed below. Therefore, assuming that the strength of the Microlattice
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sandwich is governed by failures occurring in the out-of-plane hollow struts on the core, the compressive strength is estimated to be exactly the same as those for the hollow octahedron truss, given in Lee, et al. [11]:
4π sin ω ⋅ at σc. cos 2 ω ⋅ l 2
--- (2)
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σo =
Here, the critical strength of the out-of-plane hollow struts,σ c, is taken as a minimum among
for plastic yielding.
for Euler buckling.
for local buckling.
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σ c = σ os π 2 a 2 σ c = min 2 Es 2l 1 t Es 2 3(1 − ν s ) a
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the values corresponding to the three failure modes as follows:
In the in-plane compression tests, the failure was observed to first occur on the face sheets,
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which will be discussed below. Therefore, we assume that the face sheets supported the entire load. So if the failure occurs due to the plastic yielding at the narrow regions of the face sheets
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that are divided by the holes joining the four hollow struts, the compressive strength is given by
σo =
( c − 4 a )t σ os . cl sin ω
--- (3)
If the failure occurs due to the plastic buckling of the face sheets, the strength under the inplane compression is given by:
σo =
t σ os . kl sin ω
--- (4)
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Here, k is the relative eccentricity, scaled by the thickness of the face sheets. If the failure occurs due to the elastic buckling of the face sheets, the strength is given by:
49 π 2 E s t3 36 (1 − ν 2 ) ( c − 4 a ) 2 ⋅ l sin ω .
--- (5)
See Appendix A for the derivations of Equations (3), (4), and (5).
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RESULTS
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σo =
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Out-of-plane compression
Figs. 6(a) and 6(b) show the stress–strain curves measured from the out-of-plane compression tests for the conventional Microlattices and Microlattice sandwiches with two different wall thicknesses. It is surprising that, for a given wall thickness, the stress-strain curve of the Microlattice sandwich was very similar to that of the conventional Microlattice. Namely, the
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addition of the face sheets to the bottom and top of the conventional Microlattices did not result in any reinforcement effects, contrary to our expectation. We suspect that the friction at
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the contact with the compression platens would exert a reinforcing effect on the strength and stiffness of the conventional Microlattice. Finite element analyses (FEAs) were carried out to
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simulate the effect of the friction. Figs. 7(a) and 7(b) depict the finite element models of the unit cells of the conventional Microlattice and Microlattice sandwich, respectively. Thin but perfectly rigid plates, simulating the compression platens, were placed above the top surface and under the bottom surface of the conventional Microlattice, and 10 different friction coefficients in the range of f = 0 to 0.9 were given to the contacts with the top and bottom surfaces. The vertical displacement of the top rigid plate was applied downward to simulate the
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compression test, while the bottom rigid plate was fixed. For the Microlattice sandwich, the compressive load was applied with and without the rigid plates placed above and below, as
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shown in Fig. 7(b). Namely, the vertical displacement was applied directly onto the face sheets (left) or on the top (while fixed on the bottom) via the rigid plates (right).
Fig. 8(a) shows the force versus the displacement curves that were estimated from the
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FEAs for the conventional Microlattice and the Microlattice sandwich. As mentioned above, because the architecture of the Microlattice lacks the in-plane elements, it was expected that
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its strength would be much lower than that of the one with the attached face sheets. The FEA results, however, revealed that the friction with the surface of the compression platens greatly affected the apparent strength of the Microlattice under compression. That is, when the friction coefficient was lower than f = 0.5, the strength of the Microlattice was much lower than
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that of the Microlattice sandwich, as expected. In contrast, when the friction coefficient was equal to or higher than f = 0.5, the strength of Microlattice was almost the same as that of the Microlattice sandwich, as observed in the experiments, shown in Fig. 6. Because the
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conventional Microlattice specimens tested in this work were fabricated of NiP foils with thicknesses of t = 1 to 10 µm, even the polished surfaces of the brass compression platens
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failed to provide a sufficiently smooth and hard contact, which would have resulted in the low friction. Hence, it seemed that the friction coefficient between the top (or bottom) end of the hollow tubular struts and the compression platens was higher than f = 0.5, and the high friction resulted in the barely discernible difference between the compressive behaviors of the conventional Microlattices and the Microlattice sandwich. This is why we carried out the in-
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plane compression tests to prove the strength enhancement through the attachment of the face sheets, because the high aspect minimized the effect of the friction [16].
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Fig. 8(b) shows side views of the unit cell models of the conventional Microlattice when it was compressed between the rigid platens with two different friction coefficients, f = 0.4 (left) and 0.5 (right). With f = 0.4, the Microlattice model revealed substantial horizontal slips at both
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the top and the bottom, which resulted in a low strength and stiffness, as shown in Fig. 8(a). In contrast, with the small increment of the friction coefficient to f = 0.5, the Microlattice model
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revealed no horizontal slip at all, which resulted in a strength and stiffness as high as those of the Microlattice sandwich models, as mentioned above. In contrast, for the Microlattice sandwich, the friction effect on the strength was negligible, and even with the low friction coefficient of f = 0.1, the strength was almost the same as that when only the vertical
the compression platens).
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In-plane compression
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displacement was directly applied on the face sheet surface without the rigid plates (simulating
Figs. 9(a) to 9(d) show the stress-strain curves that were measured from the in-plane
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compression tests. Each figure contains two curves for the conventional Microlattice and the Microlattice sandwich specimens of similar thicknesses in order to examine the effect of the face sheets that were attached to the Microlattice core. The stress-strain curves in Figs. 9(a) and 9(b) were measured from the two types of specimens with similar wall thicknesses of about t = 13 µm and t = 9 µm, respectively. In the figures, all of the stress-strain curves revealed
several fluctuations after the first peaks in both of the specimens with and without the face 11
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sheets. Movies S1 and S2 in Complementary Materials are the videos taken from the Microlattice sandwich specimens. Each video initially revealed the overall elastic deformation
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before the concentration of the plastic deformation (i.e., plastic buckling) at a specific section, leading to a local fracture, while the elastic deformation was recovered in the remaining area. Thereafter, the same behaviors were repeated in other sections, leaving the fluctuation and the
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large permanent deformations in the stress-strain curves, as shown in Figs. 9(a) and 9(b).
The stress-strain curves in Fig. 9(c) were measured from the specimens with and without
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the face sheets with a similar wall thickness of about t = 3 to 5 µm. Both stress-strain curves comprise only two fluctuations, and they left much lower permanent deformations than those measured from the specimens with the thicker walls that are shown in Figs. 9(a) and 9(b). Movie S3 for the Microlattice sandwich specimen initially revealed the overall elastic
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deformation before the plastic deformation was concentrated at a specific section without any elastic recovery in the remaining area, but it did not lead to the local fracture. Then, the same failure behavior occurred in another section, resulting in the lesser fluctuations and the lesser
curves in Figs. 9(c).
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permanent deformations in the stress-strain curves, as shown in the corresponding stress-strain
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The stress-strain curves in Fig. 9(d) were measured from the specimens with and without the face sheets with a similar wall thickness of about t = 1 µm. Both of the stress-strain curves showed no fluctuations and left almost no permanent deformations. Movie S4 for the Microlattice sandwich specimen revealed the overall elastic deformation that was maintained until the end without any local plastic deformation or fracture, as shown in the corresponding stress-strain curve in Fig. 9(d). 12
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Figs. 10(a) and 10(b) compare the images taken at four stages from the specimens with and without the face sheets with similar wall thicknesses of about t = 3 to 5 µm. Both of the images on the second row of the two figures revealed the local collapse at one section at the
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compressive strain of ε = 0.15, and both of the images on the third row revealed more local collapses at the highest compressive strain of ε = 0.3. After the unloading, shown on the fourth
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row, both of the images revealed some permanent deformations. For more details, see Movies S3 and S5, the latter of which is another video taken from the conventional Microlattice
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specimens with t = 3.14 µm. Overall, at a given (or similar) wall thickness, the behaviors of the conventional Microlattice specimens were pretty much the same as those of Microlattice sandwich specimens, and they varied according to the wall thickness, as shown in all of the stress-strain curves in Fig. 9. The only difference in the behaviors of the two types of the
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specimens was that the conventional Microlattices revealed much more compliant deformations, which were scaled by the initial linear portions of the stress-strain curves, because of their kinematically-indeterminate architecture, as mentioned above.
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Fig. 11(a) summarizes the effect of the added face sheets on the strengths and stiffness under the in-plane compression. For this purpose, the strengths or stiffness per the densities of
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the Microlattices sandwiches are divided by those of the conventional Microlattices, and the ratios are plotted according to the wall thickness. The figure shows that the addition of the face sheets to the conventional Microlattices increased the strength per density more than twice, if the wall thickness was higher than about t = 4 µm. Otherwise, the strength per density was not increased or even decreased. The addition of the face sheets also increased the stiffness per density up to 9 to 15 times, if the wall thickness was equal to or higher than about t = 4 µm. 13
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Otherwise, the stiffness per density was not increased by such an amount. Overall, the reinforcements owing to the addition of the face sheets were more prominent in terms of the
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stiffness than in terms of the strength. However, for both the strength and the stiffness, the reinforcing effect became much less significant, if the wall thickness was lower than about t = 4 µm. This finding means that the very thin face sheets were easily buckled over the entire area
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even at the beginning of loading and they continued to deform further as the load level was increased.
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Fig. 11(b) is a chart of the relative density versus the compressive strength that has been normalized by the yield strength of the constituent material, Ni-P foil. The measured values, denoted by the black circles, are plotted together with the estimations using the analytic solutions, denoted by the solid, dotted, and dashed lines, in the figure. In general, in the high
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density region, the measured values agreed soundly with the estimation from Equations (3) or (4), corresponding to the plastic yielding or buckling of the face sheets, respectively, whereas in the low density region, the measured values instead agreed with the estimation by Equation (5),
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corresponding to the elastic buckling of the face sheets. The transition between the elastic buckling and plastic buckling occurred at ρrel = 0.0074, which corresponded to the wall
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thickness, t = 4.13 µm. This wall thickness at the transition that was estimated by the analytic solutions agreed well with the measured one, t = 4.66 µm, that is mentioned above. In comparison with Fig. 11(a), the inferior properties observed at the thin wall were fairly well explained by the transition of the failure mode to the elastic buckling, which also agreed with deformation patterns, that are shown in Fig. 10(b) and Movies S1 to S4. Table 1 lists the measured properties including the thickness of Ni-P walls of each specimen. 14
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DISCUSSION
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As shown in Figs. 6 to 8, the high friction hindered the ends of the hollow tubular struts from horizontally sliding on the top and bottom, resulting in an overestimation of the strengths of the conventional Microlattice under the out-of-plane compression. This phenomenon seems
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to be dominant in failure of a Microlattice with hollow tubular struts of a high slenderness ratio like the specimens tested in this study. Generally, friction is not the only influencing factor. Unit
Microlattice structures [17].
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cell size and sample size (relative to unit cell size) may do play a role in node constraint for
The novel material with the ultralow density and the periodic micro architecture, which has been described in the present paper, was a type of monolithic sandwich. Also, it was shown
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that the addition of the face sheets substantially increased the strength and stiffness under the in-plane compression, if the wall was thicker than a certain limit. A sandwich panel is well known as an optimal structure, designed to support a bending load at light weights. If the
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Microlattice sandwich is to be used to support bending loads, the following issues must be addressed. First, it is likely that the face sheets composed of a foil, whose thickness is identical
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to that composed of the Microlattice core thickness, do not have a sufficient resistance against bending loads. Namely, one of the face sheets in a sandwich panel is supposed to support compressive stress, while the other supports tensile stress. The flat cross section of the face sheet in the compressed side provides a very low second moment of inertia, and it is never appropriate in the resistance against buckling. Second, the holes on the face sheets might cause a stress concentration around the fillets, as they are connected to the Microlattice core.
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Additionally, if a mass transfer is to be blocked across the sandwich panel, the face sheets with the regular-patterned holes do not provide a shield that is as effective as that of the ordinary
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panel. Therefore, the Microlattice sandwich, introduced in this study, is typically not a proper support for bending loads, especially if an ultralow density is to be achieved with a wall
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thickness of submicron order. The Microlattice sandwich could, however, be used as the core of the optimal sandwich panel, where the much higher strength and stiffness for bending loads
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are realized together with an ultra-lightness. That is, each face sheet of the Microlattice sandwich provides a large surface area that could be useful as a firm attached for an additional thin, dense, and hard face with a minimum amount of adhesive.
If it is to be used for a micro robot or bio-mimic drone as small as a beetle or a fly, the
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technique described in the present paper needs to be further improved to fabricate a sandwich with much thinner overall dimensions. Arias et al. [14] and Kolodziejska et al. [15] reported fabrication techniques of sandwich panels with the overall thicknesses of the 1 mm order;
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however, the densities of their sandwiches were too high to be used for this application. Another problem was that the facesheets and core in Arias et al.’s sandwich panel was joined
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using solder or polymer adhesive [14], and Kolodziejska et al. ’s sandwich panel was composed of polymers only [15]. Consequently, their use at a high temperature environment would be limited by melting or softening of the constituent materials or the bonding materials. On contrast, the Microlattice sandwich described in this study was monolithically composed of the Ni alloy foils without any bonding or joining with an additional material. Hence, the temperature range of its use must be much further expanded. In the authors’ research group, a
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new technique is under development for the fabrication of a novel sandwich, which is optimized to support bending loads, with an overall thickness and density of about 1 mm and
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0.01g/cc, respectively, based on this very Microlattice sandwich technique.
CONCLUSIONS
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We introduced a novel technique for the fabrication of a Microlattice-cored miniature sandwich panel with an ultralow density. This sandwich panel was made of a monolithic Ni-P
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foil without any bonding between the core and the faces. As a preliminary study, the mechanical properties were evaluated under compression in two different loading directions using experiments, and the effects of the face sheets that were added on the top and bottom were investigated compared with the conventional Microlattice without the face sheets. The
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conclusions are as follows:
i) Because the specimens were fabricated of NiP foils with thicknesses from 1 to 10 µm, even the polished surfaces of the brass compression platens failed to provide a sufficiently
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smooth and hard contact, resulting in the friction higher than f = 0.4. ii) The high friction hindered the ends of the hollow tubular struts from horizontally sliding
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on the top and bottom, resulting in an overestimation of the strengths of the conventional Microlattice under the out-of-plane compression, and almost no difference was observed between the conventional Microlattices and the Microlattice sandwiches in terms of their compressive behaviors.
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iii) Under the in-plane compression, because of the high aspect ratio, the effect of the high friction was minimized, and the intrinsic strengths of the conventional Microlattices were measured, whereby the true effects of the face sheet additions could be evaluated.
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iv) If the wall thickness was higher than about t = 4 µm, the addition of the face sheets to the conventional Microlattices increased the strength per density more than twice and the
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stiffness per density up to 9 to 15 times. Otherwise, for both the strength and stiffness, the reinforcing effect became much less significant, because the very thin face sheets were easily
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buckled across the entire area even at the beginning of the loading.
v) In the high density region, the failure of the Microlattice sandwich under the in-plane compression was governed by the plastic buckling or yielding of the face sheets, whereas in the low density region, the failure was governed by the elastic buckling of the face sheets. The
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transition between the elastic and plastic buckling at the relative density of ρrel = 0.0074, which showed a fairly sound agreement with the deformation patterns observed in the experiments,
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was estimated using analytic solutions.
ACKNOWLEDGMENTS
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This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT and Future Planning (No.2015R1A2A1A01003702).
REFERENCES
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[1] T.A. Schaedler, A.J. Jacobsen, A. Torrents, A.E. Sorensen, J. Lian, J.R. Greer, L. Valdevit, W.B. Carter, Ultralight metallic microlattices, Science 334 (2011) 962-965.
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[2] D. Jang, L.R. Meza, F. Greer, J.R. Greer, Fabrication and deformation of three-dimensional hollow ceramic nanosctructures, Nat. Mater. 12 (2013) 893-898.
[3] L.R. Meza, S. Das, J.R. Greer, Strong, lightweight, and recoverable three-dimensional ceramic
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nanolattices, Science 345 (2014) 1322-1326.
[4] X. Zheng, et al. Ultralight, ultrastiff mechanical metamaterials, Science 344 (2014) 1373-
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1377.
[5] V.S. Deshpande, N.A. Fleck, M.F. Ashby, Effective properties of the octet-truss lattice material, J. Mech. Phys. Solids 49 (2001) 1747-1769.
[6] A.J. Jacobsen, W. Barvosa-Carter, S. Nutt, Compression behavior of micro-scale truss
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structures formed from self-propagating polymer waveguides, Acta Mater. 55 (2007) 67246733.
[7] J.C. Maxwell, On the calculation of the equilibrium and stiffness of frames, Philos. Mag. 27
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(1864) 294-299.
[8] L. Valdevit, S.W. Godfrey, Compressive strength of hollow microlattices: Experimental
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characterization, modeling, and optimal design, J. Mater. Res. 28 (2013) 2461-2473. [9] S.C. Han, J.W. Lee, K. Kang, A new type of low density material; Shellular, Adv. Mater. 27 (2015) 5506-5511.
[10] S. Hyde, Z. Blum, T. Landh, S. Lidin, B.W. Ninham, S. Andersson, K. Larsson, The Language of Shape, Elsevier, 1996.
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[11] M.G. Lee, J.W. Lee, S.C. Han, K. Kang, Mechanical analyses of “Shellular”, an ultralowdensity cellular metal, Acta Mater. 103 (2016) 595-607.
https://www.thoughtco.com/plant-tissue-systems-373615 .
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[12] R. Bailey, Plant Tissue Systems, ThoughtCo, Mar. 4, 2017.
[13] L. Valdevit, A.J. Jacobsen, J.R. Greer, W.B. Carter, Protocol for the optimal design of
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multifunctional structures: from hypersonics to micro-architected materials, J. Am. Ceram. Soc. 94 (2011) s15-s34.
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[14] F. Arias, P.J.A. Kenis, B. Xu, T. Deng, O.J.A. Schueller, G.M. Whitesides, Y. Sugimura , A.G. Evans, Fabrication and characterization of microscale sandwich beams, J. Mater. Res. 16 (2001) 597-605.
[15] J.A. Kolodziejska, C.S. Roper, S.S. Yang, W.B. Carter, A.J. Jacobsen, Research Update:
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Enabling ultra-thin lightweight structures: Microsandwich structures with microlattice cores. APL Materials (2015) 3, 050701.
[16] M. Zupan, V.S. Deshpande, N.A. Fleck, The out-of-plane compressive behaviour of woven-
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core sandwich plates. Eur. J. Mech. A Solid 23 (2004) 411–421. [17] R.E. Doty, J.A. Kolodziejska, A.J. Jacobsen, Hierarchical polymer microlattice structures. Adv.
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Eng. Mater. 14 (2012) 503-507.
[18] N. Wicks, J.W. Hutchinson, Optimal truss plates. International J. Solids and Structures. 38 (2001) 5165-5183.
[19] A.C. Ugural, Stresses in Plates and Shells, McGraw-Hill, New York, 1981, p. 158.
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APPENDIX A Fig. A1 shows a schematic of the Microlattice sandwich under the in-plane compression.
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Note that only the face sheets resist against the compressive load. Therefore, the force equilibrium in the x-direction of the unit cell gives
∑F
y
=0
σ o × c × l sin ω = Fcr
- - - (A1)
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Here, σo and Fcr are the compressive strength of the Microlattice sandwich and the failure load
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of a single face sheet itself, respectively. If the failure occurs due to the plastic yielding at the narrow regions of the face sheets that are connected to the four hollow struts (i.e., at the gray regions on the front view of Fig. A1), the failure load is simply given by Fcr = ( c − 4 a )t ⋅ σ os . Substituting this into Equation (A1), the compressive strength is given by
( c − 4 a )t σ os . cl sin ω
--- (3)
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σo =
When the face sheets were relatively thick such as t = 13 µm and t = 9 µm, they failed due to plastic buckling on the wide areas of the face sheets after the overall initial elastic
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deformation of bending, as shown in Movies S1 and S2 in Complementary Materials,
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respectively. The left of Fig. A2 shows the buckled shape of the face sheets of the first section from the top just prior to a local fracture. The right of the figure is a schematic of the failure mode, that is, longitudinal collapse of a thin plate with clamped ends, which accompanies the plastic hinges at three points of the top, middle, and bottom. On the plastic buckling, the external energy supplied by the critical force, Fcr, is given by (External Engery) = Fcr × δ ,
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while the internal energy consumed by rotations at the plastic hinges is given by (Internal Engery) = 4 M p × θ .
δ = 2×
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Here, θ is the rotation angle, and δ is the displacement along the critical force, given by 2 2 c (1 − cos θ ) ≈ c1 − 1 + θ = cθ . 2 2 2
And M p is the limit moment needed to yield a plastic hinge, given as a function of the thickness,
ct 2 σ os . 4
Hence, energy balance results in
Fcr =
2t 2
θ
σ os .
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Mp =
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t, the length, c, and the yield strength of the constituent material, σos, as follows;
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The value of Fcr is a function of the initial value of θ as well as t and σos. The initial θ is related to the eccentricity, e, (due to the initial curvature existing in the face sheets) as θ =2e/c. And if e
Fcr =
ct σ os . k
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is represented as e =k t, the critical force, Fcr, is given by
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Substituting this into Equation (A1), the compressive strength due to the plastic buckling is given by
σo =
t σ os . kl sin ω
--- (4)
Here, k is the relative eccentricity, scaled by the thickness of the face sheets. If the failure occurs due to elastic buckling on the wide areas of the face sheets, the failure stress in the face sheets at the buckling, σcr, is given in Wicks and Hutchinson [17] as follows: 22
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σ cr =
49 × 3 π 2 Es t 432 (1 −ν 2 ) c − 4a
2
Interestingly, Equation (A2) is very similar to the solution of Ugural [18] for the minimum
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buckling load of the lateral shell, as follows:
π 2 Es t . 3(1 −ν 2 ) c − 4a 2
σ cr =
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Therefore, the following equation applies: 49 × 3 π 2 Es t Fcr = ct. 432 (1 −ν 2 ) c − 4a 2
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- - - (A2)
The above equations are based on the assumption that the boundaries of a flat rectangular sheet are simply supported, allowing for a free rotation. However, the face sheets of the specimen are not flat because of the fillets that are connected to the hollow tubular struts.
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Consequently, the fillets played a constraining role at the boundaries: therefore, considering this constraint, Equation (A2) is slightly modified as follows: 49 × 3 × 4 π 2 Es t Fcr = ct. 432 (1 − ν 2 ) c − 4a 2
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- - - (A3)
By substituting Equation (A3) into (A1), the compressive strength is given by
49 π 2 E s t3 . 36 ( 1 − ν 2 ) ( c − 4 a )2 ⋅ l sin ω
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σo =
- - - (5)
Supplementary Material Movies S1-S5
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FIGURE CAPTIONS Fig. 1. A 2D schematic of deformation pattern of the Microlattice subjected to out-of-plane
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compression. Fig. 2. (a) a schematic of leaf anatomy, reproduced by courtesy of Richard Wheeler [12], and (b) optical image taken from a leaf of magnolia grandiflora (scale bar: 500 µm).
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Fig. 3. A schematic of the fabrication process of a Microlattice sandwich.
Fig. 4. Photos for: (a) A UV exposure system used to form the out-of-plane truss elements of a
sandwich specimen.
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template, and (b) a template (assembled with glass slides) along with a Microlattice
Fig. 5. (a) A photo of a Microlattice sandwich specimen, and (b) CAD images of a Microlattice sandwich with the dimensions.
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Fig. 6. Stress-strain curves measured from out-of-plane compression tests for the conventional Microlattice and the Microlattice sandwich with two different wall thicknesses: (a) about t = 3 µm and (b) about t = 1 µm.
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Fig. 7. Finite element models of the unit cells of: (a) a conventional Microlattice and (b) a Microlattice sandwich.
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Fig. 8. (a) Force versus displacement curves estimated from an FEA for a conventional Microlattice and a Microlattice sandwich, and (b) side views of the unit cell models of the conventional Microlattice compressed between the rigid plates with two different friction coefficients, f = 0.4 (left) and f = 0.5 (right).
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Fig. 9. Stress-strain curves measured from in-plane compression tests for the conventional Microlattice and Microlattice sandwich specimens with similar thicknesses: (a) about t =
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13 µm, (b) t = 9 µm, (c) about t = 3 to 5 µm, and (d) about t = 1 µm. Fig. 10. Images taken at four stages from specimens with similar wall thicknesses of about t = 3 to 5 µm under in-plane compression: (a) conventional Microlattice and (b) Microlattice
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sandwich specimens.
Fig. 11. (a) Ratio of the strengths or stiffness per the densities of Microlattices sandwiches to
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those of the conventional Microlattices plotted according to the wall thickness, and (b) a chart of the relative density versus the compressive strength normalized by the yield strength of the constituent material, Ni-P foil.
Fig. A1. A schematic of the unit cell of a Microlattice sandwich under in-plane compression.
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Fig. A2. Buckled shape of the face sheets of the first section from the top of the specimen with t = 9 µm under in-plane compression just prior to a local fracture, and a schematic of the failure mode, that is, longitudinal collapse of a thin plate with clamped ends,
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accompanying three plastic hinges.
CAPTIONS for Movies S1 to S5 Movie S1. In-situ video (played at 10x speed) of a Microlattice sandwich specimen under inplane compression with the wall thickness of t = 13.27 µm. Movie S2. In-situ video (played at 10x speed) of a Microlattice sandwich specimen under inplane compression with the wall thickness of t = 9.15 µm.
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Movie S3. In-situ video (played at 10x speed) of a Microlattice sandwich specimen under inplane compression with the wall thickness of t = 4.66 µm. Movie S4. In-situ video (played at 10x speed) of a Microlattice sandwich specimen under in-
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plane compression with the wall thickness of t = 1.37 µm.
Movie S5. In-situ video (played at 10x speed) of a conventional Microlattice specimen under in-
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plane compression with the wall thickness of t = 3.14 µm.
Table 1. Properties measured for the conventional Microlattice and Microlattice sandwich specimens. Wall Thickness Relative Density (µ µm)
Strength (MPa)
Young's Modulus (MPa)
Conventional
3.11
3.99E-03
0.2167
12.60
Sandwich
3.56
6.40E-03
0.2485
12.98
Conventional
1.14
1.46E-03
0.02835
2.109
Sandwich
1.00
1.80E-03
0.02801
2.002
Conventional
13.35
1.93E-02
0.2304
3.400
Sandwich
13.27
2.39E-02
0.6171
60.90
Conventional
8.78
1.27E-02
0.1322
1.706
Sandwich
9.15
1.65E-02
0.3719
23.69
Conventional
3.14
4.55E-03
0.01998
0.288
Sandwich
4.66
8.39E-03
0.09271
5.050
Conventional
1.28
1.87E-03
0.002924
0.048
Sandwich
1.37
2.47E-03
0.003742
0.467
Conventional
1.08
1.57E-03
0.001719
0.041
Sandwich
0.80
1.43E-03
0.001226
0.144
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Out-of-plane
Specimen Type
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Load Direction
In-plane
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Fig. 1. A 2D schematic of deformation pattern of the Microlattice subjected to out-of-plane compression.
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Fig. 2. (a) a schematic of leaf anatomy, reproduced by courtesy of Richard Wheeler [12], and (b) optical image taken from a leaf of magnolia grandiflora (scale bar: 500 µm).
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Fig. 3. A schematic of the fabrication process of a Microlattice sandwich.
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Fig. 4. Photos for: (a) A UV exposure system used to form the out-of-plane truss elements of a template, and (b) a template (assembled with glass slides) along with a Microlattice sandwich specimen.
(Scale bar: 2 mm)
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Fig. 5. (a) A photo of a Microlattice sandwich specimen, and (b) CAD images of a Microlattice sandwich with the dimensions. (Scale bar: 2 mm)
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Fig. 6. Stress-strain curves measured from out-of-plane compression tests for the conventional Microlattice and the Microlattice sandwich with two different wall thicknesses: (a) about t = 3 µm and (b) about t = 1 µm.
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Fig. 7. Finite element models of the unit cells of: (a) a conventional Microlattice and (b) a Microlattice sandwich.
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Fig. 8. (a) Force versus displacement curves estimated from an FEA for a conventional Microlattice and a Microlattice sandwich, and (b) side views of the unit cell models of the conventional Microlattice compressed between the rigid plates with two different friction factors, f = 0.4 (left) and f = 0.5 (right).
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Fig. 9. Stress-strain curves measured from in-plane compression tests for the conventional Microlattice and Microlattice sandwich specimens with similar thicknesses: (a) about t = 13 µm, (b) t = 9 µm, (c) about t = 3 to 5 µm, and (d) about t = 1 µm.
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Fig. 10. Images taken at four stages from specimens with similar wall thicknesses of about t = 3 to 5 µm under in-plane compression: (a) conventional Microlattice and (b) Microlattice sandwich specimens.
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Fig. 11. (a) Ratio of the strengths or stiffness per the densities of Microlattices sandwiches to those of the conventional Microlattices plotted according to the wall thickness, and (b) a chart of the relative density versus the compressive strength normalized by the yield strength of the constituent material, Ni-P foil.
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Fig. A1. A schematic of the unit cell of a Microlattice sandwich under in-plane compression.
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Fig. A2. Buckled shape of the face sheets of the first section from the top of the specimen with t = 9 µm under in-plane compression just prior to a local fracture, and a schematic of the failure mode, that is, longitudinal collapse of a thin plate with clamped ends, accompanying three plastic hinges.