Effect of heterogeneity on crushing failure of disordered staggered-square honeycombs

Journal Pre-proofs Effect of heterogeneity on crushing failure of disordered staggered-square honeycombs Deepak Kumar, Anuradha Banerjee PII: DOI: Reference:

S0263-8223(19)33541-X https://doi.org/10.1016/j.compstruct.2020.112055 COST 112055

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Composite Structures

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Please cite this article as: Kumar, D., Banerjee, A., Effect of heterogeneity on crushing failure of disordered staggered-square honeycombs, Composite Structures (2020), doi: https://doi.org/10.1016/j.compstruct. 2020.112055

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Effect of heterogeneity on crushing failure of disordered staggered-square honeycombs Deepak Kumara , Anuradha Banerjeea,∗ a

Department of Applied Mechanics, IIT Madras, Chennai 600036, India

Abstract Uni-axial compressive failure of silica-epoxy based heterogeneous honeycombs is investigated in detail for a range of volume fractions. Introduction of heterogeneity in compression of staggered-square honeycomb is seen to result in damage initiation at multiple locations and subsequent damage growth to be more stable compared to pure epoxy in which damage was observed to be localized until peak load when catastrophic failure of the honeycomb specimen occurs. The increase in stiffness and comparative stability of the response is accompanied with reduction in strength, however, between 0-5% the total work of compressive failure is comparable. From the elastic-plastic analysis it is evident that the non-linearity in the response of pure honeycombs, prior to peak load, is largely due to formation of plastic hinges near corners of cells, whereas in case of heterogeneous honeycomb the non-linearity is mostly due to debonding of hard filler particles and matrix cracking leading to damage growth in cell walls. Keywords: Staggered square honeycomb, Silica particle-reinforced composite, Failure mechanism, Damage, Elastic plastic analysis 1. Introduction Cellular solids, by virtue of their high specific mechanical properties and multifunctionality, find wide applications from cheap daily necessities to critical components in ∗

Corresponding author Email address: [email protected] (Anuradha Banerjee)

Preprint submitted to Composite Structures

February 2, 2020

industries such as automobile and aerospace etc. Inspired by the concept of cellular architecture in biological materials such as bone and wood, cellular structures have been successfully designed to exhibit better mechanical properties, such as high specific stiffness, resistance to fracture and energy absorption, than their bulk counterparts [1, 2]. However, presence of inhomogeneities, or microstructural defects in the bulk material can seriously compromise the reliability and performance of these architectured solids in structural applications [3–5]. To ensure better design and safe use of cellular solids, it is of immense relevance to develop a deeper understanding of the factors that play a vital role in the the complex processes during failure of these solids. Macroscopic behaviour of cellular solids is known to be strongly dependent not only on the parent material and cellular geometry but also on the irregularities in the cell morphology and inherent defects [6–15]. Initial investigations based on micromechanical modelling of a unit cell, for both 3-dimensional foams and 2-dimensional honeycombs, have focused on establishing the effective properties for a regular shaped cell [16–18] in terms of the material and geometric parameters. However, in presence of disorder such as irregularities in shape of cell and cell wall thickness, experimental data and FEM based analysis have shown that elastic as well as crushing response, in terms of initiation strain, plateau stress, densification strain energy etc., differ as disorder brings changes in load transfer paths and failure mechanisms [19, 20]. Depending on the ductility of the parent material, the primary mechanism of energy absorption during compression is either due sequential plastic collapse of cell walls [16] or due to fracture and crushing of brittle cell walls [21–23]. Brittle materials, such as ceramics, in spite of their high stiffness- and strength- to weight ratio, were rarely used in structural applications of cellular solids due to their flaw sensitivity and poor resistance to rapid propagation of damage [24, 25]. However, using advanced lithography, cellular materials with specific micro- and nano-architecture have been developed that exploit

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the size-dependent strengthening effects as well as optimize competing failure modes, fracture and buckling of members, to result in ductile-like plateau region and excellent resilience post compression in excess of 50% [26, 27]. The ease of manufacturing and cost effectiveness of such material systems, however, remains a concern for macroscale components. More recently, reinforced composites have also been used as parent material to fabricate cellular structure by advanced additive manufacturing methods to enhance their specific mechanical properties and multifunctionality [4, 28–32]. Compton et al. [4] introduced a new fiber reinforced (silicon carbide whisker and milled carbon fiber) epoxy-based ink that enables the 3D printing of composite honeycombs by controlling the aspect ratio of fiber and its matrix. For the same density of printed triangular honeycombs, SiC/C-filled were found to have higher elastic modulus and strength than only SiC-filled ones. Muth et al. [29] introduced one more level of hierarchy in honeycomb structure by creating an architected porous ceramics by direct writing of particle-stabilized foam (PSF) ink. These architected ceramic honeycombs showed high specific stiffness compared to micro- and nano-scale lattices of similar relative densities fabricated by other methods. In these studies, the primary focus was to develop new types of ink and techniques that enable to create complex architected structures to enhance their mechanical properties like elastic modulus, strength and energy absorption. However, 3D printing method has limitations in the choice of possible materials as well as reinforcements that can be used for printing. Computationally, an isotropic heterogeneous lattice structure is analyzed which has both stretch dominated triangular as well as bending dominated hexagonal architecture. This hybrid structure is shown to exhibit both high stiffness as well as damage tolerance as a consequence of the heterogeneity [14]. In the present paper, we examine the specific role of heterogeneity on the deforma-

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tion and crushing failure of heterogeneous honeycomb by establishing how particulate silica-epoxy heterogeneous honeycombs fail differently from pure epoxy honeycombs under quasi-static uni-axial compression. Honeycomb specimens were prepared by specially designed mold for a range of volume fraction of silica micro-filler particles. The specific role of heterogeneity on the deformation and crushing failure of heterogeneous honeycombs was evaluated based on the macroscopic response and the failure mechanisms that were captured in the digital images during testing. Elastic-plastic properties of bulk particulate epoxy composites were determined from tensile test of rectangular dogbone specimens to establish the changes as a result of introduction of filler particles. A two-dimensional plane stress finite element analysis was performed assuming power law strain hardening behaviour of the bulk to reveal the possible underlying mechanisms responsible for the large deformation and crushing failure of heterogeneous honeycombs.

2. Experimental procedures 2.1. Specimen preparation Honeycomb architecture in biological materials is predominantly hexagonal but there are instances of square staggered as well as combination of hexagonal and squarestaggered such as in balsa, cedar and yellow pine trees [1, 33, 34]. While there are several investigations based on hexagonal architecture, comparatively fewer studies focused on staggered-square honeycomb. In the present study, pure staggered square honeycomb specimens were prepared by casting method using epoxy resin, LY556 (Bisphenol A Diglycidyl Ether, DGEBA) mixed with curing agent, HY951 (Triethylene tetraamine, TETA). Particulate composite honeycombs were prepared by addition of irregularly shaped micron-sized silicon dioxide (SiO2 , Sigma-Aldrich, India). The quoted mechanical properties of the epoxy resin and silica particles [36] are shown in Table 1 in the 4

Figure 1: Schematic diagram of staggered square honeycomb specimen and its unit cell highlighted by dotted line square.

(a)

(b)

Figure 2: (a) SEM image of polished surface of 30% silica reinforced epoxy and (b) size distribution of silica particle.

Appendix A. Epoxy resin LY556 is a thermosetting polymer and fails in a brittle manner upon mechanical loading at room temperature which is well below its glass transition. For preparing heterogeneous honeycombs first epoxy resin was heated and maintained at 600 C then silica filler particles were mixed into resin in regular intervals using a mechanical stirrer for an hour. Mixture was cooled to room temperature before curing

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agent was poured into the mixture such that 100:12 resin/hardener ratio by weight was maintained. The mixture was slowly stirred for 10 minutes to avoid any air entrapment and then mixture was poured into a wax coated mild steel mold of honeycomb structure. The curing of mixture was done in two steps: first, it was kept at room temperature for 3 hours to gel the matrix resin (pre-curing stage), followed by curing the mixture in an oven at 600 C for 8 hours. Cured honeycomb specimen was then machined to the required dimensions of 38.5 mm x 38.5 mm x 15 mm as shown in Fig. 1. The choice of the out-of-plane thickness to be 15 mm was to avoid the heterogeneity in the thickness direction from playing a dominant role in the crushing behaviour. Any partial damage of the cell wall, thus, would manifest itself as serrations in the macroscopic response as seen in the results section. The surfaces of cell walls of the specimens were polished with 1200 and 1500 grit sand papers successively to improve the surface finish to achieve better contrast between the cell walls and void regions in the digital images. The cell size (l) and cell wall thickness (t) of all specimens were 7.5 mm and 1 mm respectively such that the relative density (ρ∗ /ρs ) of the honeycomb was 0.25. Filler distribution in epoxy resin was examined by cutting cuboids of 10 mm x 5 mm x 5 mm dimensions from the dogbone along longitudinal direction. The surfaces of cuboids were polished by sand papers of very fine grade followed by polishing them with 0.05 micron alumina suspension on micro-cloth. Samples were cleaned by keeping them in acetone in a sonicator to remove loose alumina particles and any debris. Representative SEM image of 30% volume fraction of silica reinforced epoxy is shown in Fig. 2-(a). Silica particles of different sizes are clearly seen on the polished surface. Particles of larger size having irregular shapes and sharp edges are clearly visible whereas smaller sized particles appears as white spots in epoxy resin. The distribution of normalized particle diameter obtained from particle size analyser (PSA) with average particle diameters of 2.917 micron is presented in Fig. 2-(b).

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2.2. Mechanical testing Uni-axial compression tests of honeycombs were performed on a servo-hydraulic DMG fatigue testing machine. Specimens were placed between two flat platens. All the samples were compressed at the rate of 0.5 mm/min in displacement controlled mode. Digital images of successive deformed honeycombs were captured using a chargecoupled device (CCD) camera connected with a Nikon lens of 50 mm focal length. The bulk mechanical properties of the particulate composite materials were also obtained from tensile testing of dogbone specimens as per ASTM-D638. The data is presented in the Appendix B.

3. Results 3.1. Stress-strain response of heterogeneous honeycomb

Figure 3: Macroscopic response from compression testing of heterogeneous honeycomb specimens.

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Figure 4: Normalized Young’s modulus of heterogeneous honeycomb.

Figure 5: Normalized critical strength of heterogeneous honeycomb.

To establish the role of heterogeneity on the deformation and failure process, mesoscopic heterogeneous honeycombs of volume fractions, of silica ranging between 0 to 30% were tested. The representative nominal stress-strain curves of all volume frac-

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Figure 6: Normalized critical strain of heterogeneous honeycomb.

tions are shown in Fig. 3. All curves exhibit an initial linear elastic response primarily as a consequence of bending of horizontal walls of unit cells. As shown in Fig. 3, with increase in percentage of stiffer silica particles, as expected from the response of silica dogbones, shown in Fig. B.1 in the Appendix B, the effective elastic modulus of the honeycomb also increases monotonically. The sudden drop in load is due to breakage of cell walls. There is only one major load-drop event in case of pure epoxy honeycomb which is seen in Fig. 3. In case of heterogeneous honeycombs, however, there are several major load drop events and the number of events increases with increase in silica percentage. Comparison between the effective mechanical behaviour of the bulk composite material and honeycomb structure is summarized in Figs. 4- 6. The normalized elastic modulus, for the range of filler volume fraction considered, shows a nearly linear increase for both bulk as well as the honeycomb specimen. The experimental data is well bounded by the Voigt and Reuss model estimates for the bulk, however, is not closely reproduced by either model to be utilized in the scaling relations for the effective modulus of the honeycomb (Eq. C.2). The critical strength, defined as the maximum stress resisted 9

by the structure, and critical strain, defined as the strain corresponding to the critical strength, show an initial increase and then decrease in case of bulk, however, decrease in case of honeycombs as volume fraction of silica increases. 3.2. Deformation and failure mechanism of heterogeneous honeycomb

Figure 7: Different stages of deformation and failure of pure epoxy staggered square honeycomb. The stages are highlighted on the stress-strain response in Fig. 9-(a)

.

To gain an insight into the effect of heterogeneity effect on the deformation and failure mechanisms, details related to pure epoxy and 30% volume fraction heterogeneous honeycombs were examined further. The digital images obtained using CCD camera show different stages of progressive failure of pure epoxy and heterogeneous (30% volume fraction) honeycombs in Fig. 7 and Fig. 8 respectively. With increasing 10

Figure 8: Different stages of deformation and failure of 30% volume fraction silica-epoxy staggered square heterogeneous honeycomb. The stages are highlighted on the stress-strain response in Fig. 9-(a)

.

deformation under compression of pure epoxy honeycomb, near an intersection between a horizontal cell wall and vertical cell wall, a white spot is observed that corresponds to a preexisting manufacturing surface defect as visible in Fig. 7-(a). The white spot becomes enlarged with increasing deformation as shown in Figs. 7 (c)-(e) leading to through thickness failure of cell wall in Fig. 7-(f). The event corresponds to the ma-

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jor load drop observed in the macroscopic response as it triggers catastrophic failure of several cell walls simultaneously, shown in Fig. 7-(g), of the remaining honeycomb sample. In Figs. 7 (h)-(i) the final crushing event is presented when the sample loses its structural integrity. Heterogeneous honeycomb having 30% silica volume fraction, with increasing deformation, exhibits damage initiation at a junction between cell walls (refer Fig. 8-(b)), further increase in load results in damage initiation at another location as shown in Figs. 8 (c)-(d). Breakage of cells occurs in Figs. 8 (e)-(l) in a progressive manner where the spread of damage is more stable compared to the pure epoxy honeycomb. For instance the first failure event, in Fig. 8-(e), involved breakage of 3 cell walls whereas a similar event for pure epoxy, in Fig. 7-(g), resulted in breakage of 14 cell walls. Thus, in case of heterogeneous honeycomb, damage spreads in stages and at more than one location whereas damage was localized in case of pure epoxy honeycomb. For gaining an insight into the manifestations of the observed damage growth, the different stages shown in Fig. 7 and Fig. 8 are highlighted on the macroscopic response in Fig. 9-(a). Localized damage in cell walls, such as associated with events (b)-(e) in Fig. 7 and (b)-(d) in Fig. 8, leads to minor fluctuations in the overall response with negligible change between the tangent modulus before and after. Breakage of a cell wall, however, results in larger drop in load as well as significant changes to the elastic modulus. As expected the extent of load drop is significantly higher for pure epoxy as it was associated with catastrophic failure of several cell walls. 3.3. Elastic plastic analysis of honeycomb structure using FEM The role of heterogeneity on the failure mode is further examined by comparing the experimentally obtained macroscopic response with predictions from finite element analysis of the honeycomb specimens. Two-dimensional plane stress analysis of the honeycomb specimen was performed assuming power-law strain hardening constitutive 12

(a)

(b) Figure 9: (a) Stress-strain response of pure and 30% volume fraction silica-epoxy staggered square honeycomb, and (b) Stress-strain response of pure and 30% volume fraction silica-epoxy dogbone specimens.

relation for the cell wall material as per:

σ , σ ≤ σy E  1 σy σ n , σ > σy = E σy 13

ǫ=

(1)

Figure 10: The contours of von Mises stress of deformed pure epoxy honeycomb structure (a) before and, (b) after the plastic collapse.

Figure 11: The comparison of deformed shape of pure epoxy honeycomb structure before collapse (a) in elastic plastic analysis and, (b) in the experiment.

Table 1: Mechanical properties used in elastic plastic analysis of honeycomb structure.

Material Pure epoxy 30% silica composite

E (GPa) 3.59 5.65

σy (MPa) 25 42.4

n 0.638 0.605

where E is the elastic modulus, σy is the proportional limit and n is the power law exponent. Thus, initially the elastic-plastic properties of the pure epoxy and particulate 14

composite material used for study were determined from the tensile response of dogbone specimens reported in the Appendix Fig. B.1 and tabulated in Table 1. Plane stress analysis of uni-axial tensile bar was shown to well reproduce the experimental elasticplastic data as evident in Fig. 9-(b). In the elastic-plastic (EP) analysis of the honeycomb,13852 number of 4-noded plane stress elements were used to model the in-plane geometry. From the FE analysis, the obtained elastic-plastic response in comparison to experimental data is shown in dashed lines in Fig. 9-(a). The peak-stress of the stress-strain curve corresponds to the plastic collapse strength which also marks the onset of localization of strains. It is interesting to note that in case of pure epoxy honeycomb, barring a few events that damage the cell walls partially, the response is essentially elastic-plastic and the the peak-stress is close to the plastic collapse strength. In contrast, the heterogeneous honeycomb response is well reproduced by the elastic-plastic simulation initially but deviations occur early as damage growth occurs well below the plastic collapse strength. The plastic collapse load was also determined under the assumption of elastic-perfectly plastic (EPP) response computationally as well as theoretically (scaling relations are presented in Appendix C) and summarized in Table 2. It is evident that the hardening behaviour strongly influences the plastic collapse load in FEM simulations and thus scaling laws obtained based on assumption of elastic-perfectly plastic behaviour can be inaccurate for the materials considered in the present study. The theoretical and elastic-perfectly plastic collapse strengths still differ as the theoretical estimates are based on the cell dimensions ignoring the cell wall thickness. The FE analysis, in contrast, incorporates the actual geometry of specimens such that the beam dimensions differ by ∼ 13% as t/l = 0.133. This results in a difference in the plastic collapse strength by ∼ 28%. Further, the theoretical estimates also exclude filleting details which are known to cause higher strength as reported in Simone and Gibson [35]. The contours of the von Mises stress

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before and after the collapse of honeycomb structure are shown in Fig. 10. Table 2: Peak stress of honeycomb (HC) structures.

Exp. Pure epoxy HC 30% silica HC

3.10 1.70

Peak Stress (MPa) Theoretical EPP (σpl ) Elastic-brittle (σcr ) 0.89 1.27 1.50 1.21

FE Analysis EP EPP 3.53 1.20 5.39 2.02

Further, the deformed shape of a quarter of the honeycomb sample highlighted in Fig. 10, that corresponded to the peak load, is compared with the counterpart in the experiment in Fig. 11. The correspondence of arched horizontal cell walls, tilting of the vertical cell walls etc. confirms that the elastic-plastic deformation has a dominant role in the non-linearity that was observed in the macroscopic response of pure epoxy honeycomb prior to failure. In case of 30% silica honeycomb sample, on the basis of elastic solution for the maximum flexural moment being the cause of strut failure, the crushing strength is evaluated to be σcr = 43 ( tl )2 σcbulk . The estimates of crushing strength are close to that observed in experiments suggesting the mode of failure in this case is due to brittle failure of cell walls in bending. 3.4. Work of crushing failure To compare the work of crushing during compressive failure of the heterogeneous honeycombs of differing compositions, the number of cell walls broken for pure epoxy honeycomb upto complete load drop ( 24 and 21 for the two pure epoxy samples tested) was chosen as a reference, no = 23. As the number of cell walls broken after the first complete load drop, n1 , for the heterogeneous honeycombs were typically less than the reference, they were further compressed until the total number of cell walls broken were higher than the reference. The response to the subsequent compression is presented in Fig. 12. Work of fracture based on the response till the first complete load drop, 16

Figure 12: Macroscopic response of heterogeneous honeycomb specimens compressed subsequently after first complete load drop.

Figure 13: Work of fracture of heterogeneous honeycombs.

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Table 3: Work of compressive failure of honeycomb structure

Specimen 0S5A 0S5B 1S5A 1S5B 3A5A 3A5B 5S5A 5S5B 10S5A 20S5A 30S5A

Vol. frac. of silica (%) 0 0 1 1 3 3 5 5 10 20 30

First compression n1 W1 (J) 24 1.216 21 1.527 12 0.813 20 1.127 7 0.753 8 0.508 8 0.619 12 0.7 12 0.488 8 0.511 10 0.388

Subsequent compression n2 W2 (J) 10 0.494 3 0.048 21 0.484 32 0.659 15 0.510 9 0.444 12 0.323 16 0.324 14 0.381

Total WOF W1 + W2 (J) 1.216 1.527 1.307 1.175 1.237 1.167 1.129 1.144 0.811 0.835 0.769

W1 , and for the subsequent compression until the breakage of no total cell walls, W2 , is tabulated in Table 3. It is evident that the total work of fracture to break the no number of walls, as resulted from the compression of pure epoxy, requires subsequent compression of the heterogeneous honeycombs. The contribution from the subsequent compression becomes higher with increasing heterogeneity upto 5% such that the total work of fracture is comparable for volume fraction is the range 0-5% . Beyond 5% there is an overall decrease in total work of fracture.

4. Discussion Earlier investigations on the effect of hard filler particles have concluded that with increasing volume fraction of the filler particles, typically the elastic modulus has a monotonic increase. However, on the failure behaviour, represented by measured peak strength or fracture toughness, the effect is significantly more complex as there are strong dependences on the shape, size, relative mechanical properties of the filler particles as well as interface integrity [36–38]. To identify the differences in fracture mechanisms between pure and silica filled 18

Figure 14: SEM image of fracture surfaces of (a) Pure epoxy, and (b) 30% silica reinforced epoxy.

epoxy composite materials, the fracture surface is examined in detail. SEM images of fracture surfaces of pure epoxy and 30% silica composite specimens are shown in Fig. 14. The fracture surface of pure epoxy has typical three distinct regions: first, where crack starts at a defect followed by a flat featureless mirror zone (region A), called crack initiation point which spreads to a transition zone (region B) where the surface roughness steadily increases and a final propagation zone (region C) with tail or conical marks [39] as evident in Fig. 14-(a). In Fig. 14-(b), we observe that in a silica filled epoxy composite material, crack initiation started from the edge of the specimen. As we move away from the crack initiation point a relatively smooth surface where crack accelerates can be seen in which debonded and smooth zones (region A and B) are present, at higher magnification. As we move further radially a highly rough surface zone (region C) is seen that covers most of the fracture surface in Fig. 14-(b). In this zone new fracture surfaces forms by crack branching [40]. These differences in the propagation of damage at microscopic length scales are the probable cause of the damage nucleating at multiple sites and spreading in a comparatively stable manner in heterogeneous honeycombs, resulting in a large number of micro as well as macro-events. The effect of the size of the specimen, when the considered size is comparable to the cell size, has been well established, experimentally, computationally as well as theoret19

ically, to result in more compliance in tension, higher stiffness in shear and less tensile strength compared to an infinite honeycomb [41–45]. The results of the present study obtained from 5x5 sized samples, while provide an insight into the role of heterogeneity on the comparative compressive response, are not representative of the bulk honeycomb (as per scaling relations in Appendix C). Further testing with larger samples would be needed to provide a more accurate representation of the bulk behaviour, however, it is outside the scope of the present study. The main focus of the present work is to establish the role of heterogeneity on the crushing failure mechanisms. Details of crushing mechanisms in brittle cellular solids are best identified in a mesoscale specimen as the individual failure events and their effect on the macroscopic response are easily distinguishable [10, 46, 47], unlike cellular solids with more ductile cell wall materials in which the local mesoscale (cell level) rotations/constraints are highly influenced by the proximity of stiff or compliant boundary layers in smaller samples [48–53].

5. Conclusion The response of mesoscopic honeycomb samples, despite the size effect, is known to contain certain signatures of the macroscopic honeycomb response. Here, based on deformation and failure response of mesoscale staggered square silica-epoxy composite honeycombs, the following conclusions can be made on the role of heterogeneity: • In contrast to the toughening behaviour of the bulk silica-epoxy composite material at lower volume fraction of silica particles, the failure response of the heterogeneous honeycomb shows a monotonic decrease in the compressive peak strength with addition of filler material. • As a consequence of heterogeneity, damage is seen to nucleate at multiple sites and spread in a comparatively stable manner in heterogeneous honeycombs, resulting

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in a large number of micro as well as macro- events. While there is a decrease in the work of crushing of the heterogeneous honeycomb after the first complete load drop, further reloading is possible since the failure is not catastrophic as in case of pure epoxy. As a result the total work of fracture for a similar number of broken cell walls is comparable for compositions between 0-5%. • Elastic-plastic behaviour of the pure epoxy honeycomb is well approximated with a power law hardening constitutive relation before failure and its load bearing capacity is shown to be close to the plastic collapse strength suggesting that even though the epoxy doesn’t have enough ductility to show a plastic collapse, the breakage process maybe induced by plastic instability. Non-linearity in heterogeneous honeycombs, however, is largely as a result of damage growth in cell walls at multiple sites. Based on the present study, it is evident that addition of filler particles has increased, apart from the stiffness, the comparative stability of the response also, with minimal change (increase or decrease) in work of fracture up to 5% of volume fraction, however, it is at the cost of reduced strength. With further fine-tuning of filler size, mechanical properties of constituents and interface properties one could achieve more control over the desired response of the honeycomb. 6. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. Acknowledgment Authors would like to acknowledge the financial support by the Exploratory Research Project Fund (Project no.: APM1718843RFERANUR) at IIT Madras. 21

Appendix A. Mechanical properties of epoxy resin and silica particles Table A.1: Mechanical properties of epoxy resin and silica particles [36]

Material .

Silica particles Epoxy resin

Density (g/cc) 2.2 1.15-1.2

Young’s modulus Poisson’s ratio (GPa) 72.1 0.18 3.59 0.26

Appendix B. Bulk mechanical properties of silica-epoxy composite

Figure B.1: Macroscopic response obtained from quasi-static tension test conducted on dogbone specimens of particulate epoxy composite.

In the present study, to find the effect of silica particulate fillers, the macroscopic response of silica-epoxy composite materials having volume fraction ranging between 0-30% was characterized. All the corresponding stress-strain curves, shown in Fig. B.1, exhibit largely linear elastic behaviour, followed by minimal plasticity prior to brittle fractures. On the elastic behaviour, addition of silica filler shows a monotonic increase in the effective modulus with increasing volume fraction. On the failure behaviour, 22

however, there is an initial increase in the load bearing capacity as well as the associated strain followed by an overall decrease.

Appendix C. Scaling relations

Figure C.2: (a) Loading direction, and (b) deformed horizontal cell wall of unit cell of staggered square honeycomb.

The elastic deformation of an infinitely wide staggered square honeycomb cell is, shown in shown in Fig. C.2-(a), primarily due to bending of the horizontal walls, shown in Fig. C.2-(b). From the classical beam theory:

δ=

(σlb) l3 192Es I

23

(C.1)

Such that the elastic modulus of the honeycomb is:

σl = 16Es E = δ  3 E∗ t = 16 Es l ∗

 3 t l

(C.2)

The plastic collapse load is evaluated by equating fully plastic moment of cell wall (= σy bt2 /4) to the maximum moment in the beam (= σ ∗ bl2 /8) to give:

∗ σpl =2 σy

 2 t l

(C.3)

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