thermoset-matrix syntactic foams subject to hydrostatic loading

Accepted Manuscript Micromechanical explanation of failure modes dependent on surface treatment failure of glass-microballoons/thermoset-matrix syntactic foams subject to hydrostatic loading Lorenzo Bardella, Giovanni Perini, Andrea Panteghini, Noel Tessier, Nikhil Gupta, Maurizio Porfiri PII:

S0997-7538(17)30809-4

DOI:

10.1016/j.euromechsol.2018.01.007

Reference:

EJMSOL 3537

To appear in:

European Journal of Mechanics / A Solids

Received Date: 9 November 2017 Revised Date:

20 January 2018

Accepted Date: 22 January 2018

Please cite this article as: Bardella, L., Perini, G., Panteghini, A., Tessier, N., Gupta, N., Porfiri, M., Micromechanical explanation of failure modes dependent on surface treatment failure of glassmicroballoons/thermoset-matrix syntactic foams subject to hydrostatic loading, European Journal of Mechanics / A Solids (2018), doi: 10.1016/j.euromechsol.2018.01.007. 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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Failure of glass-microballoons/thermoset-matrix syntactic foams subject to hydrostatic loading

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Department of Civil, Environmental, Architectural Engineering and Mathematics University of Brescia, Via Branze, 43 — 25123 Brescia, Italy

Department of Mechanical and Aerospace Engineering, Tandon School of Engineering New York University, Six MetroTech Center, Brooklyn, NY 11201, USA (c)

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Lorenzo Bardella(a)∗, Giovanni Perini(a,b) , Andrea Panteghini(a) , Noel Tessier(c) , Nikhil Gupta(b) , Maurizio Porfiri(b)

CMT Materials, 107 Frank Mossberg Dr, Attleboro, MA 02703, USA

Abstract

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This study focuses on a lightweight syntactic foam constituted by an epoxy matrix filled with polydispersed Glass Microballoons (GMs) up to 75% volume fraction. We present experimental results on hydrostatic loading which demonstrate the possibility of different failure modes depending on whether the surface of the composite is painted/coated or not. In order to explain this surprising behaviour, we propose a three-dimensional Finite Element (FE) micromechanical model. First, we develop a cubic MultiParticle Unit Cell (MPUC) which includes 100 randomly placed GMs and accounting for their polydispersion, in terms of both size and radius ratio. This model is validated by subjecting it to effective uniaxial stress and comparing its predictions of the elastic moduli with experimental findings and an analytical homogenisation technique. Second, towards modelling failure, we implement a structural criterion proposed by our group, which posits that any GM undergoes brittle failure when its average elastic energy density reaches a critical value. We then utilise our measurements of the effective strength under uniaxial compressive stress to identify different critical values for selected types of GMs. Third, on the basis of the MPUC, we reach our goal by developing a larger FE model, including 300 GMs, which enables the study of the stress diffusion from the external surface through an appropriately thick layer of composite, under macroscopic uniform pressure. This micromechanical model allows us to demonstrate the influence of the paint/coating on the syntactic foam failure mode through a detailed analysis of the collapsed GMs and the matrix stress state. Keywords: Syntactic foam; Micromechanics; Glass microballoons; Stress diffusion; Surface coating; Failure mode; Finite Element method.

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Corresponding author: Tel.: +39 030 3711238. E-mail address: [email protected]

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Introduction

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Syntactic foams are particulate composites in which hollow particles are dispersed in a matrix material (Shutov, 1986). The filler is often purposely constituted by a material that is stiffer than the matrix, such that the enclosed porosity (constituted by the so-called reinforced voids) can be leveraged to reduce moisture absorption (Sauvant-Moynot et al., 2006) and mass density (Gladysz et al., 2006), without compromising mechanical properties, such as stiffness or strength (De Runtz and Hoffman, 1969; Shutov, 1986; Bardella, 2000; Gupta et al., 2010). Syntactic foams find a natural application in buoyancy modules, where a steady surge in the demand of high performance syntactic foams is observed for deep water vehicles and gas pipes, to name a few examples. Other engineering applications can be found in Gupta et al. (2014) and references therein. This investigation is concerned with a deep water buoyancy syntactic foam for 4000 m service depth, referred to as SF-4K, constituted by an epoxy resin filled with hollow glass microspheres (Glass Microballoons, GMs). Commercial applications of deep water buoyancy syntactic foams often demand the composites to be coated or painted. We present surprising experimental results which demonstrate that the failure mode of SF-4K samples under hydrostatic pressure (tested by following the procedure proposed in Walden et al., 2010) is different, both under monotonic and cyclic loading, depending on whether the samples undergo a surface treatment or not. In our study, the surface treatment consists of epoxy-based painting or polyurethane coating. In both these cases, samples will be henceforth referred to as PC samples, where PC stands for painted/coated; bare foam samples, without surface treatment, will be referred to as BF samples. The pioneering experimental studies of De Runtz and Hoffman (1969) on the failure of syntactic foams subject to hydrostatic loading highlighted a large-scale breakage of GMs with no macroscopic fractures and large inelastic deformations. De Runtz and Hoffman (1969) tested several syntactic foam samples with different shapes and coated with a water-based modelling clay to prevent fluid penetration, whereas the pressurising fluid was hydraulic oil. They described the failure as: “a statistically uniform crushing leaving a permanent volume change of thirty-forty percent”. The presence of macroscopic inelastic deformations is a feature common to the failure mode that we have observed on PC samples. On the contrary, in our tests, BF samples exhibit a “spider web pattern” failure mode, in which there is no evidence of macroscopic inelastic deformations and GMs collapse along the web patterns, where damage is localised. As categorised in Bardella et al. (2014), various failure mode have been experimentally observed in literature on syntactic foams constituted by GMs filling a thermoset matrix (see, e.g., De Runtz and Hoffman, 1969; Rizzi et al., 2000; Bardella, 2000; Gupta et al., 2004; Adrien et al., 2007; Gupta et al., 2010). Because of the quasi-brittle nature of these syntactic foams’ failure, differences in failure mode are not only due to main factors such as the filler composition and volume fraction, but also to the precise way the effective (i.e., macroscopic) boundary conditions are applied. The objective of our work is the micromechanical explanation of the dependence of the failure mode on the sample surface (BF or PC). To this purpose, we propose three-dimensional (3D) Finite Element (FE) micromechanical models, which extend recent efforts by our group to investigate the effective moduli and strength of syntactic foams (Bardella et al., 2012, 2

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2014; Panteghini and Bardella, 2015; Shams et al., 2017). The first step in developing this type of micromechanical model consists of describing the relevant features of the syntactic foam microstructure in a cubic MultiParticle Unit Cell (MPUC). This fundamental step lays the basis to obtain reliable 3D local (i.e., microscopic) stress and strain fields to explain the observed macroscopic behaviour under any suitable macroscopic field. In this investigation, we consider effective uniaxial stress state and effective hydrostatic stress state. Remarkably, the proposed FE models are based on a MPUC which includes GMs up to 69% volume and they display macroscopic isotropy albeit accounting for filler polydispersion in terms of size and radius ratio, the latter being the ratio between the inner and outer radii of each GM. To the best of our knowledge, such a detailed description of the microstructure for highly filled syntactic foams has never been attempted. We demonstrate the potential of such microstructure description by employing the new FE model to compute the effective elastic constants and comparing our predictions with an accurate analytical homogenisation technique (Bardella and Genna, 2001; Bardella et al., 2012) and our experimental data for uniaxial compressive stress. Then, we model the composite failure by adopting the GMs’ failure criterion developed in Bardella et al. (2014), such that a GM undergoes brittle failure when its average elastic energy density reaches a critical value. This approach allows us to smooth out over the GM Beltrami’s criterion for brittle materials (Beltrami, 1889). The accuracy of this modelling strategy was previously demonstrated through comparison with experimental findings on glass/epoxy syntactic foams with 30 to 60% filler volume (Gupta et al., 2010; Panteghini and Bardella, 2015). In particular, with respect to the microscopic pointwise application of Beltrami’s criterion, this approach has two notable features. First, it allows us to avoid a very expensive, perhaps even unfeasible, numerical model to accurately capture the local stress peaks. Second, it does not require the knowledge of the glass tensile strength, which could vary with the GMs wall thickness, as shown by the probabilistic analysis of Kschinka et al. (1986) and by the 3D Eshelby-like FE model including a flawed GM of Bardella and Genna (2005). The relevance of the wall thickness polydispersion for the fillers under investigation is demonstrated in section 2.1. Given the very high filler volume fraction of the syntactic foam under investigation and our previous studies (Bardella et al., 2014; Panteghini and Bardella, 2015), we assume linear elastic behaviour of the epoxy matrix. Hence, we neglect the epoxy nonlinear time-dependent behaviour, which may be relevant in the effective properties of certain syntactic foams under specific boundary conditions: see Bardella (2001) and Bardella and Belleri (2011) for quasistatic behaviour and Shams et al. (2017) for strain-rate sensitivity at high strain rates. The assumption of linear elastic epoxy behaviour is here critically assessed by evaluating the stress state in the matrix, through the Mohr-Coulomb criterion, at the collapse of each GM. Also, we assume a perfect filler-matrix interface 1 and we do not account for the possibly modified properties of the epoxy matrix in the interphase region, which may be due to different epoxy curing close to the GMs, acting as heat sinks (Palumbo and Tempesti, 1998), or to the use of chemical agents on the GMs surface in order to establish a strong bond between filler and polymeric matrix (Al-Moussawi et al., 1993). The critical values of the average elastic energy density for selected types of GMs constituting the filler employed in the SF-4K are identified to match our measurements of the 1

The influence of the filler-matrix debonding on the effective stiffness of syntactic foams has been recently studied through a 3D FE micromechanical model by Cho et al. (2017).

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2.1

Experimental observations Materials

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effective strength under uniaxial compression with the predictions of the MPUC FE model, obtained by using mixed periodic boundary conditions (Suquet, 1987; Segurado and Llorca, 2002; Panteghini and Bardella, 2015). We finally provide a micromechanical explanation of the different failure modes experimentally observed on BF and PC samples by utilising the MPUC for SF-4K, validated and employed under effective uniaxial compression as above explained. To this purpose, we develop a model which incorporates 300 GMs, towards a tailored analysis that offers insight on the stress diffusion from the composite external surface (either BF or PC) through a layer of SF-4K, under macroscopic uniform pressure. By focusing on the collapse of the first few GMs, which macroscopically corresponds to incipient failure, this study helps in isolating the possible mechanisms through which the macroscopic uniform pressure determines the effective failure mode. Hence, this study does not aim at predicting the experimentally observed macroscopic crack propagation, which would require a sophisticated multi-scale analysis combining micro- and macro-mechanical models. The manuscript is organised as follows. In section 2, we report our experimental observations motivating this investigation. In section 3, we illustrate the 3D FE micromechanical model to evaluate the effective properties under macroscopic uniaxial stress state. Through comparison with experimental results and analytical homogenisation, this model is initially used to validate the microstructure description of the syntactic foam under study and, then, is employed to identify the critical average energy density corresponding to the collapse of the GMs. In section 4, we present the main results obtained from the micromechanical model for the stress diffusion, built on the basis of the model illustrated in section 3. Such a model allows us to demonstrate the influence of the surface treatment on the syntactic foams’ failure mode. This is highlighted in the concluding remarks offered in section 5.

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The SF-4K syntactic foam is constituted by an epoxy matrix filled with two types of GMs: one having effective density of 0.25 g/cm3 and average size of about 40 µm (denoted as GMs-25) and the other having effective density of 0.3 g/cm3 and average size of about 15 µm (denoted as GMs-30). Nominally, SF-4K has filler volume fraction f ranging from 0.7 to 0.75, of which 90% consists of GMs-25 and 10% consists of GMs-30. The SF-4K effective density is 0.44 g/cm3 , on the average. We manufacture “Macroscopic Samples” (MSs) for hydrostatic loading up to 4000 m service depth. Each MS is obtained by properly assembling, as explained below, four pieces cut from a molded block of SF-4K, resulting into a non-homogeneous sample. In fact, due to the manufacturing process, each molded block turns out to have non-uniform effective properties, whereby a larger percentage of heavier GMs accumulates towards the bottom region of the molded block. The molded block heterogeneity is characterised by testing, as reported in detail in section 2.1.1, six layers cut from the height of the block. Henceforth, these layers are labelled 1 to 6 from top to bottom. The assemblage of each molded block is illustrated in Fig. 1. It consists of bonding a plateshaped edge piece obtained from the bottom of the molded block to three parallelepipedic pieces of equal size, all sharing a side with the full molded block height. Such an assemblage is 4

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Figure 1: Assemblage of a “Macroscopic Sample”. Dimensions are in inches.

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designed to simulate a possible worst case practical scenario, in which the bulk modulus’ jump between the edge piece and the other pieces induces additional shear stress at the interface. The MSs subjected to surface treatment are either painted with an epoxy-based marine grade paint or coated with polyurethane. Henceforth, such MSs are referred to as PC-MSs, to distinguish them from the BF-MSs. The epoxy-based paint has thickness, as measured by Scanning Electron Microscope (SEM), of about 200 µm, while the polyurethane coating is a few millimetres thick. While the MSs are the samples to be subjected to hydrostatic loading, as described in section 2.2, the SF-4K is characterised by testing samples from all the six layers of a molded block under uniaxial compression. This is presented next, with focus on the filler polydispersion. Syntactic foam effective density, modulus, and uniaxial compressive strength

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Each layer is characterised by performing uniaxial compression tests following the ASTM D695-10 (“Standard Test Method for Compressive Properties of Rigid Plastics”). Hence, the tests are carried out on parallelepiped specimens of size 12.7 mm × 12.7 mm × 50.8 mm at 1.3 mm/min displacement rate, corresponding to nominal initial strain rate of 4.3 · 10−4 s−1 . From each of the six layers of a MS we obtain a single sample. For a representative MS, in Tab. 1 we report densities, Young moduli, and compressive strengths of the layers, whereas Fig. 2 shows a typical compressive stress-strain curve of the sample obtained from layer 2. The data of Tab. 1 clearly demonstrate that heavier GMs accumulate on the bottom (layers 5 and, mostly, 6), whereas layers 1 to 4 have very similar properties. 2.1.2

Filler characterisation

Particle size distribution and true particle density tests are performed on floated samples for both the employed fillers, GMs-25 and GMs-30. The particle size distribution is obtained by using the Sympatec’s laser diffraction sensor Helos/KR, according to the specification ISO 5

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Table 1: Syntactic foam characterisation: Results of uniaxial compression tests on samples from each of the six layers composing a MS. Layer Density Modulus Peak strength g/cm3 MPa MPa 1 0.44 2436 50.88 2 0.43 2408 50.53 3 0.43 2405 51.14 4 0.44 2420 51.85 5 0.45 2548 54.64 6 0.52 3056 74.53

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13320. This sensor has absolute accuracy within ±1% with respect to the standard metre and its measuring range is from 0.1 µm to 8750 µm. GMs are floated on water to discard broken particles. The GMs are extracted from each layer by carving appropriate pieces of syntactic foam prior to gelation. The layers are then washed and soaked with acetone to remove the not yet completely cured resin. In order to gain insight on the relevance of broken particles, we measure the effective density before and after the floating process. The GMs-25 effective density decreases from 0.258 g/cm3 to 0.210 g/cm3 after floating. The influence of broken particles on GMs-30 is less relevant, as its effective density decreases from 0.301 g/cm3 to 0.293 g/cm3 after floating. The size distribution of floated GMs is reported in Tabs. 2 and 3, referring, respectively, to the fillers of the employed commercial batches and to filler found in each of the six layers used to manufacture a MS (see Fig. 1). Given a percentage p, the data reported in Tabs.

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Table 2: GMs size distribution for the commercial batches. GMs-25 GMs-30 10% 50% 90% 10% 50% 90% µm µm µm µm µm µm 14.19 38.58 84.10 10.19 14.55 20.39

Table 3: Microballoons size distribution and true densities of each layer. Layer Effective density Size distribution 10% 50% 90% g/cm3 µm µm µm 1 0.202 12.98 33.36 84.36 2 0.197 13.01 34.41 86.44 3 0.196 13.25 35.54 85.50 4 0.194 13.36 34.44 82.85 5 0.210 12.87 31.31 75.44 6 0.277 12.42 23.19 56.19

2 and 3 denote the size d such that p of the filler volume consists of GMs having size lower than d. Tabs. 2 and 3 demonstrate that the employed fillers are polydispersed systems, in which 6

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Figure 2: Typical uniaxial compressive stress-strain curve characterising the layer 2 of the molded block. both size and effective density may be different for every particle. This implies a polydispersion in the GMs’ wall thickness and radius ratio as well. The radius ratio η is the most important geometric parameter characterising the filler (Bardella and Genna, 2001; Gupta et al., 2004) and can be computed for each GM through the relation η = (1 − ρf /ρg )1/3 , (1) in which ρf is the GM effective density and ρg is the glass density. Eq. (1) may be used 7

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to compute an average radius ratio by assigning to ρf the average effective density of a certain polydispersed filler. The average radius ratio may be enough to evaluate the syntactic foam effective elastic constants (Bardella and Genna, 2001; Porfiri and Gupta, 2009), while the polydispersion characterisation is needed in order to estimate the effective strength of syntactic foams filled with GMs (Aureli et al., 2010; Bardella et al., 2014; Panteghini and Bardella, 2015; Shams et al., 2017). In particular, it has been shown that lighter GMs collapse first (d’Almeida, 1999; Adrien et al., 2007; Tagliavia et al., 2011; Bardella et al., 2014; Panteghini and Bardella, 2015). In fact, even if lighter GMs carry lower load, they absorb larger average elastic strain energy density (and, hence, are possibly subject to larger peak stress). Therefore, the GMs-25 polydispersion, further characterised in the following, is the most relevant to study the SF4K failure behaviour, because GMs-25 are expected to collapse much earlier than GMs-30, thereby triggering the entire quasi-brittle failure mode. The polydispersion is obtained for both the GMs-25 commercial batch and for the GMs extracted from layer 2, by following the procedure adopted by Aureli et al. (2010). Such a procedure consists in measuring the inclusions’ size by sieving and, then, measuring the effective densities of the sieved GMs. To this purpose, three sieves with openings equal to 90 µm, 63 µm, and 25 µm are employed. The results are collected in Tabs. 4 and 5.

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Table 4: GMs-25 commercial batch polydispersion. Particles larger than Weight True densities Volume Percentage µm g g/cm3 % 90 0.0789 0.2082 6.40 63 0.4323 0.1823 40.04 25 0.7878 0.2484 53.56 Average/Total 1.299 0.2194 100.00

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Table 5: Layer 2 polydispersion. Particles larger than Weight True densities Volume Percentage µm g g/cm3 % 90 0.0737 0.1848 13.61 63 0.2882 0.1805 54.49 25 0.2030 0.2172 31.90 Average/Total 0.5649 0.1928 100.00

The GMs-25 commercial batch polydispersion (Tab. 4) is obtained from particles that are not floated; however, particles smaller than 25 µm are assumed to mostly consist of broken parts from larger GMs, such that they are unaccounted for. This hypothesis is confirmed by the layer 2 polydispersion (Tab. 5), obtained from floated GMs, that is characterised by a negligible amount of inclusions smaller than 25 µm. By comparing the particle size distribution of layer 2 reported in Tab. 3 with the GMs-25 and GMs-30 size distributions of Tab. 2, we may deduce that the percentage of GMs-30 in layer 2 should be lower than 10% and that the polydispersion of Tab. 5 is representative of that of GMs-25 in layer 2. Given that layer 2 is representative of the whole SF-4K, except for the stiffer and stronger bottom layer, we employ the polydispersion reported in Tab. 5 to 8

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model the failure mode of the syntactic foam under study. 2.1.3

Epoxy matrix characterisation

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In order to evaluate the Young modulus and strength of the employed epoxy resin we have performed three uniaxial compressive stress tests on cylindrical specimens by using the Instron 4467 test system. The tests are conducted by imposing a displacement rate equal to 0.01 mm/s, corresponding to initial 3.9 · 10−4 s−1 strain rate. The results are reported in Tab. 6. Fig. 3 shows a typical stress-strain curve obtained from these tests. The resin density is equal

to 1.13 g/cm3 .

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Hydrostatic tests

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Table 6: Matrix characterisation: Results of uniaxial compression tests on cylindrical specimens of plain epoxy matrix. Specimen Modulus Peak strength Peak strain MPa MPa 1 3586 144.5 0.063 2 3576 144.9 0.064 3 3565 144.5 0.063 Average 3575 144.6

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Following Walden et al. (2010), deep water buoyancy syntactic foams should be undamaged and gain less weight than 3% of the MS initial weight after being subjected for 24 h to a pressure pT larger than 10% the service pressure, the latter being equal to 45 MPa. This test is followed by five cycles up to the pressure pT . In each cycle the loading ramp lasts 6 min while the unloading ramp lasts 10 s. The peak pressure is maintained for half hour in the first four cycles and for one hour in the fifth cycle. The total weight gain after these five additional cycles gives information on the long term water absorption. The MSs are also monotonically tested to determine their hydrostatic strength and to analyse their critical behavior. We have performed these tests at the Woods Hole Oceanographic Institution test facility, consisting of the Dunegan SE9125-M acoustic emission transducer connected with a pressure test chamber. A frequency counter records the acoustic events occurring during the test. A sudden increase of the acoustic events detects the foam failure, whereby the hydrostatic strength is determined by a pressure transducer. In all the tests the pressurising fluid is water. 2.2.1

Qualitative observations on the samples after cyclic testing

While the BF-MSs successfully pass the cyclic test without exhibiting any macroscopic damage, this is not the case of the PC-MSs, as illustrated in Fig. 4. In particular, polyurethane coated MSs (figure 4(a)) show large inelastic deformations in several regions without exhibiting macroscopic fractures. De Runtz and Hoffman (1969) also observed relevant inelastic deformation in hydrostatic tests up to failure, on syntactic foam samples coated with a water-based modelling clay to prevent fluid penetration. Painted MSs (figure 4(b)) exhibit implosions and macroscopic inelastic deformation below the paint in multiple regions, along with major cracks near the bonded surfaces. 9

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loading. Moreover, the SEM analysis of BF-MSs, conducted by using the Hitachi S3400-N SEM and documented in Fig. 5, demonstrates that the composite is largely undamaged far from the external surface (figure 5(a)), while its outer part experiences matrix failure (figures 5(b) and 5(c)) that may allow water to leak in. This explains the (small) weight gain measured after the cyclic tests. For the sake of brevity, we omit the illustration of the SEM analysis of the PC-MSs subjected to cyclic tests, as it is analogous to that concerned with the monotonic tests up to failure, presented next. 2.2.2

Monotonic tests up to failure

BF- and PC-MSs are monotonically loaded up to failure, in the so-called “hydrostatic crush test,” and the hydrostatic strength is determined by an acoustic emission system. Fig. 6 shows a graph obtained from a typical “hydrostatic crush acoustic emission test,” where the total counted acoustic events are plotted as a function of the hydrostatic pressure. The hydrostatic strength pu is determined in correspondence to the sudden increase of the acoustic events. For the painted MS, pu = 54.33 MPa, while for the BF-MS pu = 54.74 MPa. Even though these values are very close each other, the two MSs exhibit significantly different failure modes, as is clear from the comparison of Fig. 7 and also from the acoustic events profiles in Fig. 6. Bare foam samples’ failure mode As documented in figures 7(a) and 7(c), the BF samples exhibit a spider web pattern failure mode, in which the damage is localised along 11

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Figure 5: Micrograph of a BF-MS after cyclic testing: (a) detail of a region far from the sample external surface; (b) surface orthogonal to the sample external surface, on the left side; (c) surface orthogonal to the sample external surface, on the right side. The samples analysed by SEM are obtained by cutting the specimens with a diamond saw and they are polished prior to examination.

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Figure 6: Acoustic events recorded in the hydrostatic crush test for both BF- and painted MSs. Here and henceforth compressive pressure is taken as positive. inclined lines, there is no evidence of macroscopic inelastic deformations, and, microscopically, the GMs collapse along the (macroscopically visible) web patterns. The web patterns are very well visible in figure 7(c), illustrating a sample obtained from a single layer of a molded block. These patterns have similarities to those observed by Gupta et al. (2010) on specimens failed under uniaxial compression and find agreement with the results of the X-ray tomography for the damage characterisation performed by Adrien et al. (2007).

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Painted samples’ failure mode Painted MSs after hydrostatic crush test, as displayed in figure 7(b), present a failure mode similar to that observed in the foregoing cyclic tests, even if more pronounced. Such a failure mode is characterised by extended fractures near the bonds and evident macroscopic inelastic deformation in the composite parts constituted by the upper layers of the molded block (containing the lightest GMs). As one can infer from cyclic testing in Fig. 4, the main difference between the painted and polyurethane coated MSs’ failure modes concerns the cracks. Specifically, polyurethane coated MSs are almost free from cracks, while fractured surfaces are observed in painted MSs at the bonds between layers of different density. These fractures should propagate in the unloading stage due to self-equilibrated stresses, initially needed to ensure compatibility between adjacent lighter and heavier composite regions when inelastic deformations occur. In fact, in the loading stage, MS’s pieces of different density suffer different macroscopic inelastic deformation, microscopically consisting in the collapse of GMs. The polyurethane coating, much thicker (and, then, stiffer) than the paint, can partly withstand the above mentioned self-equilibrated stresses, thus preventing the formation of cracks. Moreover, the SEM analysis presented in Fig. 8 illustrates that the failure of painted MSs propagates from the surface to inner composite regions and involves a large scale breakage of GMs without preferential directions.

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Figure 8: Micrographs of cut surfaces of painted MSs: Failed regions near the external surface.

Micromechanical modelling - Part I: Description of the microstructure, validation, and identification of the glass microballoons’ strength Modelling the SF-4K microstructure: The MultiParticle Unit Cell geometry

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As a first step, we develop a cubic MPUC representative of the SF-4K microstructure, by adapting the algorithm provided by Donev et al. (2004). The MPUC contains 100 hollow spheres of two different sizes: 35 of them model the GMs-25 and have external radius equal to 0.16225`, with ` denoting the MPUC side; the remaining 65 particles model the GMs-30 and have external radius equal to 0.06165`. The filler volume fraction in the MPUC is 69%. The MPUC is geometrically periodic, such that any inclusion cut by a cube face has its complementary part sharing the same cut with the opposite cube face. We compute the radius ratio ηGMs-30 for the GMs-30 by adopting filler effective density ρf = 0.293 g/cm3 and glass density ρg = 2.54 g/cm3 ;

hence, through relation (1), we obtain ηGMs-30 =0.95997. For the GMs-25, that are weaker and larger, we consider the radius ratio polydispersion by assigning three different thicknesses, on the basis of the filler characterisation reported in section 2.1. We consider both the radius ratio polydispersions reported in Tabs. 4 and 5, obtained from the GMs-25 commercial batch and from the GMs extracted from layer 2 of the molded block, respectively. For both cases, the actual radius ratios in the MPUC, ηj with j = 1, 2, 3, are assigned by following the procedure proposed in Bardella et al. (2014). Such a procedure accounts for the (integer) number of hollow spheres in the MPUC, for the actual filler effective density, and for the structural criterion adopted for the GMs failure. The same criterion is employed in section 3.3.2 to identify the strength of the GMs. The employed radius ratios are reported in Tab. 7, indicating also the values directly obtained from the experimental data, ηjexp , and the number of hollow spheres in the MPUC for any assigned radius ratios, Nj . 15

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Table 7: GMs-25 polydispersion in the MPUC. Layer 2 ηj 0.97513 0.97512 0.97573 0.97571 0.97064 0.97061 ηjexp

Nj 5 19 11

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Commercial batch ηjexp ηj Nj 0.97189 0.97193 2 0.97548 0.97550 14 0.96628 0.96634 19

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Finally, assigning position and radius ratio to the particles in the MPUC requires a lengthy trial and error procedure to ensure that the MPUC is macroscopically isotropic, the latter being demonstrated in section 3.3.1. In the appendix we report all the relevant data for the MPUC geometries of the two microstructures considered, which, in light of the results discussed in the following, constitute a central achievement of the present investigation.

Finite Element discretisation of the MultiParticle Unit Cell

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On the basis of the MPUC geometry illustrated in detail in section 3.1, we develop a micromechanical FE model for the SF-4K, as depicted in Fig. 9. The MPUC FE model is implemented

Figure 9: MPUC FE model for the SF-4K: Mesh of the matrix. in the commercial FE code ABAQUS (Dassault Syst`emes, 2013) by using a Python script. The matrix is discretised with 716701 fully integrated 4-noded tetrahedron continuum elements (C3D4), detailed in Fig. 9. The hollow spheres are discretised with 152656 3-noded shell elements (S3) overall, as depicted in Fig. 10 for small (GMs-30) and large (GMs-25) inclusions. The average FE size 0.015`, both for continuum and shell elements. 16

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Figure 10: FE discretisation for single GMs-30 (left) and GMs-25 (right) inclusions. We assume a perfect interface between matrix and filler, such that the shell degrees of freedom are condensated out of the system by constraining them to those of the matrix via a “tie constraint” in which the matrix hole surface is the “master” surface. We note that a fine discretisation of both the matrix holes and the microballoons, as that shown in Figs. 9 and 10, is important to obtain accurate interpolation in the condensation of the shell rotational degrees of freedom, thereby leading to a faithful representation of a perfect interface.

MultiParticle Unit Cell under effective uniaxial stress state

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The MPUC is first utilised to simulate the effective behaviour of a macroscopic inner point (far from the external surface). Hence, this micromechanical model does not distinguish between BF- and PC-MSs. We focus on effective uniaxial compressive stress, both to validate the model in the linear elastic regime and to identify, by comparison with experimental results, the GMs’ strength to be used in the micromechanical model for the failure of SF-4K subject to hydrostatic stress. In order to represent an inner macroscopic material point subject to uniaxial compressive stress, we apply mixed periodic boundary conditions to the MPUC (Suquet, 1987; Segurado and Llorca, 2002), as explained in detail in Panteghini and Bardella (2015) and summarised below. In the Cartesian reference system (x, y, z), the MPUC occupies the space region {(x, y, z) : 0 ≤ x ≤ `, 0 ≤ y ≤ `, 0 ≤ z ≤ `}. By denoting the vectorial displacement field with u, the MPUC is subject to u(x = 0, y, z) − u(x = `, y, z) = ∆u(x) (2) u(x, y = 0, z) − u(x, y = `, z) = ∆u(y)

(3)

u(x, y, z = 0) − u(x, y, z = `) = ∆u(z)

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where ∆ux is the shortening displacement (applied along the x direction on the face of (x) (x) outward normal pointing as x), whereas ∆uy , ∆uz , ∆u(y) , and ∆u(z) constitute a set of eight constant scalars to be obtained as result of the FE analysis, thus ensuring antiperiodic traction vector on the MPUC surface.

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By denoting with hg(x, y, z)i the average over the unit cell of any local (microscopic) (x) field g, the effective longitudinal strain hεxx (x, y, z)i reads ∆ux /`. The effective longitudinal stress hσxx (x, y, z)i is in principle obtained from the average of the nodal reaction forces on one of the faces of outward normal along x; in our numerical model, it is actually computed by dividing by `2 the reaction force of the sole degree of freedom left on those faces after all the remaining degrees of freedom along x are condensated out of the system, to impose the first scalar boundary condition in Eq. (2). Moreover, the imposed boundary conditions are such that hσyy (x, y, z)i = hσzz (x, y, z)i = hσxy (x, y, z)i = hσyz (x, y, z)i = hσzx (x, y, z)i = 0. (x) Hence, the effective Young modulus is given by E = hσxx (x, y, z)i`/∆ux . Moreover, among (y) the macroscopic solution variables, the effective transverse strains ∆uy /` ≡ hεyy (x, y, z)i and (z) ∆uz /` ≡ hεzz (x, y, z)i turn out to be almost identical, as later shown in Tab. 8 and required by the overall isotropy of the micromechanical model. Therefore, the effective Poisson’s ratio (y) (x) (z) (x) is given by ν = −∆uy /∆ux ≈ −∆uz /∆ux . MultiParticle Unit Cell Finite Element model validation

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We validate the MPUC FE model by estimating the effective elastic moduli under macroscopic uniaxial stress (see Fig. 11) and comparing the results with both analytical homogenisation

Figure 11: MPUC characterised with Layer 2 polydispersion and macroscopic uniaxial compressive stress imposed along the x-direction. Countour of the ux displacement component. and experimental data. The employed analytical homogenisation procedure is the Composite Sphere-based Self Consistent Scheme (CS-SCS), developed by Bardella and Genna (2001) on the basis of the Morphologically Representative Pattern theory of Bornert et al. (1996), thus extending the model of Herv´e and Pellegrini (1995) to rigorously account for the filler polydispersion and,

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if necessary, for the presence of unreinforced voids. 2 The Self-Consistent Scheme of Herv´e and Pellegrini (1995) is, in turn, the extension to the case of hollow spherical inclusions of the linear elastic homogenisation procedure that Christensen and Lo (1979) proposed as the “Generalized Self-Consistent Scheme.” In this homogenisation theory the representative volume element is a Composite Sphere Assemblage (Hashin, 1962). Specifically, we account for the four different radius ratios of the hollow spheres included in the MPUC (three for the GMs25 and one for the GMs-30). The accuracy of the CS-SCS in predicting the effective elastic moduli of syntactic foams has been demonstrated in Bardella et al. (2012) and Panteghini and Bardella (2015). The Young modulus and Poisson’s ratio of the glass are assumed to be equal to Eg = 68.9 GPa and νg = 0.276 ,

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respectively, whereas the analogous constants for the epoxy matrix are Ee = 3575 MPa and νe = 0.35 .

Tab. 8 summarise, for both the GMs-25 polydispersions considered in this investigation, the Table 8: GMs-25 polydispersion: Comparison between FE and analytical homogenisation.

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Commercial batch ν G [-] [MPa] 0.3074 1024.85 0.3078 1025.55 0.3082 1026.36 0.3078 1025.59 0.3137 1025.71 1.88% 0.01%

K [MPa] 2318.59 2325.90 2333.87 2326.12 2411.07 3.52%

E [MPa] 2492.95 2493.12 2494.24 2493.43 2500.06 0.27%

Layer 2 ν G [-] [MPa] 0.3072 953.54 0.3075 953.39 0.3076 953.76 0.3074 953.57 0.3125 952.39 1.63% -0.12%

K [MPa] 2155.12 2158.48 2160.39 2158.00 2222.45 2.90%

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E [MPa] 2679.72 2682.41 2685.44 2682.52 2694.97 0.46%

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results on the overall isotropy of the FE model and the comparison with the CS-SCS analytical estimate. The overall isotropy is assessed by comparing the elastic constants obtained from the application of the effective uniaxial stress state along different directions (x, y, z). The results are provided in terms of the effective Young modulus, E, Poisson’s ratio, ν, shear modulus, G, and bulk modulus, K. In the FE analyses we compute E and ν, as explained above, and use the standard relations for isotropic materials to obtain G and K. The results reported in Tab. 8 demonstrate that the MPUC FE models are very close to be macroscopically isotropic and accurate in estimating the effective elastic moduli. The next step consists of the validation against experimental results, reported in Tab. 1 for layer 2. Accordingly, we consider only the micromechanical model built on the basis of the GMs-25 layer 2 polydispersion. We remark that the samples’ filler volume fraction ranges from 0.70 to 0.75, and we may consider such a large volume fraction in the analytical CS-SCS estimate only, since the MPUC filler volume fraction is fixed to 0.69. Tab. 9 demonstrates that the employed models are successful in closely predicting the experimentally measured Young modulus as well. 2 Unreinforced voids are also denoted as interstitial voids and are due to suboptimal manufacturing (see, e.g., Bardella, 2000; Gupta et al., 2010; Gladysz and Chawla, 2015).

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3.3.2

[-] NA 0.3074 0.3125 0.3122 0.3106

[-] 0.7–0.75 0.69 0.69 0.70 0.75

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Experimental results (average) FE average CS-SCS CS-SCS CS-SCS

[MPa] 2408 2493.43 2500.06 2484.8 2408.06

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Table 9: Effective moduli of SF-4K with GMs-25 from layer 2 polydispersion: Comparison among experimental, FE, and analytical results. E ν filler volume fraction

Identification of the filler structural strength

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The GMs’ failure criterion is the sole constitutive prescription that we use to model failure in the present study. Any microballoon fails when its average strain energy density reaches a critical value U0 , as originally proposed by Bardella et al. (2014) and later extended to include matrix nonlinearities in Panteghini and Bardella (2015) and Shams et al. (2017). For most thermoset/glass syntactic foams, the effective strength is controlled by the first (brittle) GM collapse, followed by prominent softening (Panteghini and Bardella, 2015; Shams et al., 2017). Hence, we evaluate the effective strength as proposed in Bardella et al. (2014), by a series of linear elastic analyses, such that each analysis ends when a GM collapses according to the above structural criterion; collapsed GMs are one-by-one removed from the FE model. This implies that each analysis has one more GM than the following one. Here, U0 is identified on the basis of the comparison between experimental data reported in Tab. 1 and FE analyses on the MPUC, still under macroscopic uniaxial stress. For GMs25, we set U0 = 6.93 MPa to attain an effective uniaxial strength equal to 51.1 MPa. For GMs-30, we set U0 = 12.62 MPa in order to achieve an effective strength equal to 74.53 MPa on samples with much larger content of GMs-30, which characterise the bottom layer, of effective density ≈ 0.52 g/cm3 . Note that U0 for GMs-30 is estimated through a FE analysis on the MPUC with layer 2 polydispersion in which the GMs-25 filler is not allowed to fail. This approximation is acceptable because GMs-30 fail deep in the softening regime, after most GMs-25 in the MS have already collapsed. Hence, their larger U0 does not contribute the failure mode on the basis of the first GMs that fail in the composite, that is the purpose of the micromechanical model presented in next section 4. Even if the same U0 were assigned to GMs-25 and GMs-30, the latter would collapse much later because of the lower radius ratio. This is demonstrated in Fig. 12, presenting a typical effective stress-strain curve that can be obtained as a result of our MPUC FE analyses. In particular, in Fig. 12, where each symbol corresponds to the failure of a single GM, we show the negligible difference in the effective stress-strain curves depending on whether U0 for GMs-30 is set to 12.62 MPa or 6.93 MPa as for the GMs-25. In both analyses the first nine failed GMs are exactly the same GMs-25 particles.

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Figure 12: Effective stress-strain curves from the MPUC FE analyses. Black circles refer to U0GMs-30 = 12.62 MPa, whereas red squares refer to U0GMs-30 = 6.93 MPa.

4 4.1

Micromechanical modelling - Part II: Stress diffusion under effective hydrostatic loading and failure modes Micromechanical model for stress diffusion under effective hydrostatic loading

Towards understanding the origin of the different failure modes experienced by BF- and PC-

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Figure 13: MPUC3 FE models for SF-4K: (a) bare foam model; (b) geometry of the SF-4K external surface; (c) painted foam model.

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MSs under hydrostatic loading, we focus on the stress diffusion from the external surface into an appropriately thick surface layer of composite. To accomplish this task, we develop two models to study the micromechanics of a foam region next to the external surface, under macroscopic hydrostatic stress state. The two models share the same syntactic foam microstructural description, consisting of three MPUCs one next to the other along the direction, x, normal to the external surface. Hence, these models are referred to as MPUC3 models and each of them involves 300 hollow spheres, including 105 GMs-25 and 195 GMs30. Distinction between the two models consists of the different treatment of the square face representing the external surface. We comment that none of the two models is expected to represent any average macroscopic behaviour. Instead, they should be considered as individual realisations that we systematically compare to pinpoint the onsent of different failure modes. The first model, concerned with the bare foam, is illustrated in figure 13(a). In this case, the applied pressure follows the holes left by the microballoons on the external surface (represented in figure 13(b)), thus pointwise acting along a direction dependent on the microstructure. Moreover, in this model, the cut GMs on the external surface are not allowed to fail. In the second model, describing the painted foam model and represented in figure 13(c), the pressure is applied on the paint such that it always acts along the direction normal to the external surface. Moreover, the paint fills the hole left by a cut GM on the external surface only if the GM centre falls outside the parallelepiped defining the space occupied by the MPUC3. Such a choice is made on the basis of the SEM analysis. SEM pictures also allow the determination of the average paint thickness utilised in the analyses, equal to 0.8`. The paint is modelled as a linear elastic material of Young modulus equal to 1192 MPa, 22

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that is one-third of the matrix Young modulus, and Poisson’s ratio equal to 0.35, as for the matrix. In order to establish the reliability of the proposed model and to better understand the micromechanics of stress diffusion, we have also simulated paint layers as stiff as the matrix. The comparison with the results obtained by employing different paint layers is only briefly commented in section 4.2.

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Paint discretisation The paint layer is discretised by 176847 fully integrated linear 4noded tetrahedron continuum elements (C3D4). The mesh is finer near the interface (see figure 13(c)) to ensure an accurate description of stress diffusion. The paint and the MPUC3 meshes are bonded together via a “tie constraint,” in which the interface on the paint side is the “master” surface.

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Mixed periodic boundary conditions A crucial aspect shared by the bare and painted foam models is that mixed periodic boundary conditions are imposed on the lateral faces (normal to the external surface) only, such that Eqs. (3) and (4) still hold. On the external surface the pressure is applied as above explained, whereas on the face opposite to the external surface, henceforth referred to as “bottom surface,” we impose vanishing normal displacement component. In fact, it is very difficult to a priori establish the correct boundary conditions for the bottom surface. Hence, the purpose of the cubic region (MPUC) opposite to the external surface, henceforth referred to as “bottom MPUC,” is to smooth out the non-physical stress concentrations associated with the simple boundary condition applied on the bottom surface. This allows us as to simulate a reliable stress field in the remaining region of the MPUC3, consisting of two MPUCs. As a consequence, GMs are not allowed to fail in the bottom MPUC, such that the failure mode is studied by evaluating which GMs fail among the 200 included in the “upper” part of the model. Contrary to the case of the boundary conditions illustrated in section 3.3 to model effective uniaxial stress, in which the FE analysis may be displacement-driven, in this case, the FE analysis must be conducted by applying the overall pressure. Indeed, the application of three equal shortening displacements along the three normals to the MPUC3 faces would result in an imperfect hydrostatic macroscopic stress state, due to the lack of exact macroscopic isotropy. This challenges the possibility to model any nonlinearity because of the prominent softening observed in the macroscopic response after very few GMs have collapsed. We address this issue by evaluating the effective failure mode, as proposed in Bardella et al. (2014) and described in section 3.3.2, by a series of linear elastic analyses. Including nonlinearities, such as those governing the matrix (Panteghini and Bardella, 2015; Shams et al., 2017) or the interface behaviours, would at least require very expensive numerical techniques, such as the Riks (arc-length) method to reach convergence in the softening regime. This is a key issue in our investigation because we aim at discriminating between different failure modes (experienced by bare and painted samples, respectively) through the analysis of the positions of the first inclusions that fail, corresponding to the onset of the softening regime. We note that the MPUC3 micromechanical model for stress diffusion is not developed to obtain any effective stress-strain curve, while it models the external region of the MSs. For instance, it would not be possible to evaluate the effective stiffness from the paint model, because of the influence of the paint compliance on that region of composite.

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4.2

Results and discussion

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Matrix stress state Since we assume linear elastic matrix behaviour, it is crucial to monitor the matrix stress state and the validity of this hypothesis. To accomplish this task, we evaluate the matrix stress state, at incipient failure of each GM, by using the Mohr-Coulomb criterion. In order to apply it, we employ epoxy uniaxial compressive and tensile strengths equal to 144.6 MPa and 40 MPa, respectively, the former being experimentally measured (see Tab. 6), the latter being a conservative estimate. From these data, the cohesion coefficient c and the friction angle φ are equal to 53.78 MPa and 0.60241 in the following form of the Mohr-Coulomb criterion: σI − σIII σI + σIII ψ= + sin φ − c cos φ (5) 2 2 in which σI and σIII are the maximum and minimum principal stress components, respectively, and ψ < 0 indicates the appropriateness of the assumed linear elastic behaviour, while ψ ≥ 0 denotes material points that should experience inelastic deformation or even develop cracks. We have also carried out analyses by assuming 80 MPa for the epoxy tensile strength, in order to establish the reliability of our results in terms of the model sensitivity to this parameter. The results of such analyses are here omitted for the sake of brevity.

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Fig. 14 shows the matrix stress state in terms of ψ given by Eq. (5) in both the bare foam (figures 14(a) and 14(c)) and the painted (figures 14(b) and 14(d)) models. Figures 14(a) and 14(b) refer to the stress state at incipient failure of the first GM, whereas figures 14(c) and 14(d) correspond to the last analysis step which we deem to be reliable on the basis of the Mohr-Coulomb criterion in Eq. (5). These last steps correspond to six failed GMs for the bare foam model (figure 14(c)) and to four failed GMs for the painted foam model (figure 14(d)). Until these states, the matrix regions where nonlinearities are expected to occur on the basis of the Mohr-Coulomb criterion may be considered to be negligible. The hydrostatic strength pult is reached at failure of the first GM, in both cases, bare and painted. The bare foam model predicts pult = 45.41 MPa versus the experimental value of 54.73 MPa, whereas the painted foam model predicts pult = 44.84 MPa versus the experimental value of 54.33 MPa. Hence, both models and experiments indicate very similar hydrostatic strengths for the bare and painted cases, though predictions underestimate experimental data of about 17%. This should be considered an adequate match, given the well known difficulties in predicting quasi-brittle failure along with the fact that we have identified the GMs’ critical average energy density through comparison with experimental results on uniaxial compression. We also note that the prediction of the effective hydrostatic strength is not the central objective of our study, which instead seeks the simplest micromechanical model explaining the origin of the different failure modes experimentally observed. Hence, delving into the failure modes, figure 15(a) shows, in red, the first six GMs failed in the bare foam model; note that three of them are among the GMs split in two parts in the MPUC and one of them, in the external surface region, is split in four parts (see Tab. 10 in the appendix). In particular, the first two GMs fail on the external surface region. These collapses do not lead to a global failure, while the effective failure is triggered by the 3rd to 6th failed GMs, that are those in the central region of the model. Figure 15(b) displays, in red, the four first GMs failed in the painted model; note that two of them are among the GMs split in two parts in the MPUC. These collapses trigger the 24

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Figure 14: Contour of ψ in the matrix: (a) bare foam model after the failure of the first GM; (b) painted foam model after the failure of the first GM; (c) bare foam model after the failure of six GMs; (d) painted foam model after the failure of four GMs.

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Figure 15: Failure predicted by the MPUC3 FE models under hydrostatic loading: (a) first 6 GMs failed in the bare foam and (b) first 4 GMs failed in the painted foam.

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Concluding remarks

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failure in painted samples, in which the paint overloads the GMs close to the external surface because of the diffusion of the pressure loading on the lateral paint faces, also due to the fact that the paint is softer than the SF-4K. This agrees well with the SEM images reported in Fig. 8. We note that the use of a significantly stiffer paint layer, for instance as stiff as the matrix, alters the failure mode. In this case, the pressure loading the lateral paint faces is not transferred to the foam, causing the first relevant GMs’ collapses to occur close to the bottom MPUC, that is, far from the external surface. On the same line of arguments, a model for the polyurethane coating, similar to the foregoing MPUC3 FE model for painted samples, would not improve our understanding of the failure. In fact, the polyurethane coating is thicker and softer than the paint, such that the effect of overloading the GMs close to the external surface would be even emphasised, thereby leading to the same results of the paint foam model. Also, modelling the differences in the experimentally observed failure modes of painted and polyurethane coated MSs, which consist of the presence of macroscopic fractures in the former (see section 2.2), is beyond the purpose of the proposed MPUC3 micromechanical model. A closing comment should touch on the statistical value of our analysis. While the specific microstructure realisation examined in this paper is unlikely to represent every portion of the macroscopic sample, it is tenable that the micromechanics in some of the sample outer surface will be closely simulated. Therefore, the onset of failure identified in our analysis will be elicited in some areas of the sample and ultimately concur in triggering macroscopic failure. As a result, our approach offers insight on the effect of the way the macroscopic hydrostatic pressure is actually applied (that is, absence or presence of surface treatment) through a reductionist lens that avoids large-scale computational materials science. Large-scale simulations would be instead needed towards the study of a larger portion of the sample or the systematic analysis of different microstructural realisations that could support statistical inference. This knowledge coupled with stochastic multi-scale computation, similar to recent efforts on micro-resonators Wu et al. (2016), could eventually shed light on the mechanics of failure in the whole sample.

This investigation has focused on the failure mode of a deep water buoyancy syntactic foam for 4000 m service depth, denoted as SF-4K, composed of an epoxy resin filled with glass microballoons (GMs), up to 75% volume fraction. We have presented experimental results illustrating the significant change in failure mode due to painting or polyurethane coating the SF-4K samples, versus bare foam samples. In order to explain experimental observations, we have established a three-dimensional Finite Element (FE) micromechanical model, characterised by 69% filler volume fraction and filler polydispersion, both in size and radius ratio. The basic geometry of the SF-4K microstructure is described in a cubic MultiParticle Unit Cell (MPUC) including 100 hollow spheres. First, the FE model consisting of a single MPUC has been subjected to macroscopic uniaxial stress ensuing from mixed periodic boundary conditions. This allows for the estimate of the effective elastic moduli, towards the validation of the model against experimental results and analytical homogenisation. Through this analysis, we have also established that the proposed FE model is successful in accurately predicting isotropic effective elastic moduli.

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Acknowledgements

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Second, by adopting the criterion proposed by our group in Bardella et al. (2014), the FE micromechanical model for effective uniaxial stress state has been employed to identify, by comparison with experimental data on the uniaxial strength, the critical average energy density governing the GMs failure. Finally, we have presented a novel micromechanical model for the stress diffusion under hydrostatic pressure, with the purpose to explain the effect of the surface treatment on the failure mode. Geometrically, these models consist of three MPUCs, one next to the other along the normal to the external surface. The macroscopic hydrostatic stress state is obtained by applying mixed periodic boundary conditions on the rectangular lateral faces, while the square external surface is subject to boundary conditions which depend on whether there is surface treatment (painting/coating) or not. This micromechanical modelling scheme has allowed us to demonstrate that the presence of painting or coating causes an overload of the GMs that are close to the external surface. This overload is responsible for triggering a very different failure mode, with respect to that exhibited by bare foam samples, in agreement with the experimental findings. Our computational framework is expected to aid in the design and optimisation of syntactic foams for deep water applications, by affording improved physical understanding of the role of surface treatment.

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Work funded by the Italian Ministry of Education, University, and Research (MIUR), and by the Office of Naval Research Grant N00014-10-1-0988 with Dr. Y.D.S. Rajapakse as the program manager. G. Perini thanks the H2CU Center, the University of Brescia, and CMT Materials for the support during his stay at the NYU Tandon School of Engineering.

Appendix: The MultiParticle Unit Cell geometries

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In Tab. 10 we report all the relevant data for the MPUC geometries of the two microstructures considered, in a nondimensional MPUC of side ` = 1. The position (xc , yc , zc ) of each GM centre is given in terms of a Cartesian coordinate system whose origin lies at a MPUC corner. The positions of GMs-25 and GMs-30 are identical for the two microstructures, whereas they differ for the radius ratios’ assignment of the GMs-25. Note that any GM whose centre lies at a distance lower than the GM radius from the MPUC border is split in an appropriate number of parts (from two, for GMs located near the centre of a MPUC face, up to six, for GMs close to MPUC corners). Such parts are labelled with integer numbers after the underscore in the first column of Tab. 10. Also, since the filler GMs-30 is modelled as a monodispersed system, the subscript j for its radius ratio is NA (Not Applicable) in Tab. 10. Table 10: SF-4K MPUC microstructure data: Position, radius ratio, and type of GM for each GM. Inclusion number 1 2 3

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Table 10: SF-4K MPUC microstructure data: Position, radius ratio, and type of GM for each GM.

0.59671 0.70284 0.43854 0.42579 0.31966 0.78891 0.33241 0.58395 0.26972 0.79647 0.64072 0.68352 0.62226 0.79872 0.57785 0.22657 0.47216 0.80241 0.55247 0.77283 0.70402 0.75533 0.68115 0.50013 0.64823 0.75223 0.63616 0.38116 0.48850 0.85812 0.55537 0.57772 0.36746 0.37381 0.87864 0.93651 0.61160 0.61771 0.49284 0.64188 0.20024 0.21180

0.66403 0.39974 0.55790 0.38698 0.65128 0.82730 0.82220 0.49311 0.88031 0.12758 0.83034 0.86317 0.12332 0.77257 0.71720 0.49648 0.16708 0.18826 0.88697 0.41060 0.81384 0.91787 0.59727 0.90321 0.45278 0.80624 0.26620 0.60559 0.33563 0.83716 0.80101 0.70700 0.43065 0.82549 0.92093 0.47970 0.91332 0.09426 0.78096 0.82816 0.14644 0.32866

0.51633 0.35816 0.25203 0.52908 0.68725 0.77114 0.41020 0.79338 0.62936 0.84440 0.23179 0.57572 0.82623 0.51354 0.74598 0.18934 0.58684 0.32295 0.44873 0.13905 0.36385 0.24104 0.23602 0.27757 0.57962 0.17230 0.85128 0.47439 0.28974 0.10767 0.32276 0.15061 0.75043 0.16817 0.21825 0.37272 0.34179 0.15041 0.21762 0.08918 0.76300 0.48420

TE D

AC C

GM type 25 25 25 25 25 25 25 25 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

ηj in terms of j Batch Layer 2 1 1 2 2 3 3 2 1 3 2 2 3 3 1 3 2 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA

RI PT

zc

SC

yc

M AN U

xc

EP

Inclusion number 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

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Table 10: SF-4K MPUC microstructure data: Position, radius ratio, and type of GM for each GM.

0.81178 0.11437 0.75121 0.70058 0.65684 0.54490 0.87169 -0.12831 0.87169 0.16149 1.16149 0.49058 0.49058 0.09114 1.09114 0.09114 1.09114 0.83739 -0.16261 0.83739 0.83739 0.83739 0.21352 0.21352 0.21352 0.99057 -0.00943 0.17424 0.17424 0.00333 1.00333 0.00333 1.00333 0.40028 0.40028 0.40028 0.40028 0.81965 0.81965 0.97782 -0.02218 0.97782

0.57033 0.76692 0.91255 0.17414 0.70998 0.88399 0.36147 0.36147 0.36147 0.54514 0.54514 0.92833 -0.07167 0.00670 0.00670 1.00670 1.00670 0.12139 0.12139 1.12139 0.12139 1.12139 0.91557 -0.08443 0.91557 0.26809 0.26809 0.71607 0.71607 0.43901 0.43901 0.43901 0.43901 0.04514 1.04514 0.04514 1.04514 0.99104 -0.00896 0.09718 0.09718 1.09718

0.81116 0.36442 0.10796 0.91457 0.29249 0.15949 0.87092 0.87092 -0.12908 0.42295 0.42295 0.67449 0.67449 0.26780 0.26780 0.26780 0.26780 0.09594 0.09594 0.09594 1.09594 1.09594 0.84541 0.84541 -0.15459 0.43570 0.43570 0.14590 1.14590 0.15865 0.15865 1.15865 1.15865 0.08319 0.08319 1.08319 1.08319 0.44846 0.44846 0.71276 0.71276 0.71276

TE D

AC C

GM type 30 30 30 30 30 30 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25

ηj in terms of j Batch Layer 2 NA NA NA NA NA NA NA NA NA NA NA NA 2 3 2 3 2 3 3 1 3 1 2 2 2 2 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 2 3 2 2 2 2 2 3 2 3 2 3 2 3 2 3 3 3 3 3 3 3 3 2 2 2 2 3 2 3 2 3 2

RI PT

zc

SC

yc

M AN U

xc

EP

Inclusion number 46 47 48 49 50 51 52 1 52 2 52 3 53 1 53 2 54 1 54 2 55 1 55 2 55 3 55 4 56 1 56 2 56 3 56 4 56 5 57 1 57 2 57 3 58 1 58 2 59 1 59 2 60 1 60 2 60 3 60 4 61 1 61 2 61 3 61 4 62 1 62 2 63 1 63 2 63 3

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Table 10: SF-4K MPUC microstructure data: Position, radius ratio, and type of GM for each GM.

-0.02218 0.89719 -0.10281 0.05536 1.05536 0.88444 -0.11556 0.53192 0.53192 0.04260 1.04260 0.04260 1.04260 0.30690 0.30690 0.13598 1.13598 0.13598 1.13598 0.14874 1.14874 0.76555 0.76555 0.47782 0.47782 0.57751 0.57751 0.25487 0.25487 0.04269 1.04269 0.04269 1.04269 0.94026 -0.05974 0.94026 -0.05974 0.90829 0.90829 0.51402 0.51402 0.63384

1.09718 0.70331 0.70331 0.80944 0.80944 0.53239 0.53239 0.12268 1.12268 0.63852 0.63852 0.63852 0.63852 0.48036 0.48036 0.20331 0.20331 0.20331 0.20331 0.37423 0.37423 0.62577 0.62577 0.75741 0.75741 0.31838 0.31838 0.10993 1.10993 0.02864 0.02864 1.02864 1.02864 0.92026 0.92026 0.92026 0.92026 0.79641 0.79641 0.98701 -0.01299 0.99552

0.71276 0.31682 0.31682 0.58111 0.58111 0.59387 0.59387 0.37092 0.37092 0.85816 0.85816 -0.14184 -0.14184 0.96430 -0.03570 0.97705 0.97705 -0.02295 -0.02295 0.70000 0.70000 0.02909 1.02909 0.95154 -0.04846 0.06857 1.06857 0.54184 0.54184 0.49625 0.49625 0.49625 0.49625 0.00376 0.00376 1.00376 1.00376 0.98022 -0.01978 0.89325 0.89325 0.84207

TE D

AC C

GM type 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 30 30 30 30 30 30 30 30 30 30 30 30 30

ηj in terms of j Batch Layer 2 3 2 2 2 2 2 3 3 3 3 2 2 2 2 3 3 3 3 3 2 3 2 3 2 3 2 2 3 2 3 3 2 3 2 3 2 3 2 3 2 3 2 2 2 2 2 3 2 3 2 2 3 2 3 3 2 3 2 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA

RI PT

zc

SC

yc

M AN U

xc

EP

Inclusion number 63 4 64 1 64 2 65 1 65 2 66 1 66 2 67 1 67 2 68 1 68 2 68 3 68 4 69 1 69 2 70 1 70 2 70 3 70 4 71 1 71 2 72 1 72 2 73 1 73 2 74 1 74 2 75 1 75 2 76 1 76 2 76 3 76 4 77 1 77 2 77 3 77 4 78 1 78 2 79 1 79 2 80 1

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Table 10: SF-4K MPUC microstructure data: Position, radius ratio, and type of GM for each GM.

0.63384 0.98293 -0.01707 0.84959 0.84959 0.70637 0.70637 0.98383 -0.01617 0.98383 0.98383 -0.01617 -0.01617 0.20296 0.03586 1.03586 0.03586 1.03586 0.31652 0.31652 0.98160 -0.01840 0.95494 -0.04506 0.72462 0.72462 0.71505 0.71505 0.63477 0.63477 0.35676 0.35676 0.81781 0.81781 0.07610 0.07610 0.07610 0.03577 1.03577 0.64782 0.64782 0.72757

-0.00448 0.85190 0.85190 0.02324 1.02324 0.86931 0.86931 0.02758 0.02758 1.02758 0.02758 1.02758 0.02758 0.92853 0.82471 0.82471 0.82471 0.82471 0.05207 1.05207 0.90854 0.90854 0.73444 0.73444 0.04974 1.04974 0.00528 1.00528 0.96121 -0.03879 0.26074 0.26074 0.92751 0.92751 0.94586 -0.05414 0.94586 0.22119 0.22119 0.97958 -0.02042 0.03744

0.84207 0.11957 0.11957 0.88806 0.88806 0.99106 -0.00894 0.94311 0.94311 0.94311 -0.05689 0.94311 -0.05689 0.07162 0.99713 0.99713 -0.00287 -0.00287 0.31821 0.31821 0.87972 0.87972 0.09081 0.09081 0.77763 0.77763 0.65198 0.65198 0.09344 0.09344 0.03418 1.03418 0.97557 -0.02443 0.03513 0.03513 1.03513 0.21235 0.21235 0.23538 0.23538 0.91294

TE D

AC C

GM type 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

ηj in terms of j Batch Layer 2 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA

RI PT

zc

SC

yc

M AN U

xc

EP

Inclusion number 80 2 81 1 81 2 82 1 82 2 83 1 83 2 84 1 84 2 84 3 84 4 84 5 84 6 85 1 86 1 86 2 86 3 86 4 87 1 87 2 88 1 88 2 89 1 89 2 90 1 90 2 91 1 91 2 92 1 92 2 93 1 93 2 94 1 94 2 95 1 95 2 95 3 96 1 96 2 97 1 97 2 98 1

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Table 10: SF-4K MPUC microstructure data: Position, radius ratio, and type of GM for each GM. zc

0.72757 0.57791 0.57791 0.63628 0.63628 0.63628 0.63628

1.03744 0.09045 0.09045 0.97095 -0.02905 0.97095 -0.02905

0.91294 0.94955 -0.05045 0.96637 0.96637 -0.03363 -0.03363

GM type 30 30 30 30 30 30 30

ηj in terms of j Batch Layer 2 NA NA NA NA NA NA NA NA NA NA NA NA NA NA

RI PT

yc

SC

xc

M AN U

Inclusion number 98 2 99 1 99 2 100 1 100 2 100 3 100 4

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