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Mechanical Performance of 3D Printed Interpenetrating Phase Composites with Spinodal Topologies

The mechanical response of interpenetrating phase composites (IPCs) with stochastic spinodal topologies is investigated experimentally and numerically. Model polymeric systems are fabricated by Polyjet multi-material printing, with the reinforcing phase taking the topology of a spinodal shell, and the remaining volume filled by a softer matrix. We show that spinodal shell IPCs have comparable compressive strength and stiffness to IPCs with two well-established periodic reinforcements, the Schwarz P triply periodic minimal surface (TPMS) and the octet truss-lattice, while exhibiting far less catastrophic failure and greater damage resistance, particularly at high volume fraction of reinforcing phase. The combination of high stiffness and strength and a long flat plateau after yielding makes spinodal shell IPCs a promising candidate for energy absorption and impact protection applications, where the lack of material softening upon large compressive strains can prevent sudden collapse. Importantly, in contrast with all IPCs with periodic reinforcements, spinodal shell IPCs are amenable to scalable manufacturing via self-assembly techniques.


Introduction
Composite materials attain their superior combinations of properties from the synergistic contribution of each constituent phase. In the vast majority of structural composites, only one phase (the reinforcement) is stiff and strong and responsible for load bearing, while the other phase (the matrix) provides load transfer between elements of the reinforcement phase and guarantees structural stability of the composite. In these traditional designs, only the matrix phase is continuous throughout the component, whereas the reinforcement elements (fibers, whiskers, particulates…) are topologically isolated. While these designs can be very efficient under specific loading conditions (e.g., unidirectional fiber-reinforced laminae under tensile uniaxial load), their efficiency significantly suffers under more complex multi-axial states, where the lack of a continuous reinforcement phase inevitably introduces soft and weak directions.
Interpenetrating phase composites (IPCs), composed of two phases that are topologically interconnected throughout the microstructure [1], represent an exciting alternative to traditional designs and have drawn a lot of research interest in recent years. Multiple studies, both numerical [2]- [10] and experimental [11]- [18], have consistently demonstrated that the combination of reinforcement and matrix in such a way that both phases are independently self-supporting and load-bearing results in improved mechanical properties compared to traditional discontinuously reinforced composite materials as the topological interconnectivity allows each constituent phase to contribute its most desirable properties to the overall properties of the composite. The synergistic role of two phases can be engineered to result in intriguing combinations of properties. As a few examples, (i) in Al2O3/Al IPCs, the stiffness of the composite was shown to be superior than that of Al, while its toughness and structural integrity were superior than that of monolithic Al2O3 [19]; (ii) while traditional aluminum foams undergo irrecoverable plastic deformation throughout their stress plateau, aluminum foam-polyurethane IPCs display extensive recoverable deformation under cyclic loadings, thanks to the stabilizing effect of the soft elastomeric phase [17]; (iii) metal-polymer interpenetrating phase nanocomposites based on Ti nanoporous can be tuned to match the elastic modulus of human bones, showing great potential for implant applications [18]; (iv) finally, molecular dynamics simulations have indicated that silicon carbide-aluminum IPCs display unique toughening mechanisms [10]. All these studies have consistently shown that IPCs are promising candidates for structural and multi-functional applications. It is worth noting that the vast majority of IPCs in these studies are manufactured through conventional methods, such as powder metallurgy [12] and infiltration processes [13], [20]. While efficient and generally scalable, these approaches randomly distribute controlled amounts of the two phases, and do not allow full topological control of the composite architecture.
Recent advancements in additive manufacturing have made it possible to manufacture IPCs with controlled and complex topologies [21]- [25]. Notable examples that have been recently extensively investigated are IPCs based on triply periodic minimal surfaces (TPMS). TPMS are mathematically defined periodic surfaces which have zero mean curvature everywhere on the surface, resulting in locally minimal surface area in each defined unit cell. Examples of TPMS include the Schwarz P surface, the Schwarz D surface and the gyroid surface [26]. In recent years, the mechanical performance of TPMS-based structures has drawn much research interest. For example, cellular materials based on TPMS have been shown to be mechanically efficient compared to truss-based cellular materials, by virtue of their smooth and regular topology, resulting in low local stress concentrations and hence efficient load transfer [26] , [28]. Recent studies have shown that similar benefits in mechanical properties extend to TPMS-based IPCs, where either the interface between the two solid phases is a TPMS or the reinforcement phase is a thickened TPMS shell structure, embedded in a softer matrix. In particular, excellent combinations of high stiffness, strength and energy absorption have been demonstrated [29]- [34], and TPMS-based IPCs have been shown to possess better mechanical properties than their truss-based counterparts [31]. In addition, TPMS-based IPCs have been shown to possess multifunctional extrema [35], [36]. However, due to the periodic nature of TPMS, all TPMS-based IPCs are generally difficult to manufacture in a scalable fashion, thus limiting their potential applications. Far better scalability can be achieved by incorporating a stochastic reinforcement phase that can be self-assembled through spinodal decomposition of two materials followed by material conversion.
Spinodal decomposition is a thermodynamic transformation where a homogeneous (solid or liquid) solution separates spontaneously into two coexisting phases [37]. The result is an interpenetrating phase composite with very large interfacial area between the phases, and hence a characteristic length scale (domain size) much smaller than the sample size; while thermodynamics tends to reduce the interfacial area between the two phases (hence increasing the characteristic length scale), this growth can be arrested by reducing the temperature of the system (in the case of solid phases) [38] or by jamming the interface with particles that are immiscible in both phases (in the case of liquid phases) [39], [40]. A number of materials conversion techniques can be subsequently used to (i) eliminate one of the phases and converting the remaining phase to the desired material (hence producing a cellular material with spinodal solid topology) [41], (ii) converting both phases to the desired materials (resulting in an IPC with spinodal solid topology) [42], [43], or (iii) eliminating one phase, coating the other phase with the desired material and finally eliminate the second phase as well (resulting in a cellular material with spinodal shell topology) [40] . As a notable example, in a recent study bicontinuous interfacially jammed emulsion gels (bijels) are formed and processed into sacrificial porous nickel scaffolds for chemical vapor deposition to produce freestanding three-dimensional turbostratic graphene (bi-3DG) monoliths with spinodal shell topologies, possessing exceptionally high specific surface area and exceeding 100,000 unit cells [44]. In all cases, the inherent self-assembly of spinodal topologies provides a route to fabricate micro-or nano-architected materials with macroscopic dimensions, with a level of scalability unmatched by any additive manufacturing technique [44], [45]. As spinodal topologies are bicontinuous, such micro/nano-architected materials could be infiltrated with a second phase via deposition or infiltration processes, potentially providing a uniquely scalable fabrication process for shell-based IPCs.
Previous studies have shown that cellular materials with spinodal shell topologies are exceptionally efficient from the perspective of specific stiffness and strength, when compared to both cellular materials with spinodal solid topologies and truss-based lattice materials [46], and perform on par with cellular materials with triply periodic minimal surface (TPMS) topologies [47], [48]. The similarity with TPMS topologies can be attributed to the very tight distribution of curvatures in spinodal topologies, with the vast majority of surface patches possessing near-zero mean curvature and negative Gaussian curvature, the signature feature of TPMSs. Remarkably, the intrinsically stochastic nature of spinodal shell topologies (and hence the deviation from minimal surface characteristics) not only does not significantly depress mechanical properties, but rather results in considerable imperfection insensitivity [46]. Whether the unique combination of scalable manufacturing and mechanical performance shown by cellular materials with spinodal topologies also translates to interpenetrating phase composites with spinodal topologies remains to be demonstrated.
In this paper, we investigate the mechanical properties of interpenetrating phase composites with spinodal shell reinforcement, with emphasis on stiffness, yield strength and energy absorption under non-linear deformations.
Spinodal shell-based IPCs are compared with composites with other reinforcement topologies, namely (1) spinodal solids, where both phase topologies are directly obtained by spinodal decomposition [42], [43]; (2) octet lattices, the most widely studied cellular architecture [49]- [51]; and (3) Schwarz P surfaces, one of the most mechanically efficient TPMS topologies [30], [52]. While a wide range of truss lattice and TPMS topologies exist, and many have been characterized mechanically, the octet lattice and the Schwartz P surface approach upper bounds of performance for the two classes of periodic reinforcement topologies, respectively, and are thus ideal candidates for assessing the mechanical efficiency of spinodal shell reinforcements. In order to accurately control all topologies, all IPCs in this study are produced by multi-material jetting, a recently developed additive manufacturing technique. While the resulting materials are polymer/polymer composites with stiffness and strength far below those of any structural material, the significant difference in mechanical properties between the two constituents allows extraction of mechanistic understanding that can be readily extend to other classes of composite materials, including ceramic/metal composites.

Spinodal solid topology
The spinodal solid topology was generated numerically with the approach reported in [46] and detailed in Appendix A. A short synopsis of the procedure is presented here. A 50% dense spinodal solid cellular topology is generated by solving the Cahn-Hilliard evolution equation [37], one of the classic evolution models for spinodal decomposition.
The Cahn-Hilliard equation can be written as: where ( , , , ) is the concentration of the material and void phases (−1 ≤ ≤ 1, with = −1 indicating solid material and = 1 indicating void space) at a coordinate (x,y,z,), t is the evolution time, ( ) = 1 4 ( 2 − 1) 2 is a double-well free energy function, θ is the width of the interface between the two phases and Δ is the Laplacian operator.
The equation was solved in space and time over a cubic domain with edge length of N, via a finite difference algorithm.
The evolution time controls the characteristic feature size (λ), a measure of the domain size of the topology. As time progresses, the features coarsen to reduce interfacial energy, thus increasing λ. We choose to extract our topologies at the evolution time corresponding to λ = 1 5 N (loosely corresponding to a sample with 5 × 5 × 5 unit cells), as this number of unit cells was shown in previous studies to provide a homogenized response [46]. To produce spinodal solid topologies with volume fractions different from 50%, a thresholding technique is used. See Appendix A for details.

Spinodal shell topology
The spinodal shell topology is extracted from the interface between the solid and void phases, produced as explained in sec. 2.1.1. The volume fraction of reinforcement, , is chosen by assigning the appropriate thickness to the shell, , so that = / , with A the surface area of the shell and V the sample volume. The procedure is schematically depicted in Fig. 1 and described in more detail in Appendix A. Spinodal shell topologies extracted from a 50% dense spinodal solid topology have negative Gaussian curvature throughout the shell, with mean curvature close to zero everywherethus approaching the geometrical characteristics of TPMSs. While spinodal shell topologies with nonzero mean curvature can be generated by extracting the surface of solid spinodal topologies with densities different from 50%, previous studies have demonstrated that their mechanical properties are inferior [46]. Hence in this work we will limit our attention to spinodal shell topologies with near-zero mean curvature throughout.

Schwarz P TPMS topology
The Schwarz P surface was created using the level set approximation technique from the implicit surface: cos( ) + cos( ) + cos( ) = 0 where (x,y,z) is a coordinate point in three-dimensional Euclidean space. The minimal surface corresponding to the above equation is generated using Minisurf, a software used to generate minimal surface CAD files [53], [54]. Cubic samples with 3 × 3 × 3 unit cells were generated. As explained in sec. 2.1.2, the volume fraction of reinforcement, , is chosen by assigning the appropriate thickness to the shell, , so that = / , with A the surface area of the shell and V the sample volume.

Octet lattice topology
The octet lattice was built using the commercially available CAD modeling software package SolidWorks. Cubic samples with 3 × 3 × 3 unit cells were generated. The volume fraction of reinforcement, , is chosen by assigning the appropriate diameter to the bars, d, according to = 6√2 ( ) 2 with being the length of the strut [49].

Numerical generation of the IPC architectures
After the reinforcement phase topologies were generated, they were imported into Geomagic Design X, a reverse engineering software that can also be used to handle CAD files. After the appropriate shell thicknesses and bar diameters were generated to result in the desired volume fraction of reinforcement, each CAD model was subtracted from a cube of the same size with a Boolean cut operation to create the complementary matrix phase. Finally, the CAD files are converted into STL files and transferred to the GrabCAD software, the control software for the 3D printer. This process, along with the printed samples, is illustrated in Fig 2.

Fabrication approach
All samples were manufactured with a PolyJet 3D printer (Objet260 Connex3, Stratasys), which allows 3 different materials to be printed simultaneously, with intermediate (digital) formulations obtained by mixing appropriate amounts of the 3 constituent materials at any voxel location. The reinforcement phase in IPCs was printed with VeroWhitePlus, a glassy photopolymer, while the soft matrix phase was printed with Agilus30, a rubbery polymer.
All samples are 30x30x30 mm cubes. Selected reinforcement-only cellular materials were printed by replacing Agilus30 with support material, which was subsequently removed by water jetting. All 3D printed samples were left at room temperature for 4 days for curing.

Mechanical characterization
All mechanical tests were performed with an Instron 8800 mechanical test frame equipped with a 100kN load cell.
The constituent materials were characterized under both uniaxial tensile and compression loading, using dog bone and cylindrical samples, respectively. Engineering stress and strain were extracted as 0 = / 0 and 0 = /ℓ 0 , with and the raw force and displacement measurements, and 0 and ℓ 0 the original cross-section and length of the sample, respectively. True stress and true strains were obtained as = 0 (1 + 0 ) and = ln (1 + 0 ), respectively. The Young's modulus was extracted as the slope of the initial linear region.
All IPC composites and cellular materials were tested in compression only. ASTM 695-15 for compressive properties of rigid polymer was followed in all compression tests [55]. Engineering stress and strain were extracted as explained above. To minimize the influence of the anisotropic nature of 3D printing introduced by the layer-by-layer printing process, all compression tests were conducted in the direction parallel to the printing direction, which is known to be the strongest. A quasi-static strain rate of 0.001s -1 was used in all tests. All samples, except for those subjected to cyclic tests, were compressed to 50% strain to measure Young's modulus, 0.2% offset yield strength, and energy absorption. The latter was extracted as the area under the stress-strain curve up to 50% strain. The cyclic test samples were compressed to 10% strain for three cycles using the same quasi-static strain rate as all in other tests.

Finite Elements Modeling
Finite element meshes of spinodal shell and Schwartz P shell composite samples generated through SimpleWare Since tension/compression asymmetry is small up to ~25% strain, the matrix material, Agilus30, is modeled as a hyperelastic material, with a Marlow strain energy potential fitted to the uniaxial compression engineering stressstrain curve in Fig. 3b.

Mechanical properties of constituent materials
Constituent materials were first tested in tension and compression as described in sec. 2.4. Stress-strain curves for the two materials are provided in Fig. 3. Notice that the reinforcement phase material is approximately three orders of magnitude stiffer than the matrix phase material; consequently, it is expected that the reinforcement phase will take the vast majority of the load in the composite samples. The matrix phase, which is three orders of magnitude softer than the reinforcement phase, still plays an important role on deformation mechanisms and damage evolution. See Appendix C for details. It is worth noting that both constituent materials show some tension-compression asymmetry, with the reinforcement phase material about 20% stiffer and 32% stronger in compression than in tension. In previous studies, VeroWhitePlus has been shown to exhibit significant size effects on strength [56]. While we observe similar size effects on the tensile properties of VeroWhitePlus dog bone specimen and the compressive properties of VeroWhitePlus cellular samples with spinodal topology (Appendix B), we did not observe any significant size effect in our experimental results on fully dense IPC samples. We attribute this lack of size effects to the fact that all samples experience similar degree of curing regardless of feature size, as the matrix and the reinforcement phases are printed at the same time in fully dense samples, and all samples are of the same size. The conclusion is that size effects on materials properties can be ignored in the following analysis of the mechanical performance of IPC composites. For more details on size effect, refer to Appendix B.

Spinodal IPCs: the difference between solid and shell reinforcement topologies
We start by comparing the mechanical response of IPCs with spinodal shell and spinodal solid reinforcement topologies, for volume fractions of reinforcement between 20 and 50%. Compressive stress-strain curves are obtained as explained in sec. 2.4, and depicted in Fig. 4a,b. While the response at low relative density is qualitatively similar, IPCs with spinodal shell topologies display a much more gradual failure as the relative density is increased, with much reduced load drops and softening at large strains. Young's modulus (E), yield strength ( ) and energy absorption (U) are presented as a function of the volume fraction of reinforcement in Fig. 4c for solid topologies [46], in good agreement with the scalings found herein for composites, confirming that the reinforcement material dominates the mechanical response. The implication is that IPCs with spinodal shell reinforcement topologies are consistently superior in all metrics, with the advantage increasing significantly at lower volume fractions of reinforcement, where the thin shells with nearly zero mean curvature and negative Gaussian curvatures behave in a predominantly stretching dominated manner and deform uniformly, with little stress intensification. As is increased, the topological difference between solid and shell spinodal reinforcement blurs, and the properties of the two IPCs converge. Given the consistent superior mechanical response of IPCs with spinodal shell reinforcement topology, IPCs with spinodal solid reinforcement topologies are not investigated further.

The mechanical advantage of spinodal shell IPC compared to IPCs with regular reinforcement topologies
It is instructive to compare the mechanical response of IPCs with spinodal shell reinforcement topology, which is intrinsically stochastic, with that of IPCs with periodic reinforcement topologies, in particular the octet lattice topology and the Schwartz P shell topology (Fig. 2). Compressive stress-strain curves obtained over a wide range of volume fractions of reinforcement ( = 5 − 50%) are presented in Fig. 5a,b,c. Young's modulus (E), yield strength ( ) and energy absorption (U) are presented as a function of in Fig. 5d,e,f. While repeat tests at any single density were not conducted, the clear power-law behavior for all properties over the entire density range, with all data points narrowly banded around the average trend, confirms repeatability of the results (e.g., in the case of strength, all data fit within a 98% confidence interval). Two key results clearly emerge: (i) The mechanical properties of IPCs with these three very different reinforcement topologies (periodic truss and shell and stochastic shell) are nearly identical over the entire range of , with the spinodal shell IPC performing slightly worse than the others in stiffness and strength (we attribute this to manufacturing defects, as the reinforcement shell of spinodal IPCs is thinner than that of the other two geometries at most volume fractions and is close to the resolution of the 3D printer at 5% volume fraction), and slightly better in energy absorption at high ; (ii) At > 35%, while the plastic and failure response of octet lattice and Schwartz P shell-based IPCs is qualitatively similar, and characterized by a sharp stress drop immediately after the ultimate strength, the stochastic spinodal shell-based IPC displays a much more gradual failure mechanism, characterized by a nearly flat stress plateau over the entire strain range.
The first result clearly reveals that the bicontinuous nature of the phases is more important than the specific reinforcement topology in determining stiffness, strength and failure initiation of the IPC composite. As many applications require materials to remain within the elastic regime, the implication is that the design space for stiff and strong IPCs is very broad: a wide range of topologies will result in very similar mechanical behavior. Conversely, the post-yielding deformation and failure behavior of the IPC composite (particularly at high volume fraction of reinforcement) is strongly affected by the topological arrangement of the reinforcement and matrix phases, suggesting that topology optimization would play a substantial role in design of IPCs for energy absorption and impact protection.
To better understand this deformation and failure behavior, we compare both the stress-strain curve and deformation response of the IPCs with the three reinforcement topologies, at = 50% (Fig. 6). Notice that the two types of IPCs with periodic reinforcements experience more catastrophic failure events than the spinodal shell IPC. Soon after the ultimate strength (at ~0.1), cracks in the reinforcement are clearly visible in the Schwartz P shell IPC; at a strain as low as ~0.2, these cracks have multiplied and aligned along a shear band, inducing catastrophic failure soon after.
While cracks are not visible at the surface of the octet lattice IPC at strains as low as ~0.1, presumably those cracks exist at the interior of the sample; at ~0.2, alignment of cracks along a shear band is clearly visible, inducing the same catastrophic failure mechanism as in the Schwartz P shell IPC. In fact, the stress-strain curves of these two IPCs are essentially identical, throughout the entire strain range. By contrast, the stochastic spinodal shell IPC does not exhibit any visible reinforcement cracking until ~0.3; even then, the cracks appear stochastically distributed across the microstructure, and not banded as for the two periodic topologies. This substantial difference in the failure mechanism results in a nearly flat stress-strain curve up to ~0.3, followed by a very gentle stress drop up to ~0.4, when the test was interrupted.
To further investigate the damage resistance of the spinodal shell IPC, loading-unloading compression experiments were performed, whereby IPCs with the three reinforcement topologies, at = 50, were compressed to 10% strain for three cycles (Fig. 7). As the number of cycles increases, all IPCs show increasingly visible cracks in the reinforcement phase. The Schwartz P shell IPC and the octet lattice IPC both experience decreasing load-bearing capacity as a result. On the contrary, even with increasing visible fracture sites, the load-bearing capacity of the spinodal shell IPC is largely unaffected by the cyclic loading, demonstrating superior resistance to damage.
We attribute these dramatic differences in deformation and failure behavior to (i) the larger surface area and (ii) the stochastic nature of the spinodal shell topology. The larger surface area of the spinodal shell compared to the other two periodic geometries results in increased reinforcement/matrix interaction, with the matrix preventing/arresting crack propagation in the reinforcement phase. For a more detailed discussion on the influence of surface area, refer to Appendix D. In addition, the stochastic nature and complex shape of the spinodal shell reinforcement distribute the loads more efficiently throughout the topology, maintaining load-bearing capacity after fracture initiation and preventing the formation of catastrophic failure-inducing crack bands. In periodic geometries like Schwartz P or octet truss, failures first occur around stress concentration locations, and subsequently band along specific directions, leading to catastrophic failure and loss of load bearing capacity. Conversely, for the stochastic spinodal shell topology, stress is more uniformly distributed across the entire structure. Even after fracture of some shell sections, the complex topology still allows the spinodal shell-reinforced composite to maintain nearly unchanged load bearing capacity.
The difference in failure response between the spinodal and the two types of periodic reinforcement IPCs is less pronounced at low volume fractions of the reinforcement (Fig. 5), as the increased volume fraction of the matrix helps stabilize even periodic structures against crack banding of the reinforcement.
To better understand and quantify the differences in deformation and failure behavior, finite element analyses were performed on the IPCs with the three reinforcement topologies, at a volume fraction of reinforcement = 30%. The results are presented in Fig. 8. Notice that the computational prediction of the stress-strain response is in good agreement with the experimental results, well capturing the initial stiffness, the yield and ultimate strength and the beginning of the post-failure behavior (Fig. 8a,b,c). This agreement validates the computational model and provide

Conclusions
In summary, we have fabricated and mechanically investigated a new type of interpenetrating phase composite (IPC) with spinodal shell reinforcement topology, comparing their mechanical response to that of well-established mechanically efficient periodic IPCs with octet lattice and Schwartz P shell reinforcement topologies. Polymeric experiments performed in this work [59], [60]. This strongly suggests that spinodal shell IPCs would be excellent performers for impact protection.
Most importantly, unlike periodic IPCs, spinodal shell IPCs can in principle be scalably manufactured at various length scales, using self-assembly approaches followed by material conversion techniques. Possible self-assembly approaches include spinodal decomposition of block copolymers [45], interfacially jammed colloidal suspensions (bijels) [40], [44] and selective etching of bimetallic alloys [41]. These approaches allow ready fabrication of polymeric, metallic or ceramic macro-scale samples with domain sizes at the micro or nano-scale, resulting in architected materials with enormous surface area and further improving mechanical properties by virtue of wellestablished size effects on the constituent materials [61]- [63]. As spinodal topologies are bicontinuous, such micro/nano-architected materials could be infiltrated with a second phase via deposition or infiltration processes, potentially providing a uniquely scalable fabrication process for shell-based IPCs.
Finally, we emphasize that, while the multi-material additive manufacturing approach used in this study resulted in polymer-polymer composites with absolute mechanical properties far inferior to those of any structural material, the self-assembly-based processes envisioned above could be used to produce ceramic-polymer, ceramic-metal, metalpolymer and metal-metal composites that can in principle outperform most existing structural materials.
Demonstration and characterization of such advanced spinodal shell-based IPCs will be the subject of future studies.

Appendix A. Generation of spinodal topologies
The generation of spinodal topologies follows an approach discussed in detail in [42,51], and summarized here for completeness. Spinodal decomposition can be described by the Cahn-Hillard evolution equation [37]: be sufficiently small to achieve convergence ( =0.005 was used here), and m the time step. After discretization with a finite difference scheme, equation (A1) can be written as: where is the thickness of the interface between the two phases and Δ is the Laplacian operator. A randomly generated initial condition, ( , , , 0) = 0 ( , , , 0) ∈ [−5,5] × 10 −4 ≠ 0, was used as a perturbation to exit unstable equilibium and start the decomposition kinetics. As the solution progresses, the system phase separates at early times, and subsequently continues to coarsen; during the coarsening phase, the curvature of the interface between solid and void decreases and the size of the single-phase domains increases. A cutoff is defined to separate phase A from phase B, with the phase at a point ( , , ) and a time = defined as: where (•) represents the Heaviside function. To achieve a 50% volume fraction ( ) of phase A, the cutoff is adjusted so that satisfied the distribution given by: Here the decomposition time that controls the characteristic feature size (λ) has been set to provide λ = 1/5 N, a geometrical condition that has been shown to provide the best overall mechanical performance for spinodal shellbased cellular materials [46].
The spinodal shell topologies are subsequently derived by extracting the surface from spinodal solid cellular topologies with volume fraction, V=50%, resulting in a shell topology with negative Gaussian curvature and near-zero mean curvature throughout the domain. While shell-topologies with non-zero mean curvature can be derived from spinodal solid topologies with ≠ 50% , previous studies have demonstrated that near-zero mean curvature topologies provide the best overall mechanical performance for spinodal shell-based cellular materials [46].

Appendix B. Size effect on the mechanical properties of 3D printed consituent materials
The mechanical properties of the photopolymers used in this study, namely the glassy VeroWhitePlus and the rubbery Agilus30, have not been as thoroughly investigated as other commonly used structural materials for additive manufacturing. As several materials for additive manufacturing are known to exhibit size-dependent mechanical properties, we investigated the tensile properties of dog bone samples printed with VeroWhitePlus (the reinforcement material in our study), as a function of sample size. The results are reported in Fig. B.1a, and show a clear size effect for sample dimensions lower than ~1mm, with smaller samples being less stiff and strong, but far more ductile. The results are consistent with a recent study, which has shown a well-defined size threshold for the strength of VeroWhitePlus: above this threshold, no size effect is observed, whereas below this threshold, a smaller-is-weaker effect is present [56]. This size threshold was estimated to be approximately 50 times the printing layer thickness, which is in good agreement with the threshold identified in our dog bone experiments. The size threshold is shown as a horizontal line in Fig. B.2, overlaid with the reinforcement feature sizes of IPC samples tested in this study.
According to Fig. B.2, we would expect to see no size effect for most IPCs examined in this study, with the exception of IPCs with spinodal shell topologies for reinforcement volume fractions smaller than 30%. To verify this assumption, we proceeded to test in compression two 30% dense spinodal shell cellular materials, with identical topology but different shell thicknesses, 1 = 0.75 and 2 = 1.5 . The latter was printed with two times the overall dimension of the former, so to have the same topology and volume fraction but different shell thickness. As shown in can be explained as follows. A recent study on size effect in 3D printing polymers has shown that the size effect observed on thinner samples emanate from the layer-by-layer printing process [64]. As the sample thickness increases, previously printed layers are exposed longer to UV light, which leads to higher curing degree and better mechanical properties. In agreement with this study, we believe that the absence of size effect in our IPC composites stems from the fact that the reinforcing phase and the matrix phase are printed on the same layer in the same scan over the entire sample, so that printing composites is qualitatively similar to printing solid blocks. As all composite samples have similar size and therefore similar number of layers, the polymer in them is cured to a similar degree.
The important implication is that size effects on the base material properties, while generally important, can be ignored in the analysis of the mechanical properties of the IPC composites discussed in this study.

Appendix C. Comparison of the deformation mechanisms in composite and cellular (reinforcement only) samples
Past research has shown that the soft phase in a 3D printed composite helps stabilize the reinforcing cellular materials which leads to more uniform deformation and increased load bearing capacity [34]. while larger surface area leads to increased support from the matrix, incompressibility promotes membrane deformation of the shell members, while preventing bending and buckling. The overarching conclusion is that the matrix, which is three orders of magnitude softer than the reinforcement, barely contributes to initial stiffness and strength of the composite, but has a pronounced effect on deformation mechanisms and damage evolution.

Appendix D. Influence of surface area of the reinforcement phase in IPCs
One possible reason for the unique properties of spinodal shell IPCs is the increased surface area of the spinodal shell relative to the other reinforcement topologies, which leads to increased interaction and support from the matrix. As discussed in Appendix A, the surface area and curvature of spinodal geometry is affected by decomposition time t, with increasing t resulting in deceased surface area and curvature. The decomposition time t can be controlled to generate different ratio between the characteristic length over the cubic length, / (Fig. D. 2). In all spinodal samples tested in the manuscript, / = 1/5 was used as it has been shown to provide the best overall mechanical properties to spinodal shell cellular materials [46]. The surface area of the 3D printing models for all three topologies was extracted in Netfabb, a 3D printing software, and plotted in Fig. D.1. Notice that the spinodal shell topology has significantly more surface area than both Schwartz P and octet lattice topologies, especially at lower volume fractions of reinforcement. As seen in Fig. 6, even at high volume fraction, where the octet lattice has similar surface area, spinodal shell IPCs are still better at avoiding catastrophic failures and have better energy absorption as a result. This is attributed to the stochastic nature and complex shape of the spinodal shell topology.
The influence of curvature and surface area on mechanical properties of spinodal shell IPCs deserves further investigation. Some preliminary results are shown in Fig. D. 3. Samples with / = 1/8 have the highest surface area and curvature, while samples with / = 1/3 have the lowest. It can be seen that samples with / = 1/8 seem to perform slightly better mechanically than samples with / = 1/5, and significantly better than samples / = 1/3.
One possible explanation is that samples with / = 1/3 are more susceptible to defects from manufacturing. It has been found in our previous study that samples with / = 1/8 are less sensitive to imperfection than samples with / = 1/3 because of its increased surface area and more stochastic nature [46]. Further experiments and simulation are needed to better understand this phenomenon.

Appendix E. Supplementary Data
Movie S1. Numerical simulation of deformation and fracture of spinodal shell-reinforced IPC, under 10% compressive strain. Contours represent the Von Mises stress distribution.
Movie S2. Numerical simulation of deformation and fracture of Schwartz P shell-reinforced IPC, under 10% compressive strain. Contours represent the Von Mises stress distribution.
Movie S3. Numerical simulation of deformation and fracture of octet lattice-reinforced IPC, under 10% compressive strain. Contours represent the Von Mises stress distribution.
Strategic Academic Customer Program between UC Irvine and SIMULIA. All fabrication was carried out at the Institute for Design and Manufacturing Innovation at UC Irvine. Figures   Fig 1. Schematic of approach for numerically generating the spinodal shell reinforcement phase for IPCs.