The near-isotropic elastic properties of interpenetrating composites reinforced by regular ﬁbre-networks

two times larger than the Voigt limit, and a Poisson’s ratio with a positive or negative or zero value. Moreover, they are easy to manufacture, their Young’s moduli are in general 1.0–3.0 times those of the conventional particle or short ﬁbre reinforced composites and other types of IPCs including those reinforced by the triply periodic minimal surface (TPMS) shells, and the type of IPCs with the largest Young’s modulus has been identiﬁed. Crown Copyright (cid:1) 2022 Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Interpenetrating phase composites (IPCs) are a class of composites in which the constituent phases are continuous in their geometry and each phase is interconnected in terms of its architecture or microstructure to bear loads independently and hold the completeness of the structure, even if the other phases of the composhttps://doi.org/10.1016/j.matdes.2022.1109230264-1275/Crown Copyright Ó 2022 Published by Elsevier Ltd.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

⇑ Corresponding author.
E-mail address: zhuh3@cardiff.ac.uk (H.Zhu).ites fail [1].The co-continuous structure of all the phases provides the IPCs with various advantages in tuneable stiffness, strength, coefficient of thermal expansion, high impact resistance and enhanced ionic conductivity [2][3][4][5][6][7][8][9][10][11], compared to their conventional, discretely reinforced counterparts, such as particle, whisker or unidirectional fibre reinforced composites [12][13][14][15][16][17].IPCs reinforced by a regular cubic or body centred cubic open-cell foam structure have been studied by experimental measurements [10,18,19], finite element (FE) simulation [20,21] and molecular dynamics [7].The effects of the constituent materials' mechanical properties, however, have not been systematically considered.Foam based IPCs have been manufactured and found to be much stiffer and stronger than those of the constituent foams [22].Triply periodic minimal surfaces (TPMS) are also regarded as promising reinforcement structures for IPCs [23][24][25] and experimental results indicated that the mechanical properties of the TPMS reinforced IPCs were much better than those of the IPCs with the idealized Schoen's I-WP reinforcement structures at 15% volume fraction [23,25].However, this conclusion may need further consideration as the connecting parts of the idealized reinforcement structure are composed of sharp connection edges, which is not considered a well-designed structure.
Notably, it has been found that the Young's Modulus of an almost isotropic composite can significantly exceed the Voigt limit [26,27], which had long been regarded as an unreachable limit of the stiffness of isotropic composite materials.Moreover, the Poisson's ratio of IPCs was proved to be tuneable to any desired value, e.g.positive, or negative, or zero when using special matrix material or specifically designed auxetic reinforcement structures [26][27][28][29].Analytical approach and FE simulation also showed that in addition to the superior mechanical properties, IPCs also performed well in terms of their thermal and electrical conductivities [30].
In general, the factors which affect the mechanical properties of composites include the geometrical structure, mechanical properties of the constituent materials (such as Poisson's ratios, Young's moduli), interphase conditions between reinforcement and matrix [27,31].Identical periodic cells, including cubic, and body-centredcubic (BCC or tetrakaidekahedral) cells have been widely used to simulate the reinforcement structure of IPCs [20,21,27,32,33].However, very little effort has been made to explore the combined effects of the micro-scale structure and the mechanical properties of the constituent phases on the effective mechanical properties of the IPCs.
This paper aims to perform numerical simulations to investigate these effects on the elastic properties of IPCs with different types of geometric structures of the reinforcement phase, and to achieve an almost isotropic Young's modulus much larger than the Voigt limit.In addition, an analytical approach will be introduced to determine the normalised Young' s moduli of the composites within the feasible design space of the Young's moduli and Poisson's ratios of the constituent materials.The results obtained in this paper will be compared to the relevant experimental and simulation results of IPCs and conventional composites, and the IPC with the largest and almost isotropic dimensionless Young's modulus will be identified.The IPCs studied in this paper can have a size from macro scale down to micro or nano scale, and they have applications in most engineering structures.

Geometric structures
This paper focuses on the two-phase interpenetrating composites made of two homogeneous and isotropic constituent materials.
Thus, the reinforcement structure in a two-phase IPC is a kind of open cell foam.As simple and easily manufacturable structures, three types of reinforcement fibre network structures are considered: (I) cross-cubic, (II) cross-cubic with space diagonals, (III) tetrakaidekahedral.All the reinforcement fibres are assumed to have the same uniform circular cross-section and the matrix phase completely and perfectly fills the porous space of the fibre-network (i.e.there is no interlayer or defect between the fibres and the matrix).Logically, the stiffer the reinforcement lattice structure, the larger the stiffness of the IPC.Obviously, the cross-cubic lattice has the largest nearly isotropic Young's modulus among all different types of open cell foams (fibre-networks), but the BCC and cross-cubic lattice with space diagonals have better isotropy.Moreover, all these three types of reinforcement fibre networks are easy to produce compared to other perfect regular open cell structures such as TPMS.
Representative volume elements (RVEs) are usually used to model the mechanical properties of composites [34].RVE models should be sufficiently small to reduce the calculation time and sufficiently large to include all the features and characteristics of the structure to represent the target material.As all three types of IPCs have periodic structures, periodic RVE models and periodic boundary conditions are used to simulate their mechanical properties, as shown in Fig. 1.The material in cyan colour represents the matrix while the material in purple colour represents the reinforcement fibre networks.
All three types of cubic RVEs are assumed to have the same edge length L, the Young's moduli and Poisson's ratios of the constituent materials (i.e.matrix and fibre-network) are denoted as E m ; E f ; m m and m f , respectively.In order to compare their effective elastic properties, the three different types of IPCs are assumed to have the same volume fraction of the reinforcement fibre network material V f .To make it easier to partition the reinforcement fibrenetwork into tetrahedral elements, the intersections of the fibres are represented by the smallest possible spheres that can enclose the intersection lines between the connected fibres.As the IPCs are made of only two constituent materials, it is obvious that The volume fractions of the reinforcement fibre network structures considered in this study range from 3% to nearly the geometrical upper limit of each type of the structures.The fibre diameters of the three types of reinforcement fibre networks d I , d II and d III should thus satisfy the following relations: where L is the edge length of the corresponding cubic RVE model.Thus, the maximum fibre volume fractions (i.e.upper limits) of the three types of IPCs considered in this study are given below in Table 1.It should be noted that if the fibre volume fraction is too small, the total number of elements of the RVE model would be too large, which would be significantly increase the computational cost.
When the volume fractions of the reinforcement fibre networks approach their upper limits, the corresponding matrix structures of the three types of IPCs are shown below in Fig. 2.

Model parameters
In order to explore how the elastic properties of the IPCs are affected by those of their constituent materials and volume fractions, the Young's moduli and Poisson's ratios of the constituent materials should be in reasonable ranges.The Young's moduli of most metals, alloys, ceramics and carbon fibres are in the range between 60 and 400 GPa; and most solid polymers have a Young's modulus from 0.1 to 10 GPa [35].In most metal matrix composites reinforced by a metal or ceramic, the ratio E f =E m stays in the range from 2 to 10, e.g.Al/Al 2 O 3 IPCs [36].In this paper, the Young's modulus of the matrix material is assumed to be E m ¼ 1, thus the Young's modulus of the fibrenetwork material is E f ¼ E f j actual =E m j actual , and the obtained effective Young's modulus of the IPC, E c , is a dimensionless modulus which has already been normalised by the Young's modulus of the matrix material.
Although the possible Poisson's ratios of isotropic materials are in the domain (À1, 0.5), most natural materials have a positive Poisson's ratio.For example, cork's Poisson's ratio is very close to 0 [37]; Beryllium and Boron are close on the periodic table and have similar Poisson's ratio between 0.02 and 0.1 [38]; Bones have a Poisson's ratio from 0.1 to 0.2, woods are anisotropic materials with a directional Poisson's ratio from 0.2 to 0.4; and the Poisson's ratios of metals such as iron, aluminium and copper lie in the interval of (0.2, 0.4).When it comes to synthetic or fabricated isotropic materials, however, the range of their Poisson's ratios is signifi-cantly widened.Solid polymer/rubber and low density regular or irregular open cell foams have a Poisson's ratio close to the upper limit 0.5 [39,40]; modern carbon fibre fabrications and their composites can have a Poisson's ratio from 0.02 to negative [38]; reentrant foams [41], auxetic composites can have an isotropic Poisson's ratio close to À1.0 [35].
In the simulations of this paper, E f =E m ¼ 2 is for two similar constituent phases (e.g.metal-ceramic IPCs) and E f =E m ¼ 10 is for two constituent phases with some larger difference in Young's moduli (such as polymer-aluminium or resin-glass IPCs).It has been found that the Young's moduli of an almost isotropic composite could be tuned to surpass the Voigt limit with disparate Poisson's ratios [26,27].Thus, m m and m f are set to be as different as possible within the possible range of isotropic materials to show the effect of Poisson's ratios of the constituent materials on the elastic performance of the IPCs.Based on the above discussion, the possible values of E m ; E f ; m m and m f of the constituent materials of the IPCs are given in Table 2.

Computational method
The RVE models of the IPCs reinforced by the three different types of fibre-networks as shown in Fig. 1 are constructed in ANSYS software [42].Both the fibre and matrix in each RVE model are discretized by solid 187 tetrahedral elements.As all three types of IPCs are periodic, periodic boundary conditions are applied to the RVE models in the finite element simulations.To obtain the effective Young's modulus E xx and Poisson's ratio v xy of the IPCs,   a tensile strain of 0.1% is applied to the RVE models in the [1 0 0] direction in the simulations.

Results
The focus of this study is to investigate how the elastic properties of the IPCs depend on the geometrical structure and the volume fraction of the reinforcement fibre-network, and the combination of the elastic properties of the two constituent materials.As all three types of RVEs have cubic symmetry, the IPCs have only three independent elastic constants [39] to be determined, namely E xx , G xy and m xy .Further, the factor of their anisotropy will be evaluated.

The normalised Young's moduli of IPCs
To study how the elastic properties of these IPCs depend on those of their constituent materials, six different combinations of the elastic properties of their constituent materials are considered and given in Table 3.For the convenience to compare the Young's moduli of the IPCs with those of other composite materials, the Young's moduli of the IPCs are normalised by the Voigt limit and expressed as where For different combinations of the constituent materials given in Table 3, Fig. 3 presents the comparison of the normalised Young's moduli of the three types of IPCs.The broken horizontal line with a dimensionless value of 1.0 stands for the Voigt limit normalised by itself.Thus, the results plotted above this line indicates that the Young's modulus of the IPC E c is larger than the Voigt limit.
As can be seen from Fig. 3, the normalised Young's moduli E n of the IPCs decrease, reach their lower limits, and then increase with the increase of the fibre volume fraction when both m f and m m are positive.Most of the values of the normalised Young's moduli of the IPCs, E n , are smaller than 1, indicating that the Young's moduli of the IPCs are smaller than the Voigt limit.However, when m f is positive and m m is negative, the normalised Young's moduli of the IPCs increase first, reach their peaks, and then decrease with the increase of the fibre volume fraction as can be seen in Fig. 3(c)  and (d).E n is larger than 1 in most of the fibre volume fraction range and can be significantly larger than 2.0 in some conditions.Fig. 3(c) shows that for IPCs with E f =E m ¼ 2 and the Poisson's ratio of their matrix being negative, their largest normalised Young's modulus can be obtained when the fibre volume fraction V f is between 20% and 30%.For similar IPCs with E f =E m ¼ 10, Fig. 3(d) indicates that the largest normalised Young's modulus can be achieved when the fibre volume fraction V f is around 10-15%.
In general, E n increases with E f =E m approaching 1 and with the increase of m f À m m .The normalised Young's modulus of the type I IPC is larger than those of the other two types of IPCs when the Poisson's ratios of both the two constituent materials are positive.However, when the difference of Poisson's ratios of the two constituent materials is large (i.e.m f is positive and m m is negative), the normalised Young's modulus of the type III IPC is larger than that of the type II, and the latter is larger than the type I in most of the cases.The possible reason is that the much larger stiffness of the type-I reinforcement fibre network (i.e.'open cell foam') dominates the stiffness of the IPC when the difference of Poisson's ratios of the two constituent materials is small.When the difference of Poisson's ratios of the two constituent materials is large, the stresses and strains in the type-III IPC are more non-uniform, and the much higher stress and strain concentration in the type-III IPC results in larger stored strain energy density and consequently larger stiffness of the IPC.This finding may provide a guideline for designing new lattice reinforced IPCs with the possible largest and almost isotropic stiffness.Thus, the largest stiffness of almost isotropic IPCs depends on the interplay among the stiffness of the reinforcement fibre-network ('open cell foam'), the combination of the constituent material properties, and the stress and strain concentration level.
Here the extreme negative Poisson's ratios of matrix materials are selected to test the structures which are hard to find in common non-porous materials.However, it has been demonstrated that with materials of easy-access and proper reinforcement structures, the Poisson's ratio of solid composites can be designed to be either positive, or negative, or zero [28].It is possible to use these structures as the matrix in all three types of IPCs in this work.For example, the type-I solid auxetic structure (composite) of reference [28] (as shown in Fig. 4(a)) can be used as the matrix of the type-I IPCs in this paper (as shown in Fig. 4(b)).The type-I solid auxetic structure (composite) in reference [28] can reach a Poisson's ratio of À0.4 with proper solid constituent materials.Fig. 4 (c) demonstrates that the normalised Young's modulus of the type-I IPCs with their matrix being a solid auxetic composite can be larger than the Voigt limit.

The Poisson's ratio of the IPCs
Fig. 5. shows the effects of fibre volume fraction on the Poisson's ratios of different types of IPCs with different combinations of the constituent material properties.As can be seen, there is no obvious difference between the Poisson's ratios of these three different types of IPCs, and they all can be positive, negative or zero depending on the constituent material properties and the fibre volume fraction.Although a negative Poisson's ratio cannot be achieved by the three types of IPCs studied here when the Poisson's ratios of both the constituent materials are positive, the effective Young's moduli of these IPCs are much larger than those of the auxetic IPCs [28].

The anisotropy factors of the IPCs
As all three types of the IPCs have cubic symmetry, they have only three independent elastic constants and the anisotropy factor can be expressed as [39] A ¼ 2ðS    and 5, and their shear moduli are also obtained by simulations.As the shear moduli of the three types of IPCs are similar to their Young's moduli shown in Fig. 3 and the main difference is their magnitudes, the results are not presented here.Thus, their anisotropy factors can be obtained from Equation ( 5) and the results are shown in Fig. 6.As can be seen, the anisotropy factors of the three types of IPCs are very close to 1.0 except for the case with the constituent material properties shown in Fig. 6(b).Therefore, the elastic properties of all three types of IPCs are almost isotropic in most cases, e.g. the cases with the difference between the Young's moduli of the two constituent materials not being too large as shown in Fig. 6(a) and (c).

The feasible design space of the IPCs
The Young's modulus E xx and Poisson's ratio v xy of the IPCs reinforced by a cubic fibre-network can be theoretically obtained by solving a set of linear equations [27].To simplify the analysis, the reinforcement fibres are assumed to have a uniform square cross section of side length T. The analytical results of the normalised Young's moduli E n of the type I IPCs can be obtained by solving a set of simultaneous equations given in appendix 1. Fig. 3(c) indicates that the type I IPC with cylinder fibres can achieve a maximum normalised Young's modulus when the fibre volume fraction V f ¼ 0:2.The same value is selected in the theoretical analyse in this section.Thus T=L % 0:28714 is obtained from =L 3 ¼ 0:2: Confining the Poisson's ratio of the fibre material m f to (0, 0.5) and that of the matrix m m to (À1, 0.5), the effects of m f , m m on the normalisedYoung's moduli of the type I IPCs with a fixed fibre volume fraction V f ¼ 0:2 and different ratios E f =E m ¼ 2 and E f =E m ¼ 10 can be analytically obtained and shown in Fig. 7, where the grey horizontal plane represents the normalised Voigt limit E V .Furthermore, if the Young's modulus of the fibre material E f varies from 1 to 100 and that of the matrix E m varies from 1 to 10, the effects of E f and E m on the normalised Young's moduli (i.e.normalised by the Voigt limit and obtained by the analytic method) of the type I IPCs with the fixed fibre volume fraction V f ¼ 0:2 and two different combinations of the Poisson's ratios of their constituent materials m f ¼ m m ¼ 0:3 and m f ¼ 0:49, m m ¼ À0:8 are illustrated in Fig. 8, where the grey horizontal plane represents the normalised Voigt limit E V and the pink vertical plane represents the cases when E m ¼ E f .Obviously, the closer the Young's moduli of the two constituent materials E m and E f , the larger the effects of the Poisson's ratios of the constituent materials on the nor-malised Young's moduli of the IPCs, and the IPCs have the largest normalised Young's moduli when E m ¼ E f .

The self-similar hierarchical structure of IPCs
All the IPC structures presented in this paper have the potential to form self-similar hierarchical structures to further tune their mechanical properties.Here a two level self-similar hierarchy   acts as the matrix of the second level, the reinforcement material of the second level is the same as that of the first level.For the selfsimilarity, the fibre volume fraction of the second level V f 2 and the first level V f 1 are the same, thus 9(b) shows the comparison between the elastic moduli of the hierarchical IPCs and the single-level IPCs with the same total fibre volume fractions.As can be seen, structural hierarchy can significantly enhance the normalised Young's moduli of the IPCs.

Discussion
As the traditional particle composites and unidirectional fibre reinforced composites have been well investigated, it is necessary to compare the elastic properties of the lattice structured IPCs in this paper to those of the conventional counterparts.Chawla et al. [43] have done both experimental measurements and computational simulation on the elastic properties of SiC/Aluminium particle composites.Dekkers and Heikens [44] have tested the tensile behaviour of polystyrene-glass-bead composites.Rousseau and Tippur have studied the fracture behaviour of compositionally graded epoxy based glass particle composites [45].Comparing the conventional composites to the IPCs could prove that reinforcement with a self-connected phase significantly enhances the elastic properties of IPCs with the same constituent materials and fibre volume fraction.The elastic properties of the constituent materials in the conventional particle reinforced composites in the reference papers [43][44][45] are listed in Table 4, and their reinforcement volume fractions are listed in Table 5.
Fig. 10 shows the comparison between the normalised Young's moduli of the IPCs reinforced by the three types of regular fibrenetworks in this paper and those of the particle composites in the reference papers [43][44][45].As can be seen, the three types of IPCs in general have a normalised Young's modulus larger than those of the particle composites.
Although IPCs can be fabricated via different methods [22,23,32,36] as shown in Fig. 11, however, it is hard to fabricate IPCs, especially those whose reinforcement phase is a wellconnected lattice (or a regular) structure.Thus, the relevant experimental samples and results in literature often lack the comprehensiveness for exploring their superior mechanical properties.Here, the relevant experimental and numerical results of the IPCs in literature [22,23,32,36] are compared with the simulation results of the IPCs in this paper.The elastic properties and the volume fractions of the constituent materials in the IPCs in literature are listed in Table 6, and the normalised Young's moduli of those IPCs are compared with our results in Table 7 and Fig. 12.
As can be seen from Table 7, the Young's moduli of the IPCs in this paper are better than those of the IPCs fabricated with syntactic foam [22,32], the alumina-aluminium composites [36], and the TPMS-IPCs [23] which are relatively difficult to fabricate with metals like aluminum or steel.Al-ketan et al [23] have used 3D printing to manufacture periodic IPCs reinforced by four different types of  Fig. 10.Comparison between the normalised Young's moduli of the three types of structured IPCs and those of the traditional particle composites in literature: (a) the computational results in [43], (b) experimental results in [44] and (c) the experimental results in [45].TPMS shells and experimentally measured their elastic properties.Dalaq et al [24] have used finite element method to simulate the elastic properties of these TPMS-IPCs.The type D of shell TPMS IPCs (D-IPCs) possesses both good mechanical performance and isotropy.Four types of shell TPMS IPCs were also fabricated by 3D printing and the secondary IWP (IWP-S) IPCs have the largest Young's Moduli [46].Here, the normalised Young's moduli of the D-IPCs obtained by experiments [23] and simulations [24], as well as those of the IWP-S obtained by experiments [46] are compared with the results of our models in Fig. 12(a), (b) and (c), respectively.As can be seen, the IPCs in this paper in general have obviously larger Young's moduli than those reinforced by TPMS shells.
For IPCs made of conventional constituent materials (i.e. with constituent materials having positive Poisson's ratios), the IPC reinforced by a cubic fibre-network has a larger Young's modulus than those of all other types of IPCs including the TPMS-IPCs.
It is noted that the composites with their reinforcement phase being a regular cubic closed cell foam of uniform wall thickness [26,30] have an almost isotropic Young's modulus larger than those of all other different types of nearly isotropic composites and have a completely isotropic thermal or electric conductivity that is the same as the theoretical upper limit of all the isotropic composites.The high stiffness of the composites designed in [26] is because regular cubic closed cell foams with a uniform wall thickness have the largest almost isotropic stiffness among all the single-phase cellular materials as confirmed by literature [47], however, the composites in [26] are not IPCs.

Conclusion
In this paper, the effects of the constituent materials and the fibre volume fraction on the effective elastic properties of three different types of IPCs are investigated by finite element simulation and analytical methods.It can be concluded from the results that the elastic properties of these of IPCs are almost isotropic, their Young's moduli could be much larger than the Voigt limit, and their Poisson's ratios could be positive, or negative, or zero.This is in line with the analytical results in [26,27].The normalised Young's moduli of the IPCs depend strongly on the geometrical structure of the reinforcement fibre network, the Poisson's ratios and the ratio of the Young's moduli of the two constituent materials.
In general, the normalised Young's modulus of the IPCs in this paper increases when the ratio E f =E m approaches 1 and when the value of m f À m m increases.When the Poisson's ratios of the two constituent materials are both positive, the normalised Young's modulus of the type-I IPC is larger than those of the other two types of IPCs.In contrast, when the difference of Poisson's ratios of the two constituent materials is large (i.e.m f is positive and m m is negative), the normalised Young's modulus of the type-III IPC is larger than that of the type II, and the latter is larger than the type I in most of the cases.The possible reason is that the much larger stiffness of the type-I reinforcement fibre network (i.e. an 'open cell foam') dominates the stiffness of the IPC when the difference of Poisson's ratios of the two constituent materials is small.When the difference of Poisson's ratios of the two constituent materials is large, the stresses and strains in the Type III IPC are more nonuniform, and the much higher stress and strain concentration in the Type III IPC results in larger stored strain energy density and consequently larger stiffness of the IPC.When E f =E m is small (e.g.E f =E m ¼ 2), all three types of IPC are almost isotropic even when the Poisson's ratios of the constituent materials are very different.When E f =E m is large, the anisotropy of the IPCs is more dominated by the reinforcement fibre network.The Young's moduli of the IPCs in this paper are larger than those of their conventional counterparts like particle and short fibre reinforced composites.The theoretical background is that in IPCs, some struts/fibres can directly provide the 'springs-in-paral lel' stiffening mechanism, while the other fibres can enable and enhance the stiffening mechanism or effects of the different Poisson's ratios of the constituent materials.In the particle composites, however, the only stiffening mechanism is the 'springs-in-series' which is much weaker than the 'springs-in-parallel'.Moreover, these three types of IPCs are relatively easy to manufacture, and have an additional mechanism for warning possible failure (e.g. by application of an electric signal).When both the constituent materials are conventional isotropic materials with positive Poisson's ratios, the type-I IPC has the largest Young's modulus among all different types of IPCs including the TPMS-IPCs in [23,24].

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix I
The geometrical structure shown in Fig. A1 is a 1/8 of the RVE model of the type-I IPC (in Fig. 1a), where l denotes half the edge length of the RVE (l = L/2).To simplify the analysis, the selfconnected reinforcement fibres are assumed to have a uniform square cross-section with t being half the edge length of the cross-section (t = T/2).This 1/8 RVE structure could be used to perform theoretical analysis on the stiffness of the type-I IPCs, and the structure can be divided into 8 parallelepipeds.When the IPC is stretched by a strain e x in the Â direction, according to the structural symmetry, the normal stress components in the parallelepipeds are denoted as r x1 ; r x2 ; r x3 ; r y1 ; r y2 ; r y3 and r y4 , and the corresponding lateral strain is denoted as e y .
The force equilibrium and the deformation compatibility conditions require the following relations to be satisfied: The 7 unknown normal stress components and the lateral stress e y can be determined by solving the above 8 simultaneous equations.Thus, the effective Young's modulus and the Poisson's ratio of the IPC can be obtained.
Materials & Design 221 (2022) 110923 Contents lists available at ScienceDirect Materials & Design j o u r n a l h o m e p a g e : w w w .e l s e v i e r .c o m / l o c a t e / m a t d e s

Fig. 1 .
Fig. 1.Periodic RVE models of IPCs reinforced by three different types of self-connected regular fibre networks: (a) Type I, (b) Type II, (c) Type III.

Fig. 2 .
Fig.2.The corresponding matrix structures in the RVEs when their fibre volume fractions approach their upper limits.
, H. Zhu, R. Yuan et al. Materials & Design 221 (2022) 110923 ratios of the three types of IPCs are obtained and presented in Figs. 3

Fig. 3 .
Fig. 3. Comparison of the normalised Young's moduli of the three types of IPCs with different combinations of constituent material properties.

Fig. 4 .
Fig. 4. (a) The solid auxetic composite with a negative Poisson's ratio [28], (b) the type-I IPC with its matrix being made of the solid auxetic composite, (c) demonstration of the type-I composites with their Young's moduli larger than the Voigt limit.

Fig. 5 . 6 .
Fig. 5. Comparison of the Poisson's ratios of the three types of IPCs with different combinations of the constituent material properties.

Fig. 7 .Fig. 8 .
Fig. 7. Effects of m f and mm on the normalised Young's moduli En of the type I IPCs with a fixed volume fraction V f ¼ 0:2: (a) E f =Em ¼ 2; (b) E f =Em ¼ 10.

Fig. 9 .
Fig. 9. Effects of self-similar structural hierarchy on the Young's moduli of the type I IPCs with different fibre volume fractions V f ¼ 0:1; 0:2; 0:3; 0:4.(a) schematic illustration of the hierarchical structure; (b) the enhancement of the Young's modulus by structural hierarchy.

Table 1
Fibre volume fractions of different types of composite structures.
11 ÀS 12 Þ where S ij denote the components of the compliance matrix, v xy represents the Poisson's ratio, G xy the shear modulus and E xx the Young's modulus of the IPCs.The Young's moduli and Poisson's

Table 2
Young's moduli and Poisson's ratios of the constituent materials.

Table 3
Parameters of the constituent materials used in comparison of the normalised Young's moduli of type I, II and III IPCs.

Table 4
Summary of particle reinforced composites used for comparison with our models.

Table 5
Reinforcement volume fractions of the particle reinforced composites in reference papers.

Table 6
The elastic properties of the constituent materials and the fibre volume fractions in IPCs in literature.

Table 7
Comparison between the normalised Young's moduli of the different types of IPCs in this work and those of the IPCs in literature.
r y1 À m B r x1 À m B r y4