Abstract
We have designed a new type of composite materials which not only has a Young’s modulus much larger than the Voigt limit, but also is always nearly isotropic. Moreover, its Poisson’s ratio can be designed at a desired value, e.g. positive, or negative, or zero. We have also demonstrated that structural hierarchy can help to enhance the stiffness of this type of composite materials. The results obtained in this paper provide a very useful insight into the development of new functional materials and structures.
Introduction
Our life quality and living conditions largely rely on composite materials. In fact, the bones in our body are a nanostructured hierarchical composite material with the basic building blocks being nanosized single crystal mineral plates embedded in soft protein matrix^{1,2}. Many different types of advanced artificial composite materials are used more and more frequently in our daily lives, examples include kitchen tools, sport facilities, vehicle and airplane structures.
The Voigt limit has long been regarded as an unexceedable upper limit for the stiffness of isotropic composite materials, as can be seen from thousands of text books, e.g. reference^{3}. For a two phase composite made of two different isotropic materials A and B whose Young’s moduli are E_{A} and E_{B} and Poisson ratios are v_{A} and v_{B}, respectively, the Voigt limit for the Young’s modulus of the composite is given as
Where f_{A} and f_{B} are the volume fractions of the two materials and thus f_{A} + f_{B} = 1. The lower limit (i.e. the Reuss limit) for the Young’s modulus of the twophase composite is given as
It is relatively easier to make the Young’s modulus of an anisotropic composite equal to or larger than the Voigt limit than an isotropic composite. For example, for a laminate composite made of two isotropic materials, the inplane Young’s modulus is obviously the same as the Voigt limit if the Poisson’s ratios of the two component materials are the same and larger than the Voigt limit if the Poisson’s ratios are different. The larger the difference of the two Poisson’s ratios, the larger the stiffness of the laminate composite. In general, laminate composites may have 3 orthogonal planes of symmetry, thus they may have up to 9 independent elastic constants. If a laminate composite is inplane isotropic, the number of the independent elastic constants will reduce to 5 from 9.
Lim^{4} has investigated the outofplane modulus of semiauxetic laminates and found that outofplane stiffness can be made larger than the Voigt limit by using a combination of positive and negative Poisson’s ratios. Liu et al.^{5} have analyzed the elastic properties of inplane isotropic semiauxetic laminates and obtained all the 5 independent elastic constants. They found that both the inplane and outofplane moduli can be made larger than the Voigt limit using a combination of positive and negative Poisson’s ratios. Lim and Rajendra Acharya^{6} and Grima et al.^{7} have also studied the elastic properties of semiauxetic laminates.
As semiauxetic laminate composites are in general orthotropic, they may have 5 or a larger number of independent elastic constants. The objective of this paper is to design a new type of composite materials which not only has a Young’s modulus much larger than the Voigt limit, but more importantly, is always nearly isotropic. The Poisson’s ratio can be designed at a desired value, e.g. positive, or negative, or zero and structural hierarchy can further enhance the Young’s modulus.
Geometric and Mechanics Model
The emphasis of this paper is on the design of singlelevel twophase composite materials. The designed singlelevel composite materials are assumed to be composed of a large number of identical cubic periodic cells, as shown in Fig. 1(a) which is one representative volume element (RVE) of the composite. In the RVE, material A is a hollow cubic box which has square walls of uniform thickness t/2 and an external edge length L; material B is a solid cube which is inside the hollow cubic box of material A and has an edge length L – t. The interfaces of materials A and B are assumed to be perfectly bonded. Advanced manufacturing technology, e.g. 3D printing or prototyping^{8}, makes it possible to produce such designed composite material.
In the twophase composite, the volume fraction of material A is
and the volume fraction of material B is thus .
The designed composite material has a cubic symmetry and thus has only up to 3 independent elastic constants^{9,10}, namely E_{xx}, G_{xy} and v_{xy}. Obviously, E_{yy} = E_{zz} = E_{xx}, G_{xz} = G_{yz} = G_{xy} and v_{yz} = v_{xz} = v_{xy} and the Zener’s anisotropy factor is always very close to 1 (i.e. nearly isotropic). To obtain the effective Young’s modulus E_{xx} and the Poisson’s ratio v_{xy} for the composite material, the cubic periodic RVE shown in Fig. 1(a) is stretched to a strain ε_{x} in the x direction by an effective uniaxial tensile force/stress. The periodic boundary conditions and the symmetry of the applied load require that all the six outside planes of the cubic periodic unit RVE in Fig. 1(a) remain plane after deformation.
To simplify the analysis, the RVE is divided into 8 parallelepipeds, as can be seen in Fig. 1(b). In order to carry out analytical solution, we consider only the normal stresses within each of the 8 parallelepipeds in the RVE and the periodic conditions (i.e. compatibility conditions) on the outside surfaces of the RVE and ignore the shear stresses inside the parallelepipeds and the compatibility conditions on the interfaces between the parallelepipeds inside the RVE. Thus, the cubic periodic representative volume element (RVE) shown in Fig. 1(b) can be used as a simplified mechanics model of the twophase composite, where the 3 normal stresses in each of the 8 parallelepipeds are assumed to have constant values. When the RVE is stretched in the x direction, the normal stresses and strains on the top plane of the RVE shown in Fig. 1(b) are exactly the same as those on the right plane. According to the symmetry, we have 7 different unknown normal stresses, namely, σ_{x1}, σ_{x2} and σ_{x3} on the front surface of the RVE; and σ_{y1}, σ_{y2}, σ_{y3} and σ_{y4} on the right surface of the RVE, as shown in Fig. 1(b). From the Hooke’s law and the periodic boundary conditions of the RVE, we have following stressstrain relations
In addition, the zero total force in the normal direction of the top or right plane of the RVE in Fig. 1(b) requires
For a given value of the tensile strain ε_{x}, we have in total only 8 unknowns to be determined: σ_{x1}, σ_{x2}, σ_{x3}, σ_{y1}, σ_{y2}, σ_{y3}, σ_{y4} and ε_{y}. They can be solved from the 8 simultaneous linear Equations (4, 5, 6, 7, 8, 9, 10, 11). Thus, the effective Young’s modulus and Poisson’s ratio of the composite material can be obtained as
Results
It is well known that the range of the Poisson’s ratio of isotropic materials is from −1.0 to 0.5, i.e. (refs 11, 12, 13, 14, 15, 16). For example, solid polymer or rubber materials, or low density random irregular open cell foams^{12} have a Poisson’s ratio close to 0.5; most metal materials have a Poisson’s ratio between 0.1 and 0.4; cork has a Poisson’s ratio close to 0 (ref. 17); open cell foams with reentrant cells (i.e. auxetic foams) have a negative Poisson’s ratio^{13,14}; hierarchical laminates^{18} or auxetic materials^{19} can be designed to be isotropic and to have a Poisson’s ratio close to −1.0 (refs 13, 14, 15, 16,19).
For singlelevel twophase composite materials with the cubic periodic RVE structure shown in Fig. 1 and with E_{A} = 2.0E_{B} and , the relationship between the effective Young’s modulus E_{xx} and the volume fraction f_{A} can be obtained by solving Equations 3, 5, 6, 7, 8, 9, 10, 11, 12 and plotted in Fig. 2. The Voigt bound, the Reuss bound and the Hashin—Shtrikman^{20} upper and lower bounds are also presented for comparison. It is noted that the Young’s moduli in Fig. 2 are normalized by E_{B}. As can be seen from Fig. 2, the effective Young’s modulus of the composite material predicted from our mechanics model shown in Fig. 1(b) is larger than the Hashin—Shtrikman upper limit when the volume fraction f_{A} is smaller than 82%. As the possible effect of the Poisson’s ratios of materials A and B is completely absent in Fig. 2, the enhancement of the effective Young’s modulus (i.e. larger than the Hashin—Shtrikman upper limit) can be attributed to the geometrical structure. We have also tested cases of and other values and found that as long as , the results of the effective Young’s modulus of the composite materials obtained from Equations (3, 4, 5, 6, 7, 8, 9, 10, 11, 12) remain unchanged.
We now explore how to make the Young’s modulus of a singlelevel composite material larger than the Voigt limit. For a singlelevel twophase composite material with the cubic periodic RVE structure shown in Fig. 1, the effects of different combinations of the Young’s moduli and Poisson’s ratios of materials A and B on the relationship between the effective Young’s modulus of the composite and the volume fraction f_{A} are illustrated in Fig. 3(a–f), where the Young’s modulus of the composite is normalized by the Voigt limit . As the Voigt limit normalized by itself is constantly 1.0, a value above 1.0 in Fig. 3(a–f) indicates that the Young’s modulus of the composite material is larger than the Voigt limit.
We can see from Fig. 3(a–f) that, when E_{A} = E_{B} (i.e. when the possible effects of the difference between E_{A} and E_{B} are absent), the difference between v_{A} and v_{B} can make the normalised Young’s modulus of the composite material larger than 1.0 (i.e. exceeding the Voigt limit). Moreover, the larger the difference between v_{A} and v_{B}, the larger the Young’s modulus of the twophase composite material. Comparing Fig. 3(b,c) to (e,f), it can be found that if v_{A} is negative and v_{B} is positive, the composite material has a larger Young’s modulus than the case when v_{A} is positive and v_{B} is negative. In the case when v_{A} = −0.8 and v_{B} = 0.45, the Young’s modulus of the composite material is about 150% larger than the Voigt limit.
Figure 4(a–f) show that by properly choosing the Young’s moduli and the Poisson’s ratios of materials A and B, the Poisson’s ratio of a twophase composite material can be designed to have a desired value, e.g. positive, or negative, or zero. These results are very useful for the design of more interesting and useful functional materials or structures for applications in many different areas. For example, materials with a zero Poisson’s ratio are perfect for sealing applications^{17}.
Discussion
To validate the analytical results for the effective Young’s moduli and Poisson’s ratios of the twophase composite materials obtained from Equations (4, 5, 6, 7, 8, 9, 10, 11), we used the commercial finite element software ABAQUS to perform a number of simulations (i.e. to do numerical experiments) for the cubic periodic RVE structural model shown in Fig. 1(a). The RVE is partitioned into 8000 C3D8 elements. Periodic boundary conditions are used in all the finite element simulations and the obtained simulation results can be assumed to be the exact results. Table 1 presents the analytical results and the finite element simulation results for the twophase composite materials with different combinations among the values of E_{A}, E_{B}, v_{A}, v_{B} and f_{A}, where the effective Young’s moduli of the composites are normalized by the Voigt limit (E_{C})_{upper}.
Table 1 shows that the analytical results for the Young’s modulus of the singlelevel composite materials obtained from Equations (4, 5, 6, 7, 8, 9, 10, 11) are always smaller than the simulation results, suggesting that the analytical results always tend to underestimate the Young’s modulus of the composite materials. This is consistent with the mechanics principle because any additional restraint always makes a material or structure stiffer. In the analysis of Equations (4, 5, 6, 7, 8, 9, 10, 11), only normal stresses in the RVE and periodic conditions on the outside boundaries of the RVE are considered, while all the possible shear stresses and all the compatibility conditions inside the RVE are ignored. This could result in possible gaps or overlaps between the 8 deformed parallelepipeds inside the RVE. To remove the gaps and overlaps (i.e. to make the interfaces between the 8 deformed parallelepipeds inside the RVE perfectly bonded), additional work has to be done and this consequently increases the stored strain energy in the RVE and hence makes the composite stiffer. In contrast, all the actual normal and shear stresses and all the compatibility conditions inside and outside the RVE have already been considered in the finite element simulations using the ABAQUS software. As the finite element simulations have considered much more restraints between the interfaces of the 8 parallelepipeds than the simplified mechanics model shown in Fig. 1(b), the exact results for the effective Young’s modulus obtained from the finite element simulations are consequently always larger than the analytical results obtained from Equations (4, 5, 6, 7, 8, 9, 10, 11).
Table 1 shows that when v_{B} ≥ −0.8, the difference between the effective Young’s modulus of the composite materials obtained from Equations (4, 5, 6, 7, 8, 9, 10, 11) and that obtained from the ABAQUS finite element simulation is constantly less than 8%, indicating that the analytical results shown in Figs 3 and 4 are quite accurate and hence reliable. When v_{B} approaches −1.0, although the error of the analytical results becomes larger, the predicted trend of the effects remains correct.
Now we demonstrate how structure hierarchy could further enhance the elastic properties of a twophase composite material. The twophase hierarchical composite material is assumed to be made of isotropic materials A and B with Young’s moduli E_{A} and E_{B}, Poisson ratios v_{A} and v_{B} and volume fraction f_{B}. At each hierarchical level n, the composite material is assumed to be composed of a large number of identical RVEs, as shown in Fig. 5 and each of the cubic fillers/inclusions (i.e. equivalent to material ‘B’ in Fig. 1) in the RVEs is also made of a large number of identical lower level (i.e. level n − 1) cubic periodic RVEs. For simplicity, the hierarchical composite material is assumed to be selfsimilar in structure and the volume fraction of the cubic fillers/inclusions (i.e. material ‘B’ ) in the RVEs is assumed to remain fixed at all hierarchical levels^{21},
Where, n is the specific hierarchical level and N is the total number of the hierarchical levels.
For a given material volume fraction f_{B} and a given number of the total hierarchical levels N, the volume fraction of the cubic fillers/inclusions in the RVEs at each hierarchical level, f_{B(n)}, can be obtained from Equation (14) and the Young’s modulus E_{(n)} and Poisson’s ratio v_{(n)} at each hierarchical level can be obtained from Equations (4, 5, 6, 7, 8, 9, 10, 11, 12, 13). Figures 6 and 7 show the analytical results of the Young’s modulus E_{N} and Poisson ratio v_{N} for a few hierarchical and selfsimilar composite materials as functions of the number of the total hierarchical levels N, where the Young’s modulus is normalized by the Voigt limit . In Figs 6 and 7, the results of the case N = 1 are those of the singlelevel composites, which can also be seen from Figs 3(c,f) and 4(c,f). The results in Fig. 6(a,b) indicate that increasing the number of hierarchical levels tends to enhance the stiffness of composite materials. The results obtained in this paper provide very useful insight into the development of new functional materials and structures.
Additional Information
How to cite this article: Zhu, H.X. et al. Composite materials with enhanced dimensionless Young’s modulus and desired Poisson’s ratio. Sci. Rep. 5, 14103; doi: 10.1038/srep14103 (2015).
References
Buehler, M. J. Nature designs tough collagen: explaining the nanostructure of collagen Fibrils. Proc. Natl. Acad. Sci. 103, 12285–12290 (2006).
Gao, H., Ji, B., Jager, I. L., Arzt, E. & Fratzl, P. Materials become insensitive to flaws at nanoscale: lessons from nature. Proc. Natl. Acad. Sci. 100, 5597–5600 (2003).
Hull, D. & Clyne, T. W. An introduction to composite materials. Cambridge Press, 1996.
Lim, T. C. Outofplane modulus semiauxetic laminates, European J. Mech. A/Solids 28, 752–756 (2009).
Liu, B., Feng, X. & Zhang, S. M. The effective Young’s modulus of composites beyond the Voigt estimate due to the Poisson effect. Composites Sci. Tech. 69, 2198–2204 (2009).
Lim, T. C. & Rajendra Acharya, U. Counterintuitive modulus from semiauxetic laminates. Phys. Status Solidi B 248, 60–65 (2011).
Grima, J. N., Cauchi, R., Gatt, R. & Attard, D. Honeycomb composites with auxetic outofplane characteristics. Composite Strut. 106, 150–159 (2013).
Derby, B. Printing and prototyping of tissues and scaffolds. Science 338, 921–926 (2012).
Reid, C. N. Deformation geometry for materials scientists, Pergamon, Oxford, 1973.
Zhu, H. X., Knott, J. F. & Mills, N. J. Analysis of the elastic properties of opencell foams with tetrakaidecaheral cells. J. Mech. Phys. Solids 45, 319–343 (1997).
Greaves, G. N., Greer, A. L., Lakes, R. S. & Rouxel, T. Poisson’s ratio and modern Materials. Nature Materials 10, 823–838 (2011).
Zhu, H. X., Hobdell, J. R. & Windle, A. H. Effects of cell irregularity on the elastic properties of open cell foams. Acta Materialia 48, 4893–4900 (2000).
Lakes, R. S. Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987).
Lakes, R. S. Negative Poisson’s ratio materials. Science 238, 551 (1987).
Baughman, R. H., Shacklette, J. M., Zakhidov, A. A. & Stafstrom, S. Negative Poisson’s ratios as a common feature of cubic metals. Nature 392, 362–364 (1998).
Lakes, R. S. Lateral deformations in extreme matter. Science 288, 1976 (2000).
Gibson, L. G. & Ashby, M. F. Cellular Solids – Structures and Properties. Cambridge University Press, 1997.
Milton, G. W. Composite materials with Poisson’s ratio close to 1. J. Mech. Phys. Solids 40 1105–1137 (1992).
Evans, K. E. & Alderson, A. Auxetic materials: Functional materials and structures from lateral thinking!. Advanced Mater. 12, 617–628 (2000).
Hashin, Z. & Shtrikman, S. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127–140 (1963).
Zhang, Z., Zhang, Y. W. & Gao, H. On optimal hierarchy of loadbearing biological materials. Proc. R. Soc. B. 1–7 (2010) 10.1098/rspb.2010.1093.
Acknowledgements
TXF acknowledges the consistent financial support of National Basic Research Program of China (No. 2012CB619601). We acknowledge support from Engineering and Physical Science Research Council (UK) and the National Science Foundation of China.
Author information
Affiliations
Contributions
H.X.Z., T.X.F. and D.Z. discussed about the research ideas; H.X.Z. and T.X.F. analysed the results; H.X.Z. performed the calculations and simulations and prepared the manuscript.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Zhu, H., Fan, T. & Zhang, D. Composite materials with enhanced dimensionless Young’s modulus and desired Poisson’s ratio. Sci Rep 5, 14103 (2015). https://doi.org/10.1038/srep14103
Received:
Accepted:
Published:
Further reading

Auxetic interpenetrating composites: A new approach to nonporous materials with a negative or zero Poisson’s ratio
Composite Structures (2020)

The elastic properties of composites reinforced by a transversely isotropic random fibrenetwork
Composite Structures (2019)

Structurally Controlled Cellular Architectures for High‐Performance Ultra‐Lightweight Materials
Advanced Materials (2019)

Homogenized outofplane shear response of threescale fiberreinforced composites
Computing and Visualization in Science (2019)

The role of the soft phase in rigidity enhancements in a particulate composite
Composites Part A: Applied Science and Manufacturing (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.