Abstract

The effective Young’s modulus and Poisson’s ratio of spherical monodisperse and polydisperse core-shell particles ordered or randomly distributed in a continuous matrix were predicted using detailed three-dimensional numerical simulations of elastic deformation. The effective elastic moduli of body-centered cubic (BCC) and face-centered cubic (FCC) packing arrangements of monodisperse microcapsules and those of randomly distributed monodisperse or polydisperse microcapsules were identical. The numerical results were also compared with predictions of various effective medium approximations (EMAs) proposed in the literature. The upper bound of the EMA developed by Qiu and Weng (1991) was in good agreement with the numerically predicted effective Young's modulus for BCC and FCC packings and for randomly distributed microcapsules. The EMA developed by Hobbs (1971) could also be used to estimate the effective Young's modulus when the shell Young's modulus was similar to that of the matrix. The EMA developed by Garboczi and Berryman (2001) could predict the effective Poisson’s ratio, as well as the effective Young's modulus when the Young's modulus of the core was smaller than that of the matrix. These results can find applications in the design of self-healing polymers, composite concrete, and building materials with microencapsulated phase change materials, for example.

Schematics and Assumptions

The deformation of composite materials consisting of a (i) matrix containing monodisperse spherical microcapsules with simple cubic (SC), body-centered cubic (BCC), or face-centered cubic (FCC) packing arrangements and of (ii) monodisperse and polydisperse microcapsules randomly distributed throughout the matrix was simulated numerically. Figure 1 shows a quarter of a simulated unit cell with (a) SC, (b) BCC, and (c) FCC packing arrangements along with the associated Cartesian coordinate system. Figure 2 shows examples of simulated (a) monodisperse and (b) polydisperse microcapsules randomly distributed in a continuous matrix.

Figure 1. Schematics of core-shell particles with an (a) SC, (b) BCC, and (c) FCC packing arrangement, and the associated Cartesian coordinate system. Five of six faces are fixed, and a uniform normal displacement is applied to the final face.

Results and Discussion

Table 1 reports numerical predictions of the effective Young’s modulus E_eff and the effective Poisson’s ratio ν_eff of three-component composite structures consisting of various core and shell volume fractions of monodisperse and polydisperse microcapsules randomly distributed in a continuous matrix with a unit cell length L = 75 μm. It also shows the values of E_eff and ν_eff for equivalent composites consisting of monodisperse microcapsules with BCC and FCC packing arrangements having nearly identical core and shell volume fractions φ_c and φ_s. In the baseline case, the core, shell, and matrix Young’s moduli were taken as E_c = 55.7 MPa, E_s = 6.3 GPa, and E_m = 16.75 GPa, while their Poisson’s ratios were ν_c = 0.499, ν_s = 0.34, and ν_m = 0.2, respectively. These properties corresponded to a three-component PCM composite consisting of microencapsulated paraffin wax with a polymeric shell embedded in a cement matrix. Table 1 establishes that E_eff and ν_eff were independent of spatial and size distributions. In other words, BCC and FCC packing arrangements were representative of the elastic behavior of monodisperse or polydisperse microcapsules randomly distributed in a matrix.

Table 1. Effective Young’s modulus E_eff and effective Poisson’s ratio ν_eff of three-phase composites consisting of p microcapsules either monodisperse or polydisperse and either in ordered packing arrangements or randomly distributed in a continuous matrix with until cell length L. Here, E_c = 55.7 MPa, E_s = 6.3 GPa, E_m = 16.75 GPa, ν_c = 0.499, ν_s = 0.34, and ν_m = 0.2.

Figure 3 plots (a) the effective Young’s modulus E_eff and (b) the effective Poisson’s ratio ν_eff of a three-component core-shell-matrix composite as functions of core volume fraction φ_c ranging from 0 to 0.3 for a constant shell volume fraction φ_s of 0.043. The microcapsules either monodispere or polydisperse and either packed in SC, BCC, or FCC arrangements or randomly distributed in the matrix. Figure 3 illustrates that the predictions of the effective Young’s modulus and Poisson’s ratio were identical for monodisperse spheres in BCC or FCC packing structrues and for randomly distributed mono- and poly-dispersed microcapsules, as previously observed in Table 1. Predictions for SC packing differed from other spatial distributions. Figure 3 also plots the effective Young’s modulus and Poisson’s ratio as a function of core volume fraction predicted by several EMAs found in the literature. Among the three-component EMAs, the D-EMT model [1], the Qiu and Weng [2] upper bound, and the model by Dunn and Ledbetter [3] gave the best agreement with numerical predictions of E_eff and ν_eff for BCC and FCC packing arrangements and randomly distributed monodisperse and polydisperse microcapsules. In addition, predictions of the effective Young’s modulus by the EMA developed by Hobbs [4] for two-component systems fell with 2% of those obtained numerically for BCC and FCC packing arrangements for the cases considered by ignoring the presence of the shell. This led to acceptable results because the shell volume fraction was small and E_s was similar to E_m.

Figure 3. (a) Effective Young’s modulus E_eff and (b) effective Poisson’s ratio ν_eff of a core-shell-matrix composite as functions of the core volume fraction φ_c for constant shell volume fraction φ_s = 0.043. Here, E_c, E_s, E_m, ν_c, ν_s, and ν_m were those of the baseline case.

Figures 4a-4d plot the ratio E_eff/E_m of monodisperse core-shell particles packed in SC, BCC, or FCC structures as a function of the ratio E_c/E_m ranging from 10^-4 to 10⁴ for ratios E_s/E_m equal to 0.1, 1, 10, and 100, respectively. The ratio E_eff/E_m is shown for two values of matrix Young’s moduli E_m, namely 1 and 10 GPa. Then, the shell Young’s modulus E_s was adjusted to achieve the desired E_s/E_m ratio. In all cases, the core and shell volume fractions were 0.3 and 0.1, respectively, and the Poisson’s ratios of the core, shell, and matrix were those of the baseline case. Figure 4 indicates that the ratio E_eff/E_m was dependent only on the ratios E_c/E_m and E_s/E_m, rather than on the Young’s moduli of each constituent component. It is interesting to note that the ratio E_eff/E_m showed little dependence on E_c/E_m in the limiting cases when E_c/E_m was very small (soft core) or very large (hard core). This suggests that for microencapsulated PCM-concrete composites with a soft care, a variance in the Young’s modulus of the PCM has little effect on E_eff. Additionally, Figure 4 plots the EMAs for E_eff that gave the best agreement with numerical results. The predictions by the upper bound of the Qiu and Weng model agreed the most closely with numerical predictions of E_eff for microcapsules with BCC or FCC packings across the ranges of E_c/E_m considered. For cases when E_c was smaller than E_m, such as for microencapsulated PCM in concrete, predictions from the D-EMT EMA were in better agreement with numerical predictions than those by the Qiu and Weng model. Finally, for the case when E_s/E_m = 1, the predictions of the EMA proposed by Hobbs fell within 3% of those for BCC and FCC packing arrangements.

Figure 4. Ratio E_eff/E_m of a core-shell-matrix composite for SC, BCC, or FCC packings as a function of the ratio E_c/E_m equal to (a) 0.1, (b) 1, (c) 10, and (d) 100. Predictions of EMAs are also shown including the D-EMT EMA [1], the upper Qiu and Weng bound [2], and the Hobbs model [4] (for E_s = E_m). Here, φ_c = 0.3, φ_s = 0.1, ν_c = 0.499, ν_s = 0.34, and ν_m = 0.2.

Conclusion

• Detailed 3D simulations were performed for elastic deformations in three-component composites consisting of monodisperse or polydisperse microcapsules ordered in SC, BCC, or FCC packing or randomly distributed in a continuous matrix.
• The effective Young’s modulus E_eff and effective Poisson’s ratio ν_eff were identical for BCC, FCC, and randomly distributed microcapsule packing arrangements over a wide range of core and shell volume fractions and constituent mechanical properties.
• The ratio E_eff /E_m was found to be a function of the core and shell volume fractions φ_c and φ_s and of the ratios E_c/E_m and E_s/E_m. However, it was generally independent of ν_m and of the Poisson’s ratios ν_c/ν_m and ν_s/ν_m.
• The effective Poisson’s ratio ν_eff was found to be a function of φ_c and φ_s, the ratios E_c/E_m and E_s/E_m, and the shell and matrix Poisson’s ratios ν_s and ν_m.
• Accurate effective medium approximations (EMAs) for composites’ effective moduli were identified:
  • The upper bound of the EMA by Qiu and Weng [2] predicted accurately the numerical results of E_eff for BCC and FCC packings and randomly distributed microcapsules.
  • The D-EMT EMA developed by Garboczi and Berryman [1] gave excellent predictions of E_eff for cases when E_c/E_m was less than 1.
  • The EMA developed by Hobbs [4] for two- component composites gave accurate predictions of E_eff when the shell Young’s modulus E_s was similar to that of the matrix E_m.
  • The D-EMT EMA gave the best agreement with numerical predictions of the effective Poisson’s ratio for BCC and FCC packing arrangements in the range of parameters considered.

These results can be used to inform the selection of materials for PCM composites for building applications and for other three-phase core-shell- matrix composites such as self-healing polymer composites. In the meantime, efforts are underway to confront these numerical results with experimental measurements for concrete containing microencapsulated PCMs.

References

[1] E.J. Garboczi and J.G. Berryman, "Elastic moduli of a material containing composite inclusions: effective medium theory and finite element computations", Mechanics of Materials, vol. 33 (8) 455-470, 2001.

[2] Y. P. Qiu and G. J. Weng, "Elastic moduli of thickly coated particle and fiber-reinforced composites", Journal of Applied Mechanics, 58, 388-398, 1991.

[3] M. L. Dunn and H. Ledbetter, "Elastic moduli of composites reinforced by multiphase particles", Journal of Applied Mechanics, 62, 1023-1028, 1995.

[4] D. W. Hobbs, "The dependence of the bulk modulus, Young’s modulus, creep, shrinkage and thermal expansion of concrete upon aggregate volume concentration", Materiaux et Construction, 4 (2) 107-114, 1971.

Publication

B. Young, A. Fujii, A. Thiele, A. Kumar, G. Sant, E. Taciroğlu, and L. Pilon, 2016. Effective Elastic Moduli of Core-Shell-Matrix Composites, Mechanics of Materials, Vol. 92, pp. 94-106. doi: 10.1016/j.mechmat.2015.09.006 pdf