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Research Papers

The Overall Elastic Dielectric Properties of Fiber-Strengthened/Weakened Elastomers

[+] Author and Article Information
Victor Lefèvre

Department of Civil and
Environmental Engineering,
University of Illinois,
Urbana-Champaign, IL 61801-2352
e-mail: vlefevre@illinois.edu

Oscar Lopez-Pamies

Department of Civil and
Environmental Engineering,
University of Illinois,
Urbana-Champaign, IL 61801-2352
e-mail: pamies@illinois.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 13, 2015; final manuscript received July 24, 2015; published online September 10, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(11), 111009 (Sep 10, 2015) (21 pages) Paper No: JAM-15-1313; doi: 10.1115/1.4031187 History: Received June 13, 2015; Revised July 24, 2015

By employing recent results (Lopez-Pamies, O., 2014, “Elastic Dielectric Composites: Theory and Application to Particle-Filled Ideal Dielectrics,” J. Mech. Phys. Solids, 64, p. 6182 and Spinelli, S. A., Lefèvre, V., and Lopez-Pamies, O., “Dielectric Elastomer Composites: A General Closed-Form Solution in the Small-Deformation Limit,” J. Mech. Phys. Solids, 83, pp. 263–284.) on the homogenization problem of dielectric elastomer composites, an approximate solution is generated for the overall elastic dielectric response of elastomers filled with a transversely isotropic distribution of aligned spheroidal particles in the classical limit of small deformations and moderate electric fields. The solution for such a type of dielectric elastomer composites is characterized by 13 (five elastic, two dielectric, and six electrostrictive) effective constants. Explicit formulae are worked out for these constants directly in terms of the elastic dielectric properties of the underlying elastomer and the filler particles, as well as the volume fraction, orientation, and aspect ratio of the particles. As a first application of the solution, with the objective of gaining insight into the effect that the addition of anisotropic fillers can have on the electromechanical properties of elastomers, sample results are presented for the case of elastomers filled with aligned cylindrical fibers. These results are confronted to a separate exact analytical solution for an assemblage of differential coated cylinders (DCC), wherein the fibers are polydisperse in size, and to full-field simulations of dielectric elastomer composites with cylindrical fibers of monodisperse size. These results serve to shed light on the recent experimental findings concerning the dielectric elastomers filled with mechanically stiff fibers. Moreover, they serve to reveal that high-permittivity liquid-like or vacuous fibers—two classes of filler materials yet to be explored experimentally—have the potential to significantly enhance the electrostriction capabilities of dielectric elastomers.

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Figures

Grahic Jump Location
Fig. 1

Schematic of the microstructure of a transversely isotropic dielectric elastomer composite with initial axis of symmetry N

Grahic Jump Location
Fig. 2

Schematic of the microstructure of a dielectric elastomer filled with aligned spheroidal particles of aspect ratio ω distributed with spheroidal symmetry of the same aspect ratio ω. The lower half of the figure shows the limiting cases of (a) an isotropic distribution of spherical particles (corresponding to ω = 1) and (b) a transversely isotropic distribution of aligned cylindrical fibers with circular cross section (corresponding to ω=∞).

Grahic Jump Location
Fig. 3

Schematic of (a) the undeformed and (b) the deformed configurations of a dielectric elastomer filled with a transversely isotropic distribution of cylindrical fibers (N=e3) with circular cross section subjected to a Lagrangian electric field of magnitude E=Φ/L2 in the orthogonal direction to the fibers e2. The electrostriction H¯ undergone by the composite is described by Eqs. (42) and (43) with Eq. (36).

Grahic Jump Location
Fig. 4

Electrostriction ratio H¯11/H11 m of an incompressible (λ = ∞) dielectric elastomer with permittivity ε = 3.2ε0 filled with a transversely isotropic distribution of rigid (μf=λf=∞) cylindrical fibers with circular cross section; see Fig. 3. Results are shown for fibers with permittivities εf = 4ε0 and 104ε0, as functions of the volume fraction of fibers c. The solid line corresponds to the theoretical result (45)1. The dashed line corresponds to the response of a DCC assemblage of polydisperse fibers (see Appendix A), while the solid circles correspond to FE simulations of a dielectric elastomer composite with monodisperse fibers (see Appendix B).

Grahic Jump Location
Fig. 5

Electrostriction ratio H¯11/Ĥ11 m under e1-e2 plane-strain conditions of an incompressible (λ=∞) dielectric elastomer with permittivity ε = 3.2ε0 filled with a transversely isotropic distribution of rigid (μf=λf=∞) cylindrical fibers with circular cross section, see Fig. 3. Results are shown for fibers with permittivities εf = 4ε0 and 104ε0, as functions of the volume fraction of fibers c. The solid line corresponds to the theoretical result (47). The dashed line corresponds to the response of a DCC assemblage of polydisperse fibers (see Appendix A), while the solid circles correspond to FE simulations of a dielectric elastomer composite with monodisperse fibers (see Appendix B).

Grahic Jump Location
Fig. 6

Electrostriction ratios H¯11/H11 m and H¯33/H33 m of an incompressible (λ=∞) dielectric elastomer with permittivity ε = 3.2ε0 filled with a transversely isotropic distribution of liquid-like (μf=λf=∞) cylindrical fibers with circular cross section, see Fig. 3. Results are shown for fibers with permittivity εf=102ε0, as functions of the volume fraction of fibers c. The solid line corresponds to the theoretical results (48). The dashed line corresponds to the response of a DCC assemblage of polydisperse fibers, while the solid circles correspond to FE simulations of a dielectric elastomer composite with monodisperse fibers.

Grahic Jump Location
Fig. 7

Electrostriction ratios H¯11/H11 m, H¯22/H22 m, H¯33/H33 m of an incompressible (λ=∞) dielectric elastomer with permittivity ε = 3.2ε0 containing a transversely isotropic distribution of vacuous (μf=λf=∞, εf=ε0) cylindrical pores with circular cross section, as functions of the volume fraction of pores c, see Fig. 3. The solid line corresponds to the theoretical results (49). The dashed line corresponds to the response of a DCC assemblage of polydisperse cylindrical pores, while the solid circles correspond to the FE simulations of a dielectric elastomer composite with monodisperse cylindrical pores.

Grahic Jump Location
Fig. 8

Unit cells with N = 60 circles randomly distributed for three volume fractions of fibers: (a) c = 0.05, (b) c = 0.15, and (c) c = 0.25. Parts (d), (e), and (f) show three progressively refined meshes, with approximately (d) 8,000, (e) 40,000, and (f) 200,000 elements, of a unit cell containing a random distribution of N = 60 circles at volume fraction c = 0.15.

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