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Research Papers

Numerical Modeling of the Nonlinear Elastic Response of Filled Elastomers via Composite-Sphere Assemblages

[+] Author and Article Information
Taha Goudarzi

e-mail: goudarz2@illinois.edu

Oscar Lopez-Pamies

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

An extension of this approach to nonlinear, though convex, problems appears to have been first carried out by Barrett and Talbot [7] in the context of two-phase dielectrics.

The analogous kinematically admissible approximation is discussed in the Appendix.

In other words, the composite sphere is a composite structure and not a composite material.

The initial shear moduli of standard reinforcing fillers (e.g., silica) are typically four orders of magnitude larger than those of standard elastomers (e.g., silicone).

Manuscript received July 21, 2012; final manuscript received January 18, 2013; accepted manuscript posted January 28, 2013; published online July 19, 2013. Assoc. Editor: Martin Ostoja-Starzewski.

J. Appl. Mech 80(5), 050906 (Jul 19, 2013) (10 pages) Paper No: JAM-12-1339; doi: 10.1115/1.4023497 History: Received July 21, 2012; Revised January 18, 2013; Accepted January 28, 2013

This paper proposes an effective numerical method to generate approximate solutions for the overall nonlinear elastic response of isotropic filled elastomers subjected to arbitrarily large deformations. The basic idea is first to idealize the random microstructure of isotropic filled elastomers as an assemblage of composite spheres and then to generate statically admissible numerical solutions, via finite elements, for these material systems directly in terms of the response of a single composite sphere subjected to affine stress boundary conditions. The key theoretical strengths of the method are discussed, and its accuracy and numerical efficiency assessed by comparisons with corresponding 3D full-field simulations. The paper concludes with a discussion of straightforward extensions of the proposed method to account for general classes of anisotropic microstructures and filler-elastomer interphasial phenomena, features of key importance in emerging advanced applications.

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Figures

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Fig. 1

(a) Electron micrograph of a styrene-butadiene rubber filled with an isotropic distribution of silica particles [14] and (b) its idealization as a composite-sphere assemblage (CSA) in the undeformed configuration. All the composite spheres in the assemblage are homothetic in that they have the same ratio of inner-to-outer radius Ri/Ro = c1/3.

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Fig. 2

Schematic illustrating that the overall response of a CSA subjected to affine stress boundary conditions can be variationally approximated by the overall response of a corresponding single composite sphere subjected to the same affine stress boundary conditions. Specifically, the approximation is such that the total elastic energy W¯ of the CSA is bounded from below by the total elastic energy W¯S of the single composite sphere.

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Fig. 3

Three representative meshes in the undeformed configuration for a composite sphere with particle concentration c = 0.15: (a) coarse mesh with 28,400 elements, (b) fine mesh with 102,600 elements, and (c) very fine mesh with 260,800 elements

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Fig. 4

Contour plots of the maximum principal logarithmic strain for a composite sphere with c = 0.15, Neo-Hookean matrix, and 104–times stiffer Neo-Hookean particle subjected to affine uniaxial stress (24) with S¯ = diag(s¯1 > 0,0,0); the undeformed configuration is also depicted for comparison purposes. The overall stretch in the direction of applied stress is λ¯1 = 3.5.

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Fig. 5

The normalized initial effective shear modulus μ¯/μ of isotropic incompressible elastomers filled with random isotropic distributions of rigid particles. Plots are shown for: (i) the CSA approximation μ¯S, (ii) full-field FE simulations, (iii) the Hashin–Shtrikman lower bound μ¯HS/μ = (2+3c)/(2-2c), and (iv) the Reuss lower bound μ¯R/μ = 1/(1-c), as functions of the concentration of particles c.

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Fig. 6

Macroscopic response of Neo-Hookean rubber filled with an isotropic distribution of rigid particles of concentration c = 0.15. Part (a) displays the entire effective stored-energy function in terms of the macroscopic principal stretches λ¯1 and λ¯2, whereas part (b) shows the energy along axisymmetric loading conditions with λ¯1 = λ¯, λ¯2 = λ¯-1/2. Results are shown for the CSA approximation W¯S in both parts, and for corresponding full-field FE simulations for isotropic distributions of spherical particles in part (b).

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Fig. 7

Macroscopic response of filled Neo-Hookean rubber with various values of concentration of particles c under: (a) uniaxial compressive, (b) uniaxial tensile, (c) pure shear, and (d) simple shear loading conditions. Plots are shown for the CSA approximation and corresponding full-field FE simulations for isotropic distributions of spherical particles.

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Fig. 9

(a) Electron micrograph of a magnetorheological elastomer with iron particles distributed anisotropically in chainlike structures [9] and (b) its idealization as an ellipsoidal assemblage of possibly nonspherical particles (CEA). All the composite ellipsoids in the assemblage are homothetic in that they are scaled-up or scaled-down versions of each other. Part (b) also illustrates schematically the straightforward incorporation of bound rubber into the CEA idealization.

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Fig. 10

Comparisons between the kinematically admissible approximation (A1) and (A2), denoted as CSA-F¯, and the statically admissible approximation Eqs. (17)–(20), denoted as CSA-S¯, for the overall response of filled Neo-Hookean rubber. Part (a) shows results for the normalized initial shear modulus μ¯/μ as a function of particle concentration c, while part (b) shows stress-deformation results for c = 0.15 under uniaxial tension.

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Fig. 8

Macroscopic response of filled silicone rubber with various values of concentration of particles c under: (a) uniaxial compressive, (b) uniaxial tensile, (c) pure shear, and (d) simple shear loading conditions. Plots are shown for the CSA approximation and corresponding full-field simulations for isotropic distributions of spherical particles.

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