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Research Papers

Some Remarks on the Effect of Interphases on the Mechanical Response and Stability of Fiber-Reinforced Elastomers

[+] Author and Article Information
Katia Bertoldi

 School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138bertoldi@seas.harvard.edu

Oscar Lopez-Pamies

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

The development that follows can be easily generalized to interphases that are not homogeneous, such as for instance graded interphases.

Here, h is required to be convex in order to automatically ensure strong ellipticity of W(3) .

In a recent set of experiments [4], blocks of a transparent elastomer reinforced by cylindrical nitinol rods were axially compressed up to the point at which buckling of the rods was observed. The surfaces of the rods were not treated before fabrication of the composites resulting in fairly weak bonding between the elastomer and the rods.

J. Appl. Mech 79(3), 031023 (Apr 05, 2012) (11 pages) doi:10.1115/1.4006024 History: Received October 03, 2011; Revised January 10, 2012; Posted February 13, 2012; Published April 04, 2012; Online April 05, 2012

In filled elastomers, the mechanical behavior of the material surrounding the fillers -termed interphasial material-can be significantly different (softer or stiffer) from the bulk behavior of the elastomeric matrix. In this paper, motivated by recent experiments, we study the effect that such interphases can have on the mechanical response and stability of fiber-reinforced elastomers at large deformations. We work out in particular analytical solutions for the overall response and onset of microscopic and macroscopic instabilities in axially stretched 2D fiber-reinforced nonlinear elastic solids. These solutions generalize the classical results of Rosen (1965, “Mechanics of Composite Strengthening,” Fiber Composite Materials, American Society for Metals, Materials Park, OH, pp. 37–75), and Triantafyllidis and Maker (1985, “On the Comparison between Microscopic and Macroscopic Instability Mechanisms in a Class of Fiber-Reinforced Composites,” J. Appl. Mech., 52 , pp. 794–800), for materials without interphases. It is found that while the presence of interphases does not significantly affect the overall axial response of fiber-reinforced materials, it can have a drastic effect on their stability.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

(a) Schematic of two unit cells (or repeat lengths) of a fiber-reinforced elastomer with interphases in the undeformed configuration Ω0. Materials r = 1, 2, 3 characterize the matrix, fibers, and interphases, respectively. The initial fiber direction and repeat length are denoted by N and L0 . (b) Unit cell in the deformed configuration Ω of the axially stretched fiber-reinforced elastomer before the occurrence of an instability.

Grahic Jump Location
Figure 2

Macroscopic response of fiber-reinforced materials with interphases under uniaxial stress in the fiber direction: t¯1=0. The results correspond to c0(2)=30% volume fraction of fibers, interphase shear modulus μ(3)  = 0.1, various volume fractions of interphases c0(3), and are shown in terms of the nominal stress t¯2 as a function of the applied axial stretch λ¯2. Part (a) displays the results for tension λ¯2≥1, and part (b) for compression λ¯2≤1.

Grahic Jump Location
Figure 3

Macroscopic response of fiber-reinforced materials with interphases under uniaxial tensile stretch: λ¯1=1 and λ¯2≥1. The results correspond to c0(2)=30% volume fraction of fibers, interphase shear modulus μ(3)  = 0.1, and various volume fractions of interphases c0(3). Part (a) shows results for the nominal stress t¯2versusλ¯2, whereas part (b) shows t¯1versusλ¯2.

Grahic Jump Location
Figure 4

Critical stretches λ¯2cr and stresses t¯2cr at which instabilities develop in fiber-reinforced materials subjected to uniaxial stress in the fiber direction: t¯1=0. The results are shown as functions of the volume fraction of fibers c0(2) for various values of interphase volume fraction c0(3) and interphase shear modulus μ(3) .

Grahic Jump Location
Figure 5

Onset-of-instability curves in (λ¯1,λ¯2)-deformation and (t¯1,t¯2)-stress spaces. The results correspond to materials with c0(2)=30% volume fraction of fibers and c0(3)=2% volume fraction of interphases with shear moduli μ(3)  = 0.01, 0.05, 0.1.

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