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TECHNICAL PAPERS

# The Viscoelastic Fiber Composite with Nonlinear Interface

[+] Author and Article Information
Mayue Xie, Alan J. Levy

Department of Mechanical and Aerospace Engineering,  Syracuse University, Syracuse, NY 13244-1240

Note that $κ=λ+μ=k+μ∕3$ where $λ$, $μ$ are Lame moduli

For antiplane shear, it is the mode multiplier in the single-mode approximation [see Eq. $(282)$].

The difference between the applied boundary strain and the mean strain of the composite cylinder.

For antiplane shear, this is only approximately true (6).

An additional constraint is that the force length ratio $ρ$ remains constant for each composite cylinder (4).

This is exactly true for equibiaxial and axial tension load. For antiplane shear, bounds are required (see below).

Note that for viscoelastic composite response, $τ̂$ is taken to be $τ∕μ1+$.

J. Appl. Mech 73(2), 268-280 (Jul 22, 2005) (13 pages) doi:10.1115/1.2083807 History: Received January 11, 2005; Revised July 22, 2005

## Abstract

Effective viscoelastic response of a unidirectional fiber composite with interfaces that may separate or slip according to uniform Needleman-type cohesive zones is analyzed. Previous work on the solitary elastic composite cylinder problem leads to a formulation for the mean response consisting of a stress-strain relation depending on the interface separation∕slip discontinuity together with an algebraic equation governing its evolution. Results for the fiber composite follow from the composite cylinders representation of a representative volume element (RVE) together with variational bounding. Here, the theory is extended to account for viscoelastic matrix response. For a solitary elastic fiber embedded in a cylindrical matrix which is an $nth$-order generalized Maxwell model in shear relaxation, a pair of nonlinear $nth$-order differential equations is obtained which governs the relaxation response through the time dependent stress and interface separation∕slip magnitude. When the matrix is an $nth$-order generalized Kelvin model in shear creep, a pair of nonlinear $nth$-order differential equations is obtained governing the creep response through the time dependent strain and interface separation∕slip magnitude. We appeal to the uniqueness of the Laplace transform and its inverse to show that these equations also apply to an RVE with the composite cylinders microstructure. For a matrix, which is a standard linear solid $(n=2)$, the governing equations are analyzed in detail paying particular attention to issues of bifurcation of response. Results are obtained for transverse bulk response and antiplane shear response, while axial tension with related lateral Poisson contraction and transverse shear are discussed briefly. The paper concludes with an application of the theory to the analysis of stress relaxation in the pure torsion of a circular cylinder containing unidirectional fibers aligned parallel to the cylinder axis. For this problem, the redistribution of shear stress and interface slip throughout the cross section, and the movement of singular surfaces, are investigated for an interface model that allows for interface failure in shear mode.

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## Figures

Figure 1

The viscoelastic composite cylinder

Figure 2

The normal interface force law f and Df

Figure 3

Stress response under relaxation boundary condition; ε0=0.012

Figure 4

Interface separation under relaxation boundary condition; ε0=0.012

Figure 5

Interface force under relaxation boundary condition; ε0=0.012

Figure 6

Bifurcation under relaxation boundary condition; ρ=0.002, ε0=0.0054

Figure 7

Strain response under creep boundary condition; σ̂=0.03

Figure 8

Interface separation under creep boundary condition; σ̂=0.03

Figure 9

Interface force under creep boundary condition; σ̂=0.03

Figure 10

Bifurcation under creep boundary condition; ρ=0.01,σ̂=0.058

Figure 11

The shear interface force law g and its first mode f and Df

Figure 12

Notation

Figure 13

Stress relaxation of a viscoelastic circular cylinder ρ=0.05, T̂(t̂)=T(t̂)∕μ1+

Figure 14

Stress relaxation of a viscoelastic circular cylinder ρ=0.02, T̂(t̂)=T(t̂)∕μ1+

Figure 15

Stress relaxation of a viscoelastic circular cylinder ρ=0.005, T̂(t̂)=T(t̂)∕μ1+

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