Editorial Type:
Article Category: Research Article
 | 
Online Publication Date: 01 Jan 2016

Constitutive Relation for Large Deformations of Fiber-Reinforced Rubberlike Materials with Different Response in Tension and Compression

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Page Range: 51 – 72
DOI: 10.2346/tire.16.440104
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ABSTRACT

Fiber-reinforced rubberlike materials commonly used in tires undergo large deformations and exhibit different responses in tension and compression along the fiber direction. Assuming that the response of a fiber-reinforced rubberlike material can be modeled as transversely isotropic with the fiber direction as the axis of transverse isotropy, we express the stored energy function in terms of the five invariants of the right Cauchy-Green strain tensor and account for different response in tension and compression along the fiber direction. The constitutive relation accounts for both material and geometric nonlinearities and incorporates effects of the fifth strain invariant, I5. It has been shown by Merodio and Ogden that in shear dominated deformations, I5 makes a significant contribution to the stress-strain curve. We have implemented the proposed constitutive relation in the commercial software, LS-DYNA. The numerical solutions of a few boundary value problems studied here agree with their analytical solutions derived by using Ericksen's inverse approach, in which part of the solution is assumed and unknowns in the presumed solution are found by analyzing the pertinent boundary value problem. However, computed results have not been compared with experimental findings. When test data become available, one can modify the form of the strain energy density and replace the proposed constitutive relation by the new one in LS-DYNA.

Keywords: fiber-reinforced rubberlike material model; user defined subroutine; finite plane strain bending; different response in tension and compression
FIG. 1
FIG. 1

Schematic sketch of the simple extension of a cube made of a fiber-reinforced rubberlike material with fibers inclined at angle α to the X1X3- plane; (left) fiber-reinforced body, (right) equivalent homogenized transversely isotropic body.

FIG. 2
FIG. 2

Normalized Cauchy stress component vs. the axial stretch for α = 0°, 30°, and 90°.

FIG. 3
FIG. 3

Schematic sketch of the simple shearing of a cube made of a fiber-reinforced rubberlike material with fibers inclined at angle α to the X1X3-plane.

FIG. 4
FIG. 4

For α = 0°, 30°, and 90°, normalized Cauchy stress components vs. the shear strain k.

FIG. 5
FIG. 5

Schematic sketch of the bending of a straight rectangular beam into a circular beam; (1) reference configuration, (2) deformed configuration.

FIG. 6
FIG. 6

For α = 0° and 90°, the through-the-thickness variation of the hoop stress and the Almansi-Hamel hoop strain.

FIG. 7
FIG. 7

For α = 0°, the through-the-thickness variation of the hydrostatic pressure.

Contributor Notes

1 Department of Biomedical Engineering & Mechanics, M/C 0219, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA
2 Corresponding author. Department of Biomedical Engineering & Mechanics, M/C 0219, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA. Email: rbatra@vt.edu
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