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On the Normal Stresses in Simple Shearing of Fiber-Reinforced Nonlinearly Elastic Materials

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Abstract

We are concerned with a particular aspect of the simple shear problem within the framework of nonlinear elasticity for a class of incompressible transversely-isotropic fiber-reinforced materials. It is well known that, for isotropic hyperelastic materials, the normal stress effect characteristic of nonlinear elasticity is crucial in order to maintain a homogeneous deformation state in the bulk of the specimen. For the fiber-reinforced materials of concern here, we show that the confining traction that needs to be applied to the top and bottom faces of a block in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy and on the angle of orientation of the fibers. Inclusion of the second invariant in the isotropic part of the strain-energy used is shown to be of crucial importance in assessing the nature of the confining traction. In the absence of such an applied traction, an unconfined sample tends to bulge outwards or contract inwards perpendicular to the direction of shear. The character of the normal component of traction on the inclined faces is also investigated. The results are relevant to the development of accurate shear test protocols for the determination of constitutive properties of fiber-reinforced rubber-like materials and fibrous biological soft tissues.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Virginia, Charlottesville, VA 22904, USA

    Cornelius O. Horgan

  2. Department of Mechanical Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland

    Jeremiah G. Murphy

Authors
  1. Cornelius O. Horgan
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  2. Jeremiah G. Murphy
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Correspondence to Cornelius O. Horgan.

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Dedicated to the memory of Don Carlson.

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Horgan, C.O., Murphy, J.G. On the Normal Stresses in Simple Shearing of Fiber-Reinforced Nonlinearly Elastic Materials. J Elast 104, 343–355 (2011). https://doi.org/10.1007/s10659-011-9310-0

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