The Eigenvalues for a Self-Equilibrated, Semi-Infinite, Anisotropic Elastic Strip

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Abstract
The linear theory of elasticity is used to study an homogeneous anisotropic seminfinite strip, free of tractions on its long sides and subject to edge loads or displacements that produce stresses that decay in the axial direction. If one seeks solutions for the (dimensionless) Airy stress function of the form φ = e−γxF(y), γ constant, then one is led to a fourth-order eigenvalue problem for F(y) with complex eigenvalues γ. This problem, considered previously by Choi and Horgan (1977), is the anisotropic analog of the eigenvalue problem for the Fadle-Papkovich eigenfunctions arising in the isotropic case. The decay rate for Saint-Venant end effects is given by the eigenvalue with smallest positive real part. For an isotropic strip, where the material is described by two elastic constants (Young’s modulus and Poisson’s ratio), the associated eigencondition is independent of these constants. For transversely isotropic (or specially orthotropic) materials, described by four elastic constants, the eigencondition depends only on one elastic parameter. Here, we treat the fully anisotropic strip described by six elastic constants and show that the eigencondition depends on only two elastic parameters. Tables and graphs for a scaled complex-valued eigenvalue are presented. These data allow one to determine the Saint-Venant decay length for the fully anisotropic strip, as we illustrate by a numerical example for an end-loaded off-axis graphite-epoxy strip.

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