A theory of rods 2 or, equivalently, a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a certain curve and the time as the only independent variables. The formidable mathematical obstacles presented by three-dimensional continuum theories dictate the need for tractable, accurate, and illuminating one-dimensional models.

The Theory of Rods

The boundary initial value problems of elastodynamics are formulated as boundary integral equations. It is shown that these integral equations may be solved by time-stepping numerical methods for the unknown boundary values. A specific numerical scheme is presented for antiplane strain problems and a numerical example is given.

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A numerical boundary integral equation method for elastodynamics. I

The antiplane strain model for scattering of plane SH waves by dipping layers of arbitrary shape is investigated by using the indirect boundary integral method. The dipping layers are of finite length perfectly bonded together. The material of the layers is assumed to be homogeneous, linearly elastic, and isotropic. 

Presented numerical results incorporate variation in the number of layers, the angle of incidence, the frequency of the incident wave, the material properties, and the shape of the layers.

Scattering of plane harmonic SH wave by dipping layers or arbitrary shape

Minimum and maximum principles, which have been derived for the creeping flows of a yield stress fluid, are extended to flows when wall slip is present. In addition, the minimum principle is shown to lead to a variational inequality when wall slip is absent, and the latter is generalised to include inertial effects in specific cases, including flows in pipes and the flow past a body at rest due to a uniform flow at infinity. Moreover, the variational inequality is extended to deal with problems where wall slip may be present. Finally, the squeezing flow between two co-axial and parallel disks is re-examined as an application of the variational principles obtained here.

Variational principle and variational inequality for a yield stress fluid in the presence of slip

Large buckled states of rotating rods

We have investigated solutions of equation (3) when λ2 is an eigenvalue of the linearized operator (13) and when it is not. In Section 4 we have shown that for θ ≡ 0 and λ2 = μ i 2 we have exactly two nontrivial solutions which bifurcate to the right of μ i 2 ; these solutions are shown to exist in an interval (μ i 2 , μ i 2 + δ0). The method of Section 3 may then be used to extend these two solutions to the right of μ i 2 +δ0 providing that λ2= μ i 2 +δ0 is not an eigenvalue of the linear operator (13) evaluated at θ= ±ϑ1. Either a solution can be uniquely extended, or there exists a value of λ2where the bifurcation method must be applied again3.

B. Zwahlen

Papers

Remarks on the bifurcation of solutions of a nonlinear eigenvalue problem