Michell solution

Abstract: From the biharmonic equation of the plane problem in the polar coordinate system and taking into account the variable-separable form of the partial solutions, a homogeneous ordinary differential equation (ODE) of the fourth order is deduced. Our study is based on the investigation of the behavior of the coefficients of the above fourth order ODE, which are functions of the radial coordinate r. According to the proposed investigation additional terms, ϕ -m (r, θ) (1≤m≤n) other than the usually tabulated in the Michell solution (1899, "On the Direct Determination of Stress in an Elastic Solid, With Application to the Theory of Plates, " Proc. Lond. Math. Soc., 31, pp. 100-124) are found. Finally the stress and the displacement fields due to each one additional term of ϕ -m (r, θ) are determined.

Abstract: A cylindrically symmetric layout of two opposite families of logarithmic spirals is shown to define the layout of minimum-weight, symmetrically loaded wheel structures, where different materials are used for the tension and compression members, respectively; referred to here as dual-material structures. Analytical solutions are obtained for both structure weight and deflection. The symmetric solutions are shown to form the basis for torsion arm structures, which when designed to accept the same total load, have identical weight and are subjected to identical deflections. The theoretical predictions of structure weight, deflection, and support reactions are shown to be in close agreement to the values obtained with truss designs, whose nodes are spaced along the theoretical spiral layout lines. The original Michell solution based on 45 deg equiangular spirals is shown to be in very close agreement with layout solutions designed to be kinematically compatible with the strain field required for an optimal dual-material design.

Abstract: High energy beams of elementary particles play a key role in laboratories working in fundamental research on particle physics. For several reasons (beam dump, secondary particle production etc.), these beams may be driven against solid structures. During such interactions, dynamic phenomena, very similar to those taking place after a mechanical impact, might occur in the hit solids. The studies of such dynamic thermo-mechanical problems are usually made via numerical methods. However, an analytical approach is also needed to provide reference solutions for the numerical results. In this paper a general introduction to these thermo-mechanical phenomena is first presented, followed by an example of the analytical solution for a graphite rod used as a beam target to produce secondary particles. The method allows the computation of the dynamic transient elastic stresses induced by a fast proton beam hitting off-axis the target. An exact solution for the temperature field is first obtained, using Fourier-Bessel series expansion. Quasi-static thermal stresses are then computed as a function of the calculated temperature distribution, making use of the thermoelastic displacement potential and of the Michell solution for the equivalent isothermal two-dimensional stress problem. Finally, the contribution of dynamic stresses due to longitudinal and bending stress waves is determined by means of the modal summation method, in the hypothesis of plane strain behaviour.