Results in Second-Order Elasticity with Live Loads (obtained with Mathematica)

Abstract

We present a generalization of Signorini’s method to the case of live loads which allows us to derive approximate solutions to some pure traction-value problems in nite elastostatics. The boundary value problems and the corresponding compatibility conditions are formulated in order to determine the displacement of the system up to the second-order approximation. In particular, we consider the case of homogeneous and isotropic elastic bodies and we solve the following two pure traction-value problems with live loads: (i) a sphere subjected to the action of a uniform pressure eld; (ii) a hollow circular cylinder whose inner and outer surfaces are subjected to uniform pressures. Then, starting from these solutions, we suggest experiments to determine the second-order constitutive constants of the elastic body. Expressions of the second-order material constants in terms of displacements and Lame coefcients are determined. Further we apply the eneralized Signorini’s perturbation scheme to analyze radial expansion/contraction of an hollow cylinder made of an isotropic functionally graded elastic material, whose material moduli depend upon the radial coordinate only. We study the case of an hollow circular cylinder under uniform internal and external pressure. The displacement of the sysii tem and its state of stress are determined up to the second-order approximation. We consider both compressible and incompressible functionally graded elastic bodies. The quantitative analysis of all the described problems has been carried out with the aid of the software Mathematica by Wolfram Research. The programs which have allowed to nd the solutions to these problems are presented and their characteristics are discussed.