On the Successive Approximation Method for Three-Dimensional Analysis of Radially Inhomogeneous Tubes With an Arbitrary Cylindrical Anisotropy

On small azimuthal shear deformation of fibre-reinforced cylindrical tubes

“…Most of the numerical results shown next are related with relatively thick tubes and they are mainly produced by solving equation (5-9) 1 on the basis of the SAM outlined in Section 5.4. It is worth noting that corresponding numerical results obtained on the basis of the power series method (Section 5.3) are practically identical to those based on SAM and, hence, in line with the conclusions made in [Shuvalov and Soldatos 2003], the two methods are found to be computationally equivalent. However, due mainly to its slow convergence, the power series method seems to be computationally reliable for relatively thin tubes only.…”

“…Solution of the boundary value problem (5-9) and (5-10) is next achieved analytically, via the power series method, and computationally, with use of the successive approximation method (SAM) introduced in [Soldatos and Hadjigeorgiou 1990] (see also [Shuvalov and Soldatos 2003;Ye 2003;Soldatos 2003]).…”

“…Upon choosing a suitably large value of N , each individual layer becomes itself a sufficiently thin elastic tube and, as a result, an approximate solution of the form (5-20) is considered satisfactory for the study of its behaviour. The approximate solutions thus obtained for all fictitious layers are then suitably connected by means of appropriate continuity conditions imposed on their fictitious interfaces, thus providing an arbitrarily close solution to that of the exact system (5-17) -see [Shuvalov and Soldatos 2003]. For an illustration of the relevant algorithm, consider the j-th such individual fictitious layer ( j = 1, 2, .…”

“…Moreover, micromechanics considerations reveal that, due to the natural appearance of an intrinsic material length parameter which is of the fibre thickness scale, the manner in which fibres are supported on the tube boundaries can also be accounted for, with use of appropriate boundary conditions. This is the case discussed and resolved completely in section 5, where the principal governing differential equation of the problem is solved exactly with use of the power-series method as well as the successive approximate method introduced in [Soldatos and Hadjigeorgiou 1990]; see also [Shuvalov and Soldatos 2003]. Relevant numerical results are presented in section 6, where the differences between conventional linear elasticity and the new developments introduced in section 5 are also discussed in detail.…”

Calculation of the dispersion curves of a functionally graded hollow cylinder

“…The two approaches lead to seeking an exact solution of an actually modified problem and add some questions of accuracy and validity of the results in numerical computing. The Peano expansion method [3] of keeping the continuity and property variation of the authentic problem has been demonstrated to be an exact solution for a graded plate [4,5]. The objective of this paper is to apply this method on functionally graded cylindrical structures.…”