Issue 40

Ch. F. Markideset alii, Frattura ed Integrità Strutturale, 40 (2017) 108-128; DOI: 10.3221/IGF-ESIS.40.10 109 layered ring. For the analytic solution of the problem, the procedure proposed by Savin [3] for an infinite plate with a hole strengthened by rings is adopted as it was done also in ref. [1]. The specific procedure is based on the complex potentials technique introduced by Muskhelishvili [4]. The analytic solution achieved provides full-field closed-form expressions for both the stresses and the displacements. However, the specific expressions, in spite of their “elegance” and their numerous advantages (related mainly to their generality and the fact that they are expandable to rings that are made up of any number of concentric layers), are somehow “lengthy” and cumbersome. As a result, detailed parametric analyses of the (quite a few) factors influencing the final outcomes become rather difficult. To overcome the above mentioned difficulty, the problem is revisited, also, numerically using the Finite Element method and the commercially available software ANSYS. For the validation of the numerical model, the data obtained from the analytic solution for the case of a ring made up of three concentric layers are used. Then the validated numerical model is used for a parametric study of the role of some critical parameters (i.e., the elastic modulus and thickness of the inter- mediate layer and also the length of the arcs loaded by the parabolic distribution of radial stresses) on the overall stress- and displacement-fields developed in the ring. T HEORETICAL CONSIDERATIONS The problem and the basic assumptions onsider a multi-layered hollow cylinder of length w consisting of n concentric hollow cylinders of different thicknesses, perfectly joined together without any gaps at all. The cylinders are assumed to be made of homogeneous, isotropic and linearly elastic materials. The multi-layered cylinder as a whole is in equilibrium under the simultaneous action of three different kinds of loading: inner and outer uniform pressure all along its inner and outer lateral surfaces and a parabolic pressure acting along two finite parts of its outer periphery, antisymmetric with respect to the section’s center. All three types of loading act in the body’s normal cross-section and remain constant along its length. Ignoring its weight, stresses and displacements are to be determined at any point of the multi-layered cylinder. Clearly, the as above described configuration corresponds to a 1 st fundamental problem of plane linear elasticity for the body’s cross- section, i.e., the multi-layered circular ring. In this context, Muskhelishvili’s method of complex potentials [4] will be employed for the analytic solution. Mathematical formulation Under the above assumptions, the multi-layered ring is considered lying in the z = x +i y = r e i θ complex plane, Fig.1. The origin of the Cartesian reference coincides with the center of the ring. The n constituent concentric rings are numbered in such a way so that they are encountered in the order 1, 2,…, n as one moves from the origin towards the outer perimeter of the multi-layered ring. The arbitrary ring is denoted by the index j (1≤ j ≤ n ) and its boundaries are L j and L j +1 corresponding to the radii r = R j and r = R j +1 ( R j +1 > R j ), respectively. In general, it holds that ( R 2 – R 1 )≠( R 3 – R 2 )≠…( R n +1 – R n ). Adjacent constituent rings are perfectly joined together along their common interfaces. The inner boundary of the innermost ring, L 1 , for r = R 1 , is subjected to a uniform pressure of magnitude p I >0. The outer boundary of the outermost ring, L n +1 , for r = R n+ 1 , is under the simultaneous action of a uniform pressure of magnitude p E >0 exerted all over L n +1 , and a parabolic pressure of magni- tude p ( θ )>0 acting along two finite arcs of L n +1 , antisymmetric with respect to the ring’s center. Each one of these arcs has a length equal to 2 ω o R n +1 , where ω o corresponds to the half loaded rim. Particularly, p ( θ ) is here considered equal to:                               2 2 2 1 2 sin 1 sin sin , max sin 2 2 cos 2 o c o o c n o o o F p P P p R w (1) where ϕ o is the arbitrary angle formed by the axis of symmetry of the parabolic pressure and x -axis (measured from x -axis in the anticlockwise direction) and F ( F >0) is the resultant force due to p ( θ ). Obviously, for the as above described loading conditions, the multi-layered ring as a whole and each constituent j -ring separately, are in equilibrium and for that configuration the stress- and displacement-fields are to be determined for each j -ring. According to Muskhelishvili, the latter can be implemented by obtaining on each j -ring two analytic functions of the complex variable z , the complex potentials φ j ( z ) and ψ j ( z ), in terms of which stresses and displacements are expressed as [4]: C