Axisymmetric residual stresses in a solid cylinder of finite length

Abstract

A solution technique is presented for the determination of residual stresses in a finite-length solid cylinder subject to non-uniform axisymmetric distributions of incompatible residual strains. The problem is reduced to the sequential solving of three individual problems: a problem on the determination of residual stresses in an infinitely long cylinder (the basic state) and two auxiliary problems for evaluating the stresses induced by the end-face effects (the disturbed states). The variational method of homogeneous solutions is implemented in order to determine the disturbed states within the framework of two latter problems. The solution technique is verified numerically for typical distribution profiles of incompatible strains depending on both the radial and axial coordinates. The approach can be used to evaluate residual stresses in a solid cylinder of finite length due to the intense thermal treatment.

Identification of force loadings on the inner circumference of a finite‐length elastic cylinder

“…The efficiency of solution to the inverse problem of such kind strongly relies on the analytical appearance of a solution to a direct problem for arbitrary boundary conditions on the entire surface of a cylinder. The family of dominant analytical methods for constructing solutions meeting the above requirements is presented by the method of cross-wise superposition [36], the direct integration method [37], and the method of homogeneous solutions [38,39]. The latter method is a natural extension of the classical method of variable separation and allows for constructing the ultimate solution by superposing the so-called homogeneous solutions which meet actual boundary conditions on one segment of the boundary and the homogeneous boundary conditions on the rest of the boundary.…”

“…In this paper, the solution method [38,39] is used to solve the Cauchy problem on the identification of the unknown normal and tangential loading on the inner circumference of a hollow cylinder of finite length within the framework of the axially symmetric formulation. The force loadings on the outer surface and the end faces of the cylinder are assumed to be given along with the auxiliary information about the radial and axial displacements on the same surface.…”

“…By following the strategy suggested by Chekurin and Postolaki [38] and Postolaki and Tokovyy [39], we represent the introduced Love function in the form as follows:…”

“…By following the strategy suggested by Chekurin and Postolaki [38] and Postolaki and Tokovyy [39], we represent the introduced Love function in the form as follows: $$$$\begin{equation}\chi (r,z) = \frac{1}{2}\sum_{k = 1}^\infty {\sum_{\mu = 1}^4 {C_k^{[\mu ]}\chi _k^{[\mu ]}(r,z)} } ,\end{equation}$$$$where $$C_k^{[\ell ]} = C_k^{\{ \ell \} }$$, $$C_k^{[\ell + 2]} = \bar{C}_k^{\{ \ell \} }$$, $$C_k^{\{ \ell \} }$$ and $$\bar{C}_k^{\{ \ell \} }$$, $$\ell = 1,2$$, $$k = 1,2,\ldots$$, are unknown complex constants (where the overline denotes the complex conjugation).…”