**8. Conclusions**

A new theoretical solution for the distribution of residual stresses and strains in an open-ended, thick-walled cylinder subjected to internal pressure followed by unloading has been proposed. A distinguished feature of this solution is that the cylinder is initially anisotropic. In particular, the paper is concentrated on a common type of anisotropy: polar orthotropy of elastic and plastic properties. The elastic response of the cylinder is controlled by the generalized Hooke's law, and the plastic response by the Tsai–Hill yield criterion and its associated flow rule. The flow theory of plasticity is employed. It has been shown that using the strain rate compatibility equation facilitates the solution. In particular, numerical techniques are only necessary to solve the linear differential Equation (46), and to evaluate ordinary integrals along characteristic curves.

The solution found can be directly used for the analysis and design of the process of autofrettage. It is worthy of note that in this case, there is no need to construct the field of strain in the entire cylinder, which is the most difficult part of the numerical solution. It follows from Equation (57) that *ϕ* = *ϕa* is a characteristic curve, and this curve corresponds to the inner surface of the cylinder. The circumferential strain along this curve can be immediately found from Equation (56). Therefore, the radius of the cylinder after unloading is determined. The circumferential stress at the inner radius of the cylinder at the end of loading follows from Equation (20) at *ϕ* = *ϕ<sup>a</sup>*. Then, the corresponding residual stress is immediate from Equations (61), (62), and (64).

An illustrative example is given in Section 7. In this case, it is assumed that the elastic properties are isotropic. As a result, the effect of the ratio *Y*/*X* on the distribution of stresses and strains has been revealed. This effect is especially significant in the range *Y*/*X* < 1.25 (Figures 5–8 and Figures 10–13). An exception is the distribution of the radial stress at the end of loading and after unloading. (Figures 4 and 9). This is because the boundary conditions on *σr* and Δ*σr*, from Equations (2) and (94), dictate that this stress vanishes at the inner radius of the cylinder.

**Author Contributions:** All three authors participated in the research and in the writing of this paper.

**Funding:** S.A. acknowledges support from the Russian Foundation for Basic Research (Project 16-08-00469).

**Acknowledgments:** This work was initiated while M.R. was a visiting researcher at Beihang University, Beijing, China. The publication has been prepared with the support of the "RUDN University Program 5-100".

**Conflicts of Interest:** The authors declare no conflict of interest.
