Enhancement of Fatigue Life of Thick-Walled Cylinders Through Thermal Autofrettage Combined with Shrink-Fit

Abstract

Thick-walled cylindrical components are used in many industries, e.g. oil and chemical industries, artillery industries and nuclear power plants for withstanding high pressure or thermal gradient. Such components are subjected to autofrettage prior to their use in service, which increases their load carrying capacity as well as fatigue life. The fatigue life of the cylinder is important when the cylinder is subjected to a fluctuating or repeated pressure. Thermal autofrettage is a potential process capable of increasing the pressure carrying capacity as well as the thermal gradient capacity of thick-walled cylinders. This is achieved by employing a radial thermal gradient across the wall thickness of the cylinder. Due to the beneficial compressive residual stresses generated at and around the inner wall of the cylinder as a result of unloading of the thermal gradient, the thermally autofrettaged cylinder enhances the load carrying capacity as well as the fatigue life. Further enhancement in the fatigue life can be achieved by combining thermal autofrettage with shrink-fit. In this work, the fatigue life analysis of the thermally autofrettaged cylinder with shrink-fit is carried out. The analysis of thermal autofrettage is based on the assumptions of a generalized plane  strain condition and Tresca yield criterion.

Appendices

Appendix 1: Residual Stress Distributions During Thermal Autofrettage of a Thick-Walled Cylinder

A thick-walled cylinder with inner radius a and outer radius b is considered. The cylinder is subjected to a temperature difference (T _b  − T _a ) for achieving thermal autofrettage. As (T _b  − T _a ) crosses the temperature difference required for initial yielding, the cylinder undergoes the first stage of elastic–plastic deformation causing the material at the inner wall to deform plastically. If (T _b  − T _a ) is increased further, at certain temperature difference, the second stage of elastic–plastic deformation occurs in the cylinder causing the simultaneous plastic deformation both at the inner and outer wall. The two stages of deformations in the cylinder with different elastic and plastic zones developed as per Tresca yield criterion are shown in Fig. 1.1. After the release of temperature difference, (T _b  − T _a ), residual stresses are generated in the cylinder. Referring to Fig. 1.1a, the residual stresses generated in different zones during the first stage of elastic–plastic deformation under the condition of generalized plane strain are given as follows [10] (Fig. 1.7):

Fig. 1.7

Different elastic and plastic zones in the cylinder during a the first stage and b second stage of elastic–plastic deformation in thermal autofrettage. With permission from Kamal and Dixit [10]. Copyright 2015, ASME

Plastic zone I, a ≤ r ≤ c:

$$\sigma_{r}^{\text{I}} = k_{1} \sigma_{Y} \ln r + C_{3} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {\ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 - \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right\},$$

(1.14)

$$\sigma_{\theta }^{\text{I}} = k_{1} \sigma_{Y} \left( {1 + \ln r} \right) + C_{3} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {1 + \ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right\},$$

(1.15)

$$\sigma_{z}^{\text{I}} = k_{1} \sigma_{Y} \left( {1 + \ln r} \right) + C_{3} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {1 + 2\ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\frac{{2b^{2} }}{{b^{2} - a^{2} }}} \right\}.$$

(1.16)

Plastic zone II, c ≤ r ≤ d:

$$\begin{aligned} \sigma_{r}^{\text{II}} & = C_{5} r^{{ - 1 + \sqrt {2\left( {1 - \nu } \right)} }} + C_{6} r^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} + \frac{{k_{1} \sigma_{Y} }}{{\left( {2\nu - 1} \right)}}\, + \frac{{E\alpha T_{a} }}{{\left( {2\nu - 1} \right)}} \\ & \quad + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {\ln \left( {\frac{r}{a}} \right) - \frac{2\nu + 1}{2\nu - 1}} \right\} \\ & \quad - \frac{{E\varepsilon_{0} }}{{\left( {2\nu - 1} \right)}} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}} \\ & \quad \left\{ {\ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 - \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right\}, \\ \end{aligned}$$

(1.17)

$$\begin{aligned} \sigma _{\theta }^{{\text{II}}} & = C_{5} \sqrt {2\left( {1 - \nu } \right)} r^{{\sqrt {2\left( {1 - \nu } \right)} - 1}} - C_{6} \sqrt {2\left( {1 - \nu } \right)} r^{{ - \sqrt {2\left( {1 - \nu } \right)} - 1}} + \frac{{k_{1} \sigma _{Y} }}{{\left( {2\nu - 1} \right)}} + \frac{{E\alpha T_{a} }}{{\left( {2\nu - 1} \right)}} \\ & \quad + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {\ln \left( {\frac{r}{a}} \right) - \frac{2}{{2\nu - 1}}} \right\} - \frac{{E\varepsilon _{0} }}{{\left( {2\nu - 1} \right)}} \\ & \quad + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {1 + \ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right\}, \\ \end{aligned}$$

(1.18)

$$\begin{aligned} \sigma_{z}^{\text{II}} & = C_{5} r^{{ - 1 + \sqrt {2\left( {1 - \nu } \right)} }} + C_{6} r^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} + \left( {\frac{2\nu }{2\nu - 1}} \right)k_{1} \sigma_{Y} + \frac{{E\alpha T_{a} }}{{\left( {2\nu - 1} \right)}} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}} \\ & \quad \left\{ {\ln \left( {\frac{r}{a}} \right) - \frac{2\nu + 1}{2\nu - 1}} \right\} - \frac{{E\varepsilon_{0} }}{{\left( {2\nu - 1} \right)}} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}} \\ & \quad \left\{ {1 + 2\ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\frac{{2b^{2} }}{{b^{2} - a^{2} }}} \right\}. \\ \end{aligned}$$

(1.19)

Elastic zone, d ≤ r ≤ b:

$$\begin{aligned} \sigma_{r}^{\text{el}} & = \frac{E\alpha }{{2\left( {1 - \nu } \right)}}\frac{{\left( {T_{b} - T_{a} } \right)}}{{\ln \left( {\frac{b}{a}} \right)}}\left[ {\ln \left( {\frac{b}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 - \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right. \\ & \quad \left. { - \left\{ {\frac{{d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}\left\{ {\ln \left( {\frac{b}{a}} \right)\left( {2\nu - 1} \right) - \nu - \ln \left( {\frac{d}{a}} \right)} \right\}\left( {1 - \frac{{b^{2} }}{{r^{2} }}} \right)} \right] \\ & \quad + \left\{ {\frac{{d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}k_{1} \sigma_{Y} \left( {1 - \frac{{b^{2} }}{{r^{2} }}} \right) + \left\{ {\frac{{d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}E\alpha T_{a} \\ & \quad - \left\{ {\frac{{d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}E\varepsilon_{0} \left( {1 - \frac{{b^{2} }}{{r^{2} }}} \right), \\ \end{aligned}$$

(1.20)

$$\begin{aligned} \sigma_{\theta }^{\text{el}} & = \frac{E\alpha }{{2\left( {1 - \nu } \right)}}\frac{{\left( {T_{b} - T_{a} } \right)}}{{\ln \left( {\frac{b}{a}} \right)}}\left[ {\ln \left( {\frac{b}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right. \\ & \quad \left. { - \left\{ {\frac{{d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}\left\{ {\ln \left( {\frac{b}{a}} \right)\left( {2\nu - 1} \right) - \nu - \ln \left( {\frac{d}{a}} \right)} \right\}\left( {1 + \frac{{b^{2} }}{{r^{2} }}} \right)} \right] \\ & \quad + \left\{ {\frac{{d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}k_{1} \sigma_{Y} \left( {1 + \frac{{b^{2} }}{{r^{2} }}} \right) + \left\{ {\frac{{d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}E\alpha T_{a} \left( {1 + \frac{{b^{2} }}{{r^{2} }}} \right) \\ & \quad - \left\{ {\frac{{d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}E\varepsilon_{0} \left( {1 + \frac{{b^{2} }}{{r^{2} }}} \right), \\ \end{aligned}$$

(1.21)

$$\begin{aligned} \sigma _{z}^{{{\text{el}}}}& = \frac{{E\alpha }}{{2\left( {1 - \nu } \right)}}\frac{{\left( {T_{b} - T_{a} } \right)}}{{\ln \left( {\frac{b}{a}} \right)}}\left[ {1 + 2\ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\frac{{2b^{2} }}{{b^{2} - a^{2} }} + 2\nu \ln \left( {\frac{b}{r}} \right)} \right. \\ & \quad \left. { - \nu - \left\{ {\frac{{2\nu d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}\left\{ {\ln \left( {\frac{b}{a}} \right)\left( {2\nu - 1} \right) - \nu - \ln \left( {\frac{d}{a}} \right)} \right\}} \right] \\ & \quad + \left\{ {\frac{{2\nu d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}k_{1} \sigma _{Y} + \left\{ {\frac{{d^{2} - b^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}E\alpha T_{a} \\ & \quad - \left\{ {\frac{{d^{2} - b^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}E\varepsilon _{0} - E\alpha \left( {T_{b} - T_{a} } \right)\frac{{\ln \left( {\frac{r}{a}} \right)}}{{\ln \left( {\frac{b}{a}} \right)}}. \\ \end{aligned}$$

(1.22)

In the above equations, a is the inner radius, b is the outer radius, c is the interface radius between the plastic zone I and II and d is the interface radius between the plastic zone II and the elastic zone. The radii, c and d are obtained by using the boundary conditions of vanishing radial stress at the inner radius and $\sigma_{\theta }^{{^{{\left( {{\text{plastic}}\,{\text{zone}}\,{\text{II}}} \right)}} }} = \sigma_{z}^{{^{{\left( {{\text{plastic}}\,{\text{zone}}\,{\text{II}}} \right)}} }}$ at r = c as described in Kamal and Dixit [10].

The constants C₃, C₅ and C₆ are given by

$$C_{5} = Q + \frac{{2\nu b^{2} }}{{\left( {2\nu - 1} \right)\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}}d^{{1 - \sqrt {2\left( {1 - \nu } \right)} }} \left\{ {\frac{{1 - \nu + \nu \sqrt {2\left( {1 - \nu } \right)} }}{{2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\}E\varepsilon_{0} ,$$

(1.23)

$$C_{6} = P - \frac{{2\nu b^{2} }}{{\left( {2\nu - 1} \right)\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}}\left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\}E\varepsilon_{0} ,$$

(1.24)

where

$$\begin{aligned} P & = \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\} \\ & \quad \left\{ {\ln \left( {\frac{d}{a}} \right) - \frac{2\nu + 1}{2\nu - 1} - \frac{{\left( {1 + \nu } \right)}}{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\}\, \\ & \quad - \frac{E\alpha }{{2\left( {1 - \nu } \right)}}\frac{{\left( {T_{b} - T_{a} } \right)}}{{\ln \left( {\frac{b}{a}} \right)}}\left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\} \\ & \quad \left[ {\ln \left( {\frac{b}{d}} \right) + \left\{ {\frac{{b^{2} - d^{2} }}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}\left\{ {\ln \left( {\frac{b}{a}} \right)\left( {2\nu - 1} \right) - \nu - \ln \left( {\frac{d}{a}} \right)} \right\}} \right] \\ & \quad + \frac{{2\nu b^{2} }}{{\left( {2\nu - 1} \right)\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}}\left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\}k_{1} \sigma_{Y} \\ & \quad + \frac{{2\nu b^{2} }}{{\left( {2\nu - 1} \right)\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}}\left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\}E\alpha T_{a} , \\ \end{aligned}$$

(1.25)

and

$$\begin{aligned} Q & = - Pd^{{ - 2\sqrt {2\left( {1 - \nu } \right)} }} \left\{ {\frac{{1 - \nu + \nu \sqrt {2\left( {1 - \nu } \right)} }}{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\} - \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}} \\ & \quad \frac{{\left( {1 + \nu } \right)}}{{d^{{ - 1 + \sqrt {2\left( {1 - \nu } \right)} }} \left\{ {1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} } \right\}}}. \\ \end{aligned}$$

(1.26)

$$C_{3} = R + \left[ \begin{aligned} \frac{{2\nu b^{2} c^{{ - 1 + \sqrt {2\left( {1 - \nu } \right)} }} }}{{\left( {2\nu - 1} \right)\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}}d^{{1 - \sqrt {2\left( {1 - \nu } \right)} }} \left\{ {\frac{{1 - \nu + \nu \sqrt {2\left( {1 - \nu } \right)} }}{{2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\} \hfill \\ - \frac{{2\nu b^{2} c^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} }}{{\left( {2\nu - 1} \right)\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}}\left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\} - \frac{1}{{\left( {2\nu - 1} \right)}} \hfill \\ \end{aligned} \right]E\varepsilon_{0} ,$$

(1.27)

where

$$\begin{aligned} R & = Qc^{{ - 1 + \sqrt {2\left( {1 - \nu } \right)} }} - k_{1} \sigma_{Y} \ln c + Pc^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} + \frac{{k_{1} \sigma_{Y} }}{{\left( {2\nu - 1} \right)}} + \frac{{E\alpha T_{a} }}{{\left( {2\nu - 1} \right)}} \\ & \quad + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}}\ln \left( {\frac{c}{a}} \right) - \frac{{2E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)^{2} \ln \left( {\frac{b}{a}} \right)}} - \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}}, \\ \end{aligned}$$

(1.28)

The constant axial strain ε₀ is given by

$$\begin{aligned} \varepsilon_{0} & = \frac{{k_{1} \sigma_{Y} }}{AE}\left\{ {\ln c\frac{{c^{2} }}{2} - \ln a\frac{{a^{2} }}{2} + \frac{1}{4}\left( {c^{2} - a^{2} } \right) + \frac{{\nu \left( {d^{2} - c^{2} } \right)}}{2\nu - 1} + \frac{{\nu d^{2} \left( {b^{2} - d^{2} } \right)}}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\} \\ & \quad + \frac{R}{AE}\left( {\frac{{c^{2} - a^{2} }}{2}} \right) + \frac{Q}{AE}\left\{ {\frac{{d^{{1 + \sqrt {2\left( {1 - \nu } \right)} }} - c^{{1 + \sqrt {2\left( {1 - \nu } \right)} }} }}{{1 + \sqrt {2\left( {1 - \nu } \right)} }}} \right\} \\ & \quad + \frac{P}{AE}\left\{ {\frac{{d^{{1 - \sqrt {2\left( {1 - \nu } \right)} }} - c^{{1 - \sqrt {2\left( {1 - \nu } \right)} }} }}{{1 - \sqrt {2\left( {1 - \nu } \right)} }}} \right\} + \frac{{\alpha T_{a} }}{A}\left[ {\frac{{d^{2} - c^{2} }}{{2\left( {2\nu - 1} \right)}} - \frac{{\left( {b^{2} - d^{2} } \right)^{2} }}{{2\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}}} \right] \\ & \quad + \frac{{\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)A}}\left\{ {\ln \left( {\frac{d}{a}} \right)\frac{{d^{2} }}{2} - \ln \left( {\frac{c}{a}} \right)\frac{{c^{2} }}{2} - \frac{1}{2}\left( {d^{2} - c^{2} } \right)\frac{6\nu + 1}{{2\left( {2\nu - 1} \right)}}} \right\} \\ & \quad + \frac{\alpha }{{2\left( {1 - \nu } \right)}}\frac{{\left( {T_{b} - T_{a} } \right)}}{{\ln \left( {\frac{b}{a}} \right)A}}\left[ \begin{aligned} &- \nu d^{2} \ln \left( {\frac{b}{d}} \right) - \left\{ {\frac{{\nu d^{2} \left( {b^{2} - d^{2} } \right)}}{{b^{2} + d^{2} \left( {2\nu - 1} \right)}}} \right\}\left\{ {\ln \left( {\frac{b}{a}} \right)\left( {2\nu - 1} \right) - \nu - \ln \left( {\frac{d}{a}} \right)} \right\} \hfill \\ &- 2\left( {1 - \nu } \right)\left\{ {\ln \left( {\frac{b}{a}} \right)\frac{{b^{2} }}{2} - \ln \left( {\frac{d}{a}} \right)\frac{{d^{2} }}{2} - \frac{1}{4}\left( {b^{2} - d^{2} } \right)} \right\} \hfill \\ \end{aligned} \right], \\ \end{aligned}$$

(1.29)

where A is defined as

$$\begin{aligned} A & = \frac{{d^{2} - c^{2} }}{{2\left( {2\nu - 1} \right)}} - \frac{{\left( {b^{2} - d^{2} } \right)^{2} }}{{2\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}} - \left( {\frac{{c^{2} - a^{2} }}{2}} \right) \\ & \quad \left\{ {\frac{{2\nu b^{2} }}{{\left( {2\nu - 1} \right)\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right. \\ & \quad \left. {\left( \begin{aligned} \left( {\frac{c}{d}} \right)^{{ - 1 + \sqrt {2\left( {1 - \nu } \right)} }} \left( {1 - \nu + \nu \sqrt {2\left( {1 - \nu } \right)} } \right) \hfill \\ - \left( {\frac{c}{d}} \right)^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} \left( {1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} } \right) \hfill \\ \end{aligned} \right) - \frac{1}{{\left( {2\nu - 1} \right)}}} \right\} \\ & \quad - \left\{ {\frac{{d^{{1 + \sqrt {2\left( {1 - \nu } \right)} }} - c^{{1 + \sqrt {2\left( {1 - \nu } \right)} }} }}{{1 + \sqrt {2\left( {1 - \nu } \right)} }}} \right\}\frac{{2\nu b^{2} }}{{\left( {2\nu - 1} \right)\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}} \\ & \quad d^{{1 - \sqrt {2\left( {1 - \nu } \right)} }} \left\{ {\frac{{1 - \nu + \nu \sqrt {2\left( {1 - \nu } \right)} }}{{2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\} \\ & \quad + \left\{ {\frac{{d^{{1 - \sqrt {2\left( {1 - \nu } \right)} }} - c^{{1 - \sqrt {2\left( {1 - \nu } \right)} }} }}{{1 - \sqrt {2\left( {1 - \nu } \right)} }}} \right\}\frac{{2\nu b^{2} }}{{\left( {2\nu - 1} \right)\left\{ {b^{2} + d^{2} \left( {2\nu - 1} \right)} \right\}}} \\ & \quad \left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\}. \\ \end{aligned}$$

(1.30)

During the second stage of elastic–plastic deformation, the residual stresses in the plastic zones I and II are given in Eqs. (1.1) and (1.2). The residual stresses in the elastic zone and two outer plastic zones as shown in Fig. 1.1b are given as follows [10]:

Elastic zone, d ≤ r ≤ e:

$$\begin{aligned} \sigma_{r}^{\text{el}} & = \frac{E\alpha }{{\left( {2\nu - 1} \right)}}T_{a} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left[ {\frac{1}{{\left( {2\nu - 1} \right)}}\left\{ {\ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right\}} \right. \\ & \quad \left. { + \frac{1}{2} - \frac{{e^{2} }}{{2r^{2} }} - \ln \left( {\frac{b}{a}} \right)\left( {1 - \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right] \\ & \quad + \frac{{k_{1} \sigma_{Y} }}{2\nu - 1}\left( {1 + \frac{{e^{2} }}{{2d^{2} }}} \right) + \frac{{e^{2} }}{{2r^{2} }}k_{1} \sigma_{Y} + \frac{{E\varepsilon_{0} }}{{\left( {1 - 2\nu } \right)}}, \\ \end{aligned}$$

(1.31)

$$\begin{aligned} \sigma_{\theta }^{\text{el}} & = \frac{E\alpha }{{\left( {2\nu - 1} \right)}}T_{a} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left[ {\frac{1}{{\left( {2\nu - 1} \right)}}\left\{ {\ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right\}} \right. \\ & \quad \left. { + \frac{1}{2} + \frac{{e^{2} }}{{2r^{2} }} - \ln \left( {\frac{b}{a}} \right)\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right] \\ & \quad + \frac{{k_{1} \sigma_{Y} }}{2\nu - 1}\left( {1 + \frac{{e^{2} }}{{2d^{2} }}} \right) - \frac{{e^{2} }}{{2r^{2} }}k_{1} \sigma_{Y} + \frac{{E\varepsilon_{0} }}{{\left( {1 - 2\nu } \right)}}, \\ \end{aligned}$$

(1.32)

$$\begin{aligned} \sigma_{z}^{\text{el}} & = \frac{E\alpha }{{\left( {2\nu - 1} \right)}}T_{a} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left[ {\frac{2\nu }{{\left( {2\nu - 1} \right)}}\left\{ {\ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right\}} \right. \\ & \quad\left. { + 1 - \ln \left( {\frac{b}{a}} \right)\frac{{2b^{2} }}{{b^{2} - a^{2} }}} \right] + \frac{2\nu }{2\nu - 1}k_{1} \sigma_{Y} \left( {1 + \frac{{e^{2} }}{{2d^{2} }}} \right) + \frac{{E\varepsilon_{0} }}{{\left( {1 - 2\nu } \right)}}. \\ \end{aligned}$$

(1.33)

Plastic zone III, f ≤ r ≤ b:

$$\sigma_{r}^{\text{III}} = - k_{1} \sigma_{Y} \ln r + C_{7} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {\ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 - \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right\},$$

(1.34)

$$\sigma_{\theta }^{\text{III}} = - k_{1} \sigma_{Y} \left( {1 + \ln r} \right) + C_{7} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {1 + \ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right\},$$

(1.35)

$$\sigma_{z}^{\text{III}} = - k_{1} \sigma_{Y} \left( {1 + \ln r} \right) + C_{7} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {1 + 2\ln \left( {\frac{r}{a}} \right) - \ln \left( {\frac{b}{a}} \right)\frac{{2b^{2} }}{{b^{2} - a^{2} }}} \right\}.$$

(1.36)

Plastic zone IV, e ≤ r ≤ f:

$$\begin{aligned} \sigma_{r}^{\text{IV}} & = \frac{E\alpha }{{\left( {2\nu - 1} \right)}}T_{a} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left[ {\frac{1}{{\left( {2\nu - 1} \right)}}\left\{ {\ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right\}} \right. \\ & \quad \left. { + \ln \left( {\frac{r}{e}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 - \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right] \\ & \quad + k_{1} \sigma_{Y} \left\{ {\frac{1}{2\nu - 1}\left( {1 + \frac{{e^{2} }}{{2d^{2} }}} \right) + \frac{1}{2} - \ln \left( {\frac{r}{e}} \right)} \right\} + \frac{{E\varepsilon_{0} }}{{\left( {1 - 2\nu } \right)}}, \\ \end{aligned}$$

(1.37)

$$\begin{aligned} \sigma_{\theta }^{\text{IV}} & = \frac{E\alpha }{{\left( {2\nu - 1} \right)}}T_{a} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left[ {\frac{1}{{\left( {2\nu - 1} \right)}}\left\{ {\ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right\}} \right. \\ & \quad \left. { + 1 + \ln \left( {\frac{r}{e}} \right) - \ln \left( {\frac{b}{a}} \right)\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\frac{{b^{2} }}{{b^{2} - a^{2} }}} \right] \\ & \quad + \left\{ {\frac{1}{2\nu - 1}\left( {1 + \frac{{e^{2} }}{{2d^{2} }}} \right) - \frac{1}{2} - \ln \left( {\frac{r}{e}} \right)} \right\}k_{1} \sigma_{Y} + \frac{{E\varepsilon_{0} }}{{\left( {1 - 2\nu } \right)}}, \\ \end{aligned}$$

(1.38)

$$\begin{aligned} \sigma_{z}^{\text{IV}} & = \frac{E\alpha }{{\left( {2\nu - 1} \right)}}T_{a} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left[ {\frac{2\nu }{{\left( {2\nu - 1} \right)}}\left\{ {\ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right\}} \right. \\ & \quad\left. { + 2\nu \ln \left( {\frac{r}{e}} \right) + 1 - \ln \left( {\frac{b}{a}} \right)\frac{{2b^{2} }}{{b^{2} - a^{2} }}} \right] \\ & \quad + k_{1} \sigma_{Y} \left\{ {\frac{2\nu }{2\nu - 1}\left( {1 + \frac{{e^{2} }}{{2d^{2} }}} \right) - 2\nu \ln \left( {\frac{r}{e}} \right)} \right\} + \frac{{E\varepsilon_{0} }}{{\left( {1 - 2\nu } \right)}}. \\ \end{aligned}$$

(1.39)

In the above equations, e is the interface radius between the elastic zone and plastic zone III and f is the interface radius between the plastic zone III and plastic zone IV. In the second stage of elastic–plastic deformation, radii c, d, e and f are evaluated by using the boundary conditions of vanishing radial stress at the inner and outer radii, $\sigma_{\theta }^{{^{{\left( {{\text{plastic}}\,{\text{zone}}\,{\text{II}}} \right)}} }} = \sigma_{z}^{{^{{\left( {{\text{plastic}}\,{\text{zone}}\,{\text{II}}} \right)}} }}$ at r = c and $\sigma_{\theta }^{{^{{\left( {{\text{plastic}}\,{\text{zone}}\,{\text{IV}}} \right)}} }} = \sigma_{z}^{{^{{\left( {{\text{plastic}}\,{\text{zone}}\,{\text{IV}}} \right)}} }} ,$ at r = f [10]. The various constants involved in the residual stress equations of different zones in the second stage of elastic–plastic deformation are given by

$$\begin{aligned} C_{6} & = \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\} \\ & \quad \left\{ {\ln \left( {\frac{d}{a}} \right) - \frac{2\nu + 1}{2\nu - 1} - \frac{{\left( {1 + \nu } \right)}}{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\}\, \\ & \quad - \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\} \\ & \quad \left[ {\frac{1}{{\left( {2\nu - 1} \right)}}\left\{ {\ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right\} - \ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right] \\ & \quad - \left\{ {\frac{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}{{d^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} 2\nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\}\left( {\frac{2\nu }{2\nu - 1}} \right)k_{1} \sigma_{Y} \frac{{e^{2} }}{{2d^{2} }}, \\ \end{aligned}$$

(1.40)

$$C_{5} = - C_{6} d^{{ - 2\sqrt {2\left( {1 - \nu } \right)} }} \left\{ {\frac{{1 - \nu + \nu \sqrt {2\left( {1 - \nu } \right)} }}{{1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} }}} \right\} - \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}}\frac{{\left( {1 + \nu } \right)}}{{d^{{ - 1 + \sqrt {2\left( {1 - \nu } \right)} }} \left\{ {1 - \nu - \nu \sqrt {2\left( {1 - \nu } \right)} } \right\}}},$$

(1.41)

$$C_{3} = N + \frac{{E\varepsilon_{0} }}{{\left( {1 - 2\nu } \right)}},$$

(1.42)

where

$$\begin{aligned} N & \quad = C_{5} c^{{ - 1 + \sqrt {2\left( {1 - \nu } \right)} }} + C_{6} c^{{ - 1 - \sqrt {2\left( {1 - \nu } \right)} }} + \frac{{k_{1} \sigma_{Y} }}{{\left( {2\nu - 1} \right)}} + \frac{{E\alpha T_{a} }}{{\left( {2\nu - 1} \right)}} \\ & \quad + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)}}\left\{ {\ln \left( {\frac{c}{a}} \right) - \frac{2\nu + 1}{2\nu - 1}} \right\} - k_{1} \sigma_{Y} \ln c. \\ \end{aligned}$$

(1.43)

$$C_{7} = M + \frac{{E\varepsilon_{0} }}{{\left( {1 - 2\nu } \right)}},$$

(1.44)

where

$$\begin{aligned} M & = \frac{E\alpha }{{\left( {2\nu - 1} \right)}}T_{a} + \frac{{E\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left[ {\frac{1}{{\left( {2\nu - 1} \right)}}\left\{ {\ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right\} - \ln \left( {\frac{e}{a}} \right)} \right] \\ & \quad + \left\{ {\frac{1}{2\nu - 1}\left( {1 + \frac{{e^{2} }}{{2d^{2} }}} \right) + \frac{1}{2} - \ln \left( {\frac{f}{e}} \right) + \ln f} \right\}k_{1} \sigma_{Y} . \\ \end{aligned}$$

(1.45)

$$\,C_{9} = \, - \frac{{e^{2} \alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left( {\nu^{2} - 1} \right) - e^{2} \left( {1 - \nu^{2} } \right)\frac{{k_{1} \sigma_{Y} }}{E}.$$

(1.46)

$$\begin{aligned} C_{8} & = \frac{{f^{2} \alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)}}\left( {\nu^{2} - 1} \right) + \left( {1 - \nu^{2} } \right)\frac{{k_{1} \sigma_{Y} }}{E}f^{2} + C_{9} \\ & \quad - \frac{1 - 2\nu }{E}\left[ {\frac{{f^{2} }}{2}\left( {C_{7} - k_{1} \sigma_{Y} \ln f} \right)} \right] - \frac{3 - 2\nu }{4E}f^{2} k_{1} \sigma_{Y} \\ & \quad - \frac{1}{2}f^{2} \alpha T_{a} - \frac{{\alpha \left( {T_{b} - T_{a} } \right)}}{{2\ln \left( {\frac{b}{a}} \right)}}f^{2} \left\{ {\ln \left( {\frac{f}{a}} \right) - \frac{3}{2}} \right\} + \frac{{f^{2} }}{2}\varepsilon_{0} , \\ \end{aligned}$$

(1.47)

$$\begin{aligned} \varepsilon_{0} & = \frac{{k_{1} \sigma_{Y} }}{BE}\left[ \begin{aligned} & \ln c\left( {\frac{{c^{2} }}{2}} \right) - \ln a\left( {\frac{{a^{2} }}{2}} \right) + \frac{1}{4}\left( {c^{2} - a^{2} } \right) + \frac{\nu }{2\nu - 1}\left( {d^{2} - c^{2} } \right) + \frac{\nu }{2\nu - 1}\left( {1 + \frac{{e^{2} }}{{2d^{2} }}} \right)\left( {f^{2} - d^{2} } \right) \\ & - \nu \left\{ {\ln \left( {\frac{f}{e}} \right)f^{2} - \frac{1}{2}\left( {f^{2} - e^{2} } \right)} \right\} - \frac{1}{4}\left( {b^{2} - f^{2} } \right) - \ln b\left( {\frac{{b^{2} }}{2}} \right) + \ln f\left( {\frac{{f^{2} }}{2}} \right) \\ \end{aligned} \right] \\ & \quad + \frac{N}{2BE}\left( {c^{2} - a^{2} } \right) + \frac{{C_{5} }}{BE}\left\{ {\frac{{d^{{1 + \sqrt {2\left( {1 - \nu } \right)} }} - c^{{1 + \sqrt {2\left( {1 - \nu } \right)} }} }}{{1 + \sqrt {2\left( {1 - \nu } \right)} }}} \right\} \\ & \quad + \frac{{C_{6} }}{BE}\left\{ {\frac{{d^{{1 - \sqrt {2\left( {1 - \nu } \right)} }} - c^{{1 - \sqrt {2\left( {1 - \nu } \right)} }} }}{{1 - \sqrt {2\left( {1 - \nu } \right)} }}} \right\} + \frac{{\alpha T_{a} }}{{2\left( {2\nu - 1} \right)B}}\left( {f^{2} - c^{2} } \right) \\ & \quad + \frac{{\alpha \left( {T_{b} - T_{a} } \right)}}{{\left( {2\nu - 1} \right)\ln \left( {\frac{b}{a}} \right)B}}\left\{ {\ln \left( {\frac{d}{a}} \right)\frac{{d^{2} }}{2} - \ln \left( {\frac{c}{a}} \right)\frac{{c^{2} }}{2} - \frac{3}{4}\left( {d^{2} - c^{2} } \right) - \frac{{d^{2} - c^{2} }}{2\nu - 1}} \right\} \\ & \quad + \frac{M}{2BE}\left( {b^{2} - f^{2} } \right) + \frac{{\alpha \left( {T_{b} - T_{a} } \right)}}{{2\left( {1 - \nu } \right)\ln \left( {\frac{b}{a}} \right)B}} \\ & \quad \left[ \begin{aligned} \frac{\nu }{{\left( {2\nu - 1} \right)}}\left\{ {\ln \left( {\frac{d}{a}} \right) + \frac{1}{2} - \frac{{e^{2} }}{{2d^{2} }}} \right\}\left( {f^{2} - d^{2} } \right) - \left\{ {\ln \left( {\frac{e}{a}} \right)e^{2} - \ln \left( {\frac{d}{a}} \right)d^{2} - \frac{1}{2}\left( {e^{2} - d^{2} } \right)} \right\} \hfill \\ + \nu \left\{ {\ln \left( {\frac{f}{e}} \right)f^{2} - \frac{1}{2}\left( {f^{2} - e^{2} } \right)} \right\} - \left\{ {\ln \left( {\frac{f}{a}} \right)f^{2} - \ln \left( {\frac{e}{a}} \right)e^{2} - \frac{1}{2}\left( {f^{2} - e^{2} } \right)} \right\} \hfill \\ \end{aligned} \right], \\ \end{aligned}$$

(1.48)

where

$$B = \left\{ {\frac{{b^{2} - a^{2} }}{{2\left( {2\nu - 1} \right)}}} \right\}.$$

(1.49)

Appendix B: Stress Analysis in the shrink-fitting of cylinders

An outer cylindrical layer of inner radius less than the outer radius b of the inner cylinder by δ, i.e. (b − δ) is considered, where δ is the shrink-fit allowance. The radius of the outer cylinder is z. A compound cylinder is formed by shrink-fitting the outer cylinder to the inner cylinder. The geometry of the compound cylinder is shown in Fig. 1.2. Due to shrink-fit, the contact pressure p_sh is generated between the two cylinders. The contact pressure p_sh acts as an external pressure on the inner thermally autofrettaged cylinder and as an internal pressure on the outer cylinder. The stresses in the inner and outer cylinders due to shrink-fit can be obtained as follows.

The radial stress σ _r and hoop stress σ _θ at any point in the wall cross section of a thick-walled cylinder are given by Lame’s equations as

$$\sigma_{r} = A + \frac{B}{{r^{2} }},$$

(1.50)

$$\sigma_{\theta } = A - \frac{B}{{r^{2} }},$$

(1.51)

where A and B are constants. Now, for the inner cylinder, the contact pressure p_sh due to shrink-fit acts as an external pressure. Hence, we get for the inner cylinder

$$\left. {\left( {\sigma_{r} } \right)} \right|_{r = a} = 0,$$

(1.52)

$$\left. {\left( {\sigma_{r} } \right)} \right|_{r = b} = p_{\text{sh}} .$$

(1.53)

Using Eq. (1.50) in Eqs. (1.52) and (1.53) and then solving, them one can obtain the constants A and B. Substituting A and B in Eqs. (1.50) and (1.51), the resulting stresses in the inner cylinder are obtained as

$$\sigma_{r}^{{{\text{sh}}_{i} }} = - p_{\text{sh}} \frac{{b^{2} }}{{b^{2} - a^{2} }}\left( {1 - \frac{{a^{2} }}{{r^{2} }}} \right),$$

(1.54)

$$\sigma_{\theta }^{{{\text{sh}}_{i} }} = - p_{\text{sh}} \frac{{b^{2} }}{{b^{2} - a^{2} }}\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right).$$

(1.55)

For the outer cylinder, p_sh acts as an internal pressure. Thus, for the outer cylinder

$$\left. {\left( {\sigma_{r} } \right)} \right|_{r = b} = - p_{\text{sh}} ,$$

(1.56)

$$\left. {\left( {\sigma_{r} } \right)} \right|_{r = z} = 0.$$

(1.57)

Using Eq. (1.50) in Eqs. (1.56) and (1.57), the constants A and B can be evaluated. The resulting stresses in the outer cylinder are obtained as

$$\sigma_{r}^{{{\text{sh}}_{o} }} = p_{\text{sh}} \frac{{b^{2} }}{{z^{2} - b^{2} }}\left( {1 - \frac{{z^{2} }}{{r^{2} }}} \right),$$

(1.58)

$$\sigma_{r}^{{{\text{sh}}_{o} }} = p_{\text{sh}} \frac{{b^{2} }}{{z^{2} - b^{2} }}\left( {1 + \frac{{z^{2} }}{{r^{2} }}} \right),$$

(1.59)

As the cylinder is open-ended, the axial stress component is zero for the inner as well as for the outer cylinder. The contact pressure p_sh can be evaluated as follows.

The radial displacement at any point in the wall of the cylinder can be obtained by using strain–displacement relation along with the generalized Hook’s law,

$$\varepsilon_{\theta } = \frac{u}{r} = \frac{1}{E}\left( {\sigma_{\theta } - \nu \sigma_{r} } \right).$$

(1.60)

Evaluating the value of σ _r and σ _θ at the inner wall of the outer cylinder, i.e. at r = b from Eqs. (1.58) and (1.59) and then substituting them in Eq. (60), the radial displacement u at the contact surface of the outer cylinder is obtained as

$$\left( {\left. u \right|_{r = b} } \right)_{{{\text{outer}}\,{\text{cylinder}}}} = \frac{{p_{\text{sh}} b}}{{E_{1} }}\left( {\frac{{z^{2} + b^{2} }}{{z^{2} - b^{2} }} + \nu_{1} } \right),$$

(1.61)

where E₁ and ν₁ are the Young’s modulus of elasticity and Poisson’s ratio of the outer cylinder. Similarly, the radial displacement at the contact surface of the inner cylinder can be obtained as

$$\left( {\left. u \right|_{r = b} } \right)_{{{\text{inner}}\,{\text{cylinder}}}} = - \frac{{p_{\text{sh}} b}}{{E_{2} }}\left( {\frac{{b^{2} + a^{2} }}{{b^{2} - a^{2} }} - \nu_{2} } \right),$$

(1.62)

where E₂ and ν₂ are the Young’s modulus of elasticity and Poisson’s ratio of the inner cylinder. The total interference δ at the contacting surface can be obtained as

$$\delta = \left( {\left. u \right|_{r = b} } \right)_{{{\text{outer}}\,{\text{cylinder}}}} - \left( {\left. u \right|_{r = b} } \right)_{{{\text{inner}}\,{\text{cylinder}}}} ,$$

$$\delta = p_{\text{sh}} b\left\{ {\frac{1}{{E_{1} }}\left( {\frac{{z^{2} + b^{2} }}{{z^{2} - b^{2} }} + \nu_{1} } \right) + \frac{1}{{E_{2} }}\left( {\frac{{b^{2} + a^{2} }}{{b^{2} - a^{2} }} - \nu_{2} } \right)} \right\}.$$

(1.63)

Equation (1.63) provides the value of the contact pressure p_sh as

$$p_{\text{sh}} = \frac{{\left( {\frac{\delta }{b}} \right)}}{{\left\{ {\frac{1}{{E_{1} }}\left( {\frac{{z^{2} + b^{2} }}{{z^{2} - b^{2} }} + \nu_{1} } \right) + \frac{1}{{E_{2} }}\left( {\frac{{b^{2} + a^{2} }}{{b^{2} - a^{2} }} - \nu_{2} } \right)} \right\}}}.$$

(1.64)

If both the inner and outer cylinder is made of the same material, then E₁ = E₂ = E and ν₁ = ν₂ = ν. Thus, Eq. (1.64) reduces to

$$p_{\text{sh}} = \frac{{\left( {\frac{\delta }{b}} \right)}}{{\frac{1}{E}\left\{ {\left( {\frac{{z^{2} + b^{2} }}{{z^{2} - b^{2} }}} \right) + \left( {\frac{{b^{2} + a^{2} }}{{b^{2} - a^{2} }}} \right)} \right\}}}.$$

(1.65)