# A Theory of Autofrettage for Open-Ended, Polar Orthotropic Cylinders

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## Abstract

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## 1. Introduction

## 2. Statement of the Problem

_{0}and outer radius b

_{0}by a uniform internal pressure P

_{0}, followed by unloading. The external pressure is zero. It is natural to solve this boundary value problem in a cylindrical coordinate system $\left(r,\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}z\right)$ whose $z-$axis coincides with the axis of symmetry of the cylinder. It is assumed that the cylinder is sufficiently long to make the stresses and strains independent of the z-coordinate. The ends of the cylinder are not loaded. The inner pressure at the end of loading is high enough so that the annulus contained by the inner radius and some internal radius $r={r}_{c}$ is plastic, while the outer annulus contained by the surface $r={r}_{c}$ and the outer radius is elastic. The surface $r={r}_{c}$ is the elastic/plastic boundary. Let ${\sigma}_{r}$, ${\sigma}_{\theta}$, and ${\sigma}_{z}$ be the stress components referred to the cylindrical coordinate system. These stresses are the principal stresses. Moreover, ${\sigma}_{z}=0$ for the open-ended cylinder. The boundary conditions at loading are

## 3. Purely Elastic Solution

## 4. Elastic/Plastic Stress Solution

## 5. Elastic/Plastic Strain Solution

## 6. Unloading

## 7. Numerical Example

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Geometric interpretation of two different mechanisms of plastic collapse (localization of plastic deformation at the inner radius of the cylinder and occurrence of the plastic region over the entire disc).

**Figure 2.**Effect of constitutive parameters on the magnitude of pressure at which plastic deformation is localized at the inner radius of the cylinder.

**Figure 5.**Variation of the circumferential stress with ρ in an a = 0.4 cylinder at several values of Y/X.

**Figure 6.**Variation of the total radial strain with ρ in an a = 0.4 cylinder at several values of Y/X.

**Figure 7.**Variation of the total circumferential strain with ρ in an a = 0.4 cylinder at several values of Y/X.

**Figure 8.**Variation of the total axial strain with ρ in an a = 0.4 cylinder at several values of Y/X.

**Figure 9.**Variation of the residual radial stress with ρ in an a = 0.4 cylinder at several values of Y/X.

**Figure 10.**Variation of the residual circumferential stress with ρ in an a = 0.4 cylinder at several values of Y/X.

**Figure 11.**Variation of the residual radial strain with ρ in an a = 0.4 cylinder at several values of Y/X.

**Figure 12.**Variation of the residual circumferential strain with ρ in an a = 0.4 cylinder at several values of Y/X.

**Figure 13.**Variation of the residual axial strain with ρ in an a = 0.4 cylinder at several values of Y/X.

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**MDPI and ACS Style**

Rynkovskaya, M.; Alexandrov, S.; Lang, L.
A Theory of Autofrettage for Open-Ended, Polar Orthotropic Cylinders. *Symmetry* **2019**, *11*, 280.
https://doi.org/10.3390/sym11020280

**AMA Style**

Rynkovskaya M, Alexandrov S, Lang L.
A Theory of Autofrettage for Open-Ended, Polar Orthotropic Cylinders. *Symmetry*. 2019; 11(2):280.
https://doi.org/10.3390/sym11020280

**Chicago/Turabian Style**

Rynkovskaya, Marina, Sergei Alexandrov, and Lihui Lang.
2019. "A Theory of Autofrettage for Open-Ended, Polar Orthotropic Cylinders" *Symmetry* 11, no. 2: 280.
https://doi.org/10.3390/sym11020280