**6. Unloading**

It is assumed that the process of unloading is purely elastic. This assumption should be verified *a posteriori*. The general elastic solution of Equation (13), in which the stress components are replaced with their increments, is valid in the entire cylinder. Then,

$$\frac{\Delta\sigma\_{\tau}}{X} = \mathbb{C}\_{3}\rho^{\tau-1} + \mathbb{C}\_{4}\rho^{-\tau-1}, \quad \frac{\Delta\sigma\_{\theta}}{X} = \tau \left(\mathbb{C}\_{3}\rho^{\tau-1} - \mathbb{C}\_{4}\rho^{-\tau-1}\right) \tag{61}$$

where *C*3 and *C*4 are new constants of integration. These constants are found from the boundary conditions of Equations (3) and (4). As a result,

$$\mathcal{C}\_3 = -\mathcal{C}\_4 = \frac{p\_m}{(a^{-\tau-1} - a^{\tau-1})}.\tag{62}$$

Here, Equation (12) has been taken into account. Substituting Equation (62) into (61) supplies the radial distribution of Δ*σr* and Δ*σθ* in the form

$$\frac{\Delta x\_{\tau}}{X} = \frac{p\_m}{(a^{-\tau-1} - a^{\tau-1})} \left(\rho^{\tau-1} - \rho^{-\tau-1}\right)\_{\prime} \quad \frac{\Delta x\_{\theta}}{X} = \frac{\tau p\_m}{(a^{-\tau-1} - a^{\tau-1})} \left(\rho^{\tau-1} + \rho^{-\tau-1}\right). \tag{63}$$

The variation of the residual stresses with *ρ* is found as

$$
\sigma\_r^{res} = \sigma\_\mathcal{r} + \Delta\sigma\_\mathcal{r} \text{ and } \sigma\_\theta^{res} = \sigma\_\theta + \Delta\sigma\_\theta. \tag{64}
$$

It is understood here that *σr* and *σθ* are known from the stress solution given in Section 4, at *p*0 = *pm*. The process of unloading is purely elastic if the yield criterion is not violated in the entire cylinder. Using Equation (6), this condition can be represented as

$$
\left(\frac{\sigma\_\theta^{\rm res}}{X}\right)^2 - \left(\frac{\sigma\_\theta^{\rm res}}{X}\right)\left(\frac{\sigma\_r^{\rm res}}{X}\right) + \left(\frac{\sigma\_r^{\rm res}}{X}\right)^2 \frac{X^2}{Y^2} \le 1\tag{65}
$$

in the range *a* ≤ *ρ* ≤ 1. The radial distribution of the strain increments is determined from the generalized Hooke's law in Equations (5) and (62), as

$$\begin{split} \frac{\Delta \epsilon\_{r}^{\tau}}{k} &= \frac{p\_{m}}{\left(\frac{1-\tau-1-\varrho^{\tau-1}}{a^{-\tau-1}-a^{\tau-1}}\right)} \Big[ \left(1+\frac{\tau a\_{r\theta}}{a\_{rr}}\right) \rho^{\tau-1} - \rho^{-\tau-1} \left(1-\frac{\tau a\_{r\theta}}{a\_{rr}}\right) \Big], \\ \frac{\Delta \epsilon\_{\theta}^{\tau}}{k} &= \frac{p\_{m}}{\left(a^{-\tau-1}-a^{\tau-1}\right)} \Big[ \left(\frac{a\_{r\theta}}{a\_{rr}} + \frac{\tau a\_{\theta\theta}}{a\_{rr}}\right) \rho^{\tau-1} - \left(\frac{a\_{r\theta}}{a\_{rr}} - \frac{\tau a\_{\theta\theta}}{a\_{rr}}\right) \rho^{-\tau-1} \Big], \\ \frac{\Delta \epsilon\_{\theta}^{\tau}}{k} &= \frac{p\_{m}}{\left(a^{-\tau-1}-a^{\tau-1}\right)} \Big[ \left(\frac{a\_{rr}}{a\_{rr}} + \frac{\tau a\_{\theta\theta}}{a\_{rr}}\right) \rho^{\tau-1} - \left(\frac{a\_{\tau\theta}}{a\_{rr}} - \frac{\tau a\_{\theta\theta}}{a\_{rr}}\right) \rho^{-\tau-1} \Big]. \end{split} \tag{66}$$

The variation of the residual strains with *ρ* is found as

$$
\varepsilon\_r^{\rm res} = \varepsilon\_\mathcal{I} + \Delta\varepsilon\_\mathcal{r}, \; \varepsilon\_\theta^{\rm res} = \varepsilon\_\theta + \Delta\varepsilon\_\theta \text{ and } \varepsilon\_z^{\rm res} = \varepsilon\_z + \Delta\varepsilon\_z \tag{67}
$$

It is understood here that *εr*, *εθ*, and *εz* are known from the strain solution given in Section 5 at *p*0 = *pm*.

### **7. Numerical Example**

This section illustrates the effect of plastic anisotropy on the distribution of stress and strain in an *a* = 0.4 cylinder, assuming that the elastic properties are isotropic. In particular, it is assumed that Poisson's ratio is equal to 0.3 (i.e., *arθ* = −0.3). The value of *k* is immaterial, because all strains are proportional to *k*. The solution given in Section 4 has been used to calculate the radial distribution of the radial and circumferential stress corresponding to *ρc* = 0.8. It is seen from Figure 1 that the solution without the localization of plastic deformation at the inner radius of the cylinder exists only if *Y*/*X* > 0.8. Therefore, the stress solution has been found at *Y*/*X* = 0.85, *Y*/*X* = 1 (isotropic material), *Y*/*X* = 1.25, and *Y*/*X* = 1.5. This solution is illustrated in Figure 4 (radial stress) and Figure 5 (circumferential stress). The associate strain solution has been found using the approach described in Section 5. This strain solution is illustrated in Figure 6 (total radial strain), Figure 7 (total circumferential strain), and Figure 8 (total axial strain). It can be seen from these figures that the effect of the ratio *Y*/*X* on the distribution of the strains is very significant in the range *Y*/*X* < 1.25. In this range, the magnitude of strains is very large in the vicinity of the inner surface of the cylinder, which indicates the tendency towards the localization of plastic deformation. Since the solution found is for small strains, it is necessary to verify for each combination of material and geometric parameters that the assumption of small strain is acceptable. The distribution of the residual stresses has been determined using the stress distributions depicted in Figures 4 and 5, in conjunction with the solution provided in Section 6. This solution is illustrated in Figure 9 (residual radial stress) and Figure 10 (residual circumferential stress). The associate strain solution has been found using the approach described in Section 6. This solution for residual strains is illustrated in Figure 11 (residual radial strain), Figure 12 (residual circumferential strain), and Figure 13 (residual axial strain). As in the case of the strain distribution at the end of loading, it is seen from these figures that the solution is very sensitive to the value of *Y*/*X* in the range *Y*/*X* < 1.25. The residual circumferential stress is of special

significance for autofrettage. It is seen from Figure 10 that the magnitude of this stress at the inner surface of the cylinder is significantly affected by plastic anisotropy.

**Figure 4.** Variation of the radial stress with *ρ* in an a = 0.4 cylinder at several values of *Y*/*X*.

**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*.
