3.6 Numerical Simulations
3.6.1 Case of Aluminum Disk
An aluminum disk is considered with inner radius, a=10 mm and outer radius, b=20 mm. The material properties of aluminum are as follows: Young’s modulus of elasticity, E=69 GPa, yield stress, σY=50.3 MPa and coefficient of thermal expansion, α=22.2×10−6/ºC. The temperature difference required for initial yielding at inner wall of the disk is calculated using Eq. (3.15) and is obtained as 53.66 ºC.
For achieving autofrettage in the aluminum disk, two temperature differences are considered— a temperature difference of 80 ºC to cause the first stage of elasto- plastic deformation and a temperature difference of 100 ºC to cause the second stage of elasto-plastic deformation in the wall of the disk. The radius of elastic-plastic interface, c is obtained as 11.21 mm when the temperature difference is 80 ºC. The estimates of c and d are obtained as 11.86 mm and 18.89 mm when the temperature difference is 100 ºC. The hardening coefficient K and strain hardening exponent n for the disk material are taken as 58.18 MPa and 0.48 in Ludwik’s hardening law given by Eq. (3.20). The various stresses generated in the disk due to thermal autofrettage are simulated and are shown as a function of radial position.
3.6.1.1 Elasto-plastic thermal stress pattern
The elasto-plastic thermal stresses for 80 °C temperature difference are obtained using Eqs. (3.16), (3.17), (3.19) and (3.22). For the temperature difference of 100
°C, the elasto-plastic stresses are estimated using Eqs. (3.38), (3.39), (3.41), (3.19), (3.47) and (3.48). The resulting elasto-plastic stress distribution for temperature difference of 80 °C and 100 °C is shown in Figure 3.4. It is observed that the magnitude of radial stresses is quite small compared to hoop stresses. The radial stresses are always tensile, whilst the hoop stresses change from tensile to compressive along the positive radial direction. This trend gets reversed when the temperature difference is removed. It is also observed that strain hardening effect is not very significant in the present case.
(a)
(b)
Figure 3.4 Elasto-plastic stress distributions in aluminum disk for (a) (Tb−Ta)=80 °C and (b) (Tb−Ta)=100 °C
3.6.1.2 Residual stress pattern
On removal of the temperature difference, residual thermal stresses are generated in the disk. The residual stress variation with the radial position is shown in Figure 3.5 for the temperature differences of 80 °C and 100 °C. It is observed that the residual compressive hoop stresses are generated at and around the inner wall of the disk, whilst residual tensile stresses are generated at the outer region of the disk. For the temperature difference of 100 °C, the magnitudes of compressive residual stresses
are higher compared to the case of temperature difference of 80 °C. The maximum magnitude of residual stress for the temperature difference of 80 °C is about 23 MPa (compressive), generated at the inner radius of the disk. For the case of 100 °C temperature difference, this magnitude is about 42 MPa (compressive). Hence, when the disk is subjected to loading by inducing internal pressure, then the resulting residual compressive stress leads to decrease in the maximum value of stress occurring in the disk.
(a)
(b)
Figure 3.5 Residual stress distributions in aluminum disk for (a) (Tb−Ta)=80 °C and (b) (Tb−Ta)=100 °C
The magnitudes of residual tensile stresses at the outer portion of the disk are small compared to residual compressive stresses for the case of 80 °C. For the 100
°C temperature difference, the magnitudes of tensile residual stresses show an increasing trend at the outer plastic zone. For the temperature difference of 80 °C, the maximum tensile residual stress is generated at the elastic-plastic interface (at r=c). For the temperature difference of 100 °C, the maximum tensile residual stress occurs at the outer radius. Moreover, its magnitude is larger in comparison to 80 °C temperature difference case. In corrosive environment, the larger tensile residual stress at the outer radius may cause stress corrosion cracking. Therefore, autofrettage by 100 °C temperature difference is not desirable in corrosive environment.
3.6.1.3 Overall stresses with and without autofrettage
Assume that the aluminum disk which is autofrettaged by inducing temperature difference between inner and outer wall is subjected to an internal working pressure of 18 MPa for both the temperature difference case. The overall stresses are then obtained by adding stresses due to internal working pressure to the residual thermal stresses. The stresses that occur in the disk due to internal working pressure are given by well-known Lame’s equations (Chakrabarty, 2006).
2 2
2i 2 1 2
r
p a b
b a r
σ = −
− , (3.74)
2 2
2p ai 2 1 b2
b a r
σθ = +
− , (3.75) where pi is the internal working pressure. If the same disk is considered only under the condition of internal working pressure without autofrettage, then the elastic stresses generated in the disk are given by Eqs. (3.74) and (3.75) only.
The resulting stresses with and without autofrettage for the case of 80 °C temperature difference is shown in Figure 3.6 (a). For 100 °C temperature difference, they are shown in Figure 3.6 (b). The equivalent Tresca stress (|σθ−σr|/2, in the present cases) is the maximum at the elastic-plastic interface (r=c), in the
autofrettaged disk during pressurization. Therefore, for the autofrettaged disk, yielding will first take place at the elastic-plastic interface (r=c).
(a)
(b)
Figure 3.6 Comparison of stresses with and without autofrettage in aluminum disk for (a) (Tb−Ta)=80 °C and (b) (Tb−Ta)=100 °C at working pressure of 18 MPa
When the temperature difference is 80 °C, the autofrettaged disk can withstand the maximum pressure of 22.2 MPa without yielding. The disk which is autofrettaged with 100 °C temperature difference can withstand the maximum pressure carrying capacity of 22.5 MPa. On the other hand, the non-autofrettaged disk can withstand a maximum pressure of 18 MPa. It is observed that the maximum equivalent Tresca stress is reduced by 13.85% in the disk when it is autofrettaged by inducing temperature difference of 80 °C as compared to the disk without
°C this reduction is observed to be 13.25%. It is always preferred to perform partial thermal autofrettage of disks in order avoid stress corrosion cracking as in the present case. Underwood and Miller (1988) have also advocated partial autofrettage even in the case of hydraulic autofrettage to circumvent the problem of stress corrosion cracking. Nevertheless, compared to hydraulic autofrettage, the level of thermal autofrettage is limited, because the maximum temperature needs to be kept well below the recrystallization temperature of the material.
It is to be noted that the wall thickness ratio b/a and temperature difference (Tb−Ta) across the wall thickness affect the pressure carrying capacity of the disk.
The effect of these parameters in thermal autofrettage for the present case is discussed in appendix B.