A Direct Methodology for Small Punch Creep Test

Abstract

Small punch creep test (SPCT) has strong advantages in practice compared with traditional uniaxial creep test because a small sheet specimen (10 × 10 × 0.5 mm) can be obtained from in-service facilities or mechanical components without damage. In this paper, a novel investigation is proposed to directly interpret SPCT results in consideration of Chakrabarty’s membrane stretching theory, which features strain analysis on thin sheet material forced by large punch ball, and it is applied to derive equivalent strain and strain rate from SPCT results. Also, the Monkman-Grant model for evaluation of creep life is investigated by using equivalent strain and strain rate data obtained from the Chakrabarty’s membrane stretching theory. To validate this methodology, both uniaxial creep test and SPCT have been performed for STS 316L stainless steel at 650 °C. Displacement and time data in SPCT were converted into equivalent strain and strain rate. The Monkman-Grant models derived from two different creep tests show a great potential such that SPCT with the proposed methodology can be substituted for uniaxial creep test.

Appendices

Appendix

Meridional Equilibrium Equation [23]

In order to derive the meridional equilibrium equation, meridional components of the circumferential and meridional stress are used. The meridional components of meridional and circumferential stress are as follows, respectively,

$$ \left({\sigma}_{\phi } Trd\theta \right) \cos \frac{d\phi }{2}-\left\{\left({\sigma}_{\phi } Trd\theta \right)+\frac{\partial \left({\sigma}_{\phi } Trd\theta \right)}{\partial r}dr\right\} \cos \frac{d\phi }{2} $$

(A1)

$$ \left({\sigma}_{\theta } TRd\phi \right) \sin \frac{d\theta }{2} \sin \left(\frac{\pi }{2}-\left(\phi +\frac{d\phi }{2}\right)\right)+\left\{\left({\sigma}_{\theta } TRd\phi \right)+\frac{\partial \left({\sigma}_{\theta } TRd\phi \right)}{\partial \theta }d\theta \right\} \sin \frac{d\theta }{2} \sin \left(\frac{\pi }{2}-\left(\phi +\frac{d\phi }{2}\right)\right) $$

(A2)

Figure 11(a) shows all stress components of infinitesimal element on the spherical thin vessel. Figure 11(b) and (c) show direction of meridional stress and circumferential stress, respectively. As geometric condition, which is exactly spherical, R and r have the relationship like equations (1) and (2). Also, for small angle of dϕ and dθ, $\sin \frac{d\phi }{2}$ and $\sin \frac{d\theta }{2}$ can be approximated to $\frac{d\phi }{2}$ and $\frac{d\theta }{2}$. Likewise, $\cos \frac{d\phi }{2}$ and $\cos \frac{d\theta }{2}$ are to be 0. Each stress component is tilted by $\frac{d\phi }{2}$ and $\frac{d\theta }{2}$, respectively, as shown in Fig. 11. The meridional stress is tilted by $\frac{d\phi }{2}$ to the meridional direction while the circumferential stress is tiled by $\frac{d\theta }{2}$ to the circumferential direction due to the curvature. Summation of equations (A1) and (A2) is equal to 0 because the element is in stress equilibrium state and, thus, the result of summation is like the left side of equation (3).

Normal Equilibrium Equation [23]

Normal equilibrium can be simply derived by manipulating of equations (A1) and (A2) because meridional direction is vertical to the normal direction, which is p direction. The normal component of meridional and circumferential stress is respectively as:

$$ \left({\sigma}_{\phi } Trd\theta \right) \sin \frac{d\phi }{2}-\left\{\left({\sigma}_{\phi } Trd\theta \right)+\frac{\partial \left({\sigma}_{\phi } Trd\theta \right)}{\partial r}dr\right\} \sin \frac{d\phi }{2} $$

(A3)

$$ \left({\sigma}_{\theta } TRd\phi \right) \sin \frac{d\theta }{2} \cos \left(\frac{\pi }{2}-\left(\phi +\frac{d\phi }{2}\right)\right)+\left\{\left({\sigma}_{\theta } TRd\phi \right)+\frac{\partial \left({\sigma}_{\theta } TRd\phi \right)}{\partial \theta }d\theta \right\} \sin \frac{d\theta }{2} \cos \left(\frac{\pi }{2}-\left(\phi +\frac{d\phi }{2}\right)\right) $$

(A4)

Summation of equations (A3) and (A4) is equal to pdϕRdθr and thus, the normal equilibrium is like the right side of equation (3).

Strain Derivation [23]

True strain calculation is derived as logarithmic expression,

$$ {\displaystyle \underset{l_0}{\overset{l}{\int }}}\frac{dx}{x}= \ln \frac{l}{l_0} $$

(A5)

Where l ₀ is the initial length, and l is the final length. Using equation (A5), the circumferential strain can be considered change of hoop circumference like:

$$ \ln \frac{2\pi r}{2\pi {r}_0} $$

(A6)

Likewise, the meridional strain is regarded as that initial length dr ₀ is changed to dr sec ϕ, and therefore, ε _ϕ in equation (6) can be obtained. ε _T in equation (6) is also simply calculated by thickness change, T ₀ to T.

The Equivalent Strain Analysis Results

Punch displacement in Fig. 6 is converted to equivalent strain and strain rate. All graphs converted to equivalent strain and strain rate are shown in Figs. 12 and 13, one is for Al ₂ O ₃ punch ball and the other is for Si ₃ N ₄ punch ball.

Appendix

Fig. 11

Stress state of infinitesimal element on the spherical thin vessel: (a) All stress components; (b) Meridional stress; (c) Circumferential stress

Fig. 12

Equivalent strain and equivalent strain rate using Al ₂ O ₃ punch ball for four different load conditions: (a) 421.83 N; (b) 470.88 N; (c) 549.36 N; (d) 598.41 N

Fig. 13

Equivalent strain and equivalent strain rate using Si ₃ N ₄ punch ball for four different load conditions: (a) 421.83 N; (b) 470.88 N; (c) 549.36 N; (d) 598.41 N