Superplastic Dome Bulge Stress Calculator
Pneumatic Sheet Forming: Calculate apex dome height ($h$), radius of curvature ($\rho$), membrane stress ($\sigma$), and pole thinning under gas blow forming.
Die Aperture & Sheet Blank
Dome Geometry & Membrane Stress
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Mechanics of Pneumatic Bulge Forming
Gas blow forming uses an inert gas (typically high-purity argon) to pneumatically expand heated superplastic sheet metal into complex female die cavities or freestanding domes at elevated temperatures.
1. Spherical Cap Geometry
For a circular die orifice of radius $a$ and apex height $h$, the spherical radius of curvature $\rho$ is:
ρ = (a² + h²) / (2 · h)
The apex pole thickness $s_{\text{pole}}$ under equibiaxial stretching is:
s_pole = s₀ / [1 + (h/a)²]²
2. Apex Membrane Stress
Equilibrium of the thin spherical shell yields equibiaxial membrane stress:
σ = (p · ρ) / (2 · s_pole)
Maintaining a constant optimal superplastic strain rate $\dot{\varepsilon}_0$ throughout the cycle requires pressure profiling where gas pressure is ramped dynamically as dome height progresses.
Frequently Asked Questions
Why is the circular bulge test fundamental in superplastic sheet forming?
The pneumatic free bulge test subjects a clamped circular sheet blank to pure equibiaxial tension without tool-sheet friction. Measuring apex dome rise rate against gas pressure allows direct determination of the biaxial stress-strain curve and dynamic strain rate sensitivity without requiring planar tensile machining.
Why does the sheet thin most severely at the dome apex (pole)?
At the clamped periphery, sheet metal is constrained by the die shoulder, experiencing plane strain deformation. The apex pole undergoes unconstrained equibiaxial stretching ($\varepsilon_1 = \varepsilon_2$). By plastic volume conservation ($\varepsilon_1 + \varepsilon_2 + \varepsilon_3 = 0$), the through-thickness thinning strain is twice the planar strain: $\varepsilon_3 = -2\varepsilon_1$, concentrating thinning at the center.
What is the relationship between apex membrane stress and curvature?
From membrane shell theory, stress is given by Laplace's law: $\sigma = \frac{p \cdot \rho}{2 s}$, where $p$ is gas pressure, $\rho = \frac{a^2 + h^2}{2h}$ is the spherical radius of curvature, and $s$ is instantaneous pole thickness. As the dome inflates, $\rho$ initially decreases until $h = a$ (hemisphere $\rho = a$), requiring precise progressive gas pressure throttling to avoid runaway stress.