z

above the free lower end, a uniform vertical filament carries the weight of length

z

. With cross-sectional area

A

F (z) = ρ A g z, σ (z) = ρ g z, σ_{m a x} = ρ gL .

(1)

The maximum stress is at the top and is independent of diameter. The diameter cancels because both mass and load-bearing area scale with

A

. Neglecting self-heating gradients, evaporation and initial fabrication stress, this top point sets the first creep-strain limit.

For a clearly labelled external reference estimate, take

ρ = 1.93 \times 1 0^{4} kg m^{- 3}

. Then

σ_{m a x} = 1.89 \times 1 0^{4} Pa = 0.0189 MPa

; the area is

3.14 \times 1 0^{- 8} m^{2}

and the total suspended mass is about

6.06 \times 1 0^{- 5} kg

. The density is independently tabulated, not invented as a supplied 2016 value.^[ref]

Use the original map's top temperature axis at

250 0^{\circ} C

and its right-hand tensile-stress axis, so no uncertain shear-stress conversion or additional shear modulus is required. The point lies in C, between the

1 0^{- 9}

and

1 0^{- 8} s^{- 1}

contours, nearer

1 0^{- 8}

. A logarithmic interpolation of the scanned contours gives about

6 \times 1 0^{- 9} s^{- 1}

, with only graphical precision. Thus

t_{f} ≃ \frac{0.05}{ε ˙ _{m a x}} ≃ 8 \times 1 0^{6} s ≃ 2.3 \times 1 0^{3} h,

(2)

or of order three months in this steady-rate, reference-density map estimate. Reading only the bracketing contours gives the appropriately broad interval

5 \times 1 0^{6}

5 \times 1 0^{7} s

; the graph does not support an exact service-life number.

For the exact prescribed input, the answer remains

t_{f} ≃ \frac{0.05}{ε ˙ ( 250 0 ^{\circ} C , ρ gL )} .

(3)

TODO: verify the density convention in the historical Data Book and the applicability of the supplied map to the filament's grain structure. Thermal expansion lowers density, and an evolved area or tertiary material creep changes the rate; a steady initial-rate estimate is not a guaranteed lifetime. In particular, the source map has its own fixed grain-size assumptions and no separately specified filament grain size. Other failure mechanisms could shorten useful life.