Strength of fibres

I wanted to compare strength of fibres, such as carbon nanowires, or cotton, to steel. The strongest steel wire I know about is Scifer. I found this to be commercially available in the following sizes/ strength values.

25 micrometre, 475 kgF/mm^2 = 475 x 9.80665 MPa = 4758 MPa
100 micrometre, 400 kgF/mm^2 = 3923 MPa

Common units to measure the strength of fibres is force per denier, per tex, or per decitex, these actually all measure the specific strength of the fibres.

Denier is a unit of measure for the linear mass density of fibers. It is defined as the mass in grams per 9,000 meters. In the International System of Units the tex is used instead. The “tex” is the mass in grams per km. The most commonly used unit is actually the decitex, abbreviated dtex, which is the mass in grams per 10,000 meters

First of all we calculate the mass in grams per unit length as a function of the diameter and density.

Volume = (PI*d*d/4) * L (m^3)
where d is diameter and L is length (meters)
Mass = Volume * density = (PI*d*d/4) * L * rho (kg)
where rho is density is in kg/m^3.

mass in g per 10 km is therefore
Mass = (PI*d*d/4) * L * density * 10,000,000 decitex

Next calculate the mass in grams this fibre can hold.

F (grams) = 1000 * (PI*d*d/4) * S / g
where S is the strength in Pa and g is acceleration due to gravity of 9.80665 m/s^2

If we combine F/M to give the specific strength we can cancel out the diameter of the fibre.

Specific strength (grams/decitex) = S / (g*rho*L)
where S is strength in Pa, g is 9.80665 m/s^2, rho is density is in kg/m^3, and L is length in m (10,000).

Specific strength in g/dtex is therefore approximately 98,100 times smaller than specific strength in N.m/kg (Pa/kg/m^3).

If we assume a density of steel (for example 9850 kg/m^3), we can calculate a values from the strength alone, as shown above. A steel strength of 475 kgF/mm^2 results in a specific strength of 6.05 g/dtex, a strength of 400 kgF/mm^2 means 5.1 g/dtex. If we take the maximum reported value of 5.5 GPa for scifer this translates to 7.14 g/dtex.

For comparison cotton fibres are graded from weak to strong, weak being below 23 g/tex and strong being 31 g/tex and above, thats 2.3 – 3.1 g/dtex.

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