4.14 More Specifically Technical

13. Variation of the Theme

How is the separation of the tines effected by the modification of their length, tilt angle, width and thickness at the location of bending? Applying equation 2 for the cantilever calculation, it was solved for various variables and the results are combined in graph 5 so that we can recognise their correlations.

Equation 2 as shown is resolved for the distortion distance δ:

δ = F×ℓ 3 ÷ 3EＩ . . . Equation 2

… which I have used extensively on the page Fountain Pen Nib Technology.

Graph 5 — Graphic Summary – Causes of Tine Widening

Now to graph 5. All calculations and their results are based on the same fictional nib. The curves have been calculated by changing only one of the variables while keeping the others constant. The variables were: tilt angle between the tines τ = 15 °, effective length l = 9mm, width at the place of bending b = 3.5mm and thickness t = 0.4mm also at the location of bending. They are lined up along the vertical black dotted line. For this set of data, the calculation showed a widening of the tines by 0.84mm as it is indicated along the horizontal black dotted line.

When keeping all other variables constant (the above set), the brown curve shows a widening with increasing tilt angle.

Changing the width of the tines at the location of bending has the least effect, the curve is the flattest. Altering the material thickness has the highest impact on the tine separation, followed by the change of the effective length. It is not an unknown fact that flex nibs are made of thin material and have comparatively longer tines.

The diagram can be used to estimate the outcome of variations. For example: Reduction of the width to 3mm increases the separation from 0.84mm to 1mm and reducing the thickness to 0.35mm enlarges the widening to 1.2mm. Application of both variations at the same time would result in a tine widening of: Δ_W=0.16mm + Δ_T=0.36mm + 0.84mm = 1.36mm.

14. Permanent Damage

Earlier on I talked about applying a force to a nib so that its tines spread beyond the point of no return. Ingeneering calls this a distortion extending beyond the Yield Strength of a material. I have written a detailed paper dealing with this topic under the heading Stresses and Strains.

Graph 6 — Stress-Strain Diagram

Within the limits of elastic deflection, the force‑deflection relationship is proportional and reliable calculations are possible using the Modulus of Elasticity, also called the Young’s Modulus. Graph 6 shows a typical pattern of a ductile material such as steel.

As long as the strain (elongation per cross‑section area) stays below the straight line up to the Yield Strength no damage occurs. Other materials such as Gold, do not have a distinct yield strength as shown in the graph. When this is the case, then the yield strength is assigned to a point where the plastic deformation equals 0.2%.

For higher karat gold, the yield strength is 200MPa (megapascal = 1N/mm²) with little variation. Nibs are mostly made of 14 karat gold of a particular alloy with a yield strength of around 320MPa, see Fountain Pen Nib Materials. The latter I use in the following Sample Calculation 4.

Just to add another useful comment: In the segment of the curve called Strain Hardening is where work hardening happens, and Necking occurs when the cross‑section of a component reduces before it breaks.

Sample Calculation 4

I use the dimensions and data of the Waterman Ideal 2, fine nib. Its tines spread by 1.5mm under a load of 3N. Effective length = 9mm, effective breadth = 7mm (both tines), thickness of tines = 0.4mm

F = σ×b×t²/ ℓ×6 = 320×7mm×0.16mm²/ 9mm×6 = 6.3N

The load at the yield limit is 6.3N for a flat nib. My recommendation for the highest load is 5N so to stay safely away from the yield strength limit.

Graph 7 — Residual Deformation

Antonios Zavaliangos, a contributor to the forum “Flex Nibs” has performed a test when measuring the correlation between the load and tine separation of a particular nib. Note that the curves are very straight. The graph shows a residual spread of the tines after the nib has been loaded beyond the yield limit. With his permission, I show graph 7. Let me go through it: As the load increased, the tines spread. After the load had reached 2.5N (250g), he reduced it back to zero; however, a residual deformation of 0.45mm remains.

In the next chapter, Nibs are not Flat, I show in drawing 7 that profiled tines bend only to a fifth of the flat profile.

15. Nibs are not Flat

Yes, most nibs are not flat but curved; but, let me say it right now: there are some beautiful exceptions of flat shapes. However, now, it is high time to attend to the matter of curvature. Looking at our familiar equation

δ = F×ℓ3 ÷ 3EＩ

… we realise that besides the effective bending length ℓ it is only the moment of inertiaＩthat influences the degree of bending of the tines; E = modulus of elasticity remains constant for a given nib. The moment of inertia considers all aspects of the nib’s bending profile.

Drawing 4 — Bending Axis

To help develop a feel for the shape characteristics of a nib, let us return to drawing 4, which shows the flat strip. The profile’s dimensions are t and b as well as the location of the bending axis x-x define its area moment of inertia.

From experience, we know, the closer one applies a force to the axis of bending the easier it is to bend or break, eventually, especially when performed repetitively. Wrestlers know this, and if you want to snap a branch, you bend it over your knee and not out in free air. If you want to break a wire, you hold it with a pair of pliers and bend it up and down, close to the pliers’ jaws. For the technically minded among the readers, I mentioned that the Parallel Axis Theorem, as well as the Radius of Gyration, consider this. All others, don’t worry, I will talk around it without mentioning those expressions ever again.

Did I admit before that higher mathematics is not my forte? Luckily, ingeneering handbooks are reliable sources for area momenta of inertia of many shapes. Alas, I could not find a formula for a circular, tubular but partial section as it presents itself in the profile of nibs.

Drawing 7 — Kinked Profile

In a way, this may be good because I decided to use the profile of our example, b = 3.5mm and t = 0.4mm and kink it in the middle by α = 7.5° as shown in drawing 7.

Remember folding a sheet of paper? Likewise, as expected, the effect of kinking was significant. In our example with the flat profile, the calculated tine widening was 0.69mm; the kinked profile opens only by 0.14mm, just to a fifth. The stiffening is caused by the shift of the axis of bending from x‑x to y‑y by the amount of Δy = 0.11mm. I won’t go further into this because to work this out I had to apply one of those unspeakable methods I mentioned before.

I am always happy when the mathematics of physics agree with what happens in my observations. At such times, my physics teacher would place a q.e.d. under his calculations with such determination that often the chalk would break because of his vehement final full stop.

Continuing on from this observation, one could construct the area momenta of inertia of the curved profile from a flat profile with several kinks. However, for now, we got the gist and therefore, let’s move on.

16. Effective Dimensions

Drawing 8 — Radii Variation – Effective Length and Breadth relating to tine bending

With regard to bending, the terms effective length and effective breadth have come up already several times. Together with the radius of curvature, they are the shape dimensions, which determine the way the tines bend and consequently, they spread. With a different emphasis, I have written about this topic in the paper Design of Fountain Pen Nibs, from where I transferred drawing 8. Now, let us explore this drawing from a different angle.

Presuming that the nib material is of consistent thickness, the golden dotted line marks the shortest width combined with the flattest radius. The endpoint of the golden line doesn’t have to be located at the breather hole but could have a position along the slit. However, I assume that the radius towards the breather hole is the flattest.

R3 is smaller than R2 and as we have proven in the previous chapter, the smaller the radius (the larger the kink angle of our approximated radius in drawing 7), the stiffer the profile. The green dotted line is the effective length of bending, the average distance from the tip to line b.

Drawing 9 — Variation of Material Thickness

In some nib designs the effective dimensions are influenced by a variation of the thickness, see drawing 9. The placement, profile and gradient of the crossover can vary. Thickening the tines as shown makes them flex in a confined area, for sure, they bend less and not at all along their length. This causes the slit to widen linearly rather than curved, which influences its capillarity.

Drawing 10 — Shapes of Tine Widening

Drawing 10 shows two nibs with their slits opened to the same degree but one with a proportional, linear widening, the other with a curved (just slightly overdone to clarify my point). One can find arguments for either being of advantage. Theoretically, the slit widening with the linear increase should hold and transport the ink better than the one with the curved expansion, but only in conjunction with the capillary action of the paper. I wonder whether reality agrees with me. Let me know about your experience.