4.5 Stresses and Strains

In this chapter, I will mainly focus on how nibs made from such significantly different materials like stainless steel and gold can perform similarly.

From the chapter Fountain Pen Nib Technology, we have learned how materials are assessed and characterised and noticed the significant difference between stainless steel and gold. In the chapter Nib Mechanics, we became familiar with the nib’s performance, its function and what the fundamental criteria in mechanical physics are.

Let us first return to Material Technology and refresh your mind about elasticity, elongation, or deformation and prestress.

In diagram 1 there is an ascending thick black line commencing at the intersection formed by the stress/force axis and the strain/deformation axis. First, it is a straight line, the region of yield, then follows a wiggly section, the start of plasticity, which continues as a curve ending with the break of the test component.

Diagram 1 — Stresses and Strain

The straight line shows the section of a component’s deformation when it is subjected to a force; the two are in a proportional relationship to each other, therefore the line is straight.

After removing the applied force, the deformation of the component returns to its original shape, no residual deformation will remain. Force and deformation follow the green line marked a – b. This is the area of elasticity or elastic deformation which ends at the wiggly bit, a point called the yield strength.

If a component, such as a compression spring, or a leaf spring or a nib of a fountain pen is constructed correctly (staying well clear of the yield strength), the elastic deformation can be repeated endlessly.

What happens when the force/load increases beyond the elastic range? This is shown with the course of the purple line. When a load/force is applied to an original component, its deformation curve starts at point e. Initially, the force/deformation curve follows up the elasticity (like the green line), once it enters the wiggly section through increasing the load, the component experiences plastic deformation, the sometimes desired, useful shaping of a component and work hardening through bending, for example. After removing the load at point f, for example, the force/deformation curve ends at point g. The distance e – g indicates the residual deformation of the component.

When forming a component through bending, we want it to keep the intended shape after relaxation. Therefore, we must overbend it — bring it through the range of elasticity into plasticity — then the residual deformation, the shape we desired will remain.

When forming a simple shape like a right-angle bracket, the degree of overbending can be calculated. When it comes to fountain pen nibs, calculations can only be an orientation but usually, the final shape is achieved through educated trial and error.

Prestress is a load inside the material. Generally, components do not show the load/stress they are under, like a coil spring with its windings close together. Its windings only separate after the load applied exceeds the preload. The tines of a nib are kept together with a small prestress load. The latter determines the writing pressure needed so that they move apart.

In diagram 1 you can follow this when you look at the orange line which follows parallel to the thick black line, only it is shifted to the left. Its cross point with the vertical, the force axis, is marked m. The prestress or its equivalent preload is depicted as the line h – m. It needs to be overcome before the component changes its shape, it deforms towards point k in the direction of the horizontal axis, dimensioned as deformation or strain.

Applying a force beyond the preload makes the deformation move up along the line k – m. After the load is taken off, the deformation returns to point m, and the preload remains unchanged, meaning, the load has caused a deformation within the range of elasticity. In regard to cyclic deformation, the same applies as said before, as long as it remains within the area of elasticity, the deformation will be elastic, etc.

When the force applied moves the deformation in or beyond the yield strength, the preload will be reduced.

Let me summarise the above with reference to nibs: In a nib, the preload is applied to the tines during the setting process at the end of its manufacturing. If the nib is only slightly mistreated through the load having passed beyond the yield point, the tines may still close, however, since the preload has been reduced the writing characteristics will have changed.

After the described treatment the tines will separate at a lower level of writing pressure or said differently: the tines separate more easily in response to writing pressure variations.

When the setting of the nib is done manually, the above procedure will be applied by the setter to adjust the preload force, so that the nib performs comfortably within the predetermined range of the product; a quality criterion to be expected from a reliable producer. Reducing the preload is easy; increasing is a bit more tricky. In any case, I would strongly recommend leaving this job to an expert.

Diagram 2 — Gold vs, Stainless Steel

Previously, we talked about material characteristics. Now, we look at nibs of the same design, but different material. In diagram 2 I have shown the stress-strain curves of stainless steel and gold. They are very different. While stainless steel behaves like typical steel, gold shows a curve of a malleable material, like lead, zinc or silver, etc, meaning: basically, it has no elastic range.

Mind you, the display of the curves in the graph is not proportional, otherwise, the curve for gold would be flat on the horizontal axis.

For further investigations, I have calculated the modulus of elasticity, the fraction of force/stress over deformation/strain within the range of elasticity. Since its line is straight, the E-modulus of stainless steel is easy to calculate: 180GPa (giga (1 billion) Pascal (a measure of force)) while for gold an approximation, a tangent to the curve is used. This is standard procedure for malleable materials. Gold’s E-modulus is 75GPa, a fraction of 2.5 of stainless steel.

Diagram 3 — zoomed into the red circle

How can a nib made of such two different materials perform similarly, have similar characteristics? Firstly, nibs operate within the red circle, the small area in the stress-strain diagram. More so, even though the circle is small, proportionally, it is far too large. Viewed more realistically, it would have only the size of a dot.

A magnification of what is going on in this dot is shown in diagram 3. It shows that nibs from different materials can operate within the same range of writing pressure by applying different levels of prestressing.

Stress_Strain_Stainless-nib field of work- forces

Diagram 4

The point where the two lines cross is shown in diagram 4 in more detail. Inside the green circle are the force-deformation-area of the nibs’ operation, the opening, and the closing of the tines. You can see that gold is brought up into this field through a significantly higher preload during the setting process. As expected, because of the very different E-moduli, the two materials behave differently in this area. It may look a bit overwhelming; however, I will guide you through it.

Here we go: Along the horizontal, deformation axis, you see the section marked as “Opening of Tines” which is expected to be the same for both nibs when applying a similar writing pressure, the central force where the two curves intersect.

From the horizontal axis, following the two vertical dotted lines (depicting the opening of the tines) upwards, they cross the gold and stainless-steel curves at two points each. Projecting the intersection points with both curves horizontally (two lines each) to the vertical axis (the force axis) shows that even though the central force is the same — to cause the same opening of the tines — the amount of force variation for the stainless-steel nib FS is much larger than for the gold nib FG. Approximately 2.5 times larger, the ratio of the E-moduli as we had calculated them around diagram 2.

We have known all along that gold nibs are more ticklish (they respond more easily to writing pressure variations) than stainless-steel ones, now we know why this is the case.

One point: referring to our examples, the force, which needs to be applied to the gold nib before it responds by widening its tines is higher than for the stainless steel. Often this is compensated by reducing the preload or reducing the material thickness, which changes the character of the nib. Now we all know why nibs behave the way they do.

The stainless-steel nib opens at a lower writing force and the widening is easier to control as it happens across a wider force range, which is my preferred behaviour.

Producing nibs of the same shape but different material, which are required to operate within the same dimensional constraints, it is work-hardening and prestressing which are the variables to move them into the same area of operation. Knowing the above takes a lot of guessing out of nib construction, therefore leaving more time and room for developing beauty into the nib’s shape.

In the following chapter, I will talk about Nib Manufacturing.