Get a grip

Putting anything on the internet is an invitation for unwanted comments. My broken toothbrush post, for example, inspired people to tell me that:

  1. I should change my toothbrush more often. Maybe so, but you can’t draw that conclusion from the fact that it failed in fatigue. Fatigue isn’t synonymous with long usage; it just means the failure occurred through incremental crack growth under cyclic loading. There could be lots of cycles (that’s called high cycle fatigue) or relatively few (that’s called, as you might guess, low cycle fatigue). A small fatigue zone, as in my toothbrush, is usually associated with low cycle fatigue, so there’s no evidence that I’m using my toothbrush too long.
  2. I’m damaging my gums by brushing too hard. This isn’t true, either, but I can’t blame readers for this interpretation. I did a bad job explaining the forces and stresses in that post.

The problem came from this breezy explanation of the forces:

My teeth push out on the brush, and I hold the handle just about where the break is. The toothbrush, therefore, acts like a cantilever beam with high tensile stresses on the brush side of the handle near my hand.

I wanted to get the fracture surface photos, so I shortened up the discussion of forces. Simplifying an explanation is fine, but oversimplifying can be misleading, and that’s what I did. Let’s fix that here.

Here’s me holding my new toothbrush.

Toothbrush grip

Flipping my hand over allows us to see where the main forces are applied.

Toothbrush with hand forces

The biggest forces (represented by the two large black arrows) come from my thumb and index finger. These apply opposing forces that are nearly equal in magnitude—the thumb force is slightly smaller. Because these forces are offset, other forces are needed to keep the toothbrush from rotating in the plane of the photo. These small balancing forces are applied along the rest of my grip, but tend to be concentrated near the end of handle, where it rests against my palm. The short black arrow there represents the small balancing forces.

Even if I’m not brushing my teeth, gripping the toothbrush tightly leads to high tensile stresses across the handle from my index finger. This is where my old toothbrush broke. The force from my teeth onto the brush (the small gray arrow) increases the stress, but that force doesn’t have to be large and doesn’t even have to be present for there to be high stresses. If the failure was my fault, it wasn’t because I brush too hard; it was because I grip the handle too hard. And maybe I could move my thumb back a bit, too.

But I’m not convinced my grip is the main, or sole, cause of the failure. After all, I’ve been brushing my teeth for over five decades, and I’ve never had a toothbrush snap in my hand before. I suspect the design of the handle—in particular, a change in shape near the fracture—is the primary culprit. But I’ll need to strip away the rubbery coating material from my old brush to be able to see that. A post for another day, perhaps.

I often find myself dissatisfied with articles that purport to explain engineering and mechanics because I feel they oversimplify to the point of being wrong. I need to read my own stuff more carefully.