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I found here: What happens when we cut objects? that it is due to fracture wood splits open when using a wedge.

Does it only apply when the wedge is already in the wood? How about the moment before a crack is made, do you need to overcome the compressive and shear stress of the wood to start a crack?

I want to compute for the forces needed to cut the wood, along the grain, using a wedge. but Im confused whether shear and compressive strength or fracture initiation is involved to start the cut. and when the wedge in the the wood should I compute for fracture propagation or the cosine of the normal force on the wedge that is pushing the grains apart. I hope I'm making sense.

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Assume the wedge is like a cone with curved end. At first, there is only one normal force between the tip and the material below it. You can assume the surface of the material as a thin film of very large surface tension. This penetrating force causes the film to deform and the cone goes slightly in the film (the film doesn't not tear). Now because of the curved tip of the cone, the tilted edge acts as a constraint, converting some of the normal force to side-ward force. As a result, we have the material to compress to sides, this means there must exist some where the material is stretching(, so that it's compressing somewhere). That point is the tip. Since the material can't tolerate too much stretch, the bonds break down and the material separates below the tip, this makes more space for the wedge to go inside, and the same process repeats. But this doesn't continue forever. That's because friction force is acting and because any material is at least poorly elastic. But why doesn't the friction stop the wedge at the start? The answer is that the magnitude of friction force depends linearly (classical view) on the normal-ward (I have to say it like this :)) force on the surface. The component of the penetrating force doesn't change much, but the materials response to stretch, which is compression, tries to get back the material to it's initial state, that is to squeeze the wedge, that is to produce a normal force that helps friction grow, and an overall upward force that want to shoot the wedge upward.

I tried to include any details. If some part is ambiguous, tell me.

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It depends on the sharpness of the wedge. For a dull edge the wood directly in front of the edge will compress and the grains will buckle starting the outward push that will eventually make space for the wedge to progress further into the wood. I believe that even for moderately dull blades the force required to do this buckling will still be less than the force required to split the wood once the crack is established, but that's just intuition, along with some limited wood splitting experience.

Once the wedge is in, there will be a portion of the normal force acting on the wedge that resists movement (sin of half the angle at the tip) and then I would imagine that the friction would play a large role as well.

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