This machine broke down big chunks of woods into smaller chunks.
This machine broke up big chunks of woods into smaller chunks.
Are both equivalent, what difference does it make if you use up instead of down?
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They sound equivalent to me, in that context. They could also have said "breaks big chunks of wood into smaller ones."
In other contexts, there could be some difference. "Broken up" puts focus on the parts that are the result of the breaking, for example,
The council was broken up into several working groups.
The phrase "broken down" sounds as if something else is the result of the breaking, for example,
In digestion, foods are broken down into simpler chemicals that can be absorbed.