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This article discusses how the robots sent to explore the Fukushima reactors have been damaged by radiation that exceeds 650 sieverts per hour:
This is the seventh such robot that has died probing the reactor. I assume the robots are well-shielded....
What exactly is damaging the robots (alpha, beta, gamma particles) and how? Is the damage permanent?
Ionizing radiation can induce a voltage by freeing electrons in a material.
If the radiation is capable of freeing electronics at a rate faster than the material is capable of dissipating it, then the voltage grows to the point that it reaches the breakdown voltage of the material, at which point the material conducts electricity. This usually "fries" the object.
Support structures don't generally care about the induced voltages, but electronics would. The article you linked stated, as @Paul mentioned, that one robot failed because the tracks got stuck; on a previous unit the camera failed due to higher-than-anticipated radiation levels.
